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DARSHAN INSTITUTE OF ENGINEERING & TECHNOLOGY Mathematics-I (3110014)
I N D E X
UNIT 1 ............................................................................................................ 1
1). METHOD â 1: 0/0 TYPE INDETERMINATE FORM ............................................................................ 1
2). METHOD â 2: â/â TYPE INDETERMINATE FORM ......................................................................... 3
3). METHOD â 3: 0 Ã â FORM ............................................................................................................................ 4
4). METHOD â 4: â â â FORM ........................................................................................................................... 5
5). METHOD â 5: 00, â0 & 1â FORM ................................................................................................................ 6
6). METHOD â 6: IMPROPER INTEGRAL OF FIRST KIND ..................................................................... 9
7). METHOD â 7: IMPROPER INTEGRAL OF SECOND KIND ............................................................. 11
8). METHOD â 8: CONVERGENCE OF IMPROPER INTEGRAL OF FIRST KIND ........................ 13
9). METHOD â 9: CONVERGENCE OF IMPROPER INTEGRAL OF SECOND KIND ................... 14
10). METHOD â 10: EXAMPLE ON GAMMA FUNCTION AND BETA FUNCTION ........................ 16
11). METHOD â 11: VOLUME BY SLICING METHOD............................................................................... 18
12). METHOD â 12: VOLUME OF SOLID BY ROTATION USING DISK METHOD ........................ 19
13). METHOD â 13: VOLUME OF SOLID BY ROTATION USING WASHER METHOD ............... 21
14). METHOD â 14: LENGTH OF PLANE CURVES .................................................................................... 23
15). METHOD â 15: AREA OF SURFACES BY REVOLUTION ................................................................ 25
UNIT 2 .......................................................................................................... 27
16). METHOD â 1: CHARACTERISTICS OF SEQUENCE AND LIMIT ................................................. 28
17). METHOD â 2: CONVERGENCE OF SEQUENCE .................................................................................. 29
18). METHOD â 3: CONTINUOUS FUNCTION THEOREM ..................................................................... 31
19). METHOD â 4: GEOMETRIC SERIES ........................................................................................................ 32
20). METHOD â 5: TELESCOPING SERIES .................................................................................................... 34
21). METHOD â 6: ZERO TEST ........................................................................................................................... 35
22). METHOD â 7: COMBINING SERIES ........................................................................................................ 36
23). METHOD â 8: INTEGRAL TEST ................................................................................................................ 37
24). METHOD â 9: DIRECT COMPARISION TEST ..................................................................................... 38
25). METHOD â 10: LIMIT COMPARISION TEST ...................................................................................... 40
26). METHOD â 11: RATIO TEST ...................................................................................................................... 42
27). METHOD â 12: RABBEâS TEST ................................................................................................................. 45
28). METHOD â 13: CAUCHYâS NTH ROOT TEST ........................................................................................ 46
29). METHOD â 14: LEIBNITZâS TEST............................................................................................................ 47
30). METHOD â 15: ABSOLUTE CONVERGENT SERIES ........................................................................ 49
DARSHAN INSTITUTE OF ENGINEERING & TECHNOLOGY Mathematics-I (3110014)
I N D E X
31). METHOD â 16: CONDITIONALLY CONVERGENT SERIES ........................................................... 50
32). METHOD â 17: POWER SERIES ............................................................................................................... 52
33). METHOD â 18: 1ST FORM OF TAYLORâS SERIES.............................................................................. 55
34). METHOD â 19: 2ND FORM OF TAYLORâS SERIES ............................................................................. 56
35). METHOD â 20: MACLAURINâS SERIES ................................................................................................. 57
UNIT 3 .......................................................................................................... 61
36). METHOD â 1: FOURIER SERIES IN THE INTERVAL (0, 2L) ....................................................... 64
37). METHOD â 2: FOURIER SERIES IN THE INTERVAL (âð, ð) ...................................................... 67
38). METHOD â 3: HALF-RANGE COSINE SERIES IN THE INTERVAL (ð, ð) ............................... 70
39). METHOD â 4: HALF-RANGE SINE SERIES IN THE INTERVAL (ð, ð) ..................................... 71
UNIT 4 .......................................................................................................... 73
40). METHOD â 1: LIMIT OF FUNCTION OF TWO VARIABLES .......................................................... 74
41). METHOD â 2: CONTINUITY OF FUNCTION OF TWO VARIABLES .......................................... 75
42). METHOD â 3: PARTIAL DERIVATIVES ................................................................................................. 78
43). METHOD â 4: CHAIN RULE ........................................................................................................................ 82
44). METHOD â 5: IMPLICIT FUNCTION....................................................................................................... 84
45). METHOD â 6: TANGENT PLANE AND NORMAL LINE .................................................................. 86
46). METHOD â 7: LOCAL EXTREME VALUES ........................................................................................... 89
47). METHOD â 8: LAGRANGEâS MULTIPLIERS ........................................................................................ 91
48). METHOD â 9: GRADIENT ............................................................................................................................ 93
49). METHOD â 10: DIRECTIONAL DERIVATIVE ..................................................................................... 95
UNIT 5 .......................................................................................................... 97
50). METHOD â 1: DOUBLE INTEGRALS BY DIRECT INTEGRATION ............................................. 99
51). METHOD â 2: TRIPLE INTEGRALS BY DIRECT INTEGRATION ............................................ 102
52). METHOD â 3: D.I. OVER GENERAL REGION IN CARTESIAN COORDINATES ................. 105
53). METHOD â 4: DOUBLE INTEGRALS AS VOLUMES ...................................................................... 109
54). METHOD â 5: D.I. BY CHANGE OF ORDER OF INTEGRATION ............................................... 110
55). METHOD â 6: D.I. OVER GENERAL REGION IN POLAR COORDINATES ............................ 113
56). METHOD â 7: AREA BY DOUBLE INTEGRATION IN CARTESIAN COORDINATES ....... 115
57). METHOD â 8: AREA BY DOUBLE INTEGRATION IN POLAR COORDINATES.................. 115
58). METHOD â 9: T.I. OVER GENERAL REGION IN CARTESIAN COORDINATES.................. 117
59). METHOD â 10: T.I. OVER GENERAL REGION IN CYLINDRICAL COORDINATES .......... 118
60). METHOD â 11: T.I. OVER GENERAL REGION IN SPHERICAL COORDINATES ............... 119
DARSHAN INSTITUTE OF ENGINEERING & TECHNOLOGY Mathematics-I (3110014)
I N D E X
61). METHOD â 12: JACOBIAN........................................................................................................................ 121
62). METHOD â 13: D.I. BY CHANGE OF VARIABLE IN CARTESIAN COORDINATES ........... 123
63). METHOD â 14: D.I. BY CHANGE OF VARIABLE IN POLAR COORDINATES ..................... 124
64). METHOD â 15: T.I. BY CHANGE OF VARIABLE OF INTEGRATION ...................................... 127
UNIT 6 ........................................................................................................ 129
65). METHOD â 1: ECHELON FORM AND RANK OF MATRIX .......................................................... 130
66). METHOD â 2: GAUSS ELIMINATION METHOD ............................................................................. 135
67). METHOD â 3: GAUSS - JORDAN METHOD ....................................................................................... 139
68). METHOD â 4: INVERSE BY GAUSS - JORDAN METHOD ............................................................ 141
69). METHOD â 5: EIGEN VALUES, EIGEN VECTORS AND EIGEN SPACE ................................. 144
70). METHOD â 6: ALGEBRAIC AND GEOMETRIC MULTIPLICITY ............................................... 148
71). METHOD â 7: DIAGONALIZATION ...................................................................................................... 150
72). METHOD â 8: THE CAYLEY - HAMILTON THEOREM................................................................. 153
FORMULAE, RESULTS & SYMBOLS ............................................................. 155
73). STANDARD SYMBOLS: .............................................................................................................................. 155
74). BASIC FORMULAE:...................................................................................................................................... 155
75). STANDARD SERIES: ................................................................................................................................... 155
76). TRIGONOMETRIC IDENTITIES AND FORMULAE: ....................................................................... 156
77). VALUES OF CIRCULAR FUNCTIONS: .................................................................................................. 157
78). INVERSE TRIGONOMETRIC FUNCTIONS: ....................................................................................... 157
79). DIFFERENTIATION: ................................................................................................................................... 158
80). INTEGRATION: ............................................................................................................................................. 158
81). RELATION WITH CARTESIAN COORDINATES: ............................................................................ 163
82). HYPERBOLIC FUNCTIONS: ..................................................................................................................... 163
83). FREQUENTLY USED LIMIT: .................................................................................................................... 163
84). LOGARITHM RULES: .................................................................................................................................. 164
85). AREA:................................................................................................................................................................. 164
86). VOLUME:.......................................................................................................................................................... 164
87). VALUE OF SOME CONSTANTS: ............................................................................................................. 165
88). TRIGONOMETRIC TABLE: ....................................................................................................................... 165
LIST OF ASSIGNMENTS............................................................................... 166
DARSHAN INSTITUTE OF ENGINEERING & TECHNOLOGY Mathematics-I (3110014)
I N D E X
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DARSHAN INSTITUTE OF ENGINEERING & TECHNOLOGY Mathematics-I (3110014)
U N I T - 1 [ 1 ]
UNIT 1
â INDETERMINATE FORMS:
â The following are indeterminate forms which we will study:
0
0 , â
â , 0 Ã â, â â â, 00, â0 & 1â.
â Lâ HOSPITALâS RULE:
If lim x â a
f(x)
g(x) leads to the indeterminate form
0
0 or
â
â then
lim x â a
f(x)
g(x) = lim
x â a
f â²(x)
gâ²(x) , provided the later limit exists.
â Procedure to find the limit using Lâ Hospitalâs rule:
(1). Differentiate numerator and denominator separately and apply the limit.
(ð). If it again reduces to indeterminate form 0
0 or
â
â then again
differentiate numerator and denominator separately and apply the limit.
(ð). Continue this process till we get finite or infinite value of the limit.
â Remark: limx â 0
log x = ââ & lim x â â
log x = â.
METHOD â 1: 0/0 TYPE INDETERMINATE FORM
C 1 Evaluate lim
x â 0 ( ex â 1 â x
x2 ) .
ðð§ð¬ð°ðð«: ð/ð.
C 2 Evaluate lim
x â Ï 2
( 2x â Ï
cos x ).
ðð§ð¬ð°ðð«: â ð.
H 3
Evaluate the examples: (ð). lim x â 0
( x â tan x
x3 ) (ð). lim
x â 0
sin x â x + x3
6
x5 .
ðð§ð¬ð°ðð«: (ð). â ð/ð (ð).ð/ððð.
DARSHAN INSTITUTE OF ENGINEERING & TECHNOLOGY Mathematics-I (3110014)
U N I T - 1 [ 2 ]
T 4 Evaluate lim
x â 0
2x â x cos x â sin x
2x3 .
ðð§ð¬ð°ðð«: ð/ð.
C 5 Evaluate lim
x â 0 ( (sin x)2 â x2
x2(sin x)2 ).
ðð§ð¬ð°ðð«: â ð/ð.
T 6 Evaluate the examples: (ð). lim
x â 0 ( tan x â x
sin x â x ) (ð). lim
x â 0 ( tan x â sin x
(sin x)3 )
(ð). lim x â 0
( x(cos x â 1)
sin x â x ) (ð). lim
x â 0 ( 2â1 + x â 2 â x
2 sin2 x ).
ðð§ð¬ð°ðð«: (ð). â ð (ð). ð/ð (ð).ð (ð). â ð/ð.
C 7 Evaluate lim
x â 0 ( ex + eâx â x2 â 2
(sin x)2 â x2 ).
ðð§ð¬ð°ðð«: â ð/ð.
H 8 Evaluate lim
x â 0 ( ex â esin x
x â sin x ).
ðð§ð¬ð°ðð«: ð.
T 9 Evaluate lim
x â 1 2
( cos2 Ïx
e2x â 2ex ).
ðð§ð¬ð°ðð«: ðð/ðð.
C 10 Evaluate lim
x â y ( xy â yx
xx â yy ).
ðð§ð¬ð°ðð«: (ð â ð¥ðš ð ð²) / (ð + ð¥ðš ð ð²).
H 11 Evaluate lim
x â 0 { ln cos âx
x }.
ðð§ð¬ð°ðð«: â ð/ð.
H 12 Evaluate lim
x â 0 ( xex â log(1 + x)
x2 ).
ðð§ð¬ð°ðð«: ð/ð.
W-19 (4)
DARSHAN INSTITUTE OF ENGINEERING & TECHNOLOGY Mathematics-I (3110014)
U N I T - 1 [ 3 ]
T 13 Evaluate lim
x â 0 ( ax â bx
x).
ðð§ð¬ð°ðð«: ð¥ðšð (ð/ð).
T 14 Evaluate lim
x â 1 (
xx â x
x â 1 â log x ).
ðð§ð¬ð°ðð«: ð.
METHOD â 2: â/â TYPE INDETERMINATE FORM
C 1 Evaluate lim
x â a ( log(ex â ea)
log(x â a) ).
ðð§ð¬ð°ðð«: ð.
H 2 Evaluate lim
x â 0 ( cot 2x
cot x ).
ðð§ð¬ð°ðð«: ð/ð.
T 3 Evaluate lim
x â Ï 2 (3 sec x
1 + tan x ).
ðð§ð¬ð°ðð«: ð.
C 4 Evaluate lim
x â â ( xn
ex ) , n > 1.
ðð§ð¬ð°ðð«: ð.
C 5 Evaluate lim x â 0
( logtanx tan 2x ).
ðð§ð¬ð°ðð«: ð.
T 6 Evaluate the given examples: (ð). lim
x â â ( log x
xn ) (ð). lim
x â Ï 2 ( ln(cos x)
sec x )
(ð). lim x â â
( (ln x)2(3 + ln x)
2x ).
ðð§ð¬ð°ðð«: (ð). ð (ð).ð (ð). ð.
DARSHAN INSTITUTE OF ENGINEERING & TECHNOLOGY Mathematics-I (3110014)
U N I T - 1 [ 4 ]
â 0 Ã â TYPE INDETERMINATE FORM:
In this case we write f(x) â g(x) as f(x)
{ 1
g(x) } or
g(x)
{ 1
f(x) } which leads to the form
0
0 or
â
â ,where Lâ Hospitalâs rule is applicable.
â Procedure to find the limit of indeterminate form 0 Ã â:
(ð). Transform f(x) â g(x) into f(x)
{ 1
g(x) } or
g(x)
{ 1
f(x) }
i. e. transform 0 Ãâ into 0
0 or
â
â .
(ð). Remember: Donât put the logarithm function in the denominator in step (1).
(ð). Apply Lâ Hospitalâs rule to find the value of given limit.
METHOD â 3: 0 Ã â FORM
C 1 Evaluate lim
x â â { (a
1 x â 1) x } .
ðð§ð¬ð°ðð«: ð¥ðšð ð.
H 2 Evaluate lim x â 1
{ (1 â x) tan ( Ïx
2) } .
ðð§ð¬ð°ðð«: ð/ð.
C 3 Evaluate lim
x â â { (âx + 1 â âx) log (
1
x ) } .
ðð§ð¬ð°ðð«: ð.
H 4 Evaluate lim x â 0
{ (sin x) (ln x) }.
ðð§ð¬ð°ðð«: ð.
T 5 Evaluate the given examples:
(ð). lim x â a
{ ln (2 â x
a ) cot(x â a) } (ð). lim
x â 1 2
{ ln (3
2 â x) cot (x â
1
2 ) } .
ðð§ð¬ð°ðð«: (ð).âð/ð (ð).âð.
DARSHAN INSTITUTE OF ENGINEERING & TECHNOLOGY Mathematics-I (3110014)
U N I T - 1 [ 5 ]
â â - â TYPE INDETERMINATE FORM:
In this case, we write f(x) â g(x) = {
1 g(x)
â 1
f(x) }
{ 1
f(x) g(x) }
which leads to the form 0
0 or
â
â .
where, Lâ Hospitalâs rule is applicable.
â Procedure to find the limit of indeterminate form â ââ:
(ð). Take L. C.M. in f(x) â g(x) which will transform â ââ into 0
0 or
â
â .
(ð). Apply Lâ Hospitalâs rule to find the value of given limit.
METHOD â 4: â â â FORM
C 1 Evaluate lim
x â 0 {
1
sin x â 1
x } .
ðð§ð¬ð°ðð«: ð.
H 2 Evaluate lim x â 0
( cosec x â cot x).
ðð§ð¬ð°ðð«: ð.
T 3 State Lâ²Hospitalâ²s Rule. Use it to evaluate lim
x â 0 { 1
x2 â
1
(sin x)2 } .
ðð§ð¬ð°ðð«: â ð/ð.
W-18 (4)
C 4 Evaluate lim
x â 0 { 1
x2 â (cot x)2 } .
ðð§ð¬ð°ðð«: ð/ð.
C 5 Use Lâ²Hospitalâ²s rule to find the limit of lim
x â 1 {
x
x â 1 â
1
log x } .
ðð§ð¬ð°ðð«: ð/ð.
S-19 (3)
H 6 Evaluate lim
x â 0 { 1
x â
1
ex â 1 } .
ðð§ð¬ð°ðð«: ð/ð.
T 7 Evaluate lim
x â 2 {
1
x â 2 â
1
log(x â 1) } .
ðð§ð¬ð°ðð«: â ð/ð.
DARSHAN INSTITUTE OF ENGINEERING & TECHNOLOGY Mathematics-I (3110014)
U N I T - 1 [ 6 ]
T 8 If an electrostatic field E acts on a liquid or a gaseous polar dielectric, the net
dipodal moment P per unit volume is P(E) =eE + eâE
eE â eâE â1
E .
Show that lim E â 0+
P(E) = 0.
S-19 (3)
â 00, â0 & 1â TYPE INDETERMINATE FORM:
â In this case, we will write y = lim x â â
{ f(x) g(x) }. After that we will take logarithm on both
the sides, then we get log y = lim x â â
g(x) â log f(x), which leads to indeterminate form
0 Ã â or â Ã 0. We know that how to deal with it.
â Remember: It gives us value of log y = L(any value), but we want the value of y. So,
use y = eL to find the value of given limit i.e. y.
â Procedure to find the limit of indeterminate form 00, â0 & 1â:
(1). Assume y = lim x â â
{ f(x) g(x) } .
(2). Apply logarithm function both the side of above assumption.
(3). Simplify R.H.S. to convert function into indeterminate form 0
0 or
â
â .
(4). Find the value of R.H.S. using above concept of indeterminate form i.e. value of
log y.
(5). Find the value of y i.e. value of given limit.
METHOD â 5: 00, â0 & 1â FORM
C 1 Evaluate lim
x â 0 (1
x )tan x
.
ðð§ð¬ð°ðð«: ð.
H 2 Evaluate lim
x â 0 (1
x )1â cosx
.
ðð§ð¬ð°ðð«: ð.
DARSHAN INSTITUTE OF ENGINEERING & TECHNOLOGY Mathematics-I (3110014)
U N I T - 1 [ 7 ]
C 3 Evaluate lim x â 0
{ (cos x)cot x }.
ðð§ð¬ð°ðð«: ð.
T 4 Evaluate given examples: (ð). lim x â 1
{ (x â 1)xâ1 } (ð). lim x â
Ï 2
{ (sin x)tan x }
(ð). lim x â
Ï 2
{ (tan x)cos x } .
ðð§ð¬ð°ðð«: (ð). ð (ð).ð (ð). ð.
C 5 Evaluate lim
x â 0 ( tan x
x )
1 x2 .
ðð§ð¬ð°ðð«: ðð/ð.
C 6
Evaluate lim x â 0
( ax + bx + cx
3 )
1 3x
.
ðð§ð¬ð°ðð«: (ððð)ð/ð.
H 7 Evaluate the given examples:
(ð). lim x â 0
( 1x + 2x + 3x + 4x
4 )
1 x
(ð). lim x â 0
( ax + bx + cx + dx
4 )
1 x
.
ðð§ð¬ð°ðð«: (ð). ððð/ð (ð). (ðððð)ð/ð.
T 8
Evaluate lim x â 0
( ex + e2x + e3x
3 )
1 x
.
ðð§ð¬ð°ðð«: ðð.
H 9 Evaluate lim
x â 0 (ax + x)
1 x .
ðð§ð¬ð°ðð«: ðð.
H 10 Evaluate lim
x â 1 (2 â x)
(tan Ïx 2) .
ðð§ð¬ð°ðð«: ðð/ð.
T 11 Evaluate lim x â
Ï 2
(cos x)( Ï 2 â x)
.
ðð§ð¬ð°ðð«: ð.
DARSHAN INSTITUTE OF ENGINEERING & TECHNOLOGY Mathematics-I (3110014)
U N I T - 1 [ 8 ]
T 12 Evaluate the examples: (ð). lim
x â Ï 2
{ (cosec x)(tan2 x) } (ð). lim
x â 0 (cosâx )
1 x .
ðð§ð¬ð°ðð«: (ð). âð (ð). ð/âð.
C 13 Evaluate lim
x â 0 x(
1 ln(exâ1)
).
ðð§ð¬ð°ðð«: ð.
T 14 Evaluate lim
x â 1 (1 â x2)
( 1
log(1âx) ) .
ðð§ð¬ð°ðð«: ð.
â IMPROPER INTEGRALS:
â If the limit of integral is infinite (one or both) and/or integrand function is discontinuous
for some value(s) on the interval of given integral then such integral is known as an
improper integral.
â Types of Improper Integrals:
(1). Improper integral of first kind.
(2). Improper integral of second kind.
(3). Improper integral of third kind (combination of 1st & 2nd kind).
â IMPROPER INTEGRAL OF FIRST KIND:
For â« f(x)dx
b
a
either a or b or both (a and b) are infinite, then such integral is known as an
improper integral of first kind.
â Sub-types of improper integrals of first kind are as follows:
(1). If f is continuous on [a,â) then
â« f(x)dx = lim b â â
â« f(x)dx
b
a
â
a
.
(2). If f is continuous on (ââ, b] then
DARSHAN INSTITUTE OF ENGINEERING & TECHNOLOGY Mathematics-I (3110014)
U N I T - 1 [ 9 ]
â« f(x)dx = lim a â ââ
â« f(x)dx
b
a
b
ââ
.
(3). If f is continuous on (ââ,â) then
â« f(x)dx = â« f(x)dx + â« f(x)dx
â
t
=
t
ââ
lim a â ââ
â« f(x)dx + lim b â â
â« f(x)dx
b
t
t
a
â
ââ
.
METHOD â 6: IMPROPER INTEGRAL OF FIRST KIND
C 1 Evaluate â«
dx
x2 + 1
â
0
.
ðð§ð¬ð°ðð«: ð/ð.
W-19 (4)
H 2 Evaluate â«
1
â x dx
â
1
.
ðð§ð¬ð°ðð«: â.
T 3 Evaluate â«
x
(1 + x)3 dx
â
0
.
ðð§ð¬ð°ðð«: ð.
T 4 Discuss type I & II improper integrals with example of each & find the value
of â«x + 3
(x â 1)(x2 + 1) dx.
â
2
ðð§ð¬ð°ðð«: (âð/ð) + ð¥ðšð ð + ððð§âð ð.
C 5 Evaluate â« e2x dx
0
ââ
.
ðð§ð¬ð°ðð«: ð/ð.
T 6 Evaluate â« x sin x dx.
0
ââ
ðð§ð¬ð°ðð«: ââ.
DARSHAN INSTITUTE OF ENGINEERING & TECHNOLOGY Mathematics-I (3110014)
U N I T - 1 [ 1 0 ]
C 7 Evaluate â«
1
1 + x2 dx
â
ââ
.
ðð§ð¬ð°ðð«: ð.
H 8 Evaluate â«
x
(1 + x2)2 dx.
â
ââ
ðð§ð¬ð°ðð«: ð.
T 9 Evaluate â« ex dx
â
ââ
.
ðð§ð¬ð°ðð«: â.
C 10 Evaluate â«
1
ex + eâx dx
â
ââ
.
ðð§ð¬ð°ðð«: ð/ð.
â IMPROPER INTEGRAL OF SECOND KIND:
For â« f(x)dx
b
a
, f(x) become discontinuous (infinite) at x = a or x = b or at finite
number of points in the interval (a, b), then such an integral is known as improper
integral of second kind.
â Sub-types of improper integrals of second kind are as follows:
(1). If x = a is the point of discontinuity for f(x) then the integral is defined as
â« f(x)dx = lim t â a
â« f(x) dx
b
t
b
a
.
(2). If x = b is the point of discontinuity for f(x) then the integral is defined as
â« f(x)dx = lim t â b
â« f(x) dx
t
a
b
a
.
(3). If x = c is the point of discontinuity for f(x) then the integral is defined as
DARSHAN INSTITUTE OF ENGINEERING & TECHNOLOGY Mathematics-I (3110014)
U N I T - 1 [ 1 1 ]
â« f(x)dx = â« f(x)dx +â« f(x)dx
b
c
=
c
a
lim t â c
â« f(x)dx
t
a
b
a
+ lim t â c
â« f(x)dx
b
t
.
METHOD â 7: IMPROPER INTEGRAL OF SECOND KIND
C 1
Evaluate the integrals â«1
âx â 2 dx
5
2
and â«1
(a â x)2 dx
b
a
.
ðð§ð¬ð°ðð«: ðâð & â â.
H 2 Evaluate the integrals â«
1
x2 dx
1
0
and â«1
x 2/3 dx
1
â1
.
Answer: â & 6.
H 3
Evaluateâ« sec x dx
Ï 2
0
.
ðð§ð¬ð°ðð«: â.
T 4
Evaluateâ«1
â3 â x dx.
3
0
ðð§ð¬ð°ðð«: ðâð.
T 5
Can we solve the integral â«1
(x â 2)2 dx
5
0
directly? Give the reason.
ðð§ð¬ð°ðð«: ððš, ð ð¢ð¯ðð§ ðð®ð§ððð¢ðšð§ ð¢ð¬ ð§ðšð ððšð§ðð¢ð§ð®ðšð®ð¬ ðð ð± = ð.
C 6
Evaluateâ«1
(x â 1) 2/3 dx.
3
0
ðð§ð¬ð°ðð«: ð (ðð/ð â (âð)ð/ð) ðð ð (ðð/ð + ð).
T 7 Evaluate â«
1
âa2 â x2 dx
a
âa
.
ðð§ð¬ð°ðð«: ð.
DARSHAN INSTITUTE OF ENGINEERING & TECHNOLOGY Mathematics-I (3110014)
U N I T - 1 [ 1 2 ]
â CONVERGENCE OR DIVERGENCE OF IMPROPER INTEGRAL:
â If integral has finite value then we can say that given integral is convergent.
â If integral has infinite value then we can say that given integral is divergent.
â P-INTEGRAL TEST:
â«1
xp
â
a
dx is convergent if P > 1 & divergent if P †1.
â«1
xp
a
0
dx is convergent if P < 1 & divergent if P ⥠1.
â DIRECT COMPARISON TEST:
â If f(x) and g(x) are continuous on [a, â) and 0 †f(x) †g(x) for all x ⥠a then
(ð). If â« g(x)dx is convergent thenâ« f(x)dx
â
a
is convergent.
â
a
(ð). If â« f(x)dx is divergent thenâ« g(x)dx
â
a
divergent.
â
a
â LIMIT COMPARISON TEST:
If f(x) and g(x) are positive function and continuous on [a,â) and L = limxââ
f(x)
g(x) then
(ð). If 0 < L < â thenâ« f(x)dx and
â
a
â« g(x)dx
â
a
both convergent or divergent together.
(ð). If L = 0 andâ« g(x)dx is convergent then
â
a
â« f(x)dx
â
a
is convergent.
(ð). If L = â andâ« g(x)dx is divergent then
â
a
â« f(x)dx
â
a
is divergent.
â Convergence of improper integral of second kind can be checked by proper changes in
above Direct/Limit comparison test.
DARSHAN INSTITUTE OF ENGINEERING & TECHNOLOGY Mathematics-I (3110014)
U N I T - 1 [ 1 3 ]
METHOD â 8: CONVERGENCE OF IMPROPER INTEGRAL OF FIRST KIND
C 1 Check the convergence of â«
1
âx dx
â
1
and â«1
xâ2 dx
â
1
.
ðð§ð¬ð°ðð«: ðð¢ð¯ðð«ð ðð§ð ðð§ð ððšð§ð¯ðð«ð ðð§ð.
C 2 Check the convergence ofâ«
dx
(1 + x2)(1 + tanâ1 x)
â
0
.
ðð§ð¬ð°ðð«: ððšð§ð¯ðð«ð ðð§ð.
H 3 Check the convergence ofâ«
1
xp dx
â
1
.
ðð§ð¬ð°ðð«: ð© †ð â¹ ðð¢ð¯ðð«ð ðð§ð & ð© > ðâ¹ ððšð§ð¯ðð«ð ðð§ð.
T 4 Check the convergence of â«
sin2x
x2 dx
â
1
and â« sin3 x
x3 dx
â
1
.
ðð§ð¬ð°ðð«: ððšð§ð¯ðð«ð ðð§ð & ððšð§ð¯ðð«ð ðð§ð.
C 5 Check the convergence of â«
1
log x dx
â
2
.
ðð§ð¬ð°ðð«: ðð¢ð¯ðð«ð ðð§ð.
H 6 Check the convergence of â«
x
(1 + x)4 dx
â
2
and â«1
âx (1 + x)2 dx
â
1
.
ðð§ð¬ð°ðð«: ððšð§ð¯ðð«ð ðð§ð & ððšð§ð¯ðð«ð ðð§ð.
T 7 Check the convergence ofâ« eâx
2 dx
â
1
.
ðð§ð¬ð°ðð«: ððšð§ð¯ðð«ð ðð§ð.
C
8 Investigate the convergence of â«
5x
(1 + x2)3 dx.
â
5
ðð§ð¬ð°ðð«: ð ððððâ & ððšð§ð¯ðð«ð ðð§ð.
W-18 (3)
H 9 Prove that â«
5
ex + 3 dx
â
1
is convergent.
DARSHAN INSTITUTE OF ENGINEERING & TECHNOLOGY Mathematics-I (3110014)
U N I T - 1 [ 1 4 ]
H 10 Check the convergence ofâ«
3x + 5
x4 + 7 dx
â
4
.
ðð§ð¬ð°ðð«: ððšð§ð¯ðð«ð ðð§ð.
T 11 Investigate the convergence of â«
x10(1 + x5)
(1 + x)27 dx.
â
0
ðð§ð¬ð°ðð«: ð(ðð!)(ð!) ðð!â & ððšð§ð¯ðð«ð ðð§ð.
W-18 (4)
METHOD â 9: CONVERGENCE OF IMPROPER INTEGRAL OF SECOND KIND
C 1
Check the convergence of â«1
x2 dx.
5
0
ðð§ð¬ð°ðð«: ðð¢ð¯ðð«ð ðð§ð.
H 2
Determine whether the integral â«dx
x â 1
3
0
converges or diverges.
ðð§ð¬ð°ðð«: ðð¢ð¯ðð«ð ðð§ð.
S-19 (3)
H 3 Determine â«
1
(x â 2)2 dx
2
0
and is it convergent or divergent?
ðð§ð¬ð°ðð«: â â & ðð¢ð¯ðð«ð ðð§ð.
T 4 Check the convergence ofâ«
dx
â1 â x2
1
0
.
ðð§ð¬ð°ðð«: ð/ð & ððšð§ð¯ðð«ð ðð§ð.
C 5 Define improper integral of both the kinds and check the convergence of
â«dx
â9 â x2
3
0
.
ðð§ð¬ð°ðð«: ð/ð & ððšð§ð¯ðð«ð ðð§ð.
H 6
Check the convergence of â« cos 3x
x5/2 dx
3
0
.
ðð§ð¬ð°ðð«: ðð¢ð¯ðð«ð ðð§ð.
DARSHAN INSTITUTE OF ENGINEERING & TECHNOLOGY Mathematics-I (3110014)
U N I T - 1 [ 1 5 ]
T 7 Determineâ« ln x dx
1
0
and is it convergent or divergent?
ðð§ð¬ð°ðð«: â ð & ððšð§ð¯ðð«ð ðð§ð.
C 8 Check the convergence of â«
dx
x2
2
â2
.
ðð§ð¬ð°ðð«: ðð¢ð¯ðð«ð ðð§ð.
â GAMMA FUNCTION:
If n > 0, then Gamma function is defined by the integral â« eâxxnâ1dx
â
0
and is denoted
by Î(n). ð¢. ð. ðª(ð§) = â« ðâð± ð±ð§âð ðð±
â
ð
.
â Properties:
(ð). Reduction formula for Gamma Function Î(n + 1) = nÎ(n) ; where n > 0.
(ð). If n is a positive integer, then Î(n + 1) = n!.
(ð). Second Form of Gamma Functionâ« eâx2x2mâ1dx =
1
2
â
0
Î(m).
(ð). Î (n +1
2 ) =
(2n)!âÏ
n! 4nfor n = 0,1,2,3,âŠ
Examples: For n = 0, Î (1
2 ) = âÏ For n = 1, Î (
3
2 ) =
âÏ
2
For n = 2, Î (5
2 ) =
3âÏ
4
â BETA FUNCTION:
If m > 0, n > 0, then Beta function is defined by the integral â«xmâ1(1 â x)nâ1dx
1
0
and
DARSHAN INSTITUTE OF ENGINEERING & TECHNOLOGY Mathematics-I (3110014)
U N I T - 1 [ 1 6 ]
is denoted by β(m,n) OR B(m,n). ð¢. ð. ð(ðŠ,ð§) = â«ð±ðŠâð(ð â ð±)ð§âððð±
ð
ð
.
â Properties:
(ð). Beta function is a symmetric function. i.e. β(m, n) = β(n,m), where m > 0, n > 0.
(ð). β(m, n) = 2â« sin2mâ1 Ξ cos2nâ1 ΞdΞ
Ï 2
0
.
(ð). â« sinp Ξ cosq ΞdΞ =1
2 β ( p + 1
2, q + 1
2)
Ï 2
0
.
(ð). β(m, n) = â«xmâ1
(1 + x)m+n
â
0
dx.
(ð). Relation Between Beta and Gamma Function,β(m, n) = Î(m)Î(n)
Î(m + n) .
METHOD â 10: EXAMPLE ON GAMMA FUNCTION AND BETA FUNCTION
C 1 Find Î (
13
2).
ðð§ð¬ð°ðð«: ðððððâð/ðð.
C 2 Define Gamma function and evaluateâ« eâx
2
â
0
dx.
ðð§ð¬ð°ðð«: âð/ð .
S-19 (4)
H 3 Evaluateâ« x3eâx
â
0
dx.
ðð§ð¬ð°ðð«: ð.
T 4 Evaluate â«
x4
4x
â
0
dx.
ðð§ð¬ð°ðð«: ðð (ð¥ðšð ð)ðâ .
DARSHAN INSTITUTE OF ENGINEERING & TECHNOLOGY Mathematics-I (3110014)
U N I T - 1 [ 1 7 ]
C 5 Prove that β(m, n) = β(m+ 1, n) + β(m,n + 1).
H 6 Find β(4,3).
ðð§ð¬ð°ðð«: ð/ðð.
T 7 Prove that
β(m + 1, n)
β(m, n)=
m
m + n .
C 8 Find β (
7
2 ,5
2 ).
ðð§ð¬ð°ðð«: ðð/ððð.
W-19 (3)
H 9 Evaluate â«
x5(1 + x5)
(1 + x)15 dx
â
0
.
ðð§ð¬ð°ðð«: ð/ðððð.
C 10 Evaluate â«
x
â1 â x5 dx
1
0
.
ðð§ð¬ð°ðð«:ð
ð ð (
ð
ð ,ð
ð ).
â VOLUME BY SLICING:
â A plane intersects a solid, then plane area of the cross section be A(x) and if plane is
perpendicular to x-axis at any point between x = a and x = b, then volume of solid is
V = â« A(x) dx
b
x=a
âŠâŠâŠâŠâŠâŠâŠâŠâŠâŠ Formula S1
â A plane intersects a solid, then plane area of the cross section be A(y) and if plane is
perpendicular to y-axis at any point between y = a and y = b, then volume of solid is
V = â« A(y) dy
b
y=a
âŠâŠâŠâŠâŠâŠâŠâŠâŠâŠ Formula S2
â Procedure for finding the volume of solid:
(1). Sketch the solid with cross-section.
(2). Find the area of that cross-section (i.e. A(x) or A(y)).
DARSHAN INSTITUTE OF ENGINEERING & TECHNOLOGY Mathematics-I (3110014)
U N I T - 1 [ 1 8 ]
(3). Find the limit of integration (i.e. âaâ and âbâ).
(4). Integrate A(x) or A(y) using the standard formula.
METHOD â 11: VOLUME BY SLICING METHOD
H 1 Using method of slicing, find volume of a cone with height 4cm and radius of
base 4cm.
ðð§ð¬ð°ðð«: (ððð/ð)ððŠð.
C 2 Using slicing method find the volume of a solid ball of radius âaâ. OR
Using slicing method determine volume of a sphere of radius âaâ.
ðð§ð¬ð°ðð«: ðððð/ð.
C 3 Derive the formula for the volume of a right pyramid whose altitude is âhâ
and whose base is a square with sides of length âaâ.
ðð§ð¬ð°ðð«: ððð¡/ð.
H 4 A pyramid having 4 m height has a square base with sides of length 4m.The
pyramid having a cross section at a distance x m down from the vertex and
perpendicular to the altitude is a square with sides of length x m. Find the
volume of the pyramid.
ðð§ð¬ð°ðð«: (ðð/ð)ðŠð.
â VOLUME OF SOLID BY ROTATION USING DISK METHOD:
â The area bounded by the curve y = f(x), the ordinates
x = a, x = b and x-axis, which is revolved about x-axis
then the volume of solid is given by
V = Ï â« y2 dx
b
x=a
âŠâŠâŠâŠâŠâŠâŠâŠâŠâŠ Formula R1
DARSHAN INSTITUTE OF ENGINEERING & TECHNOLOGY Mathematics-I (3110014)
U N I T - 1 [ 1 9 ]
â The area bounded by the curve x = f(y), the ordinates
y = a, y = b and y-axis, which is revolved about y-axis
then the volume of solid is given by
V = Ï â« x2 dy
b
y=a
âŠâŠâŠâŠâŠâŠâŠâŠâŠâŠ Formula R2
â The area bounded by the curve x1 = f(y), the ordinates
y = a, y = b and x2 = c (constant), which is revolved
about ð±ð = ð then the volume of solid is given by
ð = Ï â«(x1 â x2)2
b
y=a
dyâŠâŠâŠâŠâŠâŠâŠâŠâŠâŠFormula R3
â The area bounded by the curve y1 = f(x), the ordinates
x= a, x = b and y2 = c (constant), which is revolved
about ð²ð = ð then the volume of solid is given by
ð = ð â«(ð²ð â ð²ð)ð ðð±
ð
ð²=ð
âŠâŠâŠâŠâŠâŠâŠâŠâŠâŠ ð ðšð«ðŠð®ð¥ð ðð
METHOD â 12: VOLUME OF SOLID BY ROTATION USING DISK METHOD
C 1 Find the volume of the solid generated by revolving the area bounded by the
2y = x2, x = 4, y = 0 and about (1). X-axis, (2). Y-axis.
ðð§ð¬ð°ðð«: ðððð/ð & ððð.
T 2 The region between the curve y = âx ; 0 †x †4 and the X-axis is revolved
about the X-axis to generate a solid. Find its volume.
ðð§ð¬ð°ðð«: ðð.
H 3 Find the volume of the solid formed by rotating the region between the
graph of y = 1 â x2 and y = 0 about the X-axis.
ðð§ð¬ð°ðð«: ððð/ðð.
DARSHAN INSTITUTE OF ENGINEERING & TECHNOLOGY Mathematics-I (3110014)
U N I T - 1 [ 2 0 ]
H 4 Find the volume of solid of revolution; obtain by rotating the area bounded
below the line 2x + 3y = 6 in the first quadrant, about the X-axis.
ðð§ð¬ð°ðð«: ðð.
C 5 Find the volume of the solid generated by rotating the plane region bounded
by y =1
x, x = 1 and x = 3 about the X axis.
ðð§ð¬ð°ðð«: ðð/ð.
W-19 (7)
T 6 The graph of y = x2 between x = 1 and x = 2 is rotated around the X-axis.
Find the volume of a solid so generated.
ðð§ð¬ð°ðð«: ððð/ð.
H 7 A bowl has a shape that can generated by revolving the graph y = x2/2
between y = 0 and y = 5 about the Y-axis. Find the volume of bowl.
ðð§ð¬ð°ðð«: ððð.
H 8 Find the volume of the solid generated by revolving the region between the
Y-axis and the curve x = 2ây ; 0 †y †4, about the Y-axis.
ðð§ð¬ð°ðð«: ððð.
C 9 Find the volume of the solid obtained by rotating the region bounded by y =
x3, y = 8 and x = 0 about the Y-axis.
ðð§ð¬ð°ðð«: ððð/ð.
C 10 Calculate the volume of solid generated by revolving the region between the
parabola 2x = y2 + 2 and the line x = 3, about the line x = 3.
ðð§ð¬ð°ðð«: ðððð/ðð.
T 11 Find the volume of the solid generated by revolving the arc bounded by the
parabola y2 = 4ax & latus rectum about latus rectum, where a > 0.
ðð§ð¬ð°ðð«: ððððð/ðð.
C 12 Find the volume of the solid generated by revolving the region bounded by
y = âx and the lines y = 1, x = 4 about the line y = 1.
ðð§ð¬ð°ðð«: ðð/ð.
S-19 (4)
DARSHAN INSTITUTE OF ENGINEERING & TECHNOLOGY Mathematics-I (3110014)
U N I T - 1 [ 2 1 ]
â VOLUME OF SOLID BY ROTATION USING WASHER METHOD:
â Let A be the area bounded by the two curves y1 = f1(x)
and y2 = f2(x). Let x = a and x = b be the x-coordinates
of their point of intersection. The volume of the solid
generated by area A rotating about x-axis is given by
V = Ï â«(y2 2 â y1
2) dx
b
x=a
; y2 ⥠y1âŠâŠâŠâŠâŠ Formula W1
â Let A be the area bounded by the two curves x1 = f1(y)
and x2 = f2(y). Let y = a and y = b be the y-coordinates
of their point of intersection. The volume of the solid
generated by area A rotating about y-axis is given by
V = Ï â«(x2 2 â x1
2) dy
b
y=a
; x2 ⥠x1âŠâŠâŠâŠâŠ Formula W2
METHOD â 13: VOLUME OF SOLID BY ROTATION USING WASHER METHOD
H 1 Find the volume of the solid formed by rotating the region between the
graphs of y = 4 â x2 and y = 6 â 3x about the X-axis.
ðð§ð¬ð°ðð«: ðð/ðð.
C 2 Find the volume generated by revolving the area cut off from the parabola
9y = 4(9 â x2) by the lines 4x + 3y = 12 about X-axis.
ðð§ð¬ð°ðð«: ððð/ð.
H 3 Find the volume of the solid obtained by rotating the region enclosed by the
curves y = x and y = x2 about the X-axis.
ðð§ð¬ð°ðð«: ðð/ðð.
DARSHAN INSTITUTE OF ENGINEERING & TECHNOLOGY Mathematics-I (3110014)
U N I T - 1 [ 2 2 ]
T 4 Define volume of solid of revolution by washerâs method and use it to find
the volume of solid generated when the region between the graphs
y =1
2 + x2 and y = x over the interval [0, 2] is revolved about X-axis.
ðð§ð¬ð°ðð«: ððð/ðð.
H 5 Find the volume of the solid obtained by rotating about the Y-axis the region
between y = x and y = x2.
ðð§ð¬ð°ðð«: ð/ð.
C 6 Find the volume of the solid that results when the region enclosed by the
curves y = x2 and x = y2 is revolved about the Y-axis.
ðð§ð¬ð°ðð«: ðð/ðð.
H 7 Find the volume of the solid that results when the region enclosed by
x = y2 and x = y is revolved about the line x = â1.
ðð§ð¬ð°ðð«: ðð/ðð.
C 8 Find the volume of the solid that results when the region enclosed by
y = x2 and y = x3 is revolved about the line x = 1.
ðð§ð¬ð°ðð«: ð/ðð.
C 9 Find the volume of the solid obtained by rotating the region R enclosed by
the curves y = x and y = x2 about the line y = 2.
ðð§ð¬ð°ðð«: ðð/ðð.
W-18 (7)
H 10 Determine the volume of the solid obtained by rotating the region bounded
by y = x2 â 2x and y = x about the line y = 4.
ðð§ð¬ð°ðð«: ðððð/ð.
â LENGTH OF PLANE CURVES
â If f is continuously differentiable on the closed interval [a, b], the length of the curve
y = f(x) from x = a to x = b is
L = â«â 1 + (dy
dx )2
b
a
dx = â«(1 + [f â²(x)]2)1 2
b
a
dx .
DARSHAN INSTITUTE OF ENGINEERING & TECHNOLOGY Mathematics-I (3110014)
U N I T - 1 [ 2 3 ]
METHOD â 14: LENGTH OF PLANE CURVES
C 1 Using the arc length formula, find the length of the curve y = x
3 2 , 0 †x †1.
ðð§ð¬ð°ðð«: ð
ðð [(ðð)
ð ð â ð].
H 2 Find the length of the curve y =
1
3 (x2 + 2)
3 2 from x = 0 to x = 3.
ðð§ð¬ð°ðð«: ðð.
H 3 Find the length of the curve y =
1
2 (ex + eâx) from x = 0 to 2.
ðð§ð¬ð°ðð«: ð
ð (ðð + ðâð).
C 4 Find the length of the curve x =
y3
3 +
1
4y from y = 1 to y = 3.
ðð§ð¬ð°ðð«: ðð
ð .
C 5 Find the length of the curve y = (
x
2 )
2 3 from x = 0 to x = 2.
ðð§ð¬ð°ðð«: ð
ðð [(ðð)
ð ð â ð].
T 6 Find the length of the arc of y2 = x3 between the points (1, 1) and (4, 8).
ðð§ð¬ð°ðð«:ð
ðð (ððâðð â ððâðð).
â AREA OF SURFACES BY REVOLUTION:
â Area of surface by revolution in Cartesian form:
⢠If the curve y = f(x) revolve about x-axis
from x = a to x = b then the Area of the
surface of a solid is
S = 2Ï â« yâ1 + (dy
dx )2
b
x=a
dx ⊠(1)
DARSHAN INSTITUTE OF ENGINEERING & TECHNOLOGY Mathematics-I (3110014)
U N I T - 1 [ 2 4 ]
⢠If the curve x = f(y) revolve about y-axis
from y = a to y = b then the Area of the
surface of a solid is
S = 2Ï â« xâ1 + (dx
dy )2
b
y=a
dy ⊠(2)
â Area of surface by revolution in Polar form:
If the curve r = f(Ξ), then the Area of the surface generated by revolution from Ξ = α
to Ξ = β,
(1). About Ξ = 0 (Initial line) is
S = 2Ï â« r sin Ξ âr2 + (dr
dΞ )2
β
Ξ=α
dΞ ⊠(3)
(2). About Ξ =Ï
2 is
S = 2Ï â« r cos Ξ âr2 + (dr
dΞ )2
β
Ξ=α
dΞ ⊠(4)
â Area of surface by revolution in Parametric form:
If the curve x = f(t) and y = g(t), are then the Area of the surface generated by revolution
from t = a to t = b,
(1). About x-axis is
S = 2Ï â« yâ( dx
dt)2
+ ( dy
dt)2
b
t=a
dt ⊠(5)
(2). About y-axis is
S = 2Ï â« xâ(dx
dt )2
+ ( dy
dt)2
b
t=a
dt ⊠(6)
DARSHAN INSTITUTE OF ENGINEERING & TECHNOLOGY Mathematics-I (3110014)
U N I T - 1 [ 2 5 ]
METHOD â 15: AREA OF SURFACES BY REVOLUTION
C 1 Find the surface are of a sphere obtained by revolving the semi-circle x2 +
y2 = a2, y > 0.
ðð§ð¬ð°ðð«: ðððð.
H 2 Find the area of the surface generated by revolving the curve y = 2âx from
1 †x †2 about the x-axis.
ðð§ð¬ð°ðð«: ðð
ð(ðâð â ðâð).
H 3 Find the area of the surface generated by revolving the curve y = x2 from
x = 1 to x = 2 about the y-axis.
ðð§ð¬ð°ðð«:ð
ð [(ðð)
ð ð â (ð)
ð ð ].
C 4 Find the surface area of the spindle shaped solid generated by revolving
asteroid x 2
3 + y 2
3 = a 2
3 about the x-axis.
ðð§ð¬ð°ðð«: ððððð
ð .
T 5 Find the area of the surface generated by revolving the cardioid r =
a(1 + cos Ξ), a > 0 about the initial line.
ðð§ð¬ð°ðð«: ððððð
ð .
H 6 Determine the surface area of the curve x = 2t, y = 4t , 0 †t †2 rotated
about (i) x-axis, (ii) y-axis.
ðð§ð¬ð°ðð«: ððâðð,ððâðð.
C 7 Determine the surface area of the curve x = 2 cos t, y = 2 sin t rotated about
x-axis between t = 0 and t = Ï.
ðð§ð¬ð°ðð«: ððð.
T 8 Find the area of the surface generated by revolution of the loop at the curve
x = t2, y = t â 1
3t3 about x-axis.
ðð§ð¬ð°ðð«: ðð.
â â â â â â â â â â
DARSHAN INSTITUTE OF ENGINEERING & TECHNOLOGY Mathematics-I (3110014)
U N I T - 2 [ 2 7 ]
UNIT 2
â SEQUENCE:
â A sequence is a set of numbers in a specific order.
â It is denoted by {an} ðð {an}n=1â ðð (an) ðð {an}nâ¥1.
â It is written as {an} = a1, a2, ⊠, an , âŠ
Where, a1 is called first term, a2 is called second term and in general an is the nth term of
{an}. The individual numbers which form the sequence are called
elements/terms/members of the sequence.
â Examples:
(ð). 2, 7, 12, 17,⊠= {(5n â 3)}1â
(ð). 1, 2, 3, 4, 5,⊠= {n}
(ð). {n
n + 1 }n=1
â
=1
2 , 2
3, ⊠,
n
n + 1 ,âŠ
(ð). {1
4n }nâ¥3
=1
43 ,1
44 , ⊠,
1
4n , âŠ
â BOUNDED AND UNBOUNDED SEQUENCE:
â A sequence {an} is said to be bounded above if â K â â such that an †K,ân ϵ â. Here K is
called an upper bound for {an}.
â A sequence {an} is said to be bounded below if â k â â such that k †an, ân ϵ â. Here k is
called a lower bound for {an}.
â A sequence {an} is said to be bounded if it is both bounded above and bounded below.
i.e., â k, K â â such that k †an †K, ân ϵ â.
â MONOTONIC SEQUENCE:
â A sequence {an} is said to be an increasing sequence if an †an+1, ân ϵ â.
â A sequence {an} is said to be a decreasing sequence if an ⥠an+1, ân ϵ â.
â A sequence {an} is said to be a monotonic if it is either increasing or decreasing.
DARSHAN INSTITUTE OF ENGINEERING & TECHNOLOGY Mathematics-I (3110014)
U N I T - 2 [ 2 8 ]
â THE SANDWICH THEOREM OR SQUEEZE THEOREM:
â Let {an}, {bn} & {cn} be the sequences of real numbers.
â If an †bn †cn , ân ϵ â and limnââ
an = limnââ
cn = L then limnââ
bn = L.
ððšðð: âi
n
i=1
= 1 + 2+ â¯+ n = n(n + 1)
2 .
METHOD â 1: CHARACTERISTICS OF SEQUENCE AND LIMIT
C 1 Check whether the following sequences are bounded or not:
(ð). {(2n â 1) }1â (ð). {
1
2 ,2
3 ,3
4 , ⊠} (ð). {(â1)n } (ð). {ân}.
ðð§ð¬ð°ðð«: (ð). ððšð®ð§ððð ððð¥ðšð° (ð). & (ð). ððšð®ð§ððð (ð).ððšð®ð§ððð ðððšð¯ð.
C 2 Are the following sequences monotonic? Justify your answer.
(ð). { 3n + 1
n + 1} (ð). {
1
2, 2
3, 3
4,⊠} (ð). {
n
2n } (ð). {
n
n2 + 1 } .
ðð§ð¬ð°ðð«: (ð). & (ð). ðð§ðð«ððð¬ð¢ð§ð (ð). & (ð).ðððð«ððð¬ð¢ð§ð .
C 3 Find the following limits:
(ð). limnââ
(ân + 1 â ân) (ð). limnââ
( nân
2n3) (ð). lim
nââ(n!
nn) (ð). lim
nââ( cos2014n
n).
ðð§ð¬ð°ðð«: (ð). ð (ð).ð/ð (ð). ð (ð). ð.
T 4 Find the following limits:
(ð). limnââ
(n2
n + 5 ) (ð). lim
nââ(
3n
n + 7n1 2 ) (ð). lim
nââ{ (â1)n
3n}
(ð). limnââ
( sin n
n) (ð). lim
nââ(log (
2n + 1
n)).
ðð§ð¬ð°ðð«: (ð).â (ð). ð (ð). ð (ð). ð (ð). ð¥ðšð ð.
â CONVERGENT SEQUENCE AND DIVERGENT SEQUENCE:
â A sequence {an} is said to be convergent if ð¥ð¢ðŠð§ââ
ðð§ = ð, where an is the nth term of given
sequence and âLâ is any finite real number.
DARSHAN INSTITUTE OF ENGINEERING & TECHNOLOGY Mathematics-I (3110014)
U N I T - 2 [ 2 9 ]
â A sequence {an} is said to be divergent if ð¥ð¢ðŠð§ââ
ðð§ = â, where an is the nth term of given
sequence.
â OSCILLATING SEQUENCE:
â A sequence which is neither convergent nor divergent is said to be an oscillating
sequence.
â NOTES:
â An increasing sequence which is bounded above is always convergent.
â A decreasing sequence which is bounded below is always convergent.
â Every bounded and monotonic sequence is convergent.
METHOD â 2: CONVERGENCE OF SEQUENCE
C 1 Check the convergence of following sequences: (ð). {
n2 + 1
2n2 â 1 } (ð). {4n+1}
(ð). {4 â (â1)n }.
ðð§ð¬ð°ðð«: (ð). ððšð§ð¯ðð«ð ðð¬ ððš ð. ð (ð).ðð¢ð¯ðð«ð ðð¬ ððš â (ð). ð ð¢ð§ð¢ððð¥ð² ðšð¬ðð¢ð¥ð¥ððð
ðððð°ððð§ ð ðð§ð ð.
H 2 Check the convergence of the sequences: (ð). {8 + (â1)n } (ð). {n2(â1)n }.
ðð§ð¬ð°ðð«: (ð). ð ð¢ð§ð¢ððð¥ð² ðšð¬ðð¢ð¥ð¥ððð¢ð§ð ðððð°ððð§ ð & ð
(ð). ðð§ðð¢ð§ð¢ððð¥ð² ðšð¬ðð¢ð¥ð¥ððð¢ð§ð ðððð°ððð§ ââ & â.
C 3 Check the convergence of the sequence {
2n + 5n
5n + 3n } .
ðð§ð¬ð°ðð«: ððšð§ð¯ðð«ð ðð¬ ððš ð.
C 4 Show that the sequence {
sin n
n} converges to 0.
H 5 Check the convergence of the following sequences:
(ð). { cos2012 n
n} (ð). {
cos2 n
2n} (ð). {
1
n(â1)n+1 } .
ðð§ð¬ð°ðð«: ðð¥ð¥ ðð¡ð«ðð ð¬ððªð®ðð§ððð¬ ðð«ð ððšð§ð¯ðð«ð ðð¬ ððš ð.
DARSHAN INSTITUTE OF ENGINEERING & TECHNOLOGY Mathematics-I (3110014)
U N I T - 2 [ 3 0 ]
T 6 Check convergence of the sequence {
n!
nn } .
ðð§ð¬ð°ðð«: ððšð§ð¯ðð«ð ðð¬ ððš ð.
C 7 Define the convergence of a sequence (an) and verify whether the sequence
whose nth term is an = ( n + 1
n â 1)n
converges or not.
ðð§ð¬ð°ðð«: ððšð§ð¯ðð«ð ðð¬ ððš ðð.
S-19 (3)
H 8 Check convergence of the sequence {
log n
n} .
ðð§ð¬ð°ðð«: ððšð§ð¯ðð«ð ðð¬ ððš ð.
H 9 Check the convergence of the following sequences:
(ð). {1
3n } (ð). {ân
n } (ð). {n2eân }.
ðð§ð¬ð°ðð«: ðð¥ð¥ ðð¡ð ð¬ððªð®ðð§ððð¬ ðð«ð ððšð§ð¯ðð«ð ðð§ð ðð§ð ð¢ð ððšð§ð¯ðð«ð ðð¬ ððš ð, ð &
ð ð«ðð¬ð©ðððð¢ð¯ðð¥ð².
H 10 Check the convergence of the sequences {
32n
5n} and {
93n
152n } .
ðð§ð¬ð°ðð«: ððšðð¡ ðð¡ð ð¬ððªð®ðð§ððð¬ ðð«ð ðð¢ð¯ðð«ð ðð§ð.
T 11 Check the convergence of the following sequences: (ð). {ân (ân + 1 â ân) }
(ð). {ân2 + 1 â n } (ð). {(1 +x
n )n
} (ð). {( n â 2011
n)n
} .
ðð§ð¬ð°ðð«: ðð¥ð¥ ðð¡ð ð¬ððªð®ðð§ððð¬ ðð«ð ððšð§ð¯ðð«ð ðð§ð ðð§ð ð¢ð ððšð§ð¯ðð«ð ðð¬ ððš
ð. ð, ð, ðð± & ðâðððð ð«ðð¬ð©ðððð¢ð¯ðð¥ð².
C 12 For which values of r, { rn } is convergent?
ðð§ð¬ð°ðð«: â ð < ð« †ð.
C 13 Check the convergence of the following sequences:
(ð). {0.3, 0.33,0.333,⊠} (ð). {â6,â6â6,â6â6â6,⊠} .
ðð§ð¬ð°ðð«: ððšðð¡ ðð¡ð ð¬ððªð®ðð§ððð¬ ðð«ð ððšð§ð¯ðð«ð ðð§ð ðð§ð ð¢ð ððšð§ð¯ðð«ð ðð¬ ððš
ð/ð & ð ð«ðð¬ð©ðððð¢ð¯ðð¥ð².
DARSHAN INSTITUTE OF ENGINEERING & TECHNOLOGY Mathematics-I (3110014)
U N I T - 2 [ 3 1 ]
T 14 Show that the sequence {an } is monotonic increasing and bounded, where
nth terms is an =1
1! +1
2! +1
3! + â¯+
1
n! . Is it convergent?
ðð§ð¬ð°ðð«: ððð¬.
â CONTINUOUS FUNCTION THEOREM FOR SEQUENCES:
â Let {an} be a sequence of real numbers. If an â L as n â â and if f is a continuous at L
and defined at all an, then f(an) â f(L).
METHOD â 3: CONTINUOUS FUNCTION THEOREM
T 1
Show that { â n + 1
n } â 1as n â â.
H 2 Check the convergence of the sequence {
1
2n } .
ðð§ð¬ð°ðð«: ððšð§ð¯ðð«ð ðð¬ ððš ð.
C 3 Check the convergence of the sequence {4
1 3n } .
ðð§ð¬ð°ðð«: ððšð§ð¯ðð«ð ðð¬ ððš ð.
â INFINITE SERIES:
â If u1, u2, u3, ⊠, un, ⊠is an infinite sequence of real numbers, then the sum of the terms of
the sequence, u1 + u2 + u3 +â¯+ un +⯠is called an infinite series.
The infinite series u1 + u2 + u3 +â¯+ un +⯠is denoted by âun
â
n=1
or âun.
â SEQUENCE OF PARTIAL SUMS:
â Let â un be a given infinite series. Consider S1 = u1, S2 = u1 + u2, ⊠, Sn = u1 + u2 +â¯+
un. Then {Sn} defined as above is called the sequence of partial sums of the given series.
DARSHAN INSTITUTE OF ENGINEERING & TECHNOLOGY Mathematics-I (3110014)
U N I T - 2 [ 3 2 ]
â CONVERGENCE OF AN INFINITE SERIES:
â An infinite series âun is said to be convergent if the sequence of partial sums {Sn}
converges to some finite number L.
i. e. limnââ
Sn = L â âun
â
n=1
= L,where L is the sum of series.
â If the sequence of partial sums {Sn} does not converge, then the series is said to be
divergent.
â POSITIVE TERM SERIES:
â If all terms after few negative terms in an infinite series are positive, such a series is
called a positive terms series.
â TEST FOR GEOMETRIC SERIES:
An Infinite series of the form âarnâ1â
n=1
= a + ar + ar2 + ar3 +⯠is known as
Geometric Series, where ârâ is common ratio and âaâ is the ðð¬ð term, then given series is
(ð). ððšð§ð¯ðð«ð ðð§ð if â ð < ð« < ð and itâs sum is given by ð¬ =ð
ð â ð« .
(ð). ðð¢ð¯ðð«ð ðð§ð if ð« ⥠ð.
(ð). ðšð¬ðð¢ð¥ð¥ððð¢ð§ð if ð« †âð. (Finitely oscillating if r = â1 & Infinitely oscillating if
r < â1 ).
METHOD â 4: GEOMETRIC SERIES
C 1 Test the convergence of â
32n
23n
â
n=0
.
ðð§ð¬ð°ðð«: ðð¢ð¯ðð«ð ðð§ð.
C 2 Define the Geometric series and find the sum of â
3nâ1 â 1
6nâ1
â
n=1
.
ðð§ð¬ð°ðð«: ð/ð.
DARSHAN INSTITUTE OF ENGINEERING & TECHNOLOGY Mathematics-I (3110014)
U N I T - 2 [ 3 3 ]
H 3 Test the convergence of seriesâ
4n + 5n
6n
â
n=1
.
ðð§ð¬ð°ðð«: ððšð§ð¯ðð«ð ðð§ð.
W-18 (3)
T 4 Check the convergence of (ð).â
92n
12n
â
n=1
(ð).â(1
2n +
1
3n )
â
n=0
.
ðð§ð¬ð°ðð«: (ð).ðð¢ð¯ðð«ð ðð§ð (ð). ððšð§ð¯ðð«ð ðð¬ ððš ð/ð.
C 5 Test the convergence for 5 â
10
3+ 20
9â 40
27+⯠.
ðð§ð¬ð°ðð«: ððšð§ð¯ðð«ð ðð§ð.
H 6 Prove that 1 +
2
3+ 4
9+ 8
27 + 16
81⊠is convergent and find itâ²s sum.
ðð§ð¬ð°ðð«: ð.
H 7 Check the convergence of 1 + 2 (
1
3 ) + (
1
3 )2
+ 2(1
3 )3
+ (1
3 )4
+⯠.
ðð§ð¬ð°ðð«: ððšð§ð¯ðð«ð ðð¬ ððš ðð/ð.
T 8 Find the sum of the seriesâ
1
4n .
nâ¥2
ðð§ð¬ð°ðð«: ð/ðð.
S-19 (2)
C 9 c The figure shows the first seven of a sequence of
squares. The outermost square has an area of 4m2.
Each of the other squares is obtained by joining the
midpoints of the sides of the squares before it. Find
the sum of areas of all squares in the infinite
sequence.
ðð§ð¬ð°ðð«: ð.
T 10 You drop a ball from âaâ meters above a flat surface. Each time the ball hits
the surface after falling a distance h, it rebounds a distance rh, where 0 <
r < 1. Find the total distance ball travels up and down when a = 6 m and
r =2
3 m.
ðð§ð¬ð°ðð«: ðððŠ.
DARSHAN INSTITUTE OF ENGINEERING & TECHNOLOGY Mathematics-I (3110014)
U N I T - 2 [ 3 4 ]
â TELESCOPING SERIES
â A telescoping series is a series in which many terms cancel in the partial sums.
â Procedure to check convergent/divergent of telescoping series:
(1). Find nth term un of given series if it is not directly given.
(2). Using partial fraction separate it into two or three individual terms.
(3). Find Sn (sum of first n terms) using above individual terms.
(4). Find limnââ
Sn. If above limit is finite, then the given series is converging to the limit
otherwise divergent.
METHOD â 5: TELESCOPING SERIES
C 1 Check the convergence of
1
1 â 3 +
1
3 â 5 +
1
5 â 7 + ⯠.
ðð§ð¬ð°ðð«: ððšð§ð¯ðð«ð ðð¬ ððš ð/ð.
H 2 Check the convergence of
1
1 â 2 +
1
2 â 3 +
1
3 â 4 + ⯠.
ðð§ð¬ð°ðð«: ððšð§ð¯ðð«ð ðð¬ ððš ð.
H 3 Check the convergence of
1
1 â 5 +
1
5 â 9 +
1
9 â 13 + ⯠.
ðð§ð¬ð°ðð«: ððšð§ð¯ðð«ð ðð¬ ððš ð/ð.
T 4 Find the sum of the seriesâ
4
(4n â 3)(4n + 1) .
nâ¥1
ðð§ð¬ð°ðð«: ð.
S-19 (2)
C 5 Check the convergence of
1
2! +2
3! +3
4! + ⯠.
ðð§ð¬ð°ðð«: ððšð§ð¯ðð«ð ðð¬ ððš ð.
H 6 Check the convergence of â
4
n2 + 2n
â
n=1
.
ðð§ð¬ð°ðð«: ððšð§ð¯ðð«ð ðð¬ ððš ð.
DARSHAN INSTITUTE OF ENGINEERING & TECHNOLOGY Mathematics-I (3110014)
U N I T - 2 [ 3 5 ]
C 7 Check the convergence of â tanâ1 (
1
n2 + n + 1 )
â
n=1
.
ðð§ð¬ð°ðð«: ððšð§ð¯ðð«ð ðð¬ ððš ð/ð.
T 8 Check the convergence of â[tanâ1(n) â tanâ1(n + 1)].
â
n=1
ðð§ð¬ð°ðð«: ððšð§ð¯ðð«ð ðð¬ ððš â ð/ð.
C 9 Check the convergence of log2 + log
3
2+ log
4
3+ ⯠.
ðð§ð¬ð°ðð«: ðð¢ð¯ðð«ð ðð§ð.
T 10 Check the convergence of
1
1 â 2 â 3 +
1
2 â 3 â 4 +
1
3 â 4 â 5 + ⯠.
ðð§ð¬ð°ðð«: ððšð§ð¯ðð«ð ðð¬ ððš ð/ð.
â ZERO TEST OR CAUCHYâS FUNDAMENTAL TEST FOR DIVERGENCE OR NTH TERM TEST:
â Let âun be given series where un is the nth term of the series. If ð¥ð¢ðŠð§ââ
ð®ð§ â ð or does not
exist then the given series is divergent.
METHOD â 6: ZERO TEST
C 1 Check convergence of (ð).â
n
2n + 5
â
n=1
(ð).ân2 + 1
3n2 â 1
â
n=0
(ð).â(2n + 3
3n + 12 )2â
n=1
(ð).â(1 +1
n )nâ
n=1
(ð).âân
n + 1
â
n=1
.
ðð§ð¬ð°ðð«: ðð¥ð¥ ðð¡ð ðð¢ð¯ð ð¬ðð«ð¢ðð¬ ðð«ð ðð¢ð¯ðð«ð ðð§ð.
C 2 Check the convergence ofân tan (
1
n )
â
n=1
.
ðð§ð¬ð°ðð«: ðð¢ð¯ðð«ð ðð§ð.
H 3 Check the convergence of 2 +
3
2+ 4
3+ 5
4+⯠.
ðð§ð¬ð°ðð«: ðð¢ð¯ðð«ð ðð§ð.
DARSHAN INSTITUTE OF ENGINEERING & TECHNOLOGY Mathematics-I (3110014)
U N I T - 2 [ 3 6 ]
T 4 Check the convergence of
1
1 + 2â1 +
1
1 + 2â2 +
1
1 + 2â3 + ⯠.
ðð§ð¬ð°ðð«: ðð¢ð¯ðð«ð ðð§ð.
C 5 Check the convergence of
1
â2 +
2
â5 +
4
â17 +
8
â65 + ⯠.
ðð§ð¬ð°ðð«: ðð¢ð¯ðð«ð ðð§ð.
H 6
Check the convergence of â1
3 â 2 + â
2
3 â 3 + â
3
3 â 4 + ⯠.
ðð§ð¬ð°ðð«: ðð¢ð¯ðð«ð ðð§ð.
T 7
Check the convergence of 1 + â1
2 + â
1
3
3
+ â1
4
4
+⯠.
ðð§ð¬ð°ðð«: ðð¢ð¯ðð«ð ðð§ð.
â COMBINING SERIES:
â Whenever we have two convergent series, we add them term by term, subtract term by
term or multiply them by constant to make new convergent series.
â Rules: If âan = a and âbn = b are convergent series then
(1). Sum rule: â(an + bn) = â an +âbn = a + b.
(2). Difference rule: â(an â bn) = â an ââbn = a â b.
(3). Constant multiple rule: âkan = kâan = k â a.
â If âan diverges, then âkan also diverges.
â If âan converges and âbn diverges, then â(an + bn) and â(an â bn) both diverges.
â It is not necessary that âan and âbn both diverges â â(an + bn) diverges.
METHOD â 7: COMBINING SERIES
H 1 Give an example of sum and difference rule.
H 2 Give an example of constant multiple rule.
DARSHAN INSTITUTE OF ENGINEERING & TECHNOLOGY Mathematics-I (3110014)
U N I T - 2 [ 3 7 ]
C 3 Give an example of two divergent series whose sum is convergent.
T 4 Give an example of two divergent series whose sum is divergent.
â INTEGRAL TEST:
For the positive term seriesâun
â
n=a
=âf(n),
â
n=a
if f(n) is ðððð«ððð¬ð¢ð§ð and ð¢ð§ððð ð«ððð¥ð and
(ð). If â« f(x)
â
a
dx is ðð¢ð§ð¢ðð, then the given series is ððšð§ð¯ðð«ð ðð§ð.
(ð). If â« f(x)
â
a
dx is ð¢ð§ðð¢ð§ð¢ðð, then the given series is ðð¢ð¯ðð«ð ðð§ð.
â P-SERIES:
The series â1
np
â
n=1
is known as p â series.
(ð). â1
np
â
n=1
is ððšð§ð¯ðð«ð ðð§ð, if ð© > ð.
(ð). â1
np
â
n=1
is ðð¢ð¯ðð«ð ðð§ð, if ð© †ð.
METHOD â 8: INTEGRAL TEST
C 1 Check the convergence of ân â eân
2
â
n=1
.
ðð§ð¬ð°ðð«: ððšð§ð¯ðð«ð ðð§ð.
H 2 Check the convergence of âeân
â
n=1
.
ðð§ð¬ð°ðð«: ððšð§ð¯ðð«ð ðð§ð.
DARSHAN INSTITUTE OF ENGINEERING & TECHNOLOGY Mathematics-I (3110014)
U N I T - 2 [ 3 8 ]
C 3 Test the convergence of â
2 tanâ1 n
1 + n2
â
n=1
.
ðð§ð¬ð°ðð«: ððšð§ð¯ðð«ð ðð§ð.
T 4 Check convergence of (ð).â
en
e2n + 1 (ð).â
eâân
ân
â
n=1
(ð).â etan
â1 n
n2 + 1 .
ðð§ð¬ð°ðð«: ðð¥ð¥ ðð¡ð ðð¡ð«ðð ð¬ðð«ð¢ðð¬ ðð«ð ððšð§ð¯ðð«ð ðð§ð.
H 5 Show that the ððð«ðŠðšð§ð¢ð ð¬ðð«ð¢ðð¬ â
1
n is divergent.
â
n=1
â DIRECT COMPARISON TEST:
â If two given positive term series â un and â vn with 0 †un †vn, ân ϵ â and
(1). If â vn is convergent, then â un is also convergent.
(2). If âun is divergent, then âvn is also divergent.
â REMARKS:
(ð). If bigger series is convergent, then smaller series is also convergent.
(ð). If smaller series is divergent, then bigger series is also divergent.
(ð). For the seriesâ logn
np, if p †1 then
1
np †logn
np .
(ð). For the series â logn
np, if p > 1 then
logn
npâ€
1
n( p+1 2
) .
METHOD â 9: DIRECT COMPARISION TEST
C 1 Check convergence of (ð).â
1
ân3 + 1
â
n=1
(ð).â1
2ân â 1
â
n=1
(ð).â1
3n + 1
â
n=1
.
ðð§ð¬ð°ðð«: (ð). ððšð§ð¯ðð«ð ðð§ð (ð).ðð¢ð¯ðð«ð ðð§ð (ð). ððšð§ð¯ðð«ð ðð§ð.
H 2 Show that
1
2 +1
5 +
1
10 +
1
17 + ⯠is convergent.
DARSHAN INSTITUTE OF ENGINEERING & TECHNOLOGY Mathematics-I (3110014)
U N I T - 2 [ 3 9 ]
H 3 Determine whether the series â
1
2n2 + 4n + 3
â
n=1
is convergent or divergent.
ðð§ð¬ð°ðð«: ððšð§ð¯ðð«ð ðð§ð.
C 4 Check the convergence of (ð).â
logn
n
â
n=1
(ð).â logn
n2
â
n=1
(ð).â logn
ân
â
n=1
.
ðð§ð¬ð°ðð«: (ð).ðð¢ð¯ðð«ð ðð§ð (ð). ððšð§ð¯ðð«ð ðð§ð (ð).ðð¢ð¯ðð«ð ðð§ð.
T 5 Check the convergence of the series â
( log n)3
n3
â
n=1
.
ðð§ð¬ð°ðð«: ððšð§ð¯ðð«ð ðð§ð.
S-19 (3)
â LIMIT COMPARISON TEST:
â If two positive term series â un (given) & â vn (choose) be such that
L = limnââ
un
vn and
(1). If L is finite and non-zero, then âun & â vn both are convergent or divergent
together.
(2). If ð = ð and â vn is convergent, then â un is also convergent.
(3). If ð = â and âvn is divergent, then â un is also divergent.
â Procedure to discuss about convergence of a series using limit comparison test:
(1). Find un from given series if it is not directly given.
(2). Derive vn according to un.
(3). Find the value of L = limnââ
un
vn .
(4). Write the conclusion from limit comparison test.
DARSHAN INSTITUTE OF ENGINEERING & TECHNOLOGY Mathematics-I (3110014)
U N I T - 2 [ 4 0 ]
METHOD â 10: LIMIT COMPARISION TEST
C 1 Check the convergence of â
3n2 + 5n
7 + n4
â
n=1
.
ðð§ð¬ð°ðð«: ððšð§ð¯ðð«ð ðð§ð.
H 2 State the p â series test.
Discuss the convergence of the seriesân + 1
n3 â 3n + 2
â
n=2
.
ðð§ð¬ð°ðð«: ððšð§ð¯ðð«ð ðð§ð.
W-18 (4)
C 3 Check convergence of (ð).â
1
2n â 3
â
n=1
(ð).â 3 + 2 sin n
n3(ð).â[10
1 n â 1].
ðð§ð¬ð°ðð«: (ð). ððšð§ð¯ðð«ð ðð§ð (ð). ððšð§ð¯ðð«ð ðð§ð (ð).ðð¢ð¯ðð«ð ðð§ð.
H 4 Check the convergence of (ð).â
7
7n â 2 (ð).â
1
5n â 1
â
n=1
.
ðð§ð¬ð°ðð«: (ð).ðð¢ð¯ðð«ð ðð§ð (ð). ððšð§ð¯ðð«ð ðð§ð.
H 5 Check the convergence of (ð).â
1
n2 + n + 1
â
n=1
(ð).â1
ân3 â 2011
â
n=1
.
ðð§ð¬ð°ðð«: (ð). ððšð§ð¯ðð«ð ðð§ð (ð).ðð¢ð¯ðð«ð ðð§ð.
H 6 Check convergence of (ð).â
1
n2 + 1 (ð).â
1
ân2 â 1 (ð).â
1
n(n + 1) .
ðð§ð¬ð°ðð«: (ð). ððšð§ð¯ðð«ð ðð§ð (ð).ðð¢ð¯ðð«ð ðð§ð (ð). ððšð§ð¯ðð«ð ðð§ð.
T 7 Check the convergence of (ð).ân
an2 + b (ð).â
n
1 + nân + 1 .
ðð§ð¬ð°ðð«: ððšðð¡ ðð¡ð ð¬ðð«ð¢ðð¬ ðð«ð ðð¢ð¯ðð«ð ðð§ð.
T 8 Test the convergence of the series â
ân
n2 + 1
â
n=1
.
ðð§ð¬ð°ðð«: ððšð§ð¯ðð«ð ðð§ð.
W-19 (4)
T 9
Determine convergence or divergence of the seriesâ(2n2 â 1)
13
(3n3 + 2n+ 5)14
â
n=1
.
ðð§ð¬ð°ðð«: ðð¢ð¯ðð«ð ðð§ð.
DARSHAN INSTITUTE OF ENGINEERING & TECHNOLOGY Mathematics-I (3110014)
U N I T - 2 [ 4 1 ]
C 10 Check the convergence of â
ân + 1 â ân
n & â(ân4 + 1 âân4 â 1 )
â
n=1
.
ðð§ð¬ð°ðð«: ððšðð¡ ðð¡ð ð¬ðð«ð¢ðð¬ ðð«ð ððšð§ð¯ðð«ð ðð§ð.
H 11 Check the convergence of (ð).â[ân2 + 1 â n ] (ð).â [ ân3 + 1 â ân3 ]
â
n=1
.
ðð§ð¬ð°ðð«: (ð).ðð¢ð¯ðð«ð ðð§ð (ð). ððšð§ð¯ðð«ð ðð§ð.
T 12 Check the convergence of â
np
ân + 1 + ân
â
n=1
.
ðð§ð¬ð°ðð«: ð© ⥠âð/ð â¹ ðð¢ð¯ðð«ð ðð§ð & ð© < âð/ðâ¹ ððšð§ð¯ðð«ð ðð§ð.
C 13 Check the convergence of (ð).â tan (
1
n )
â
n=1
(ð).â1
n sin (
1
n )
â
n=1
.
ðð§ð¬ð°ðð«: (ð).ðð¢ð¯ðð«ð ðð§ð (ð). ððšð§ð¯ðð«ð ðð§ð.
H 14 Check the convergence of (ð).â sin (
1
n )
â
n=1
(ð).â sin3 (1
n )
â
n=1
.
ðð§ð¬ð°ðð«: (ð).ðð¢ð¯ðð«ð ðð§ð (ð). ððšð§ð¯ðð«ð ðð§ð.
C 15 Test the convergence of â
1
1 + 22 + 32 +â¯+ n2
â
n=1
.
ðð§ð¬ð°ðð«: ððšð§ð¯ðð«ð ðð§ð.
C 16 Test the convergence of
1
12 â 3 +
3
22 â 3 +
5
32 â 3 + ⯠.
ðð§ð¬ð°ðð«: ðð¢ð¯ðð«ð ðð§ð.
H 17 Check the convergence of (ð). 2 +
3
4+ 4
9+⯠& (ð).
3
4 + 5
9+
7
16 + ⯠.
ðð§ð¬ð°ðð«: ððšðð¡ ð¬ðð«ð¢ðð¬ ðð«ð ðð¢ð¯ðð«ð ðð§ð.
H 18 Test for convergence of the series
1
1 â 2 +
1
2 â 3 +
1
3 â 4 + ⯠.
ðð§ð¬ð°ðð«: ððšð§ð¯ðð«ð ðð¬ ððš ð.
H 19 Test the convergence of
1
1 â 2 â 3 +
3
2 â 3 â 4 +
5
3 â 4 â 5 + ⯠.
ðð§ð¬ð°ðð«: ððšð§ð¯ðð«ð ðð§ð.
DARSHAN INSTITUTE OF ENGINEERING & TECHNOLOGY Mathematics-I (3110014)
U N I T - 2 [ 4 2 ]
T 20 Test the convergence of the Series
1 â 2
32 â 42 +
3 â 4
52 â 62 +
5 â 6
72 â 82 + ⯠.
ðð§ð¬ð°ðð«: ððšð§ð¯ðð«ð ðð§ð.
T 21 Check the convergence of
2
1p +
3
2p +
4
3p + ⯠.
ðð§ð¬ð°ðð«: ð© †ð â¹ ðð¢ð¯ðð«ð ðð§ð, ð© > ð â¹ ððšð§ð¯ðð«ð ðð§ð.
â DâALEMBERTâS RATIO TEST:
Let âun
â
n=a
be the given positive term series and L = limn â â
| un un+1
| then
(1). If ð > ð, then the given series is ððšð¯ðð«ð ðð§ð.
(2). If ð †ð < ð, then the given series is ðð¢ð¯ðð«ð ðð§ð.
(3). If ð = ð, then the ð«ððð¢ðš ððð¬ð ððð¢ð¥ð¬.
â Procedure to discuss about convergence using ratio test:
(1). Find un from given series if it is not directly given.
(2). Write un+1 from un.
(3). Find the value of L = limn â â
| un
un+1 |.
(4). Write the conclusion from the ratio test.
METHOD â 11: RATIO TEST
C 1 State Dâ²Alembertâ²s ratio test. Discuss the convergence of â
4n(n + 1)!
nn+1
â
n=1
.
ðð§ð¬ð°ðð«: ðð¢ð¯ðð«ð ðð§ð.
W-18 (4)
C 2 Check the convergence of (ð).â
2n
n2 (ð).â
2n â 1
3n
â
n=0
(ð).â n! 2n
nn
â
n=1
.
ðð§ð¬ð°ðð«: (ð).ðð¢ð¯ðð«ð ðð§ð (ð). ððšð§ð¯ðð«ð ðð§ð (ð). ððšð§ð¯ðð«ð ðð§ð.
DARSHAN INSTITUTE OF ENGINEERING & TECHNOLOGY Mathematics-I (3110014)
U N I T - 2 [ 4 3 ]
H 3 Check the convergence of (ð).â
2n
n! (ð).â
n!
n2 .
ðð§ð¬ð°ðð«: (ð). ððšð§ð¯ðð«ð ðð§ð (ð).ðð¢ð¯ðð«ð ðð§ð.
H 4 Check the convergence of (ð).â
n2
3n (ð).â
n
eân
â
n=1
.
ðð§ð¬ð°ðð«: (ð). ððšð§ð¯ðð«ð ðð§ð (ð).ðð¢ð¯ðð«ð ðð§ð.
H 5 Check the convergence of â
2n
n3 + 1
â
n=1
.
ðð§ð¬ð°ðð«: ðð¢ð¯ðð«ð ðð§ð.
T 6 Test for convergence of â
n10
10n .
ðð§ð¬ð°ðð«: ððšð§ð¯ðð«ð ðð§ð.
T 7 t Check convergence of (ð).â(
1
1 + n )n
(ð).ân!
nn (ð).â
n 2n(n + 1)!
3n n!
â
n=1
.
ðð§ð¬ð°ðð«: ðð¥ð¥ ðð¡ð ð¬ðð«ð¢ðð¬ ðð«ð ððšð§ð¯ðð«ð ðð§ð.
C 8 Check the convergence of 1 +
2p
2!+ 3p
3!+ 4p
4!+ ⯠.
ðð§ð¬ð°ðð«: ððšð§ð¯ðð«ð ðð§ð.
C 9 Check the convergence of
1
3+ 1 â 2
3 â 5+ 1 â 2 â 3
3 â 5 â 7+⯠.
ðð§ð¬ð°ðð«: ððšð§ð¯ðð«ð ðð§ð.
H 10 Check the convergence of
1
1 + 2 +
2
1 + 22 +
3
1 + 23 + ⯠.
ðð§ð¬ð°ðð«: ððšð§ð¯ðð«ð ðð§ð.
H 11 Check the convergence of 1 +
3
2! +5
3! +7
4! + ⯠.
ðð§ð¬ð°ðð«: ððšð§ð¯ðð«ð ðð§ð.
T 12 Check the convergence of
1
10 +
2!
102 +
3!
103 + ⯠.
ðð§ð¬ð°ðð«: ðð¢ð¯ðð«ð ðð§ð.
DARSHAN INSTITUTE OF ENGINEERING & TECHNOLOGY Mathematics-I (3110014)
U N I T - 2 [ 4 4 ]
C 13 Check convergence of (ð).â
3n + 4n
4n + 5n & (ð).â
5 â 9 â 13 â ⊠â (4n + 1)
1 â 2 â 3⊠â n.
ðð§ð¬ð°ðð«: (ð). ððšð§ð¯ðð«ð ðð§ð (ð).ðð¢ð¯ðð«ð ðð§ð.
C 14 Test the convergence of the series
x
1 â 2 +
x2
3 â 4 +
x3
5 â 6 +
x4
7 â 8 + ⯠.
ðð§ð¬ð°ðð«: ð± > ðâ¹ ðð¢ð¯ðð«ð ðð§ð & ð± †ð â¹ ððšð§ð¯ðð«ð ðð§ð.
W-19 (7)
H 15
Test the convergence ofâ1
2 x + â
2
5 x2 + â
3
10 x3 +â¯â, x > 0.
ðð§ð¬ð°ðð«: ð± ⥠ðâ¹ ðð¢ð¯ðð«ð ðð§ð & ð < ð± < ð â¹ ððšð§ð¯ðð«ð ðð§ð.
T 16 Test the convergence of the series
1
1 â 2 â 3 +
x
4 â 5 â 6 +
x2
7 â 8 â 9 +
x3
10 â 11 â 12 ⊠, x ⥠0.
ðð§ð¬ð°ðð«: ð± ⥠ðâ¹ ððšð§ð¯ðð«ð ðð§ð & ð †ð± < ð â¹ ðð¢ð¯ðð«ð ðð§ð.
W-18 (7)
â RABBEâS TEST:
Let âun
â
n=a
be the given positive term series and L = limnââ
{ n â (un un+1
â 1) } then
(1). If ð > ð, then the given series is ððšð¯ðð«ð ðð§ð.
(2). If ð < ð, then the given series is ðð¢ð¯ðð«ð ðð§ð.
(3). If ð = ð, then ððððð ððð¬ð ððð¢ð¥ð¬.
â Procedure to discuss about convergence using Rabbeâs test:
(1). Find un from given series if it is not directly given.
(2). Write un+1 from un.
(3). Find the value of L = limnââ
{ n â (un
un+1 â 1) }.
(4). Write the conclusion from the Rabbeâs test.
DARSHAN INSTITUTE OF ENGINEERING & TECHNOLOGY Mathematics-I (3110014)
U N I T - 2 [ 4 5 ]
METHOD â 12: RABBEâS TEST
C 1 Check the convergence of â
4n(n!)2
(2n)!
â
n=1
.
ðð§ð¬ð°ðð«: ðð¢ð¯ðð«ð ðð§ð.
C 2 Check the convergence of (
1
2 )3
+ (1 â 3
2 â 4 )3
+ (1 â 3 â 5
2 â 4 â 6 )3
+⯠.
ðð§ð¬ð°ðð«: ððšð§ð¯ðð«ð ðð§ð.
H 3 Check the convergence of
2
7 +
2 â 5
7 â 10 +
2 â 5 â 8
7 â 10 â 13 + ⯠.
ðð§ð¬ð°ðð«: ððšð§ð¯ðð«ð ðð§ð.
T 4 Check the convergence of (
1
3 )2
+ (1 â 4
3 â 6 )2
+ (1 â 4 â 7
3 â 6 â 9 )2
+⯠.
ðð§ð¬ð°ðð«: ððšð§ð¯ðð«ð ðð§ð.
C 5 Check the convergence of â
1 â 3 â 5âŠâŠ (2n â 1)
2 â 4 â 6âŠâŠ .2nxn
â
n=1
, x > 0.
ðð§ð¬ð°ðð«: ð± ⥠ðâ¹ ðð¢ð¯ðð«ð ðð§ð & ð < ð± < ð â¹ ððšð§ð¯ðð«ð ðð§ð.
H 6 Check the convergence of â
(n!)2
(2n)! xn
â
n=1
, x > 0.
ðð§ð¬ð°ðð«: ð± ⥠ðâ¹ ðð¢ð¯ðð«ð ðð§ð & ð < ð± < ð â¹ ððšð§ð¯ðð«ð ðð§ð.
â CAUCHYâS NTH ROOT (RADICAL) TEST:
Let âun
â
n=a
be the given positive term series and L = limnââ
{ âunn } = lim
nââ ( un )
1 n then
(1). If ð †ð < ð, then the given series is ððšð¯ðð«ð ðð§ð.
(2). If ð > ð, then the given series is ðð¢ð¯ðð«ð ðð§ð.
(3). If ð = ð, then the ð«ðšðšð ððð¬ð ððð¢ð¥ð¬.
â Procedure to discuss about convergence using root test:
(1). Find un from given series if it is not directly given.
DARSHAN INSTITUTE OF ENGINEERING & TECHNOLOGY Mathematics-I (3110014)
U N I T - 2 [ 4 6 ]
(2). Find the value of L = limnââ
{ âunn } = lim
nââ (un)
1
n .
(3). Write the conclusion from the root test.
METHOD â 13: CAUCHYâS NTH ROOT TEST
C 1 Check the convergence of â
(n + ân)n
3n nn+1
â
n=1
.
ðð§ð¬ð°ðð«: ððšð§ð¯ðð«ð ðð§ð.
H 2 Check the convergence of (ð).â(
n
2n + 5 )n
â
n=1
(ð).â1
(n + 1)n
â
n=1
.
ðð§ð¬ð°ðð«: (ð). ððšð§ð¯ðð«ð ðð§ð (ð). ððšð§ð¯ðð«ð ðð§ð.
H 3 Check for the convergence of the series â(1 +
1
n )n2â
n=1
.
ðð§ð¬ð°ðð«: ðð¢ð¯ðð«ð ðð§ð.
T 4 State Cauchyâ²s root test. Check the convergence of â
n
(logn)n
â
n=2
.
ðð§ð¬ð°ðð«: ððšð§ð¯ðð«ð ðð§ð.
W-18 (3)
C 5 Check convergence of (
22
12â 21
11)
â1
+ ( 33
23â 31
21)
â2
+ ( 44
34â 41
31)
â3
+⯠.
ðð§ð¬ð°ðð«: ððšð§ð¯ðð«ð ðð§ð.
H 6 Test the convergence of the series
1
3 + (
2
5 )2
+â¯+ (n
2n + 1 )n
+⯠.
ðð§ð¬ð°ðð«: ððšð§ð¯ðð«ð ðð§ð.
W-19 (3)
T 7 Check the convergence of (
1
2 )12
+ (2
3 )22
+ (3
4 )32
+⯠.
ðð§ð¬ð°ðð«: ððšð§ð¯ðð«ð ðð§ð.
C 8 Check the convergence of 1 +
2
3 x + (
3
4 )2
x2 + ( 4
5)3
x3 +⯠, x > 0.
ðð§ð¬ð°ðð«: ð± ⥠ðâ¹ ðð¢ð¯ðð«ð ðð§ð & ð < ð± < ð â¹ ððšð§ð¯ðð«ð ðð§ð.
H 9 Check the convergence of â(
n + 1
n + 2)n
xnâ
n=1
; x > 0.
ðð§ð¬ð°ðð«: ð± ⥠ðâ¹ ðð¢ð¯ðð«ð ðð§ð & ð †ð± < ð â¹ ððšð§ð¯ðð«ð ðð§ð.
DARSHAN INSTITUTE OF ENGINEERING & TECHNOLOGY Mathematics-I (3110014)
U N I T - 2 [ 4 7 ]
T 10 Check the convergence of
x
â1 +
x2
â2 3
+
x3
â3 4
+ ⯠, x > 0.
ðð§ð¬ð°ðð«: ð± ⥠ðâ¹ ðð¢ð¯ðð«ð ðð§ð & ð < ð± < ð â¹ ððšð§ð¯ðð«ð ðð§ð.
â ALTERNATING SERIES :
â A series with alternate positive and negative terms is known as an alternating series.
â LEIBNITZâS TEST FOR ALTERNATING SERIES:
The given alternating series â(â1)n+1 un
â
n=1
= u1 â u2 + u3 â u4 +⯠is convergent
if u1 ⥠u2 ⥠u3 ⥠u4 ⥠⯠and limn â â
un = 0.
â In above test, if limn â â
un â 0, then given series is oscillating series.
â Procedure to discuss about convergence of a series using Leibnitzâs test:
(1). Check whether the given series is alternating or not.
(2). Check whether the sequence {un} is decreasing or not.
(3). Check whether the value of limnââ
un = 0 or not.
(4). If answer of above three step is yes then given series is convergent.
METHOD â 14: LEIBNITZâS TEST
C 1 Check the convergence of 1 â
1
2â2 +
1
3â3 â
1
4â4 + ⯠.
ðð§ð¬ð°ðð«: ððšð§ð¯ðð«ð ðð§ð.
H 2 Check the convergence of 1 â
1
â2 +
1
â3 â
1
â4 + ⯠.
ðð§ð¬ð°ðð«: ððšð§ð¯ðð«ð ðð§ð.
T 3 Test the convergence of the series â
(â1)n+1
log(n + 1) .
ðð§ð¬ð°ðð«: ððšð§ð¯ðð«ð ðð§ð.
DARSHAN INSTITUTE OF ENGINEERING & TECHNOLOGY Mathematics-I (3110014)
U N I T - 2 [ 4 8 ]
C 4 Check the convergence of
1
12 + 1 â
2
22 + 1 +
3
32 + 1 + ⯠.
ðð§ð¬ð°ðð«: ððšð§ð¯ðð«ð ðð§ð.
H 5 Check the convergence of 1 â
1
2 +2
5 â
3
10 + ⯠.
ðð§ð¬ð°ðð«: ððšð§ð¯ðð«ð ðð§ð.
H 6 Check the convergence of
1
1 â 2 â
1
3 â 4 +
1
5 â 6 â
1
7 â 8 + ⯠.
ðð§ð¬ð°ðð«: ððšð§ð¯ðð«ð ðð§ð.
H 7 Check the convergence of
3
4 â5
7 +
7
10 â
9
13 + ⯠.
ðð§ð¬ð°ðð«: ðð¬ðð¢ð¥ð¥ðððšð«ð².
C 8 Test the convergence of the series
1
12 â
1
22 +
1
32 â
1
42 + â¯.
ðð§ð¬ð°ðð«: ððšð§ð¯ðð«ð ðð§ð.
W-19 (3)
C 9 Check the convergence of (
1
2 +
1
ln2 ) â (
1
2 +
1
ln 3 ) + (
1
2 +
1
ln 4 ) â ⯠.
ðð§ð¬ð°ðð«: ðð¬ðð¢ð¥ð¥ðððšð«ð².
T 10 Check the convergence of (ð).â
(â1)n n
3n â 2 (ð).â
(â1)nâ1 n
2n â 1.
ðð§ð¬ð°ðð«: (ð).ðð¬ðð¢ð¥ð¥ðððšð«ð² (ð). ðð¬ðð¢ð¥ð¥ðððšð«ð².
C 11 Check the convergence of the seriesâ(â1)n (ân+ ân â ân ) .
ðð§ð¬ð°ðð«: ðð¬ðð¢ð¥ð¥ðððšð«ð².
S-19 (4)
T 12 Check convergence of (ð).â
(â1)nâ1
nân (ð).â(â1)n( ân + 1 â ân ).
ðð§ð¬ð°ðð«: (ð). ððšð§ð¯ðð«ð ðð§ð (ð). ððšð§ð¯ðð«ð ðð§ð.
â ABSOLUTE CONVERGENT SERIES:
The given alternating series âun
â
n=0
is said to be absolute convergent, if â | un |
is convergent.
DARSHAN INSTITUTE OF ENGINEERING & TECHNOLOGY Mathematics-I (3110014)
U N I T - 2 [ 4 9 ]
â Any absolute convergent series is always convergent.
â Procedure to discuss about absolute convergence:
(1). Write new series as âwn =â | un | from the given series.
(2). Discuss the convergence of âwn using any applicable test.
(3). If âwn is convergent, then the given series is absolute convergent.
METHOD â 15: ABSOLUTE CONVERGENT SERIES
C 1 Is series 1 â
1
2 +1
3 â1
4 +1
5 â1
6 + ⯠Convergent? Does it
converge absolutely?
ðð§ð¬ð°ðð«: ððšð§ð¯ðð«ð ðð§ð ðð®ð ð§ðšð ððð¬ðšð¥ð®ðð ððšð§ð¯ðð«ð ðð§ð.
H 2 Determine the absolute convergence of 1 â
1
2â2 +
1
3â3 â
1
4â4 â ⯠.
ðð§ð¬ð°ðð«: ððð¬ðšð¥ð®ðð ððšð§ð¯ðð«ð ðð§ð.
H 3 Determine the absolute convergence of 1 +
1
22 â
1
32 â
1
42 +
1
52 â ⯠.
ðð§ð¬ð°ðð«: ððð¬ðšð¥ð®ðð ððšð§ð¯ðð«ð ðð§ð.
C 4 Determine the absolute convergence of 1 â 2x + 3x2 â 4x3 +â¯(0 †x < 1).
ðð§ð¬ð°ðð«: ððð¬ðšð¥ð®ðð ððšð§ð¯ðð«ð ðð§ð.
T 5 Determine the absolute convergence of â
(â1)n(n + 1)n
(2 n)2 .
ðð§ð¬ð°ðð«: ððšð ððð¬ðšð¥ð®ðð ððšð§ð¯ðð«ð ðð§ð.
C 6 Determine the absolute convergence of (ð).â
(â1)nâ1
np( p > 1)
(ð).â(â1)n+1 sin n
n2 .
ðð§ð¬ð°ðð«: (ð). ððð¬ðšð¥ð®ðð ððšð§ð¯ðð«ð ðð§ð (ð). ððð¬ðšð¥ð®ðð ððšð§ð¯ðð«ð ðð§ð.
H 7 Determine absolute convergence of following series:
(ð).â (â1)n+1
(n)3 (ð).â
(â1)n5n
n! (ð).â(â1)n+1
cosn
n2
.
ðð§ð¬ð°ðð«: ðð¥ð¥ ðð¡ð ð¬ðð«ð¢ðð¬ ðð«ð ððð¬ðšð¥ð®ðð ððšð§ð¯ðð«ð ðð§ð.
DARSHAN INSTITUTE OF ENGINEERING & TECHNOLOGY Mathematics-I (3110014)
U N I T - 2 [ 5 0 ]
T 8 Determine absolute convergence of following series:
(ð).â(â1)n23n
32n (ð).â
cosnÏ
n2 + 1
.
ðð§ð¬ð°ðð«: ððšðð¡ ðð¡ð ð¬ðð«ð¢ðð¬ ðð«ð ððð¬ðšð¥ð®ðð ððšð§ð¯ðð«ð ðð§ð.
â CONDITIONALLY CONVERGENT SERIES:
The given alternating series âun
â
n=0
is said to be conditionally convergent, if
(1). âun is convergent
(2). â | un | is divergent.
â Procedure to discuss about conditionally convergence:
(1). Discuss convergence of given âun using any applicable test.
(2). Write new series as âwn =â |un| from given series.
(3). Discuss convergence of âwn using any applicable test.
(4). If âun is convergent and âwn is divergent, then given series is conditionally
convergent.
â Infinite series canât be absolute convergent and conditionally convergent together.
METHOD â 16: CONDITIONALLY CONVERGENT SERIES
C 1 Check the absolute and conditional convergence of the series â
(â1)n+1
ân 3
â
n=1
.
ðð§ð¬ð°ðð«: ððšð§ðð¢ðð¢ðšð§ðð¥ð¥ð² ððšð§ð¯ðð«ð ðð§ð.
H 2 Determine the conditionally convergence of
1
2 â1
4 +1
6 â1
8 + ⯠.
ðð§ð¬ð°ðð«: ððšð§ðð¢ðð¢ðšð§ðð¥ð¥ð² ððšð§ð¯ðð«ð ðð§ð.
H 3 Determine the conditionally convergence of 1 â
1
â2 +
1
â3 â
1
â4 + ⯠.
ðð§ð¬ð°ðð«: ððšð§ðð¢ðð¢ðšð§ðð¥ð¥ð² ððšð§ð¯ðð«ð ðð§ð.
DARSHAN INSTITUTE OF ENGINEERING & TECHNOLOGY Mathematics-I (3110014)
U N I T - 2 [ 5 1 ]
C 4 Determine the absolute or conditional convergence of seriesâ
(â1)n n2
n3 + 1
â
n=1
.
ðð§ð¬ð°ðð«: ððšð§ðð¢ðð¢ðšð§ðð¥ð¥ð² ððšð§ð¯ðð«ð ðð§ð.
T 5 Check for absolute/conditional convergence of â
(â1)n
ân + ân + 1
â
n=1
.
ðð§ð¬ð°ðð«: ððšð§ðð¢ðð¢ðšð§ðð¥ð¥ð² ððšð§ð¯ðð«ð ðð§ð.
T 6 Check the convergence of â
1
1p +
1
2p â
1
3p +
1
4p â
1
5p + ⯠; p > 0.
ðð§ð¬ð°ðð«: ð© > ð â¹ ððð¬ðšð¥ð®ðð ððšð§ð¯ðð«ð ðð§ð,
0 < ð© †ð â¹ ððšð§ðð¢ðð¢ðšð§ðð¥ð¥ð² ððšð§ð¯ðð«ð ðð§ð.
â POWER SERIES:
â An infinite series of the form
âan (x â c)n =
â
n=0
a0(x â c)0 + a1(x â c)
1 + a2(x â c)2 + a3(x â c)
3 +â¯
is known as power series in standard form, where an is the coefficient of the nth term, c is
a constant and x varies around c.
â If c = 0 in above series then it is called power series in terms of x.
i. e.âan xn =
â
n=0
a0 x0 + a1 x
1 + a2 x2 + a3 x
3 +⯠.
â THEOREM: CONVERGENCE OF POWER SERIES
Let âan xn
â
0
be a power series. Then exactly one of the following conditions hold.
(1). The series always converges at x = 0.
(2). The series converges for all x.
(3). There is some positive number R such that series converges for |x| < R and
diverges for |x| > R.
DARSHAN INSTITUTE OF ENGINEERING & TECHNOLOGY Mathematics-I (3110014)
U N I T - 2 [ 5 2 ]
â RADIUS AND INTERVAL OF CONVERGENCE FOR POWER SERIES:
â An interval in which power series converges is called the interval of convergence and the
half length of the interval of convergence is called the radius of convergence.
â If a power series is convergent for all values of x, then interval of convergence will be
(ââ,â) and the radius of convergence will be â.
â In above theorem R is radius of convergence of the power series and it can be obtained
from either of the formulas as follows:
ð = ð¥ð¢ðŠð§ â â
ð
| ðð§+ð ðð§
| ðšð« ð = ð¥ð¢ðŠ
ð§ â â
ð
(ðð§)ð ð§ .
â Procedure to discuss about convergence of power series:
(1). Using above formula of R, find interval of convergence.
(2). Check the convergence at the end point of interval individually.
(3). Update the interval of convergence.
(4). Find the radius i.e. half-length of the interval of convergence.
â Note: Any absolute convergent series is always convergent and infinite series can not
absolute and conditionally convergent together.
METHOD â 17: POWER SERIES
C 1 Find the radius of convergence and interval of convergence. Also find for
what values of x series is absolute convergent and conditionally convergent.
x â x2
2+ x3
3â x4
4+ ⯠.
ðð§ð¬ð°ðð«: ð, (âð, ð], (âð, ð) & ð± = ð.
H 2 Find the radius of convergence and interval of convergence. Also find for
what values of x series is absolute convergent and conditionally convergent.
x â x2
22+ x3
32â x4
42+⯠.
ðð§ð¬ð°ðð«: ð, [âð, ð], [âð, ð] & ððšð« ð§ðš ð¯ðð¥ð®ðð¬ ðšð ð±.
DARSHAN INSTITUTE OF ENGINEERING & TECHNOLOGY Mathematics-I (3110014)
U N I T - 2 [ 5 3 ]
T 3 For the seriesâ
(â1)n (x + 2)n
n
â
n=1
find the seriesâ²s radius and interval of
convergence. For what values of x does the series converge absolutely,
conditionally?
ðð§ð¬ð°ðð«: ð, (âð,âð], (âð,âð) & ð± = âð.
T 4 Find the radius of convergence and interval of convergence of the series
1 â1
2 (x â 2) +
1
22 (x â 2)2 +â¯+ (â
1
2 )n
(x â 2)n +⯠.
ðð§ð¬ð°ðð«: |ð± â ð| < ð â ð < ð± < ð.
T 5 Find the radius of convergence and interval of convergence for the series
â (3 x â 2)n
n.
â
n=0
For what values of x does the series converge absolutely?
ðð§ð¬ð°ðð«: ð/ð, [ð/ð, ð) & (ð/ð,ð).
S-19 (7)
C 6 For the series â
(â1)nâ1 x2nâ1
2n â 1
â
n=1
, find the radius and interval of
convergence.
ðð§ð¬ð°ðð«: â ð †ð± †ð â¹ ððšð§ð¯ðð«ð ðð§ð & ðððð¢ð®ð¬ ðšð ððšð§ð¯ðð«ð ðð§ð ð¢ð¬ ð.
H 7 Test for the convergence of the series x â
x3
3+ x5
5â ⯠, x > 0.
ðð§ð¬ð°ðð«: ð±ð > ð â¹ ðð¢ð¯ðð«ð ðð§ð & ð < ð±ð †ðâ¹ ððšð§ð¯ðð«ð ðð§ð.
H 8 Find the interval of convergence of the seriesâ
(â3)n. xn
ân + 1
â
n=0
. ðð
Find radius of convergence and interval of convergence ofâ (â3)n. xn
ân + 1
â
n=0
.
ðð§ð¬ð°ðð«: ð/ð & (âð/ð, ð/ð].
T 9 Find the radius of convergence and interval of convergence of the series
âx2nâ1
(2n â 1)! .
ðð§ð¬ð°ðð«: â & (ââ,â).
DARSHAN INSTITUTE OF ENGINEERING & TECHNOLOGY Mathematics-I (3110014)
U N I T - 2 [ 5 4 ]
C 10 Test the convergence of â
[(n + 1)x]
nn+1
n
, x ⥠0.
â
n=1
ðð§ð¬ð°ðð«: ð± ⥠ðâ¹ ðð¢ð¯ðð«ð ðð§ð & ð †ð± < ð â¹ ððšð§ð¯ðð«ð ðð§ð.
H 11 Test for the convergence of the series
1
1 â 2 â 3 +
x
4 â 5 â 6 +
x2
7 â 8 â 9 + ⯠.
ðð§ð¬ð°ðð«: |ð±| > ð â¹ ðð¢ð¯ðð«ð ðð§ð & |ð±| †ð â¹ ððšð§ð¯ðð«ð ðð§ð.
T 12 Test the convergence of the series and find radius of convergence.
x
2 â 5 â
x2
22 â 10 +
x3
23 â 15 â
x4
24 â 20 + ⯠.
ðð§ð¬ð°ðð«: ð± †âð & ð < ð± â¹ ðð¢ð¯ðð«ð ðð§ð,
âð < ð± †ð â¹ ððšð§ð¯ðð«ð ðð§ð & ðððð¢ð®ð¬ ðšð ððšð§ð¯ðð«ð ðð§ðð = ð.
C 13 Test for convergence of the series 2 +
3
2x +
4
3x2 +
5
4x3 +⯠.
ðð§ð¬ð°ðð«: ð± ⥠ðâ¹ ðð¢ð¯ðð«ð ðð§ð & ð †ð± < ð â¹ ððšð§ð¯ðð«ð ðð§ð.
H 14 Find the radius of convergence and interval of convergence of the series
1
2â1 â
x2
3â2 +
x4
4â3 â
x6
5â4 + ⯠.
ðð§ð¬ð°ðð«: ð & [âð, ð].
T 15 Find the value of x for which the given series
1
2â1 +
x2
3â2 +
x4
4â3 + â¯
converge.
ðð§ð¬ð°ðð«: â ð †ð± †ð.
T 16 Find the radius of convergence and the interval of convergence of the series
1
2 â (x â 2)
22+ (x â 2)2
23â⯠.
ðð§ð¬ð°ðð«: ð & (ð, ð).
â TAYLORâS SERIES: FIRST FORM IN POWER OF (ð± â ð)
â If f(x) possesses derivatives of all order at point âaâ then Taylorâs series of given function
f(x) at point âaâ is given by
ð(ð±) = ð(ð) + (ð± â ð)
ð! ðâ²(ð) +
(ð± â ð)ð
ð! ðâ²â²(ð) +
(ð± â ð)ð
ð! ðâ²â²â²(ð) +⯠.
DARSHAN INSTITUTE OF ENGINEERING & TECHNOLOGY Mathematics-I (3110014)
U N I T - 2 [ 5 5 ]
METHOD â 18: 1ST FORM OF TAYLORâS SERIES
C 1 Expand ex in power of (x â 1) up to first four terms.
ðð§ð¬ð°ðð«: ð [ð + (ð± â ð) +ð
ð! (ð± â ð)ð +
ð
ð! (ð± â ð)ð +⯠].
C 2 Find the Taylor series generated by f(x) =
1
x at a = 2.
ðð§ð¬ð°ðð«: [ð
ð â (ð± â ð)
ðð+ (ð± â ð)ð
ððââ¯].
H 3 Express f(x) = 2x3 + 3x2 â 8x + 7 in terms of (x â 2).
ðð§ð¬ð°ðð«: ðð + ðð(ð± â ð) + ðð(ð± â ð)ð + ð(ð± â ð)ð.
W-19 (4)
H 4 Expand 3x3 + 8x2 + x â 2 in powers of x â 3.
ðð§ð¬ð°ðð«: ððð + ððð(ð± â ð) + ðð(ð± â ð)ð + ð(ð± â ð)ð.
T 5 Find the Taylorâs series expansion of order 3 generated by f(x) = âx
at a = 4.
ðð§ð¬ð°ðð«: [ð + ð± â ð
ðâ (ð± â ð)ð
ðð+ (ð± â ð)ð
ðððâ â¯].
C 6 Find the Taylor series expansion of f(x) = x3 â 2x + 4 at a = 2.
ðð§ð¬ð°ðð«: [ð + ðð(ð± â ð) + ð(ð± â ð)ð + (ð± â ð)ð].
C 7 Find the Taylorâ²s series expansion of f(x) = tan x in power of (x â Ï
4),
showing at least four nonzero terms.Hence find the value of tan 46°.
ðð§ð¬ð°ðð«: [ð + ð(ð± â ð
ð) + ð(ð± â
ð
ð)ð
+ ð
ð(ð± â
ð
ð)ð
+â¯] & ð. ðððð.
H 8 Expand log(sin x) in power of (x â 2).
ðð§ð¬ð°ðð«: [ð¥ðšð (ð¬ð¢ð§ð) + (ð± â ð)ððšðð â (ð± â ð)ðððšð¬ðððð
ð!+ â¯].
T 9 Expand log(cos x) about Ï
3 by Taylorâs series.
ðð§ð¬ð°ðð«: [âð¥ðšð ð â âð(ð± â ð
ð) â ð(ð± â
ð
ð)ð
â ðâð
ð(ð± â
ð
ð)ð
ââ¯].
DARSHAN INSTITUTE OF ENGINEERING & TECHNOLOGY Mathematics-I (3110014)
U N I T - 2 [ 5 6 ]
â TAYLORâS SERIES SECOND FORM OF ð(ð + ð±)
â If f(x) possesses derivatives of all order at point âaâ, then the Taylorâs series of given
function f(a + x) at point âaâ is given by
ð(ð + ð±) = ð(ð) +ð±
ð! ðâ²(ð) +
ð±ð
ð! ðâ²â²(ð) +
ð±ð
ð! ðâ²â²â²(ð) + ⯠.
â ð° =Ï
180 ð«ðð = ð. ðððð ð«ðð.
METHOD â 19: 2ND FORM OF TAYLORâS SERIES
C 1 Expand sin ( Ï
4+ x) in the power of x and hence find the value of sin46°.
ðð§ð¬ð°ðð«: ð¬ð¢ð§ ( ð
ð+ ð±) =
ð
âð (ð + ð± â
ð±ð
ðâ ð±ð
ð+â¯) & ð. ðððð.
H 2 Expand sin ( Ï
4+ x) in powers of x and hence find the value of sin44°
correct up to 4 decimal places.
ðð§ð¬ð°ðð«: ð¬ð¢ð§ ( ð
ð+ ð±) =
ð
âð (ð + ð± â
ð±ð
ðâ ð±ð
ð+â¯) & ð. ðððð.
H 3 Expand cos ( Ï
4+ x) in the power of x by Taylor series. Hence find the
value of cos 46°.
ðð§ð¬ð°ðð«: ððšð¬ ( ð
ð+ ð±) =
ð
âð (ð â ð± â
ð±ð
ð!+ ð±ð
ð!+ â¯) & ð. ðððð.
T 4 Find the expansion of tan ( Ï
4+ x) in ascending powers of x upto terms in
x4 and find approximately value of tan 43°.
ðð§ð¬ð°ðð«: ððð§ ( ð
ð+ ð±) = ð + ðð± + ðð±ð +
ð
ðð±ð +
ðð
ðð±ð +⯠& ð. ðððð.
T 5 Use Taylorâs series to estimate sin 38°.
ðð§ð¬ð°ðð«: ð. ðððð ðð©ð©ð«ðšð±ð¢ðŠðððð¥ð².
S-19 (4)
C 6 Find the value of â18 using Taylorâs series.
ðð§ð¬ð°ðð«: ð. ðððð ðð©ð©ð«ðšð±ð¢ðŠðððð¥ð².
DARSHAN INSTITUTE OF ENGINEERING & TECHNOLOGY Mathematics-I (3110014)
U N I T - 2 [ 5 7 ]
H 7 Using Taylorâs theorem to find â25.15 .
ðð§ð¬ð°ðð«: ð. ðððð ðð©ð©ð«ðšð±ð¢ðŠðððð¥ð².
T 8 State Taylorâs series for one variable and hence find â36.12 .
ðð§ð¬ð°ðð«: ð. ðððð ðð©ð©ð«ðšð±ð¢ðŠðððð¥ð².
T 9 Find the value of (82)
1 4 using Taylorâs series.
ðð§ð¬ð°ðð«: ð. ðððð ðð©ð©ð«ðšð±ð¢ðŠðððð¥ð².
â MACLAURINâS SERIES:
â If f(x) possesses derivatives of all order at point â0â then the Maclaurinâs series of given
function f(x) at point â0â is given by
ð(ð±) = ð(ð) +ð±
ð! ðâ²(ð) +
ð±ð
ð! ðâ²â²(ð) +
ð±ð
ð! ðâ²â²â²(ð) + ⯠.
â If we take point a = 0 in the 1st form of Taylorâs series then we get the Maclaurinâs series.
METHOD â 20: MACLAURINâS SERIES
C 1 Express 5 + 4(x â 1)2 â 3(x â 1)3 + (x â 1)4 in ascending powers of x.
ðð§ð¬ð°ðð«: ðð â ððð± + ððð±ð â ðð±ð + ð±ð.
H 2 Express 2 + 5(x â 1) + 2(x â 1)3 + (x â 1)4 in ascending powers of x.
ðð§ð¬ð°ðð«: â ð + ðð± â ðð±ð + ð±ð.
C 3 Using Maclaurinâs series expand ex and hence derive expansions of eâx.
ðð§ð¬ð°ðð«: ðð± = ð +ð±
ð! +ð±ð
ð! +ð±ð
ð! + ⯠& ðâð± = ð â
ð±
ð! +ð±ð
ð! âð±ð
ð! + ⯠.
C 4 Using Maclaurinâs series expand sin x and cos x .
ðð§ð¬ð°ðð«: ð¬ð¢ð§ ð± = ð± â ð±ð
ð!+ ð±ð
ð!â ⯠& ððšð¬ ð± = ð â
ð±ð
ð!+ ð±ð
ð!â ⯠.
H 5 Using Maclaurinâs series expand tan x.
ðð§ð¬ð°ðð«: ððð§ ð± = ð± + ð±ð
ð+ ðð±ð
ðð+ ⯠.
DARSHAN INSTITUTE OF ENGINEERING & TECHNOLOGY Mathematics-I (3110014)
U N I T - 2 [ 5 8 ]
T 6 Using Maclaurinâs series expand sec x.
ðð§ð¬ð°ðð«: ð¬ðð ð± = ð + ð±ð
ð + ðð±ð
ððâ⯠.
T 7 Expand x2 cos x in terms of x.
ðð§ð¬ð°ðð«: ð±ðððšð¬ ð± = ð±ð â ð±ð
ð!+ ð±ð
ð!â ⯠.
C 8 Expand ex sin x in power of x upto the terms containing x6.
ðð§ð¬ð°ðð«: ðð± ð¬ð¢ð§ð± = ð + ð±ð + ð±ð
ð+
ð±ð
ððð +⯠.
W-18 (3)
T 9 Expand
ex
cos x up to first four terms by Maclaurinâs series.
ðð§ð¬ð°ðð«: ðð±
ððšð¬ ð± = ð + ð± + ð±ð +
ðð±ð
ð+⯠.
C 10 Expand tanâ1 x up to the first four terms by Maclaurinâ²s series and
hence prove that tanâ1 ( â1 + x2 â 1
x) =
1
2(x â
x3
3+ x5
5+ â¯).
ðð§ð¬ð°ðð«: ððð§âð ð± = ð± â ð±ð
ð+ ð±ð
ðâ ð±ð
ð+ ⯠.
T 11 Expand loge (
1 + x
1 â x ) and then obtain approximate value of loge (
11
9).
ðð§ð¬ð°ðð«: ð¥ðšð ð (ð + ð±
ð â ð± ) = ð(ð± +
ð±ð
ð+ ð±ð
ð+ â¯) & ð. ððððð.
C 12 Using Maclaurinâs series expand log (cos x).
ðð§ð¬ð°ðð«: ð¥ðšð (ððšð¬ ð±) = â ð±ð
ðâ ð±ð
ððâ ð±ð
ððâ⯠.
H 13 Expand log(sec x) in power of x.
ðð§ð¬ð°ðð«: ð¥ðšð (ð¬ðð ð±) = ð±ð
ð+ ð±ð
ðð+⯠.
T 14 Expand log(1 + ex) in ascending power of x as far as term containing x4 .
ðð§ð¬ð°ðð«: ð¥ðšð (ð + ðð±) = ð¥ðšð ð + ð±
ð+ ð±ð
ðâ
ð±ð
ððð â ⯠.
DARSHAN INSTITUTE OF ENGINEERING & TECHNOLOGY Mathematics-I (3110014)
U N I T - 2 [ 5 9 ]
H 15 Using Maclaurinâs series expand log(1+ x) and hence derive expansions of
log(1 â x) and log2 and logy .
ðð§ð¬ð°ðð«: ð¥ðšð (ð + ð±) = ð± â ð±ð
ð+ ð±ð
ðâ ð±ð
ð+ ð±ð
ðâŠ
ð¥ðšð (ð â ð±) = âð± â ð±ð
ðâ ð±ð
ðâ ð±ð
ðâ ð±ð
ðâŠ
ð¥ðšð ð = ð âð
ð +ð
ð âð
ð +ð
ð â â¯
ð¥ðšð ð² = (ð² â ð) â (ð² â ð)ð
ð+ (ð² â ð)ð
ðâ ⯠.
H 16 Using Maclaurinâs series expand (1 + x)m and hence derive expansions of
(1 â x)m, (1 + x)â1, (1 â x)â1, (1 + x)â2 and (1 â x)â2.
ðð§ð¬ð°ðð«: (ð + ð±)ðŠ = ð +ðŠð± + ðŠ(ðŠâ ð)
ð!ð±ð +â¯
(ð â ð±)ðŠ = ð âðŠð± + ðŠ(ðŠâ ð)
ð!ð±ð ââ¯
(ð + ð±)âð = ð â ð± + ð±ð â ð±ð +â¯
(ð + ð±)âð = ð â ðð± + ðð±ð â ðð±ð +â¯
(ð â ð±)âð = ð + ð± + ð±ð + ð±ð +â¯
(ð â ð±)âð = ð + ðð± + ðð±ð + ðð±ð +⯠.
C 17 Using Maclaurinâs expansion, expand (1 + x)
1 x upto the term x2.
ðð§ð¬ð°ðð«: (ð + ð±) ð ð± = ð (ð â
ð±
ð+ðð
ðð ð±ð ââ¯) .
T 18 Using Maclaurinâs series expand
ex
ex + 1 .
ðð§ð¬ð°ðð«: ðð±
ðð± + ð =ð
ð +ð
ð ð± â
ð
ðð ð±ð +⯠.
â â â â â â â â â â
DARSHAN INSTITUTE OF ENGINEERING & TECHNOLOGY Mathematics-I (3110014)
U N I T - 3 [ 6 1 ]
UNIT 3
â BASIC FORMULAE:
â When given function is polynomial function:
⢠Leibnitzâs formula (consider polynomial function as âuâ)
â«ð® â ð¯ ðð± = ð® ð¯ð â ð®â² ð¯ð + ð®
â²â² ð¯ð â ð®â²â²â² ð¯ð +â¯,
where uâ², uâ²â², ⊠are successive derivatives of u and v1, v2, ⊠are successive integrals of
v.
â When given function is exponential function:
â«ððð± ð¬ð¢ð§ ðð± ðð± =ððð±
ðð + ðð [ð ð¬ð¢ð§ ðð± â ðððšð¬ ðð±] + ð
â«ððð± ððšð¬ ðð± ðð± =ððð±
ðð + ðð [ð ððšð¬ ðð± + ðð¬ð¢ð§ ðð±] + ð
â When given function is trigonometric function:
ð ð¬ð¢ð§ ðððšð¬ ð = ð¬ð¢ð§(ð + ð) + ð¬ð¢ð§(ð â ð)
ðððšð¬ ð ð¬ð¢ð§ ð = ð¬ð¢ð§(ð + ð) â ð¬ð¢ð§(ð â ð)
ðððšð¬ ðððšð¬ ð = ððšð¬(ð + ð) + ððšð¬(ð â ð)
ð ð¬ð¢ð§ ðð¬ð¢ð§ ð = âððšð¬(ð + ð) + ððšð¬(ð â ð)
â NOTE (FOR EVERY ð§ â â€)
â cosnÏ = (â1)n â sin nÏ = 0 â cos(2n + 1)Ï
2 = 0
â cos2nÏ = (â1)2n = 1 â sin 2nÏ = 0 â sin(2n + 1)Ï
2 = (â1)n
â DIRICHLET CONDITION FOR EXISTENCE OF FOURIER SERIES OF ð(ð±):
(1). f(x) is bounded.
(2). f(x) is single valued.
(3). f(x) has finite number of maxima and minima in the interval.
DARSHAN INSTITUTE OF ENGINEERING & TECHNOLOGY Mathematics-I (3110014)
U N I T - 3 [ 6 2 ]
(4). f(x) has finite number of discontinuities in the interval.
â NOTE:
â At a point of discontinuity, the sum of the series is equal to average of left and right hand
limits of f(x) at the point of discontinuity, say x0.
i. e. f(x0) = f(x0
â) + f(x0 +)
2
â FOURIER SERIES
â Let f(x) be periodic function of period 2L defined in (0 , 2L) and satisfies Dirichletâs
condition, then
ð(ð±) = ðð
ð+â[ðð§ ððšð¬ (
ð§ðð±
ð) + ðð§ ð¬ð¢ð§ (
ð§ðð±
ð)]
â
ð§=ð
,
where a0, a1, ⊠, an, b0, b1, ⊠, bn are Fourier coefficients of series.
â ORTHOGONALITY OF TRIGONOMETRIC FOURIER SERIES
â Orthogonality: A collection of functions {f0(x), f1(x),⊠, fm(x),⊠} defined on [a, b] is called
orthogonal on [a, b] if
(fi, fj) = â« fi(x)fj(x)
b
a
dx = {0 ; i â j
c > 0 ; i = j
e.g., The collection {1, cos x , cos 2x , ⊠} are orthogonal functions in [âÏ, Ï].
â Trigonometric Fourier series: The set of functions sin ( nÏx
L) and cos (
nÏx
L) are
orthogonal in the interval (c, c + 2L) for any value of c, where n = 1,2,3,âŠ.
â i.e.,
â« sin mÏx
L sin
nÏx
L
c+2l
c
dx = { 0 ; m â n L ; m = n
â« cos mÏx
L cos
nÏx
L
c+2l
c
dx = { 0 m â n L m = n
DARSHAN INSTITUTE OF ENGINEERING & TECHNOLOGY Mathematics-I (3110014)
U N I T - 3 [ 6 3 ]
â« sin mÏx
L cos
nÏx
L
c+2l
c
dx = 0, for all m, n.
â Hence, any function f(x) can be represented in the terms of these orthogonal functions in
the interval (c, c + 2L) for any value of c.
f(x) = a0 +âan cos nÏx
L+
â
n=1
âbn sin nÏx
L
â
n=1
This series is known as trigonometric series.
â FOURIER SERIES IN THE INTERVAL (ð, ð + ðð):
â The Fourier series for the function f(x) in the interval (c, c + 2L) is defined by
ð(ð±) = ðð
ð+â[ðð§ ððšð¬ (
ð§ðð±
ð) + ðð§ ð¬ð¢ð§ (
ð§ðð±
ð)]
â
ð§=ð
Where the constants a0, an and bn are given by
a0 =1
L â« f(x) dxc+2L
c
an =1
L â« f(x) cos (
nÏx
L) dx
c+2L
c
bn =1
L â« f(x) sin (
nÏx
L) dx
c+2L
c
.
â FOURIER SERIES IN THE INTERVAL (ð, ðð):
â The Fourier series for the function f(x) in the interval (0,2L) is defined by
ð(ð±) = ðð
ð+â[ðð§ ððšð¬ (
ð§ðð±
ð) + ðð§ ð¬ð¢ð§ (
ð§ðð±
ð)]
â
ð§=ð
,
where the constants a0, an and bn are given by
a0 =1
L â« f(x) dx2L
0
an =1
L â« f(x) cos (
nÏx
L) dx
2L
0
DARSHAN INSTITUTE OF ENGINEERING & TECHNOLOGY Mathematics-I (3110014)
U N I T - 3 [ 6 4 ]
bn =1
L â« f(x) sin (
nÏx
L) dx
2L
0
.
METHOD â 1: FOURIER SERIES IN THE INTERVAL (0, 2L)
C 1 Find the Fourier series of 2Ï â periodic function f(x) = x2, 0 < x < 2Ï
and hence deduce that Ï2
6= â
1
n2
â
n=1
.
ðð§ð¬ð°ðð«: ð(ð±) = ððð
ð+â[
ð
ð§ð ððšð¬ ð§ð± â
ðð
ð§ð¬ð¢ð§ ð§ð± ] .
â
ð§=ð
S-19 (7)
H 2 Find the Fourier series for f(x) = x2 in (0, 2).
ðð§ð¬ð°ðð«: ð(ð±) =ð
ð +â [
ð
ð§ððð ððšð¬(ð§ðð±) â
ð
ð§ð ð¬ð¢ð§(ð§ðð±) ]
â
ð§=ð
.
H 3 Find the Fourier series to represent f(x) = 2x â x2 in (0, 3).
ðð§ð¬ð°ðð«: ð(ð±) = â[ âð
ð§ððð ððšð¬ (
ðð§ðð±
ð) +
ð
ð§ð ð¬ð¢ð§ (
ðð§ðð±
ð) ] .
â
ð§=ð
T 4 Find the Fourier series of f(x) = Ï â x
2 in the interval (0, 2Ï).
ðð§ð¬ð°ðð«: ð(ð±) = âð
ð§ ð¬ð¢ð§ ð§ð±
â
ð§=ð
.
W-19 (7)
T 5 Show that Ï â x =
Ï
2 +â
sin 2nx
n
â
n=1
, when 0 < x < Ï .
C 6 Obtain the Fourier series for f(x) = eâx in the interval 0 < x < 2.
ðð§ð¬ð°ðð«: ð(ð±) = (ð â ðâð)
ð+â
(ð â ðâð)
ð + ð§ððð[ ððšð¬ ð§ðð± + (ð§ð) ð¬ð¢ð§ð§ðð± ]
â
ð§=ð
.
H 7 Find Fourier Series for f(x) = eâx, where 0 < x < 2Ï.
ðð§ð¬ð°ðð«: ð(ð±) = ð â ðâðð
ðð+â
ðâ ðâðð
ð(ð§ð + ð) [ ððšð¬ ð§ð± + ð§ð¬ð¢ð§ð§ð± ]
â
ð§=ð
.
C 8 Find the Fourier series of the periodic function f(x) = Ï sin Ïx, where
0 < x < 1, p = 2l = 1.
ðð§ð¬ð°ðð«: ð(ð±) = ð +âð
ð â ðð§ð ððšð¬(ðð§ðð±)
â
ð§=ð
.
DARSHAN INSTITUTE OF ENGINEERING & TECHNOLOGY Mathematics-I (3110014)
U N I T - 3 [ 6 5 ]
C 9 Find the Fourier Series for the function f(x) given by
f(x) = {âÏ , 0 < x < Ï
x â Ï , Ï < x < 2Ï
. Hence show that â1
(2n + 1)2 = Ï2
8
â
n=0
.
ðð§ð¬ð°ðð«: ð(ð±) = âð
ð +â[
(ð â (âð)ð§)
ð§ððððšð¬(ð§ð±) +
(âð)ð§ â ð
ð§ð¬ð¢ð§(ð§ð±) ]
â
ð§=ð
.
H 10 Find the Fourier series for periodic function with period 2 of
f(x) = {Ïx, 0 †x †1
Ï (2 â x), 1 †x †2
.
ðð§ð¬ð°ðð«: ð(ð±) =ð
ð +â
ð[(âð)ð§ â ð]
ðð§ðððšð¬(ð§ðð±)
â
ð§=ð
.
H 11 Find the Fourier series of f(x) = {
x2 ; 0 < x < Ï
0 ; Ï < x < 2Ï .
ðð§ð¬ð°ðð«: ð(ð±) = ðð
ð+
â [ ð(âð)ð§ ððšð¬ð§ð±
ð§ð+ ð
ð{â
ðð(âð)ð§
ð§+ ð(âð)ð§
ð§ðâ
ð
ð§ð } ð¬ð¢ð§ð§ð± ]
â
ð§=ð
.
T 12 For the function f(x) = {
x ; 0 †x †2
4 â x ; 2 †x †4, find its Fourier series. Hence
show that 1
12 +
1
32 +
1
52 + ⯠=
Ï2
8 .
ðð§ð¬ð°ðð«: ð(ð±) = ð +â ð [(âð)ð§ â ð]
ððð§ð
â
ð§=ð
ððšð¬ ( ð§ðð±
ð).
â ODD & EVEN FUNCTION:
â Let f(x) be defined in (âL, L), then
(1). A function f(x) is said to be Odd Function if ð(âð±) = âð(ð±).
(2). A function f(x) is said to be Even Function if ð(âð±) = ð(ð±).
â NOTE:
(ð). If f(x) is an even function defined in (â L, L), then â« f(x)
L
âL
dx = 2â« f(x)
L
0
dx.
DARSHAN INSTITUTE OF ENGINEERING & TECHNOLOGY Mathematics-I (3110014)
U N I T - 3 [ 6 6 ]
(ð). If f(x) is an odd function defined in (â L, L), then â« f(x)
L
âL
dx = 0.
â FOURIER SERIES FOR ODD & EVEN FUNCTION:
â Let, f(x) be a periodic function defined in (â L, L)
f(x) is even, bn = 0; n = 1,2,3,⊠f(x) is odd, a0 = 0 = an; n = 1,2,3, âŠ
â ð(ð±) = ðð
ð+âðð§ ððšð¬ (
ð§ðð±
ð)
â
ð§=ð
â ð(ð±) = âðð§ ð¬ð¢ð§ ( ð§ðð±
ð)
â
ð§=ð
Where,
a0 =2
L â« f(x) dxL
0
an =2
L â« f(x) cos (
nÏx
L) dx
L
0
Where,
bn =2
L â« f(x) sin (
nÏx
L) dx
L
0
Sr. No. Type of Function Examples
1. Even Function
â x2, x4, x6, ⊠i. e. xn, where n is even.
â Any constant. e.g. 1,2, ÏâŠ
â Graph is symmetric about Y â axis.
â cos ax , |x|, |x3|, |cos x|,âŠ
â f(âx) = f(x)
2. Odd Function
â x, x3, x5, ⊠i. e. xm, where m is odd.
â sin ax
â Graph is symmetric about Origin.
â f(âx) = âf(x)
3. Neither Even nor Odd â eax
â axm + bxn ; n is even & m is odd number.
DARSHAN INSTITUTE OF ENGINEERING & TECHNOLOGY Mathematics-I (3110014)
U N I T - 3 [ 6 7 ]
METHOD â 2: FOURIER SERIES IN THE INTERVAL (âð, ð)
C 1 Find the Fourier series of the periodic function f(x) = 2x,
where â1 < x < 1.
ðð§ð¬ð°ðð«: ð(ð±) = â ð(âð)ð§+ð
ð§ðð¬ð¢ð§(ð§ðð±)
â
ð§=ð
.
H 2 Find the Fourier expansion for function f(x) = x â x3 in â1 < x < 1.
ðð§ð¬ð°ðð«: ð(ð±) = â ðð(âð)ð§+ð
ð§ðððð¬ð¢ð§ð§ðð±
â
ð§=ð
.
C 3 Expand f(x) = x2 â 2 in â 2 < x < 2 the Fourier series.
ðð§ð¬ð°ðð«: ð(ð±) = âð
ð +â
ðð(âð)ð§
ð§ðððððšð¬ (
ð§ðð±
ð)
â
ð§=ð
.
H 4 Find the Fourier series for f(x) = x2 in âl < x < l.
ðð§ð¬ð°ðð«: ð(ð±) = ð¥ð
ð+â
ð ð¥ð(âð)ð§
ð§ðððððšð¬ (
ð§ðð±
ð¥) .
â
ð§=ð
T 5 Obtain the Fourier series for f(x) = x2 in the interval âÏ < x < Ï and
hence deduce that
(ð).â1
n2 = Ï2
6
â
n=1
. (ð).â (â1)n+1
n2
â
n=1
= Ï2
12 .
ðð§ð¬ð°ðð«: ð(ð±) = ðð
ð+â
ð(âð)ð§
ð§ðððšð¬ð§ð±
â
ð§=ð
.
T 6 Find the Fourier series expansion of f(x) = |x|; â Ï < x < Ï.
ðð§ð¬ð°ðð«: ð(ð±) = ð
ð+â
ð [(âð)ð§ â ð]
ðð§ðððšð¬ð§ð±
â
ð§=ð
.
C 7 Find the Fourier series of f(x) = x2 + x, where â2 < x < 2.
ðð§ð¬ð°ðð«: ð(ð±) =ð
ð +â[
ðð(âð)ð§
ð§ðððððšð¬ (
ð§ðð±
ð)+
ð(âð)ð§+ð
ð§ðð¬ð¢ð§ (
ð§ðð±
ð)] .
â
ð§=ð
H 8 Find the Fourier series of f(x) = x â x2; â Ï < x < Ï.
Deduce that: 1
12 â
1
22 +
1
32 â
1
42 + ⯠=
Ï2
12 .
ðð§ð¬ð°ðð«: ð(ð±) = â ðð
ð+â[
ð(âð)ð§+ð
ð§ðððšð¬ð§ð± +
ð(âð)ð§+ð
ð§ð¬ð¢ð§ð§ð±]
â
ð§=ð
.
DARSHAN INSTITUTE OF ENGINEERING & TECHNOLOGY Mathematics-I (3110014)
U N I T - 3 [ 6 8 ]
T 9 Find the Fourier series of f(x) = x + |x|; â Ï < x < Ï.
ðð§ð¬ð°ðð«: ð(ð±) =ð
ð +â[
ð[(âð)ð§ â ð]
ðð§ðððšð¬ð§ð± +
ð(âð)ð§+ð
ð§ð¬ð¢ð§ð§ð±] .
â
ð§=ð
C 10 Find the Fourier series to represent ex in the interval (âÏ, Ï).
ðð§ð¬ð°ðð«: ð(ð±) = ðð â ðâð
ðð+â
(ðð â ðâð) (âð)ð§
ð(ð§ð + ð)[ððšð¬ð§ð± â ð§ð¬ð¢ð§ ð§ð±]
â
ð§=ð
.
C 11 Find the Fourier series for periodic function f(x) with period 2,
where f(x) = { â1, â1 < x < 0
1, 0 < x < 1
.
ðð§ð¬ð°ðð«: ð(ð±) = â ð â ð(âð)ð§
ðð§ð¬ð¢ð§ð§ðð±
â
ð§=ð
.
H 12 Find Fourier series expansion of the function given by
f(x) = { 0, â 2 < x < 0
1, 0 < x < 2
.
ðð§ð¬ð°ðð«: ð(ð±) =ð
ð +â [
ð â (âð)ð§
ð§ðð¬ð¢ð§ (
ð§ðð±
ð)] .
â
ð§=ð
T 13 Find Fourier series of f(x) = {
x, â1 < x < 0
2, 0 < x < 1 .
ðð§ð¬ð°ðð«: ð(ð±) = ð
ð+â [
ð + (âð)ð§+ð
ð§ðððððšð¬ ð§ðð± +
ð + ð(âð)ð§+ð
ð§ðð¬ð¢ð§ð§ðð±]
â
ð§=ð
.
W-18 (7)
C 14 Find the Fourier series expansion of the function
f(x) = { âÏ, â Ï â€ x †0
x , 0 †x †Ï
. Deduce that â1
(2n â 1)2
â
n=1
= Ï2
8 .
ðð§ð¬ð°ðð«: ð(ð±) = â ð
ð+â[
(âð)ð§ â ð
ðð§ðððšð¬ð§ð± +
ð â ð(âð)ð§
ð§ð¬ð¢ð§ð§ð±] .
â
ð§=ð
DARSHAN INSTITUTE OF ENGINEERING & TECHNOLOGY Mathematics-I (3110014)
U N I T - 3 [ 6 9 ]
H 15 Obtain the Fourier Series for the function f(x) given by
f(x) = {0 ,âÏ â€ x †0
x2 , 0 †x †Ï
. Hence prove 1 â1
4 +1
9 â
1
16 + ⯠=
Ï2
12 .
ðð§ð¬ð°ðð«: ð(ð±) = ðð
ð+
â[ ð(âð)ð§ ððšð¬ ð§ð±
ð§ð+ð
ð {â
ðð(âð)ð§
ð§+ð(âð)ð§
ð§ðâð
ð§ð} ð¬ð¢ð§ð§ð±]
â
ð§=ð
.
T 16 Find the Fourier Series for the function f(x) given by
f(x) = { âÏ , â Ï < x < 0
x â Ï , 0 < x < Ï
.
ðð§ð¬ð°ðð«: ð(ð±) = â ðð
ð+â[
(âð)ð§ â ð
ðð§ðððšð¬ð§ð± +
(âð)ð§+ð
ð§ð¬ð¢ð§ð§ð±] .
â
ð§=ð
C 17 If f(x) = {
Ï + x , â Ï < x < 0
Ï â x , 0 < x < Ï & f(x) = f(x + 2Ï), for all x then
expand f(x) in a Fourier series.
ðð§ð¬ð°ðð«: ð(ð±) =ð
ð +â
ð
ðð§ð [ð â (âð)ð§] ððšð¬ ð§ð±
â
ð§=ð
.
H 18 Find Fourier series for 2Ï periodic function f(x) = {
âk , âÏ < x < 0
k , 0 < x < Ï .
Hence deduce that 1 â1
3 +1
5 â1
7 + ⯠=
Ï
4 .
ðð§ð¬ð°ðð«: ð(ð±) = â ðð€[ð â (âð)ð§]
ð§ðð¬ð¢ð§ð§ð±
â
ð§=ð
.
T 19 Find the Fourier Series for the function f(x) given by
f(x) =
{
1 +
2x
Ï ; âÏ â€ x †0
1 â 2x
Ï ; 0 †x †Ï
and prove 1
12 +
1
32 +
1
52 + ⯠=
Ï2
8 .
ðð§ð¬ð°ðð«: ð(ð±) = âð
ððð§ð [ð â (âð)ð§] ððšð¬ ð§ð±
â
ð§=ð
.
DARSHAN INSTITUTE OF ENGINEERING & TECHNOLOGY Mathematics-I (3110014)
U N I T - 3 [ 7 0 ]
â HALF-RANGE SERIES:
â If a function f(x) is defined only on a half interval (0, L) instead of (c, c + 2L), then it is
possible to obtain a Fourier cosine or Fourier sine series.
â HALF-RANGE COSINE SERIES IN THE INTERVAL (ð, ð):
â The half-range cosine series in the interval (0, L) is defined by
ð(ð±) = ðð
ð+âðð§ ððšð¬ (
ð§ðð±
ð)
â
ð§=ð
,
where the constants a0 and an are given by
a0 =2
L â« f(x) dxL
0
&
an =2
L â« f(x) cos (
nÏx
L) dx
L
0
.
METHOD â 3: HALF-RANGE COSINE SERIES IN THE INTERVAL (ð, ð)
C 1 Find a cosine series for f(x) = Ï â x in the interval 0 < x < Ï.
ðð§ð¬ð°ðð«: ð(ð±) = ð
ð+â
ð[(âð)ð§+ð + ð]
ðð§ðððšð¬ð§ð±
â
ð§=ð
.
H 2 Find Fourier cosine series for f(x) = x2; 0 < x †Ï.
ðð§ð¬ð°ðð«: ð(ð±) = ðð
ð+â
ð(âð)ð§
ð§ðððšð¬ð§ð±
â
ð§=ð
.
T 3 Find half-range cosine series for f(x) = (x â 1)2 in 0 < x < 1.
ðð§ð¬ð°ðð«: ð(ð±) = ð
ð+â
ð
ð§ððð ððšð¬(ð§ðð±)
â
ð§=ð
.
C 4 Obtain the Fourier cosine series of the function f(x) = ex in the range
(0, L).
ðð§ð¬ð°ðð«: ð(ð±) = ðð â ð
ð+â
ðð[ðð(âð) ð§ â ð]
ð§ððð + ððððšð¬ (
ð§ðð±
ð) .
â
ð§=ð
W-19 (4)
H 5 Find half-range cosine series for f(x) = eâx in 0 < x < Ï.
ðð§ð¬ð°ðð«: ð(ð±) = ð â ðð
ð+â
ð[ðâð(âð)ð§+ð + ð]
ð(ð§ð + ð)ððšð¬ð§ð±
â
ð§=ð
.
DARSHAN INSTITUTE OF ENGINEERING & TECHNOLOGY Mathematics-I (3110014)
U N I T - 3 [ 7 1 ]
C 6 Find half-range cosine series for sin x in (0, Ï).
ðð§ð¬ð°ðð«: ð(ð±) = ð
ð+â[
â(âð)ð+ð§ + ð
ð + ð§+â(âð)ðâð§ + ð
ð â ð§] ððšð¬ð§ð±
â
ð§=ð
, ðð = ð.
H 7 Find the half-range cosine series of f(x) = { 2 ; â2 < x < 00 ; 0 < x < 2
.
ðð§ð¬ð°ðð«: ð(ð±) = ð.
S-19 (4)
T 8
Find half-range cosine series for f(x) = { kx ; 0 †x <
l
2
k(l â x) ; l
2< x < l
.
And hence deduce that â1
(2n â 1)2 = Ï2
8
â
n=1
.
ðð§ð¬ð°ðð«: ð(ð±) = ð€ð¥
ð+ ðð€ð¥
ððâ
ð
ð§ð [ð ððšð¬ (
ð§ð
ð)â ð â (âð)ð§]
â
ð§=ð
ððšð¬ (ð§ðð±
ð¥).
â HALF-RANGE SINE SERIES IN THE INTERVAL (ð, ð):
â The half-range sine series in the interval (0, L) is defined by
ð(ð±) = âðð§ ð¬ð¢ð§ ( ð§ðð±
ð)
â
ð§=ð
,
where the constants bn is given by
bn =2
L â« f(x) sin (
nÏx
L) dx.
L
0
METHOD â 4: HALF-RANGE SINE SERIES IN THE INTERVAL (ð, ð)
C 1 Find the half-range sine series for f(x) = 2x, 0 < x < 1.
ðð§ð¬ð°ðð«: ð(ð±) = âð
ð§ð (âð)ð§+ð ð¬ð¢ð§ ð§ðð±
â
ð§=ð
.
H 2 Derive half-range sine series of f(x) = Ï â x, 0 †x †Ï.
ðð§ð¬ð°ðð«: ð(ð±) = âð
ð§ ð¬ð¢ð§ ð§ð± .
â
ð§=ð
W-18 (4)
DARSHAN INSTITUTE OF ENGINEERING & TECHNOLOGY Mathematics-I (3110014)
U N I T - 3 [ 7 2 ]
T 3 Find half-range sine series of f(x) = x3, 0 < x < Ï.
ðð§ð¬ð°ðð«: ð(ð±) = âð
ð§ (âð)ð§ [
ð
ð§ð â ðð] ð¬ð¢ð§ ð§ð± .
â
ð§=ð
T 4 Find half-range sine series for f(x) = ex in 0 < x < Ï.
ðð§ð¬ð°ðð«: ð(ð±) = â ðð§[ðð(âð)ð§+ð + ð]
ð(ð + ð§ð)ð¬ð¢ð§ð§ð±
â
ð§=ð
.
C 5 Find the sine series f(x) = {
x ; 0 < x <Ï
2
Ï â x ; Ï
2 < x < Ï
.
ðð§ð¬ð°ðð«: ð(ð±) =âð
ð§ðð ð¬ð¢ð§ (
ð§ð
ð) ð¬ð¢ð§ð§ð±
â
ð§=ð
.
T 6 Find half-range sine series for cos 2x in (0, Ï).
ðð§ð¬ð°ðð«: ð(ð±) = â ðð§[(âð)ð§+ð + ð]
ð(ð§ð â ð)ð¬ð¢ð§ ð§ð±
â
ð§=ðð§â ð
, ðð = ð.
â â â â â â â â â â
DARSHAN INSTITUTE OF ENGINEERING & TECHNOLOGY Mathematics-I (3110014)
U N I T - 4 [ 7 3 ]
UNIT 4
â FUNCTION OF SEVERAL VARIABLES:
â A function of two or more than two variables is said to be function of several variables.
⢠The area of rectangle is a function of two variables.
⢠The volume of a rectangular box is a function of three variables.
â LIMIT OF FUNCTION ð(ð±, ð²) AT POINT (ð, ð):
â Case-1: (a, b) = (0, 0)
(1). Find L1 = lim y â 0
{ lim x â 0
f(x, y) }
(2). Find L2 = lim x â 0
{ lim y â 0
f(x, y) }
(3). Find L3 = lim x â 0
{ lim y â mx
f(x, y) }
(4). Find L4 = lim x â 0
{ lim y â mx2
f(x, y) }
⢠If L1 = L2 = L3 = L4 = L then limit exist and the value of limit is L.
⢠If one of them is not equal to others then limit does not exist.
â Case-2: (a, b) â (0, 0)
(1). Let L = lim (x, y) â (a, b)
f(x, y).
In above expression, apply the limit and we get L is finite number then limit exist,
if not then take x â a = h and y â b = k and hence the limit (h, k) â (0, 0).
i.e. Lâ² = lim (x, y) â (a, b)
f(x, y) = lim (h, k) â (0, 0)
f(a + h, b + k)
(2). Again, apply the limit in above expression and if we get Lâ² is finite then limit exist,
if not then check four steps of case-1 for Lâ² and get the conclusion.
DARSHAN INSTITUTE OF ENGINEERING & TECHNOLOGY Mathematics-I (3110014)
U N I T - 4 [ 7 4 ]
METHOD â 1: LIMIT OF FUNCTION OF TWO VARIABLES
C 1 Evaluate lim (x, y) â (5, â2)
x4 + 4x3y â 5xy2 .
ðð§ð¬ð°ðð«: â ððð.
H 2 Evaluate lim (x, y) â (1, 2)
x3y3 â x3y2 + 3x + 2y.
ðð§ð¬ð°ðð«: ðð.
C 3 Find lim
(x, y) â (0, 0) 3x2y
x2 + y2 if it exist.
ðð§ð¬ð°ðð«: ð.
H 4 Evaluate lim
(x, y) â (0, 0) x3 â y3
x2 + y2 .
ðð§ð¬ð°ðð«: ð.
C 5 Show that the function f(x, y) =
2x2y
x4 + y2 has no limit as (x, y) approaches
to (0, 0).
S-19 (3)
H 6 Show that lim (x, y) â (0, 0)
xy
x2 + y2 does not exist.
H 7 Show that lim
(x, y) â (0, 0) x2 â y2
x2 + y2 does not exist.
T 8 By considering different paths show that the function f(x, y) =
x4 â y2
x4 + y2
has no limit as (x, y) â (0, 0).
T 9 Evaluate lim
(x, y) â (0, 0) x â y + 2âx â 2ây
âx â ây ; x â y.
ðð§ð¬ð°ðð«: ð.
â CONTINUITY OF FUNCTION OF TWO VARIABLES:
â A function f(x, y) is said to be continuous at point (a, b) if the following conditions are
satisfied:
(1). f(a, b) must be well define.
DARSHAN INSTITUTE OF ENGINEERING & TECHNOLOGY Mathematics-I (3110014)
U N I T - 4 [ 7 5 ]
(2). lim
(x, y) â (a, b) f(x, y) must be exist.
(3). lim
(x, y) â (a, b) f(x, y) = f(a, b).
â Given function is continuous at every point except at the origin means we have to discuss
continuity of that function at (0, 0).
ððšðð: lim x â 0
sin x
x= 1 & lim
x â â
sin x
x= 0.
METHOD â 2: CONTINUITY OF FUNCTION OF TWO VARIABLES
C 1 Discuss the continuity of the function f defined as
f(x, y) = x3 â y3
x2 + y2 ; (x, y) â (0, 0)
= 0 ; (x, y) = (0, 0) .
ðð§ð¬ð°ðð«: ððšð§ðð¢ð§ð®ðšð®ð¬.
W-18 (3)
H 2 Discuss the continuity of
f(x, y) =3x2y
x2 + y2 ; (x, y) â (0, 0)
= 0 ; (x, y) = (0, 0) .
ðð§ð¬ð°ðð«: ððšð§ðð¢ð§ð®ðšð®ð¬.
H 3 Show that following function is continuous at the origin.
f(x, y) =xy
âx2 + y2 ; (x, y) â (0, 0)
= 0 ; (x, y) = (0, 0) .
T 4 Examine for continuity at (0, 0) of the following function:
f(x, y) =xy
xy + (x â y) ; (x, y) â (0, 0)
= 0 ; (x, y) = (0, 0) .
ðð§ð¬ð°ðð«: ððšð§ðð¢ð§ð®ðšð®ð¬.
DARSHAN INSTITUTE OF ENGINEERING & TECHNOLOGY Mathematics-I (3110014)
U N I T - 4 [ 7 6 ]
C 5 Show that following function is not continuous at the origin.
f(x, y) =2x2y
x3 + y3 ; (x, y) â (0, 0)
= 0 ; (x, y) = (0, 0) .
H 6 Check the continuity for the following function at (0, 0)
f(x, y) =2xy
x2 + y2 ; (x, y) â (0, 0)
= 0 ; (x, y) = (0, 0) .
ðð§ð¬ð°ðð«: ðð¢ð¬ððšð§ðð¢ð§ð®ðšð®ð¬.
H 7 Discuss the continuity of the given function at (0, 0):
f(x, y) = y â x
y + x ; (x, y) â (0, 0)
= 0 ; (x, y) = (0, 0) .
ðð§ð¬ð°ðð«: ðð¢ð¬ððšð§ðð¢ð§ð®ðšð®ð¬.
H 8 Discuss the continuity of function at the origin:
f(x, y) =x
âx2 + y2 ; (x, y) â (0, 0)
= 2 ; (x, y) = (0, 0) .
ðð§ð¬ð°ðð«: ðð¢ð¬ððšð§ðð¢ð§ð®ðšð®ð¬.
T 9 Discuss the continuity of f(x, y) =
x3 â y3
x2 + y2 ; (x, y) â (0, 0)
= 5 ; (x, y) = (0, 0) .
ðð§ð¬ð°ðð«: ðð¢ð¬ððšð§ðð¢ð§ð®ðšð®ð¬.
C 10 Determine the continuity of the function at origin
f(x, y) = (x2 + y2) sin (1
x2 + y2 ) ; (x, y) â (0, 0)
= 0 ; (x, y) = (0, 0) .
ðð§ð¬ð°ðð«: ððšð§ðð¢ð§ð®ðšð®ð¬.
DARSHAN INSTITUTE OF ENGINEERING & TECHNOLOGY Mathematics-I (3110014)
U N I T - 4 [ 7 7 ]
â PARTIAL DERIVATIVE:
â Let f(x, y) be a function of two independent variables x and y.
⢠The first order partial derivative of ð with respect to ð± is denoted by
âf
âx ðšð« fx ðšð« (
âf
âx )y constant
and it is defined as âf
âx = lim
h â 0
f(x + h, y) â f(x, y)
h .
⢠The first order partial derivative of ð with respect to ð² is denoted by
âf
ây ðšð« fy ðšð« (
âf
ây )x constant
and it is defined as âf
ây = lim
k â 0
f(x, y + k) â f(x, y)
k .
â HIGHER ORDER PARTIAL DERIVATIVE:
â If f is a function of two independent variables x and y i.e. f(x, y), then the second order
partial derivatives are as follow:
â
âx (âf
âx ) =
â2f
âx2 = fxx ,
â
ây (âf
ây ) =
â2f
ây2 = fyy,
â
ây (âf
âx ) =
â2f
ây âx = fxy,
â
âx (âf
ây ) =
â2f
âx ây = fyx .
⢠fxy and fyx are usually called mixed second order partial derivatives of f.
â If f is a function of two independent variables x and y then the third order partial
derivatives are as bellow:
â
âx (â
âx (âf
âx )) =
â3f
âx3 = fxxx ,
â
ây (â
ây (âf
ây )) =
â3f
ây3 = fyyy ,
â
ây (â
âx (âf
âx )) =
â3f
ây âx âx = fxxy ,
â
âx (â
ây (âf
ây )) =
â3f
âx ây ây = fyyx ,
â
ây (â
ây (âf
âx )) =
â3f
ây âyâx = fxyy ,
â
âx (â
âx (âf
ây )) =
â3f
âx âx ây = fyxx ,
â
âx (â
ây (âf
âx )) =
â3f
âx ây âx = fxyx ,
â
ây (â
âx (âf
ây )) =
â3f
ây âx ây = fyxy .
â On the similar way we can define partial derivatives of order greater than three.
â Number of second order partial derivatives of function of two independent variables is
ð = ðð.
DARSHAN INSTITUTE OF ENGINEERING & TECHNOLOGY Mathematics-I (3110014)
U N I T - 4 [ 7 8 ]
â Number of nth order partial derivatives of function of two independent variables is ðð§.
â Number of nth order partial derivatives of function of ð€ independent variables is ð€ð§.
â MIXED DERIVATIVE THEOREM (CLAIRAUTâS THEOREM):
â If f(x, y), fx, fy, fxy, fyx are defined throughout an open region containing a point (a, b) and
fx, fy, fxy, fyx are all continuous at (a, b) then ðð±ð²(ð, ð) = ðð²ð±(ð, ð).
METHOD â 3: PARTIAL DERIVATIVES
C 1 If f(x, y) = x2y + xy2, then find fx(1, 2) and fy(1, 2) by definition.
ðð§ð¬ð°ðð«: ð & ð.
H 2 Find the values of
âf
âx and
âf
ây at the point (4,â5), if
f(x, y) = x2 + 3xy + y â 1 by definition.
ðð§ð¬ð°ðð«: â ð & ðð.
C 3 If f(x, y) = x3 + x2y3 â 2y2, find fx(2, 1) and fy(2, 1).
ðð§ð¬ð°ðð«: ðð & ð.
C 4 Find the second order partial derivative of f(x, y) = x3 + x2y3 â 2y2.
ðð§ð¬ð°ðð«: ðð±ð± = ðð± + ðð²ð , ðð±ð² = ðð±ð²ð, ðð²ð± = ðð±ð²
ð & ðð²ð² = ðð±ðð² â ð.
C 5 If f(x, y) = sin (
x
1 + y ) , calculate
âf
âx and
âf
ây .
ðð§ð¬ð°ðð«: ððšð¬ (ð±
ð + ð² )â
ð
ð + ð² & â ððšð¬ (
ð±
ð + ð² )â
ð±
(ð + ð²)ð .
T 6 Find the values of
âf
âx and
âf
ây , if f(x, y) =
2y
y + cosx .
ðð§ð¬ð°ðð«: ðð²ð¬ð¢ð§ð±
(ð² + ððšð¬ð±)ð &
ðððšð¬ð±
(ð² + ððšð¬ð±)ð .
T 7 Verify that the function u = eâα2k2t sin kx is a solution of heat conduction
equation ut = α2uxx.
S-19 (3)
DARSHAN INSTITUTE OF ENGINEERING & TECHNOLOGY Mathematics-I (3110014)
U N I T - 4 [ 7 9 ]
T 8 The amount of work done by the heartâs main pumping chamber, the left
ventricle is given by the equation: W(P, V, ÎŽ, v, g) = PV + VÎŽv2
2g, where W is the
work per unit time, P is the average blood pressure, V is the volume of blood
pumped out during the unit of time, ÎŽ is the weight density of blood, v is the
average velocity of the existing blood and g is the acceleration of the gravity
which is constant. Find partial derivative of W with respect to each variable.
ðð§ð¬ð°ðð«: ðð
ðð = ð,
ðð
ðð = ð +
ð ð¯ð
ðð , ðð
ðð = ðð¯ð
ðð &
ðð
ðð¯= ðð ð¯
ð .
C 9 If resistors of R1, R2 and R3 ohms are connected in parallel to make an R-ohm
resistor, the value of R can be found from the equation 1
R =
1
R1 +
1
R2 +
1
R3 . Find the value of
âR
âR2 when R1 = 30, R2 = 45 and R3 = 90 ohms.
ðð§ð¬ð°ðð«: ð/ð.
C 10 If u = e3xyz then show that
â3u
âx ây âz = (3 + 27xyz + 27x2y2z2)e3xyz.
T 11 Calculate fxxyz if f(x, y, z) = sin(3x + yz) .
ðð§ð¬ð°ðð«: â ð ððšð¬(ðð± + ð²ð³) + ðð²ð³ ð¬ð¢ð§(ðð± + ð²ð³).
C 12 If u = log(tan x + tan y + tan z) then prove that sin2x
âu
âx + sin2y
âu
ây +
sin2z âu
âz = 2.
H 13 If u = x2y + y2z + z2x then find out
âu
âx +âu
ây +âu
âz .
ðð§ð¬ð°ðð«: (ð± + ð² + ð³)ð.
W-18 (3)
H 14 If u = log(x2 + y2) , verify that
â2u
âx ây =
â2u
ây âx .
H 15 For u = tanâ1 (
y
x ) . Show that (ð)
â2u
âx2 +â2u
ây2 = 0 & (ð)
â2u
âx ây =
â2u
ây âx .
T 16 If z = x + yx, prove that
â2z
âx ây =
â2z
ây âx .
DARSHAN INSTITUTE OF ENGINEERING & TECHNOLOGY Mathematics-I (3110014)
U N I T - 4 [ 8 0 ]
T 17 If u =
ex+y+z
ex + ey + ez then show that
âu
âx +âu
ây +âu
âz = 2u.
T 18 If u = r2 cos 2Ξ then show that
â2u
âr2 +1
r
âu
âr +1
r2
â2u
âΞ2 = 0.
C 19 If u = log(x3 + y3 â x2y â xy2) , prove that (
â
âx +
â
ây )2
u = â4
(x + y)2 .
H 20 If u = x2y + y2z + z2x, prove that
âu
âx +âu
ây +âu
âz = (x + y + z)2 and
(â
âx +
â
ây +
â
âz )2
u = 6(x + y + z).
â TOTAL DIFFERENTIATION:
â If u = f(x, y), then the total differentiation of u is given by
ðð® =ðð
ðð± ðð± +
ðð
ðð² ðð².
Here du, dx & dy are called the total differentials (or exact differentials) of u, x & y
respectively.
â If u = f(x, y, z), then the total differentiation of u is given by
ðð® =ðð
ðð± ðð± +
ðð
ðð² ðð² +
ðð
ðð³ ðð³.
â If u = f(x1, x2, âŠâŠ , xn), then the total differential is given by
ðð® =ðð
ðð±ð ðð±ð +
ðð
ðð±ð ðð±ð +â¯+
ðð
ðð±ð§ ðð±ð§.
â CHAIN RULE:
â Case-1:
If u = f(x, y) and x = g(r, s) & y = h(r, s), then
âu
âr =âu
âx âx
âr +âu
ây ây
âr
âu
âs =âu
âx âx
âs +âu
ây ây
âs
x
y
r
s
u
DARSHAN INSTITUTE OF ENGINEERING & TECHNOLOGY Mathematics-I (3110014)
U N I T - 4 [ 8 1 ]
â Case-2:
If u = f(x, y, z) and x = g(r, s, t), y = h(r, s, t) & z = k(r, s, t), then
âu
âr =âu
âx âx
âr +âu
ây ây
âr +âu
âz âz
âr
âu
âs =âu
âx âx
âs +âu
ây ây
âs +âu
âz âz
âs
âu
ât =âu
âx âx
ât +âu
ây ây
ât +âu
âz âz
ât
â Case-3:
If u = f(x, y, z) and x = g(r, s), y = h(r, s) & z = k(r, s), then
âu
âr =âu
âx âx
âr +âu
ây ây
âr +âu
âz âz
âr
âu
âs =âu
âx âx
âs +âu
ây ây
âs +âu
âz âz
âs
â Case-4:
If u = f(x, y) and x = g(r, s, t) & y = h(r, s, t), then
âu
âr =âu
âx âx
âr +âu
ây ây
âr
âu
âs =âu
âx âx
âs +âu
ây ây
âs
âu
ât =âu
âx âx
ât +âu
ây ây
ât
â Case-5:
If u = f(x, y, z) and x = g(t), y = h(t) & z = k(t), the
du
dt =âu
âx dx
dt +âu
ây dy
dt +âu
âz dz
dt
x
y
r
s
z t
u
x
y
r
s
t
u
x
y
r
s
z
u
x
y t
z
u
DARSHAN INSTITUTE OF ENGINEERING & TECHNOLOGY Mathematics-I (3110014)
U N I T - 4 [ 8 2 ]
METHOD â 4: CHAIN RULE
C 1 Find
âw
âr and
âw
âs in terms of r and s, if w = x + 2y + z2, where x =
r
s ,
y = r2 + ln(s)& z = 2r.
ðð§ð¬ð°ðð«: (ð
ð¬ ) + ððð« & (
ð
ð¬ ) â (
ð«
ð¬ð ).
H 2 If z = xy2 + x2y, x = at2, y = 2at, find
dz
dt .
Answer: ððððð (ð + ðð).
C 3 If z = x2y + 3xy4, where x = sin 2t and y = cos t . Find
dz
dt when t = 0.
ðð§ð¬ð°ðð«: ð.
H 4 Let u = x4y + y2z,where x = rset, y = rs2eât & z = r2s sin(t). Find the value
of âu
âs , where r = 2, s = 1 & t = 0.
ðð§ð¬ð°ðð«: ððð.
T 5 For z = tanâ1 (
x
y ) , x = u cosv, y = u sinv, evaluate
âz
âu and
âz
âv at the
point (1.3, Ï/6).
ðð§ð¬ð°ðð«: ðð³
ðð® = ð &
ðð³
ðð¯ = âð.
C 6 If u = f (
y â x
xy, z â x
xz) , then show that x2
âu
âx + y2
âu
ây + z2
âu
âz = 0.
H 7 If u = f (
x
y , y
z , z
x ) , prove that x
âu
âx + y
âu
ây + z
âu
âz = 0.
T 8 If u = f(x â y, y â z, z â x), then show that
âu
âx +âu
ây +âu
âz = 0.
W-19 (7)
T 9 If u = f(x2 + 2yz, y2 + 2zx), then prove that (y2 â zx)
âu
âx + (x2 â yz)
âu
ây +
(z2 â xy)âu
âz = 0.
DARSHAN INSTITUTE OF ENGINEERING & TECHNOLOGY Mathematics-I (3110014)
U N I T - 4 [ 8 3 ]
C 10 If u = f(x, y),where x = es cost and y = es sint, show that (
âu
âx )2
+ (âu
ây )2
= eâ2s [(âu
âs )2
+ (âu
ât )2
].
C 11 t If u = f(r) and r2 = x2 + y2 + z2, then show that
â2u
âx2 +â2u
ây2 +â2u
âz2
= f â²â²(r) +2
r f â²(r).
T 12 Suppose f is a differential function of x and y and g(u, v)
= f(eu + sin v , eu + cos v). Use the following table to
calculate gu(0, 0), gv(0, 0).
Point f g fx fy
(0, 0) 3 6 4 8
(1, 2) 6 3 2 5
ðð§ð¬ð°ðð«: ð, ð.
S-19 (4)
C 13 Find
du
dt as total derivative for u = 4x + xy â y2, x = cos3t & y = sin 3t.
ðð§ð¬ð°ðð«: â ðð ð¬ð¢ð§ðð + ð ððšð¬ðð â ð ð¬ð¢ð§ðð.
H 14 If z = f(x, y) = x2 + 3xy â y2, find the differential dz.
ðð§ð¬ð°ðð«: (ðð± + ðð²)ðð± + (ðð± â ðð²)ðð².
â IMPLICIT FUNCTION:
â A function in which one variable cannot be expressed in terms of other variables is called
implicit function.
â First order differential coefficient of an implicit function:
Let f(x, y) = c be an implicit function then
ðð²
ðð± = â
ð©
ðª , where q â 0 and p =
âf
âx & q =
âf
ây .
â Second order differential coefficient of an implicit function:
Let f(x, y) = c be an implicit function then
DARSHAN INSTITUTE OF ENGINEERING & TECHNOLOGY Mathematics-I (3110014)
U N I T - 4 [ 8 4 ]
ððð²
ðð±ð = â [
ðªðð« â ðð©ðªð¬ + ð©ðð
ðªð] , where q â 0 and
p =âf
âx , q =
âf
ây , r =
â2f
âx2 , s =
â2f
âx ây & t =
â2f
ây2 .
â Also, we can use following formula to find second order differential coefficient:
q3d2y
dx2 = 2pqs â p2t â q2r = |
r s ps t qp q 0
|.
â PARTIAL DIFFERENTIATION OF IMPLICIT FUNCTION:
â If f(x, y, z)=0 is an implicit function then
âz
âx = â
âf âxâ
âf âzâ ,
âz
ây = â
âf âyâ
âf âzâ ,
ây
âx = â
âf âxâ
âf âyâ ,
ây
âz = â
âf âzâ
âf ây â,
âx
ây = â
âf âyâ
âf âx â &
âx
âz = â
âf âzâ
âf âx â .
METHOD â 5: IMPLICIT FUNCTION
C 1 By using partial derivatives find the value of
dy
dx for xey + sin(xy) + y
âlog2 = 0 at (0, log2).
ðð§ð¬ð°ðð«: â (ð + ð¥ðšð ð).
H 2 Find
dy
dx if xy + yx = ab.
ðð§ð¬ð°ðð«: â [ ð² ð±ð²âð + ð²ð± ð¥ðšð ð²
ð±ð² ð¥ðšð ð± + ð± ð²ð±âð ].
H 3 Find
dy
dx when (cosx)y = (siny)x.
ðð§ð¬ð°ðð«: [ð² ððð§ ð± + ð¥ðšð (ð¬ð¢ð§ð²)
ð¥ðšð (ððšð¬ð±) â ð± ððšð ð² ].
DARSHAN INSTITUTE OF ENGINEERING & TECHNOLOGY Mathematics-I (3110014)
U N I T - 4 [ 8 5 ]
T 4 Find
dy
dx when yx
y= sinx.
ðð§ð¬ð°ðð«: â [ð² ð±ð²âðð¥ðšð ð² â ððšð ð±
ð±ð² {ð¥ðšð ð± ð¥ðšð ð² + (ð/ð²)} ].
T 5 Using partial differentiation find
dy
dx for (cosx)y = xsiny.
ðð§ð¬ð°ðð«: â [ âð² ð¬ð¢ð§ ð± (ððšð¬ ð±)ð²âð â ð¬ð¢ð§ ð² (ð±)ð¬ð¢ð§ð²âð
ð¥ð§ ððšð¬ ð± (ððšð¬ð±)ð² â ð¥ð§ ð± ððšð¬ ð² (ð±)ð¬ð¢ð§ ð² ].
C 6 If x3 + y3 = 6xy, then find
dy
dx and
d2y
dx2 .
ðð§ð¬ð°ðð«: âð±ð â ðð²
ð²ð â ðð± & â
ððð±ð²
(ð²ð â ðð±)ð .
W-18 (4)
C 7 Find
dy
dx and
d2y
dx2 if x6 + 2y6 = 3a2x3.
ðð§ð¬ð°ðð«: ðððð±ð â ðð±ð
ðð²ð & â [
ððððð±ð(ððð + ðð±ð)
ðððð²ðð ].
H 8 Using partial derivatives find
d2y
dx2 for x5 + y5 = 5x2.
ðð§ð¬ð°ðð«: âðð±ð(ð + ð±ð)
ð²ð .
C 9 Find
âz
âx and
âz
ây if z defined implicitly as a function of x and y by the
equation x3 + y3 + z3 + 6xyz = 1.
ðð§ð¬ð°ðð«: â [ ð±ð + ðð²ð³
ð³ð + ðð±ð² ] & â [
ð²ð + ðð±ð³
ð³ð + ðð±ð² ].
H 10 Find
âz
âx and
âx
ây at (1, 2,â3) if x2 + 2y2 + z2 = 16.
ðð§ð¬ð°ðð«: ð/ð & â ð.
T 11 Find
âz
âx and
âz
ây using partial derivatives for z3 â xy + yz + y3 â 2 = 0
at (1, 1, 1).
ðð§ð¬ð°ðð«: ð/ð & â ð/ð.
DARSHAN INSTITUTE OF ENGINEERING & TECHNOLOGY Mathematics-I (3110014)
U N I T - 4 [ 8 6 ]
â TANGENT PLANE TO THE GIVEN SURFACE AT GIVEN POINT:
â The equation of the tangent plane to the surface f(x, y) = 0 at the point P(x1, y1) is given
by
(x â x1) (âf
âx )P+ (y â y1) (
âf
ây )P
= 0.
â The equation of the tangent plane to the surface f(x, y, z) = 0 at the point P(x1, y1, z1) is
given by
(x â x1) (âf
âx )P+ (y â y1) (
âf
ây )P
+ (z â z1) (âf
âz )P= 0.
â NORMAL LINE TO THE GIVEN SURFACE AT GIVEN POINT:
â The equation of the normal line to the surface f(x, y) = 0 at the point P(x1, y1) is given by
x â x1
(âf âx
)P
= y â y1
(âf ây
)P
.
â The equation of the normal line to the surface f(x, y, z) = 0 at the point P(x1, y1, z1) is
given by
x â x1
(âf âx
)P
= y â y1
(âf ây
)P
= z â z1
(âf âz
)P
.
METHOD â 6: TANGENT PLANE AND NORMAL LINE
C 1 Find the equations of tangent plane and normal line to the surface
2xz2 â 3xy â 4x = 7 at point (1,â1, 2).
ðð§ð¬ð°ðð«: ðð± â ðð² + ðð³ = ðð & ð± â ð
ð= ð² + ð
âð= ð³ â ð
ð .
H 2 Find the equation of the tangent plane and normal line to the surface
x3 + 2xy2 â 7z2 + 3y + 1 = 0 at (1, 1, 1).
ðð§ð¬ð°ðð«: ðð± + ðð² â ððð³ + ð = ð & ð± â ð
ð= ð² â ð
ð= ð³ â ð
âðð .
DARSHAN INSTITUTE OF ENGINEERING & TECHNOLOGY Mathematics-I (3110014)
U N I T - 4 [ 8 7 ]
H 3 Find the equation of the tangent plane and normal line at the point
(â2, 1, â3) to the ellipsoid x2
4+ y2 +
z2
9= 3. ðð
Find the equation of the tangent plane and normal line of the surface
x2
4+ y2 +
z2
9= 3 at (â2, 1, â3).
ðð§ð¬ð°ðð«: ðð± â ðð² + ðð³ + ðð = ð & ð± + ð
âð= ð² â ð
ð= ð(ð³ + ð)
âð.
H 4 Find the equation of the tangent plane and normal line to the surface z = 1
â1
10 (x2 + 4y2 ) at the point (1, 1, 1/2).
ðð§ð¬ð°ðð«: ð± + ðð² + ðð³ â ðð
ð= ð &
ð(ð± â ð)
ð= ð(ð² â ð)
ð= ðð³ â ð
ð .
T 5 Find the equation of the tangent plane and normal line to the surface
x2 + y2 + z2 = 3 at the point (1, 1, 1).
ðð§ð¬ð°ðð«: ð± + ð² + ð³ â ð = ð & ð± â ð = ð² â ð = ð³ â ð.
W-19 (3)
T 6 Find the equation of the tangent plane and normal line at the point (2, 0, 2)
to the ellipsoid 2z â x2 = 0.
ðð§ð¬ð°ðð«: ðð± â ð³ â ð = ð & ð± â ð
âð= ð³ â ð
ð .
T 7 Find the equations of tangent plane and normal line to the surface
cosÏx â x2y + exz + yz = 4 at point P(0, 1, 2).
ðð§ð¬ð°ðð«: ðð± + ðð² + ð³ â ð = ð & ð±
ð= ð² â ð
ð= ð³ â ð
ð .
C 8 Show that the surfaces z = xy â 2 and x2 + y2 + z2 = 3 have the same
tangent plane at (1, 1,â1).
ðð§ð¬ð°ðð«: ð± + ð² â ð³ â ð = ð.
â LOCAL EXTREME VALUES OF FUNCTION OF TWO VARIABLES:
â A point (a, b) is said to be a stationary point of f(x, y),
if (âf
âx )(a, b)
= 0 & (âf
ây )(a, b)
= 0.
DARSHAN INSTITUTE OF ENGINEERING & TECHNOLOGY Mathematics-I (3110014)
U N I T - 4 [ 8 8 ]
â A point (a, b) is said to be a critical point of f(x, y), if either point (a, b) is stationary point
or at point (a, b) the derivative of f is not exist.
â A stationary point (a, b) is said to be a saddle point of given function, if at point
(a, b) given function has neither local minima nor local maxima.
â A local maximum or local minimum values of a function is called its local extreme values.
â Procedure to find the local maxima and local minima of the function f(x, y) is as follow:
(ð). Find âf
âx and
âf
ây .
(ð). Solve âf
âx = 0 &
âf
ây = 0 to find the stationary point/ points (a1, b1), (a2, b2),⊠.
(ð). Find â2f
âx2 ,
â2f
âx ây &
â2f
ây2 .
(ð). Find r = (â2f
âx2 )(a1,b1)
, s = (â2f
âx ây )(a1, b1)
& t = (â2f
ây2 )(a1,b1)
.
(ð). Find ð«ð â ð¬ð for stationary point (a1, b1).
(ð). Conclusion for stationary point (a1, b1) from the following table.
Result: Conclusion:
rt â s2 > 0 & r > 0 (or t > 0) Function has local minimum value.
rt â s2 > 0 & r < 0 (or t < 0) Function has local maximum value.
rt â s2 < 0 Function has no local maxima or local minima
(i.e. saddle point).
rt â s2 = 0 OR r = 0 Nothing can be said about the local maxima or local
minima. It requires further investigation.
(7). Repeat step (4), (5), (6) for all other stationary points.
(8). Substitute the point (ai, bi) in the given function to find the local maximum or local
minimum value of the function at that point, if possible.
DARSHAN INSTITUTE OF ENGINEERING & TECHNOLOGY Mathematics-I (3110014)
U N I T - 4 [ 8 9 ]
METHOD â 7: LOCAL EXTREME VALUES
C 1 Define saddle point. Find the local extreme values of f(x, y) = x2 â y2 â
xy â x + y + 6, if possible.
ðð§ð¬ð°ðð«: ððšð ð©ðšð¬ð¬ð¢ðð¥ð.
C 2 Find the local extreme values of function f(x, y) = x3 + 3xy2 â 15x2 â 15y2
+72x.
ðð§ð¬ð°ðð«: ðð¢ð§ð¢ðŠð®ðŠ = ððð & ððð±ð¢ðŠð®ðŠ = ððð.
H 3 Find the local extreme values of function f(x, y) = x3 + y3 â 3x â 12y + 20.
ðð§ð¬ð°ðð«: ðð¢ð§ð¢ðŠð®ðŠ = ð & ððð±ð¢ðŠð®ðŠ = ðð.
H 4 Find the extreme values of x3 + 3xy2 â 3x2 â 3y2 + 4.
ðð§ð¬ð°ðð«: ðð¢ð§ð¢ðŠð®ðŠ = ð & ððð±ð¢ðŠð®ðŠ = ð.
W-18 (3)
T 5 Find the local extreme values of f(x, y) = x3 + y3 â 3xy.
ðð§ð¬ð°ðð«: ðð¢ð§ð¢ðŠð®ðŠ = âð & ððð±ð¢ðŠð®ðŠ ððšðð¬ ð§ðšð ðð±ð¢ð¬ð.
T 6 Discuss the Maxima and Minima of the function 3x2 â y2 + x3.
ðð§ð¬ð°ðð«: ðð¢ð§ð¢ðŠð®ðŠ ððšðð¬ ð§ðšð ðð±ð¢ð¬ð & ððð±ð¢ðŠð®ðŠ = ð.
W-19 (4)
T 7 Locate the stationary points of x4 + y4 â 2x2 + 4xy â 2y2 and determine
their nature.
ðð§ð¬ð°ðð«: (ð, ð) = ðð®ð«ðð¡ðð« ð¢ð§ð¯ðð¬ðð¢ð ððð¢ðšð§ ð¢ð¬ ð§ðððð¬ð¬ðð«ð²,
(âð ,ââð ) = ðŠð¢ð§ð¢ðŠð®ðŠ & (ââð , âð ) = ðŠð¢ð§ð¢ðŠð®ðŠ.
C 8 For what values of the constant k does the second derivative test guarantee
that f(x, y) = x2 + kxy + y2 will have a saddle point at (0, 0)? A local
minimum at (0, 0)?
ðð§ð¬ð°ðð«: ð€ â â\[âð, ð] & ð€ â (âð, ð).
S-19 (4)
â LAGRANGEâS METHOD OF UNDETERMINED MULTIPLIERS:
â This method is useful to find maxima and minima of given function with satisfying given
condition or constraint.
â Procedure of Lagrangeâs method of undetermined multipliers for finding maxima or
minima of the function f(x, y) with satisfying the condition Ï(x, y) = 0 is as follow:
DARSHAN INSTITUTE OF ENGINEERING & TECHNOLOGY Mathematics-I (3110014)
U N I T - 4 [ 9 0 ]
(1). Construct the new function F(x, y) as F(x, y) = f(x, y) + λÏ(x, y).
(ð). Solve the three equations âF
âx = 0,
âF
ây = 0 & Ï(x, y) = 0 to find the stationary
points(x1, y1), (x2, y2)⊠.
(ð). Find the value of f(x, y) at above stationary points & give conclusion about
maxima or minima.
â Procedure of Lagrangeâs method of undetermined multipliers for finding maxima or
minima of the function f(x, y, z) with satisfying the condition Ï(x, y, z) = 0 is as follow:
(1). Construct the new function F(x, y, z) = 0 as F(x, y, z) = f(x, y, z) + λÏ(x, y, z).
(ð). Solve four equations âF
âx = 0,
âF
ây = 0,
âF
âz = 0 & Ï(x, y, z) = 0 to find the
stationary points (x1, y1, z1), (x2, y2, z2)⊠.
(ð). Find the value of f(x, y, z) at above stationary points & give conclusion about
maxima or minima.
â NOTES:
(1). The distance between two points P(x, y, z) and Q(a, b, c) is
PQ = â(x â a)2 + (y â b)2 + (z â c)2 .
(2). The area of open rectangular box is xy + 2yz + 2zx and the area of the closed
rectangular box is 2xy + 2yz + 2zx.
(3). The volume of open/closed rectangular box is xyz.
(4). The drawback of this method is that it gives no information about the nature of
stationary points i.e. about maxima and minima.
(5). λ is known as Lagrangeâs multiplier.
DARSHAN INSTITUTE OF ENGINEERING & TECHNOLOGY Mathematics-I (3110014)
U N I T - 4 [ 9 1 ]
METHOD â 8: LAGRANGEâS MULTIPLIERS
T 1 Find the greatest and smallest values of the function f(x, y) = xy on the
ellipse x2
8+ y2
2= 1.
ðð§ð¬ð°ðð«: ðð«ððððð¬ð = ð & ððŠðð¥ð¥ðð¬ð = âð.
C 2 Find the maximum and minimum values of the function f(x, y) = 3x + 4y on
the circle x2 + y2 = 1 using the method of Lagrange multipliers.
ðð§ð¬ð°ðð«: ððð±ð¢ðŠð®ðŠ = ð & ðð¢ð§ð¢ðŠð®ðŠ = âð.
H 3 Find the numbers x, y and z such that xyz = 8 and xy + yz + zx is maximum,
using Lagrangeâs method of undetermined multipliers.
ðð§ð¬ð°ðð«: ð± = ð² = ð³ = ð.
T 4 Using Lagrangeâs method of undetermined multipliers, find the maximum
value of f(x, y, z) = xp yq zr subject to the condition ax + by + cz = p +
q + r.
ðð§ð¬ð°ðð«: (ð©/ð)ð© â (ðª/ð)ðª â (ð«/ð)ð«.
C 5 Divided a given number âaâ into three positive points such that their sum is
âaâ & product is maximum.
ðð§ð¬ð°ðð«: (ð/ð, ð/ð, ð/ð).
C 6 Find the point on the plane x + 2y + 3z = 13 closest to the point (1, 1, 1).
ðð§ð¬ð°ðð«: (ð/ð, ð, ð/ð).
H 7 Find the point on the plane 2x + 3y â z = 5 which is nearest to the origin,
using Lagrangeâs method of undetermined multipliers.
ðð§ð¬ð°ðð«: (ð/ð, ðð/ðð, â ð/ðð).
C 8 Find the shortest and largest distance from the point (1, 2, â1) to the
sphere x2 + y2 + z2 = 24.
ðð§ð¬ð°ðð«: ðð¡ðšð«ððð¬ð ðð¢ð¬ððð§ðð = âð & ððð«ð ðð¬ð ðð¢ð¬ððð§ðð = âðð .
W-18 (7)
H 9 Find the maximum and minimum distance from the point (1, 2, 2) to the
sphere x2 + y2 + z2 = 36.
ðð§ð¬ð°ðð«: ððð±ð¢ðŠð®ðŠ = ð & ðð¢ð§ð¢ðŠð®ðŠ = ð.
W-19 (7)
DARSHAN INSTITUTE OF ENGINEERING & TECHNOLOGY Mathematics-I (3110014)
U N I T - 4 [ 9 2 ]
T 10 Find the points on the sphere x2 + y2 + z2 = 4 that are closest to and
farthest from the point (3, 1,â1).
ðð§ð¬ð°ðð«: ðð¥ðšð¬ðð¬ð = (ð
âðð ,ð
âðð ,âð
âðð )& ð ðð«ðð¡ðð¬ð = (
âð
âðð ,âð
âðð ,ð
âðð ).
S-19 (7)
C 11 A rectangular box, open at the top, is to have a given capacity of 64 cm3. Find
the dimensions of the box requiring least material for its construction.
ðð§ð¬ð°ðð«: (ð â âð ð
, ð â âð ð
, ð â âð ð
).
H 12 You are to construct an open rectangular box from 12 ft2 of material. What
dimensions will result in a box of maximum volume?
ðð§ð¬ð°ðð«: (ð, ð, ð).
T 13 A rectangular box, open at the top, is to have a volume 32 c.c. Find the
dimensions of the box requiring least material for its construction.
ðð§ð¬ð°ðð«: (ð, ð, ð).
â VECTOR FUNCTION OF SINGLE VARIABLE:
â A vector function is function r ⶠI ⶠân, with n = 2, 3,⊠and I â â.
⢠In short, a function whose domain is a set of real numbers and whose codomain is a
set of vectors.
⢠For example: r(t) = ( t2, t + 1, 2t ) = t2 i + (t + 1) j + 2t k.
â Motion in space motivates to define vector function.
â Given Cartesian coordinate in â3, the values of vector function can be written in
components as r(t) = ( x(t), y(t), z(t) ), t â I where x(t), y(t), z(t) are the values of
three scalar functions.
â SCALAR FUNCTION:
â A function whose codomain is scalars is called scalar function.
⢠For example: f(x, y, z) = 2x2y + 4z is scalar function.
â DERIVATIVE OF A VECTOR FUNCTION:
â A vector function V(t) = ( v1(t), v2(t), v3(t) ) is said to be differentiable at a point t if
following limit exists:
DARSHAN INSTITUTE OF ENGINEERING & TECHNOLOGY Mathematics-I (3110014)
U N I T - 4 [ 9 3 ]
Vâ²(t) = limât â0
V(t + ât) â V(t)
ât .
â The vector Vâ²(t) is called the derivative of V(t).
â V(t) is differentiable at point t iff itâs all components are differentiable at t.
â The derivative Vâ²(t) is obtained by differentiating each component separately as
Vâ²(t) = ( v1â² (t), v2
â² (t), v3â² (t) ) =
âv1
ât i +
âv2 ât
j + âv3 ât
k.
â Properties of derivative for vector function:
(1). (C â V )â² = C â Vâ².
(2). (U + V )â² = Uâ² + Vâ².
(3). (U â V )â² = ( Uâ² â V ) + ( U â Vâ² ).
(4). (U â V â W )â² = ( Uâ² â V â W ) + ( U â Vâ² â W ) + ( U â V â Wâ² ).
(5). (U Ã V )â² = ( Uâ² Ã V ) + ( Vâ² Ã U ).
â GRADIENT OF A SCALAR FUNCTION:
â The gradient of a scalar function f(x, y, z) is denoted by grad f/ â f and defined as
grad f = â f = âf
âx i +
âf
ây j +
âf
âz k where del operator = â(nabla) =
â
âx i +
â
ây j +
â
âz k.
â The gradient of scalar function is a vector.
â The gradient of a scalar field f(x, y, z) is a vector normal to the surface f(x, y, z) = c and
has a magnitude equal to the rate of change of f(x, y, z) along this normal.
METHOD â 9: GRADIENT
T 1 Find gradient of Ï = 2z3 â 3(x2 + y2)z + tanâ1(xz) at (1, 1, 1).
ðð§ð¬ð°ðð«: (âðð/ð, â ð, ð/ð).
H 2 Find gradient of f(x, y, z) = 2x + yz â 3y2.
ðð§ð¬ð°ðð«: (ð, ð³ â ðð², ð²).
C 3 Find âÏ at (1,â2,â1), if Ï = 3x2y â y3z2.
ðð§ð¬ð°ðð«: (âðð, â ð, â ðð).
DARSHAN INSTITUTE OF ENGINEERING & TECHNOLOGY Mathematics-I (3110014)
U N I T - 4 [ 9 4 ]
T 4 Find grad(Ï) if Ï = log(x2 + y2 + z2) at the point (1,0,â2).
ðð§ð¬ð°ðð«: (ð/ð, ð, â ð/ð).
C 5 Evaluate âer2, where r2 = x2 + y2 + z2.
ðð§ð¬ð°ðð«: ððð«ð(ð±, ð², ð³).
T 6 If vector r = x i + y j + z k and r = | r | then show that
ârn = n rnâ2 r. Hence evaluate âr3, â (1
r ).
ðð§ð¬ð°ðð«: ðð«ï¿œï¿œ , âᅵᅵ
ð«ð .
â DIRECTIONAL DERIVATIVE:
â The directional derivative of f(x, y) at (a, b) in the direction of a unit vector u is denoted
by D u f and it is defined as ârate of change of f in the direction of u = (u1 , u2) at
point (a, b)â.
⎠ð ᅵᅵ ð = ð¥ð¢ðŠð¡ âð
ð(ð + ð¡ð®ð, ð + ð¡ð®ð) â ð(ð, ð)
ð¡ .
â Directional derivative in terms of gradient vector:
ð ᅵᅵ ð = ( ð ð«ðð ð )(ð,ð) âᅵᅵ
|ᅵᅵ| OR ð ᅵᅵ ð = ( ð ð )(ð,ð) â
ᅵᅵ
|ᅵᅵ| .
â The directional derivative of f(x, y) at (a, b) in the direction u = (u1, u2) is
ð ᅵᅵ ð(ð, ð) = ðð±(ð, ð) ð®ð |ᅵᅵ|
+ ðð²(ð, ð) ð®ð |ᅵᅵ|
.
â The directional derivative of f(x, y, z) at (x0, y0, z0) in the direction u = (u1, u2, u3) is
ð ᅵᅵ ð(ð±ð, ð²ð, ð³ð) = ðð±(ð±ð, ð²ð, ð³ð) ð®ð |ᅵᅵ|
+ ðð²(ð±ð, ð²ð, ð³ð) ð®ð |ᅵᅵ|
+ ðð³(ð±ð, ð²ð, ð³ð) ð®ð |ᅵᅵ|
.
â The directional derivative of f(x, y) in direction of unit vector u = (u1, u2) which makes
an angle α with positive x â axis and at point (x0, y0) is
ð ð ð(ð±ð, ð²ð) = ððšð¬ð ðð±(ð±ð, ð²ð) + ð¬ð¢ð§ð ðð²(ð±ð, ð²ð).
DARSHAN INSTITUTE OF ENGINEERING & TECHNOLOGY Mathematics-I (3110014)
U N I T - 4 [ 9 5 ]
METHOD â 10: DIRECTIONAL DERIVATIVE
C 1 Find derivative of f(x, y) = xey + cos xy at the point (2, 0) in direction of
A = 3 i â 4 j.
ðð§ð¬ð°ðð«: â ð.
H 2 Find derivative of f(x, y) = x2sin2y at the point (1, Ï/2) in the direction of
v = 3 i â 4 j.
ðð§ð¬ð°ðð«: ð/ð.
C 3 Define gradient of a function. Use it to find directional derivative of
f(x, y, z) = x3 â xy2 â z at P(1, 1, 0) in the direction of a = 2 i â 3 j + 6 k.
ðð§ð¬ð°ðð«: ð/ð.
W-18 (4)
H 4 Find the directional derivatives of f = xy2 + yz2 at the point (2,â 1, 1) in the
direction i + 2 j + 2 k.
ðð§ð¬ð°ðð«: â ð.
W-19 (3)
H 5 The temperature T at any point in space is given by T = xy + yz + zx.
Determine the directional derivative of T in the direction of V = 3 i â 4 k at
(1, 1, 1).
ðð§ð¬ð°ðð«: â ð/ð.
H 6 Find the directional derivative of g(x, y, z) = 3ex cos yz at P(0, 0, 0) in the
direction of A = 2 i + j â 2 k.
ðð§ð¬ð°ðð«: ð.
H 7 Find the directional derivatives of f(x, y, z) = xyz at the point P(â1, 1, 3) in
the direction of the vector a = i â 2 j + 2 k.
ðð§ð¬ð°ðð«: ð/ð.
T 8 Find the directional derivative of Ï = 4xz3 â 3x2 y2z at the point (2,â1, 2)
in the direction (2, 3, 6).
ðð§ð¬ð°ðð«: ððð/ð.
T 9 Find the directional derivative of f(x, y, z, ) = 2x2 + 3y2 + z2 at the point
P(2, 1 , 3) in the direction of a = (1, 0,â2).
ðð§ð¬ð°ðð«: â ð/âð .
DARSHAN INSTITUTE OF ENGINEERING & TECHNOLOGY Mathematics-I (3110014)
U N I T - 4 [ 9 6 ]
C 10 Find the directional derivative of x2y2z2 at (1, 1,â1) along a direction
equally inclined with co-ordinate axes.
ðð§ð¬ð°ðð«: ð/âð .
T 11 Find the directional derivative of Ï(x, y, z) = xy2 + yz2 + zx2 at point
(1, 1, 1) along the tangent to the curve x = t, y = t2, z = t3 at t = 1.
ðð§ð¬ð°ðð«: ðð/âðð .
C 12 Find the directional derivative Duf(x, y) if f(x, y) = x3 â 3xy +
4y2 and u is the unit vector given by angle Ξ =Ï
6 . What is Duf(1, 2)?
ðð§ð¬ð°ðð«: âð
ð(ðð±ð â ðð²) +
ð
ð (âðð± + ðð²),
ðð â ðâð
ð .
S-19 (3)
T 13 Find D (α) f(x0, y0), α = 30° for f(x, y) = x2 â y2 at (x0, y0) = (1, 2).
ðð§ð¬ð°ðð«: ð (ðð°) ð(ð, ð) = âð â ð.
â â â â â â â â â â
DARSHAN INSTITUTE OF ENGINEERING & TECHNOLOGY Mathematics-I (3110014)
U N I T - 5 [ 9 7 ]
UNIT 5
â MULTIPLE INTEGRALS:
â The multiple integral is a generalization of the definite integral to function of more than
one variable.
â DOUBLE INTEGRALS:
â Integral of function of two variables over a region R2 is called double integrals.
â Properties of double integrals: Let f(x, y) and g(x, y) be two continuous functions in
region R then
(ð). â¬[f(x, y) ± g(x, y)] dR
R
=â¬f(x, y) dR
R
± â¬g(x, y) dR
R
.
(ð). â¬k f(x, y) dR
R
= kâ¬f(x, y) dR,
R
where k is constant. (Linearity property).
(ð). ⬠f(x, y) dR ⥠0,
R
if f(x, y) ⥠0 on R.
(ð). ⬠f(x, y) dR
R
â¥â¬g(x, y) dR,
R
if f(x, y) ⥠g(x, y), â (x, y) â R. (Inequality property).
(ð). If R = R1 ⪠R2 and R1 and R2 are non over lapping sub domain of region R then
â¬f(x, y) dR
R
=⬠f(x, y) dR
R1
+â¬f(x, y) dR.
R2
(Additivity of region).
â FUBINIâS THEOREM FOR EVALUATING DOUBLE INTEGRALS:
â There are two cases to evaluate a double integral known as
Fubiniâs Theorem.
Case-I: The region R (i. e. ABCD) is bounded by
y = g(x), y = h(x) and two lines x = a, x = b.
DARSHAN INSTITUTE OF ENGINEERING & TECHNOLOGY Mathematics-I (3110014)
U N I T - 5 [ 9 8 ]
â Procedure for evaluating double integrals of case-I:
(1). Draw a strip PQ parallel to y-axis. (i.e. Vertical strip).
(2). According to the strip PQ, the lower limit of inner integral is obtained from the
curve where the strip starts i.e. y = g(x) and the upper limit is obtained from the
curve where it ends i.e. y = h(x).
(3). The lower limit of outer integral is the left most point i.e. x = a of the region and
upper limit is the right most point i.e. x = b of the region.
(4). Hence, we get the double integral
â¬f(x, y) dx dy
R
= â«{ â« f(x, y)
h(x)
g(x)
dy }
b
a
dx.
(5). Solve the above inner integral first then outer integral to find the value of given
double integrals.
Case II: The region R (i. e. ABCD) is bounded by
x = g(y), x = h(y) and two lines y = c, y = d.
â Procedure for evaluating double integral of Case-II:
(1). Draw the strip PQ parallel to x-axis. (i.e. Horizontal strip).
(2). According to the strip PQ, the lower limit of the inner integral
is obtained from the curve where the strip starts i.e. x = g(y)
and upper limit is obtained from the strip where it ends
i.e. x = h(y).
(3). The lower limit of outer integral is the lowest point i.e. y = c of the region and
upper limit is the highest point i.e. y = d of the region.
(4). Hence, we get the double integral
â¬f(x, y) dx dy
R
= â«{ â« f(x, y)
h(y)
g(y)
dx}
d
c
dy.
(5). Solve the above inner integral first then outer integral to find the value of given
double integrals.
DARSHAN INSTITUTE OF ENGINEERING & TECHNOLOGY Mathematics-I (3110014)
U N I T - 5 [ 9 9 ]
â NORMAL DOMAIN R2:
â Any domain D in R2 is known as normal domain if the projection of domain D on either
x-axis or y-axis is bounded by the two lines (perpendicular to x or y axis).
â If the domain D is normal with respect to the x-axis and f : D â R
is a continuous function, then
â¬f(x, y)
D
dx dy = â« dx
x=b
x=a
â â« dy
y=h(x)
y=g(x)
Where g(x) and h(x) are define between the two lines (parallel to
y-axis) which contains the domain.
â If the domain D is normal with respect to the y-axis and f : D â R is a continuous function,
then
â¬f(x, y)
D
dx dy = â« dy
y=b
y=a
â â« dx
x=h(y)
x=g(y)
Where g(y) and h(y) are define between the two lines (parallel to x-axis) which contains
the domain.
METHOD â 1: DOUBLE INTEGRALS BY DIRECT INTEGRATION
H 1 Evaluate: (ð). â« â«(1 â 6x2y2) dx dy
2
0
1
â1
(ð).â« â«(x2 + y2)
2
0
2
0
dx dy
(ð).â« â«y2 dx dy
3
2
1
0
(ð).â« â«(x + y) dx dy
1
2
1
0
.
ðð§ð¬ð°ðð«: (ð).âðð/ð (ð).ðð/ð (ð).ð/ð (ð).âð.
H 2 Evaluate: â« â«(1 â 6x2y) dx dy
2
0
1
â1
.
ðð§ð¬ð°ðð«: ð.
DARSHAN INSTITUTE OF ENGINEERING & TECHNOLOGY Mathematics-I (3110014)
U N I T - 5 [ 1 0 0 ]
C 3 Evaluate â« â« xy dy dx
2
1
1
0
.
ðð§ð¬ð°ðð«: ð/ð.
W-19 (3)
H 4
Evaluate: (ð).â« â«(x2 + y2) dy dx
x
0
1
0
(ð).â« â« (x2 + 3y + 2) dy dx
x2
x
1
0
(ð).â« â« e y x
x
0
dy dx
1
0
(ð).â« â«ex+y
x
0
1
0
dy dx.
ðð§ð¬ð°ðð«: (ð). ð/ð (ð).âð/ðð (ð). (ð â ð)/ð (ð). (ð â ð)ð/ð.
C 5 Evaluate â« â«
1
(2x + 5y)2 dx dy
1
0
2
1
.
ðð§ð¬ð°ðð«: (âð. ð)(ð¥ðšð ðð â ð¥ðšð ð â ð¥ðšð ð).
C 6 Evaluate â« â«
1
â (1 â x2)(1 â y2)
1
0
1
0
dx dy.
ðð§ð¬ð°ðð«: ðð/ð.
T 7
Evaluate â« â«dy dx
1 + x2 + y2
â1+x2
0
1
0
.
ðð§ð¬ð°ðð«: ð ð¥ðšð (ð + âð ) /ð.
H 8
Evaluate â« â« (x ex2+y) dy dx
3x2
2x2
4
1
.
ðð§ð¬ð°ðð«: (ððð â ðð)/ð â (ððð â ðð)/ð.
H 9
Evaluate: (ð).â« â« cos(x + y) dy dx
x
Ï 2
Ï 2
0
(ð).â« â« ( y â âx
1ây
0
1
0
) dx dy
(ð).â« â« xyeâx2 dx dy
y
0
1
0
.
ðð§ð¬ð°ðð«: (ð). ð (ð).âð/ðð (ð). ð/ðð.
DARSHAN INSTITUTE OF ENGINEERING & TECHNOLOGY Mathematics-I (3110014)
U N I T - 5 [ 1 0 1 ]
H 10
Evaluate: (ð).â« â« r dr dΞ
a(1+cosΞ)
0
Ï
0
(ð).â« â« r dr dΞ
a
asinΞ
2Ï
0
.
ðð§ð¬ð°ðð«: (ð). ðððð/ð (ð).ððð/ð
H 11
Evaluate â« â«r
(1 + r2)2
âcos 2Ξ
0
Ï 4
0
dr dΞ.
ðð§ð¬ð°ðð«: ð â ð/ð.
T 12
Evaluate â« â« r
sin Ξ
0
Ï
0
dr dΞ.
ðð§ð¬ð°ðð«: ð/ð.
W-19 (3)
C 13
Evaluate â«
Ï 2
0
â« r2
1âsin Ξ
0
cos Ξ dr dΞ.
ðð§ð¬ð°ðð«: ð/ðð.
H 14
Evaluate ⫠⫠r2 sin Ξ dr dΞ
1âcos Ξ
0
Ï
0
.
ðð§ð¬ð°ðð«: ð/ð.
T 15
Evaluate: ⫠⫠r2 dr dΞ
2 cosÓ©
0
Ï 2
âÏ 2
& ⫠⫠r sin Ξ dΞ dr
cosâ1(r a )
0
a
0
.
ðð§ð¬ð°ðð«: ðð/ð & ðð/ð.
â TRIPLE INTEGRALS:
â Integral of a function of three variables over a region R3 is called triple integral.
â Properties of triple Integrals: Let f(x, y, z) and g(x, y, z) be two continuous functions in V
then,
(ð). â[f(x, y, z) ± g(x, y, z)]
V
dV = âf(x, y, z) dV ± âg(x, y, z) dV
V
V
.
DARSHAN INSTITUTE OF ENGINEERING & TECHNOLOGY Mathematics-I (3110014)
U N I T - 5 [ 1 0 2 ]
(ð). âk f(x, y, z) dV
V
= kâf(x, y, z) dV,
V
where k is a constant.
(ð). âf(x, y, z) dV
V
⥠0, if f(x, y, z) ⥠0 on V.
(ð). âf(x, y, z) dV
V
â¥âg(x, y, z) dV
V
, if f(x, y, z) ⥠g(x, y, z), â(x, y, z) â V.
(ð). If V = V1 ⪠V2 and V1 and V2 are non over lapping sub domain of V then
âf(x, y, z) dV
V
= âf(x, y, z) dV
V1
+âf(x, y, z) dV
V2
.
METHOD â 2: TRIPLE INTEGRALS BY DIRECT INTEGRATION
H 1
Evaluate: (ð).â« â« â« xyz dz dx dy
1
0
3
2
2
1
(ð).â« â« â« dx dz dy
2
0
1ây
0
1
0
(ð).â« â« â« dz dy dx.
2âxây
0
2âx
0
1
0
ðð§ð¬ð°ðð«: (ð). ðð/ð (ð). ð (ð). ð/ð.
C 2 Evaluate â« â« â« xz â y3 dz dy dx.
1
0
2
0
1
â1
ðð§ð¬ð°ðð«: â ð.
C 3 c
Evaluate â« â« â« z dz dy dx.
1âxây
0
1âx
0
1
0
ðð§ð¬ð°ðð«: ð/ðð.
H 4
Evaluateâ« â« â« z dx dz dy
1âyâz
0
1ây
0
1
0
.
ðð§ð¬ð°ðð«: ð/ðð.
DARSHAN INSTITUTE OF ENGINEERING & TECHNOLOGY Mathematics-I (3110014)
U N I T - 5 [ 1 0 3 ]
T 5
Evaluate â« â« â« xyz dz dy dx
â1âx2ây2
0
.
â1âx2
0
1
0
ðð§ð¬ð°ðð«: ð/ðð.
T 6
Evaluate â« â« â« z dz dy dx
âx+y
0
.
x
0
1
0
ðð§ð¬ð°ðð«: ð/ð.
T 7
Evaluate â« â« â«(x2 + y2 + z2) dz dy dx
a
âa
b
âb
c
âc
.
ðð§ð¬ð°ðð«: ðððð(ðð + ðð + ðð)/ð.
H 8
Evaluate â« â« â« xyz dx dy dz
yz
0
2
1
2
0
.
ðð§ð¬ð°ðð«: ðð/ð.
H 9
Evaluate â« â« â« xyz dx dy dz
yz
0
z
1
2
0
.
ðð§ð¬ð°ðð«: ð/ð.
H 10 Evaluate â« â« â« y sin z dx dy dz.
Ï
0
Ï
0
1
0
ðð§ð¬ð°ðð«: ðð (ð â ððšð¬ ð )/ð.
H 11
Evaluate â« â« â« xyz dz dy dx
âxy
0
1
1 x
3
1
.
ðð§ð¬ð°ðð«: (ðð/ð) â { (ð¥ðšð ð)/ð }.
C 12
Evaluate â« â« â« ex+y+z dz dy dx.
x+log y
0
x
0
log 2
0
ðð§ð¬ð°ðð«: { (ð ð¥ðšð ð)/ð } â (ðð/ð).
DARSHAN INSTITUTE OF ENGINEERING & TECHNOLOGY Mathematics-I (3110014)
U N I T - 5 [ 1 0 4 ]
H 13
Evaluate â« â« â« (x + y + z)2 dz dx dy.
x+2y
0
y
0
1
0
ðð§ð¬ð°ðð«: ððð/ðð.
H 14
Evaluate â« â« â«1
â1 â x2 â y2 â z2
â1âx2ây2
0
â1âx2
0
dz dy dx.
1
0
ðð§ð¬ð°ðð«: ðð/ð.
C 15
Evaluate the integralâ« â« â« (x2 + y2)
ex
1
dz dx dy
log y
1
e
1
.
ðð§ð¬ð°ðð«: (âððð + ððð â ððð + ððð â ððð)/ðð.
W-18 (4)
C 16 Evaluate âdv
v
, where v is the solid region bounded by 1 †x †2,
2 †y †4 & 2 †z †5.
ðð§ð¬ð°ðð«: ð.
C 17
Evaluate ⫠⫠⫠(r2cos2Ξ + z2) r dΞ dr dz
2Ï
0
âz
0
1
0
.
ðð§ð¬ð°ðð«: ð/ð.
C 18
Evaluate â« â« â« r7 sin3 Ξ cos Ξ sin Ï cosÏ dr dΞ dÏ
a
0
Ï 2
0
Ï 2
0
.
ðð§ð¬ð°ðð«: ðð/ ðð.
H 19
⫠⫠⫠r dz dr dΞ
a2âr2 a
0
a sin Ξ
0
Ï 2
0
.
ðð§ð¬ð°ðð«: ðððð/ðð.
â DOUBLE INTEGRALS OVER RECTANGLES AND GENERAL REGION IN CARTESIAN COORDINATES:
â Procedure for evaluating double integrals over general region:
DARSHAN INSTITUTE OF ENGINEERING & TECHNOLOGY Mathematics-I (3110014)
U N I T - 5 [ 1 0 5 ]
(1). Draw all the curves and identify required region.
(2). Find the point(s) of intersection of all the curves if required.
(3). Identify the region bounded by all the curves.
(4). Take the strip parallel to x-axis or y-axis.
(5). If it is required then divide the region into two parts.
(6). Find the limit of inner integral from the strip.
(7). Find the limit of the outer integral from axis.
(8). Calculate the example as the example of double integrals.
METHOD â 3: D.I. OVER GENERAL REGION IN CARTESIAN COORDINATES
C 1 Evaluate â¬(x â 3y2) dA,
R
where R = (x, y), 0 †x †2, 1 †y †2.
ðð§ð¬ð°ðð«: â ðð.
H 2 Evaluate â¬y sin(xy) dA,
R
where R is the region bounded by
x = 1, x = 2, y = 0 and y =Ï
2 .
ðð§ð¬ð°ðð«: ð.
W-18 (3)
H 3 Evaluate â¬sinx siny dA,
R
where R = (x, y), 0 †x â€Ï
2 , 0 †y â€
Ï
2 .
ðð§ð¬ð°ðð«: ð.
H 4 Evaluate â¬(x + y) dy dx,
R
where R is the region bounded by x = 0, x = 2,
y = x, y = x + 2.
ðð§ð¬ð°ðð«: ðð.
C 5 Evaluate â¬e2x+3y dx dy,
R
where R is the triangle bounded by x = 0, y = 0,
x + y = 1.
ðð§ð¬ð°ðð«: (ððð â ððð + ð)/ð.
DARSHAN INSTITUTE OF ENGINEERING & TECHNOLOGY Mathematics-I (3110014)
U N I T - 5 [ 1 0 6 ]
H 6 Evaluate â¬(x2 â y2) dx dy
R
over the triangle with the vertices (0, 1), (1, 1)
& (1, 2).
ðð§ð¬ð°ðð«: â ð/ð.
W-19 (4)
H 7 Evaluateâ¬âxy â y2 dx dy,
s
where S is a triangle with vertices (0, 0), (10, 1)
& (1, 1).
ðð§ð¬ð°ðð«: ð.
C 8 Evaluate â¬(2x â y2) dA
R
; over the triangular region R enclosed between
the lines y = âx + 1, y = x + 1 and y = 3.
ðð§ð¬ð°ðð«: â ðð/ð.
C 9 Evaluate â¬x2
R
dA,where R is the region in the first quadrant bounded by
the hyperbola xy = 16 and the lines y = x, y = 0 and x = 8.
ðð§ð¬ð°ðð«: ððð.
T 10 Evaluate â¬
xy
â1 â y2
R
dx dy over positive quadrant of circle x2 + y2 = 1.
ðð§ð¬ð°ðð«: ð/ð.
C 11 Sketch the region of integration and evaluate the integral
â¬(y â 2x2)
R
dA,where R is the region inside the square |x| + |y| = 1.
ðð§ð¬ð°ðð«: â ð/ð.
S-19 (4)
T 12 Evaluateâ¬(6x2 + 2y)dx dy over the region R bounded between y = x2
and y = 4.
ðð§ð¬ð°ðð«: ððð/ð.
DARSHAN INSTITUTE OF ENGINEERING & TECHNOLOGY Mathematics-I (3110014)
U N I T - 5 [ 1 0 7 ]
T 13 Evaluate â¬y dx dy,
R
where R is the region in the first quadrant bounded by
x = 0, y = x2, x + y = 2.
ðð§ð¬ð°ðð«: ðð/ðð.
C 14 Evaluate â¬xy dx dy
R
over the region bounded by the parabolas x2 = y and
y2 = âx.
ðð§ð¬ð°ðð«: â ð/ðð.
H 15 Evaluate â¬xy dA,
R
where R is the region bounded by x â axis, ordinate
x = 2a & the curve x2 = 4ay.
ðð§ð¬ð°ðð«: ðð/ð.
H 16 Evaluate â¬xy dx dy,
R
where R is the region bounded by x2 + y2 â 2x = 0,
y2 = 2x and y = x.
ðð§ð¬ð°ðð«: ð/ðð.
T 17 Evaluate â¬â4x2 â y2 dx dy
R
over the area of triangle y = x , y = 0 and
x = 1.
ðð§ð¬ð°ðð«: ð/ð + âð /ð.
H 18 Evaluate â¬y dx dy
R
over the ellipse 4x2 + 9y2 = 36 in the positive
quadrant.
ðð§ð¬ð°ðð«: ð.
C 19 Evaluate â¬(x + 2y) dA,
D
where D is the region bounded by the parabolas
y = 2x2, y = 1 + x2.
ðð§ð¬ð°ðð«: ðð/ðð.
DARSHAN INSTITUTE OF ENGINEERING & TECHNOLOGY Mathematics-I (3110014)
U N I T - 5 [ 1 0 8 ]
H 20 Evaluate â¬xy dA,
D
where D is the region bounded by the line y = x â 1
over the parabola y2 = 2x + 6.
ðð§ð¬ð°ðð«: ðð.
T 21 Evaluate â¬
sin x
x
R
dA, where R is the triangle in xy â plane bounded by
x â axis, y = x and x = 1.
ðð§ð¬ð°ðð«: ð. ðð.
H 22
Sketch the region of integration and evaluate â« â« 3
2 e
y
âx dy dx.
âx
0
4
1
ðð§ð¬ð°ðð«: ð(ð â ð).
â DOUBLE INTEGRALS AS VOLUMES:
â When f(x, y) is a positive function over a rectangular region R in the xy â plane, we may
interpret the double integral of f over R as the volume of the 3 â dimensional solid region
over the xy â plane bounded below by R and above by the surface z = f(x, y). We define
this volume as
Volume = â¬f(x, y)dA
R
.
â When the solid region lies between the two surfaces, say lower surface is z1 = f1(x, y) and
upper surface z2 = f2(x, y) and R is the projection on xy â plane, then the volume of a
solid region by double integral define as
Volume =â¬[f2(x, y) â f1(x, y)]dA
R
.
â In polar coordinates, the volume can be obtained by the integral
Volume = â¬z â r drdΞ
R
.
DARSHAN INSTITUTE OF ENGINEERING & TECHNOLOGY Mathematics-I (3110014)
U N I T - 5 [ 1 0 9 ]
METHOD â 4: DOUBLE INTEGRALS AS VOLUMES
C 1 Find the volume of the prism whose base is the triangle in the xy â plane
bounded by the x â axis and the lines y = x and x = 1 and whose top in the
plane z = f(x, y) = 3 â x â y.
ðð§ð¬ð°ðð«: ð.
H 2 Find the volume under the plane x + y + z = 6 and above the triangle in xy â
plane bounded by 2x = 3y, y = 0 and x = 3.
ðð§ð¬ð°ðð«: ðð.
H 3 Find the volume of the region bounded by the surface x = 0, y = 0, z = 0
and 2x + 3y + z = 6.
ðð§ð¬ð°ðð«: ð.
T 4 Find the volume bounded by the cylinder x2 + y2 = 4 between the planes
y + z = 3 and z = 0.
ðð§ð¬ð°ðð«: ððð.
C 5 Find the volume below the surface z = x2 + y2 above the plane z = 0 and
inside the cylinder x2 + y2 = 2y.
ðð§ð¬ð°ðð«: ðð/ð.
W-18 (7)
â DOUBLE INTEGRALS BY CHANGE OF ORDER OF INTEGRATION:
â Sometimes, change of order of integration makes calculation of integral easier.
â Procedure for evaluating double integrals by change of order of integration:
(1). Draw the region from given data.
(2). Find the intersection point(s).
(3). Draw the strip parallel to x-axis if we want to integrate first with respect to x or
Draw the strip parallel to y-axis if we want to integrate first with respect to y.
(4). Using Fubiniâs theorem find the new limit points of both the integrals.
(5). Calculate the example as the example of double integrals.
DARSHAN INSTITUTE OF ENGINEERING & TECHNOLOGY Mathematics-I (3110014)
U N I T - 5 [ 1 1 0 ]
METHOD â 5: D.I. BY CHANGE OF ORDER OF INTEGRATION
H 1
Change the order only of â« â« f(x, y) dy dx
1+x
1âx
3
0
.
ðð§ð¬ð°ðð«: â« â« ð(ð±, ð²) ðð± ðð²
ð
ð²âð
+
ð
ð
â« â« ð(ð±, ð²) ðð± ðð²
ð
ðâð²
ð
âð
.
T 2
Change the order only of â« â« f(x, y) dx dy
â10ây2
y2 9
3
0
.
ðð§ð¬ð°ðð«: â« â« ð(ð±, ð²) ðð² ðð±
ðâð±
ð
ð
ð
+ â« â« ð(ð±, ð²) ðð² ðð±
âððâð±ð
ð
âðð
ð
.
H 3
Change the order only of â« â« f(x , y) dy dx
âa2âx2
0
a
0
.
ðð§ð¬ð°ðð«: â« â« ð(ð±, ð²) ðð± ðð²
âððâð²ð
ð
ð
ð
.
C 4
Change the order only of â« â« f(x, y) dy dx
1+â2xâx2
1ââ2xâx2
2
1
.
ðð§ð¬ð°ðð«: â« â« ð(ð±, ð² )ðð± ðð²
ð+âðð²âð²ð
ð
ð
ð
.
H 5
Change the order of integrations and hence evaluateâ« â« ex2 dx dy
4
4y
1
0
.
ðð§ð¬ð°ðð«: (ððð â ð)/ð.
T 6 Change the order of integrations and hence evaluateâ« â«
eây
y
â
x
â
0
dy dx.
ðð§ð¬ð°ðð«: ð.
DARSHAN INSTITUTE OF ENGINEERING & TECHNOLOGY Mathematics-I (3110014)
U N I T - 5 [ 1 1 1 ]
C 7
Evaluate â« â« ey3 dy dx.
1
â x 3
3
0
ðð§ð¬ð°ðð«: (ð â ð).
S-19 (7)
H 8 Change the order of integrations and hence evaluateâ« â«ey
2 dy dx
2
2x
1
0
.
ðð§ð¬ð°ðð«: (ðð â ð)/ð.
T 9 Change the order of integration and evaluateâ« â« sin(y2) dy dx
1
x
1
0
.
ðð§ð¬ð°ðð«: (ð â ððšð¬ ð)/ð ðšð« ð. ðð.
W-19 (7)
H 10
By changing the order of integration, evaluate â«â«x
x2 + y2 dx dy
3
y
3
0
.
ðð§ð¬ð°ðð«: ðð/ð.
W-18 (4)
T 11
Change the order of integration and evaluateâ« â«x
âx2 + y2 dy dx
â2âx2
x
1
0
.
ðð§ð¬ð°ðð«: ð â ð/âð .
C 12 Change the order of integrations and hence evaluate
â« â«ey
(ey + 1)â1 â x2 â y2 dy dx
â1âx2
0
1
0
.
ðð§ð¬ð°ðð«: (ð/ð) ð¥ðšð { (ð + ð)/ð }.
T 13 Change the order of integrations and hence evaluate
â« â« (x + y) dx dy
â4ây
1
3
0
& â« â« y2 dA.
âa2âx2
x
a
â2
0
ðð§ð¬ð°ðð«: ððð/ðð & ðð(ð + ð/ð )/ðð.
DARSHAN INSTITUTE OF ENGINEERING & TECHNOLOGY Mathematics-I (3110014)
U N I T - 5 [ 1 1 2 ]
H 14
Find â« â« dx dy
a+âa2ây2
aââa2ây2
by changing the order of integration.
a
0
ðð§ð¬ð°ðð«: ððð/ð.
H 15
Change the order of integrations and hence evaluateâ« â« dx dy
2+â4ây2
2ââ4ây2
2
0
.
ðð§ð¬ð°ðð«: ðð.
H 16
Change the order of integration and hence evaluateâ« â« dy dx
2âax
x2
4a
4a
0
.
ðð§ð¬ð°ðð«: ðððð/ð.
C 17
Change the order of integration and evaluateâ« â« dy dx
2âx
x2 4
4
0
.
ðð§ð¬ð°ðð«: ðð/ð.
H 18
Change the order of integrations and hence evaluateâ« â«xe2y
4 â y dy dx
4âx2
0
2
0
.
ðð§ð¬ð°ðð«: (ðð â ð)/ð.
C 19 Express as single integral and hence evaluate
â«â«(x2 + y2) dx dy
y
0
1
0
+â« â« (x2 + y2) dx dy
2ây
0
2
1
.
ðð§ð¬ð°ðð«: â« â« (ð±ð + ð²ð) ðð² ðð±
ðâð±
ð±
ð
ð
& ð/ð.
DARSHAN INSTITUTE OF ENGINEERING & TECHNOLOGY Mathematics-I (3110014)
U N I T - 5 [ 1 1 3 ]
â DOUBLE INTEGRALS OVER GENERAL REGION IN POLAR COORDINATES:
â In Cartesian coordinate, we have x-axis & y-axis (in two dimension). While in polar
coordinate, we have r-axis & Ξ-axis.
â Procedure for evaluating double integral over general region in
Polar co-ordinates:
(1). Draw the given curves to find the required region.
(2). Draw the strip from origin within the region.
(3). According to the strip PQ, the lower limit of inner
integral is obtained from the curve where the strip starts i.e. r = r1(Ξ) and the
upper limit is obtained from the curve where the strip ends i.e. r = r2(Ξ).
(4). The lower limit of outer integral is obtained from the angle where the region
starts i.e. Ξ = Ξ1 and the upper limit is obtained from the angle where the region
ends i.e. Ξ = Ξ2.
(5). Hence, we get the double integral
â¬f(r, Ξ) dr dΞ = â« â« f(r, Ξ) dr dΞ
r=r2(Ξ)
r=r1(Ξ)
Ξ=Ξ2
Ξ=Ξ1
.
(6). Calculate the example as the example of double integrals.
METHOD â 6: D.I. OVER GENERAL REGION IN POLAR COORDINATES
C 1 Evaluate â¬r3 sin 2
R
Ξ dr dΞ over the area bounded in the first quadrant
between the circles r = 2, r = 4.
ðð§ð¬ð°ðð«: ðð.
T 2 Evaluate â¬r3 dr dΞ
D
over the area included between the circles r = 2 sinΞ
and r = 4 sinΞ.
ðð§ð¬ð°ðð«: ððð/ð.
DARSHAN INSTITUTE OF ENGINEERING & TECHNOLOGY Mathematics-I (3110014)
U N I T - 5 [ 1 1 4 ]
H 3 Evaluate â¬râa2 â r2 dr dΞover upper half of the circle r = a cos Ξ.
ðð§ð¬ð°ðð«: {(ðð â ð)ðð}/ðð.
C 4 Evaluate â¬r sinΞ dr dΞ over the area of the cardioid r = a(1 + cosΞ) above
the initial line.
ðð§ð¬ð°ðð«: ððð/ð.
T 5 Evaluate integral â¬
rdrdΞ
âa2 + r2
R
over the region R which is one loop of
r2 = a2 cos2Ξ.
ðð§ð¬ð°ðð«: ð(ð â ð)/ð.
W-18 (3)
H 6 Find by double integrals, the volume of the cylinder x2 + y2 = 1 between
the planes z = 0 and y + z = 2.
ðð§ð¬ð°ðð«: ðð.
â AREA BY DOUBLE INTEGRATION IN CARTESIAN COORDINATES:
â The area A of a region R in the xy â plane bounded by the curves y = f1(x), y = f2(x) and
the lines x = a and x = b is given by:
Area =â¬dx dy = â« â« dy dx.
f2(x)
f1(x)
b
a
R
â The area A of a region R in the xy â plane bounded by the curves x = f1(y), x = f2(y) and
the lines y = a and y = b is given by:
Area =â¬dx dy = â« â« dx dy.
f2(y)
f1(y)
b
a
R
DARSHAN INSTITUTE OF ENGINEERING & TECHNOLOGY Mathematics-I (3110014)
U N I T - 5 [ 1 1 5 ]
METHOD â 7: AREA BY DOUBLE INTEGRATION IN CARTESIAN COORDINATES
H 1 Find the area bounded by x â 2y + 4 = 0, x + y â 5 = 0 & y = 0.
ðð§ð¬ð°ðð«: ðð/ð.
C 2 Find the area of the region bounded by the curves y = sin x, y = cos x and
the lines x = 0 and x = Ï
4 .
ðð§ð¬ð°ðð«: âð â ð.
S-19 (4)
T 3 Using the double integration, find the area of the region common to the
parabolas y2 = 8x and x2 = 8y.
ðð§ð¬ð°ðð«: ðð/ð.
H 4 Find the area of the region R enclosed by the parabola y = x2 and the line
y = x + 2.
ðð§ð¬ð°ðð«: ð/ð.
C 5 Find the area of the area included between the curve y2(2a â x) = x3 and its
asymptote.
ðð§ð¬ð°ðð«: ðððð.
â AREA BY DOUBLE INTEGRATION IN POLAR COORDINATES:
â The area A of a region R bounded by the curves r = f1(Ξ), r = f2(Ξ) and the radii vector
Ξ = Ξ1 and Ξ = Ξ2 is given by:
Area =â¬dx dy = â« â« r dr dΞ.
f2(Ξ)
f1(Ξ)
Ξ2
Ξ1
R
METHOD â 8: AREA BY DOUBLE INTEGRATION IN POLAR COORDINATES
T 1 Find the area common to the circles r = a and r = 2a cos Ξ using double
integrals.
ðð§ð¬ð°ðð«: ðð {(ðð/ð) â (âð /ð)}.
C 2 Find the area common to both of the circles r = cosΞ and r = sin Ξ.
ðð§ð¬ð°ðð«: (ð/ð) â (ð/ð).
DARSHAN INSTITUTE OF ENGINEERING & TECHNOLOGY Mathematics-I (3110014)
U N I T - 5 [ 1 1 6 ]
H 3 Find the area between the circles r = 2sin Ξ and r = 4sin Ξ , using the
double integration.
ðð§ð¬ð°ðð«: ðð.
C 4 Find the area of the region that lies inside the cardioid r = 1 + cos Ξ and
outside the circle r = 1 by double integration.
ðð§ð¬ð°ðð«: ð + (ð/ð).
H 5 Find area outside the circle r = 2a cosΞ and inside the cardioid r =
a(1 + cos Ξ).
ðð§ð¬ð°ðð«: ðð{ð â (ð/ð)}.
T 6 Find the area common to the circle r = a and the cardioid r = a(1 + cosΞ).
ðð§ð¬ð°ðð«: ðð(ðð/ð â ð).
H 7 Using double integrals, find area enclosed by the cardioid r = a(1 + cosΞ).
ðð§ð¬ð°ðð«: ðððð/ð.
C 8 Find the total area enclosed by lemniscate r2 = a2 cos 2Ξ.
ðð§ð¬ð°ðð«: ðð .
T 9 Use double integrals in polar to find the area enclosed by three petaled
rose r = sin 3Ξ.
ðð§ð¬ð°ðð«: ð/ð.
â TRIPLE INTEGRALS OVER GENERAL REGION IN CARTESIAN COORDINATES:
â Procedure for evaluating triple integrals over general region in cartesian coordinates:
(1). Suppose the solid region D in three-dimensional space
is bounded below by the surface z = f1(x, y) and
bounded above by the surface z = f2(x, y) as shown in
the figure.
(2). Draw a strip PQ parallel to z-axis.
DARSHAN INSTITUTE OF ENGINEERING & TECHNOLOGY Mathematics-I (3110014)
U N I T - 5 [ 1 1 7 ]
(3). According to the strip lower limit of inner integral is
obtained from the curve where the strip starts i. e., z =
f1(x, y) and the upper limit is obtained from the curve
where the strip ends i. e., z = f2(x, y).
(4). Draw a separate figure showing region R in xy-plane as
shown in the figure.
(5). By Fubiniâs theorem find limits for both the inner integrals.
(6). Calculate the example as the example of triple integrals.
METHOD â 9: T.I. OVER GENERAL REGION IN CARTESIAN COORDINATES
C 1 Find the volume of tetrahedron bounded by the plane x + y + z = 2 and the
planes x = 0, y = 0 and z = 0.
ðð§ð¬ð°ðð«: ð/ð.
H 2 Evaluate âx2 y z dx dy dz over the region bounded by the planes x = 0,
y = 0, z = 0 and x + y + z = 1.
ðð§ð¬ð°ðð«: ð/ðððð.
â TRIPLE INTEGRALS OVER GENERAL REGION IN CYLINDRICAL COORDINATES:
â Cylindrical coordinates r, Ξ, z are used to evaluate the integral for the region which is
bounded by cylinder along z-axis, planes passing through z-axis and planes perpendicular
to the z-axis.
â Procedure for evaluating the triple integrals over general region in cylindrical
coordinates:
(1). Substitute x = r cos Ξ, y = r sin Ξ& z = z.
âf(x, y, z)dx dy dz =âf(r cosΞ , r sin Ξ , z) |J| dr dΞ dz.
(ð). Calculate Jacobian J, where J = â(x, y, z)
â(r, Ξ, z)= r.
DARSHAN INSTITUTE OF ENGINEERING & TECHNOLOGY Mathematics-I (3110014)
U N I T - 5 [ 1 1 8 ]
âf(x, y, z)dx dy dz ==âf(r cos Ξ , r sin Ξ , z) r dr dΞ dz.
METHOD â 10: T.I. OVER GENERAL REGION IN CYLINDRICAL COORDINATES
T 1 Evaluate âxyz dx dy dz over the region bounded by the planes x = 0,
y = 0, z = 0, z = 1 and the cylinder x2 + y2 = 1.
ðð§ð¬ð°ðð«: ð/ðð.
C 2 Find the volume bounded by the cylinder x2 + y2 = 4 and the planes y +
z = 3 and z = 0.
ðð§ð¬ð°ðð«: ððð.
H 3 Use triple integration to find the volume of the solid within the cylinder x2 +
y2 = 9 between the planes z = 1 and x + z = 1.
ðð§ð¬ð°ðð«: ð.
â TRIPLE INTEGRALS OVER GENERAL REGION IN SPHERICAL COORDINATES:
â Spherical coordinates r, Ξ, Ï are used to evaluate the integral for the region bounded by
the
(1). Sphere with center at origin.
(2). Cone with vertices at origin and z-axis is as an axis of cone.
â Procedure for evaluating the triple integrals in spherical coordinates:
(1). Substitute x = r sin Ξ cosÏ, y = r sin Ξ sin Ï& z = r cos Ξ.
âf(x, y, z)dx dy dz =âf(r sin Ξ cosÏ, r sin Ξ sin Ï, r cos Ξ) |J| dr dΞ dÏ.
(ð). Calculate Jacobian J, where J =â(x, y, z)
â(r, Ξ, Ï) = r2sinΞ.
âf(x, y, z)dx dy dz =âf(r sin Ξ cosÏ, r sin Ξ sin Ï, r cos Ξ) r2sinΞ dr dΞ dÏ.
â Remark: When the given region is sphere x2 + y2 + z2 = a2 with centre (0, 0, 0) and
radius a, then limits of r, Ξ, Ï are as follows:
DARSHAN INSTITUTE OF ENGINEERING & TECHNOLOGY Mathematics-I (3110014)
U N I T - 5 [ 1 1 9 ]
Integrated region ð« ð ð
For Positive octant r = 0 to r = a Ξ = 0 to Ξ = Ï
2 Ï = 0 to Ï =
Ï
2
For Hemisphere r = 0 to r = a Ξ = 0 to Ξ = Ï
2 Ï = 0 to Ï = 2Ï
For Complete Sphere r = 0 to r = a Ξ = 0 to Ξ = Ï Ï = 0 to Ï = 2Ï
â Procedure for evaluating triple integrals:
(1). Draw given curves and identify the region of integration.
(2). Draw a volume for parallel to axis (y-axis or x-axis).
(3). The lower limit for z is the starting point of the volume and upper limit is the point
where it ends.
(4). Draw the region of projection in any plane (xy, zx or yz plane).
(5). Follow the steps of double integration to find the limit of x and y (z, x or y & z).
METHOD â 11: T.I. OVER GENERAL REGION IN SPHERICAL COORDINATES
C 1 Evaluate âxyz dx dy dz over the positive octant of the sphere
x2 + y2 + z2 = 4.
ðð§ð¬ð°ðð«: ð/ð.
W-19 (7)
T 2 Evaluate â2x dV,
E
where E is the region under the plane 2x + 3y + z = 6
that lies in first octant.
ðð§ð¬ð°ðð«: ð.
â JACOBIAN:
â If u and v be the function of two independent variables x and y then the Jacobian of u, v
with respect to x , y is denoted by J ( u, v
x, y) or
â(u, v)
â(x, y) or J(u, v) and it is defined as:
DARSHAN INSTITUTE OF ENGINEERING & TECHNOLOGY Mathematics-I (3110014)
U N I T - 5 [ 1 2 0 ]
J = J (u, v
x, y ) =
â(u, v)
â(x, y)=|
|
âu
âx
âu
ây
âv
âx
âv
ây
|
|.
Same way we can define J = J (x, y
u, v) =
â(x, y)
â(u, v)= ||
âx
âu
âx
âv ây
âu
ây
âv
||.
If u, v and w are the functions of three independent variables x, y and z then the Jacobian
of u, v,w W. R. T. x, y, z is denoted by J ( u, v, w
x, y, z) or
â(u, v, w)
â(x, y, z) and it is defined as:
J = J ( u, v, w
x, y, z) =
â(u, v, w)
â(x, y, z)= ||
âuâxâ
âuâyâ
âuâzâ
âvâxâ
âvâyâ
âvâzâ
âwâxâ
âwâyâ
âwâzâ
||.
Same way we can define J = J ( x, y, z
u, v, w ) =
â(x, y, z)
â(u, v, w)= ||
âxâuâ
âxâvâ
âxâwâ
âyâuâ ây
âvâ ây
âwâ
âzâuâ
âzâvâ
âzâwâ
||.
â RESULTS:
If J =â(u, v)
â(x, y) then Jâ² =
â(x, y)
â(u, v) ðšð« If J =
â(x, y)
â(u, v) then Jâ² =
â(u, v)
â(x, y).
If J = â(u, v, w)
â(x, y, z) then Jâ² =
â(x, y, z)
â(u, v, w) ðšð« If J =
â(x, y, z)
â(u, v, w) then Jâ² =
â(u, v, w)
â(x, y, z).
Multiplication of Jacobian J and Jâ is 1. or J â Jâ² = 1.
â CHAIN RULE OF JACOBIAN:
â If u, v are functions of r, s and r, s are functions of x, y then the Jacobian of u, v with respect
to x , y is given by the following chain rule:
J = J ( u, v
x, y) =
â(u, v)
â(x, y)= â(u, v)
â(r, s)â â(r, s)
â(x, y).
â If u, v,w are functions of r, s, t and r, s, t are functions of x, y, z then the Jacobian of u, v, w
with respect to x , y, z is given by the following chain rule:
DARSHAN INSTITUTE OF ENGINEERING & TECHNOLOGY Mathematics-I (3110014)
U N I T - 5 [ 1 2 1 ]
J = J ( u, v, w
x, y, z) =
â(u, v, w)
â(x, y, z)= â(u, v, w)
â(r, s, t)â â(r, s, t)
â(x, y, z) .
METHOD â 12: JACOBIAN
H 1 Find J =
â(u, v)
â(x, y) where u = x2 â y2, v = 2xy.
ðð§ð¬ð°ðð«: ð(ð±ð + ð²ð).
H 2 If x = a(u + v), y = b(u â v) and u = r2 cos 2Ξ, v = r2 sin 2Ξ, find
â(x, y)
â(r, Ξ).
ðð§ð¬ð°ðð«: â ðððð«ð.
H 3 If v = 2xy, u = x2 â y2 and x = r cos Ξ, y = r sin Ξ, find
â(u, v)
â(r, Ξ).
ðð§ð¬ð°ðð«: ðð«ð.
H 4 If x = u cos v , y = u sin v then prove that
â(u, v)
â(x, y). â(x, y)
â(u, v)= 1.
T 5 If x = r cosΞ and y = r sin Ξ, show that J â Jâ = 1.
C 6 If x = ev sec u& y = ev tan u, find the values of J and Jâ² to show that J â Jâ² = 1.
ðð§ð¬ð°ðð«: â ð±ðð¯ & â ð/ð±ðð¯.
H 7 The transformation from rΞz â space to xyz â space is given by the equation
x = r cos Ξ , y = r sin Ξ, z = z, find J(r, Ξ, z).
ðð§ð¬ð°ðð«: ð/ð«.
C 8 If x = Ï sinâ cosΞ , y = Ï sinâ sin Ξ , z = Ï cos â then obtain Jacobian
J = â(x, y, z)
â(Ï, â , Ξ). ðð
Find the Jacobian of transformation x = Ï sin â cosΞ , y = Ï sin â sin Ξ,
z = Ï cosâ .
ðð§ð¬ð°ðð«: ðð ð¬ð¢ð§â .
C 9 If x = âvw, y = âwu, z = âuv and u = r sin Ξ cos â , v = r sin Ξ sin â , w =
r cos Ξ find â(x, y, z)
â(r, Ξ, â ) .
ðð§ð¬ð°ðð«: (ð«ð/ð ) ð¬ð¢ð§ ð.
DARSHAN INSTITUTE OF ENGINEERING & TECHNOLOGY Mathematics-I (3110014)
U N I T - 5 [ 1 2 2 ]
C 10 If x + y + z = u, y + z = uv, z = uvw, find
â(x, y, z)
â(u, v, w) .
ðð§ð¬ð°ðð«: ð®ðð¯.
â DOUBLE INTEGRALS BY CHANGE OF VARIABLE OF INTEGRATION IN CARTESIAN COORDINATES:
The variables x, y inâ¬h(x, y) dx dy
R
are changed into u and v with the relations x = f(u, v)
and y = g(u, v) then the integral is transformed into â¬h{f(u, v), g(u, v)} |J| du dv
G
.
â where,
J = â(x, y)
â(u, v)= |
âx
âu
âx
âv ây
âu
ây
âv
| and G is the region in uv â plane corresponding to the
region R in xy-plane.
â Procedure for evaluating double integrals by change of variable of integration:
(1). Find ð± and ð² in the form of u and v using given relations.
(2). Find Jacobian J(x, y).
(3). Draw the region in xy-plane and from that draw the region in uv-plane.
(4). Find limit points for u and v.
(5). Replace the area element dx dy by | J |du dv. Hence, we get
â¬h(x, y)
R
dx dy = â¬h{f(u, v), g(u, v)} | J |du dv
G
.
(6). Calculate the example as example of double integrals.
DARSHAN INSTITUTE OF ENGINEERING & TECHNOLOGY Mathematics-I (3110014)
U N I T - 5 [ 1 2 3 ]
METHOD â 13: D.I. BY CHANGE OF VARIABLE IN CARTESIAN COORDINATES
H 1 Evaluate â¬(x + y)2
R
dx dy by changing variables, where R is the region
bounded by x + y = 0, x + y = 1, 2x â y = 0 & 2x â y = 3 using
transformations u = x + y & v = 2x â y.
ðð§ð¬ð°ðð«: ð/ð.
C 2 Evaluate the integral â« â« e
y x+y dy dx
1âx
0
1
0
by changing the variables x + y
= u and y = uv.
ðð§ð¬ð°ðð«: (ð â ð)/ð.
T 3 Evaluateâ¬(x2 + y2) dA
R
by changing the variables,where R is the region
lying in the first quadrant and bounded by the hyperbolas x2 â y2 = 9,
x2 â y2 = 1, xy = 2 and xy = 4.
ðð§ð¬ð°ðð«: ð.
H 4 Evaluateâ¬(y â x) dx dy
R
over the region R in XY â plane bounded by the
straight lines y = x â 3, y = x + 1, 3y + x = 5 & 3y + x = 7.
ðð§ð¬ð°ðð«: â ð.
C 5 Evaluateâ¬(x + y) dA,
R
where R is the trapezoidal region with vertices
(0, 0), (5, 0), (5
2 , 5
2 ) & (
5
2 , â
5
2 ) using the transformations
x = 2u + 3v and y = 2u â 3v.
ðð§ð¬ð°ðð«: ððð/ð.
T 6
Evaluate â« â« 2x â y
2 dx dy
y 2 +1
y 2
4
0
by applying transformations
u = 2x â y
2 & v =
y
2. Draw both the regions.
ðð§ð¬ð°ðð«: ð.
DARSHAN INSTITUTE OF ENGINEERING & TECHNOLOGY Mathematics-I (3110014)
U N I T - 5 [ 1 2 4 ]
T 7 Given that x + y = u, y = uv. Evaluate by change the variables to u, v in
the integralâ¬[ xy (1 â x â y) ]1 2 dx dy taken over the area of the triangle
with sides x = 0, y = 0 & x + y = 1.
ðð§ð¬ð°ðð«: ðð/ððð.
H 8 Evaluateâ¬x dx dy,
R
where R is the region bounded by triangle with vertices
at (0, 0), (1, 0) and (1, 1), using the transformations x = u and y = uv.
ðð§ð¬ð°ðð«: ð/ð.
C 9 Evaluate â¬(x2 â y2)2
ð
dA over the area bounded by lines | x | + | y | = 1
using the transformations x + y = u and x â y = v.
ðð§ð¬ð°ðð«: ð/ð.
â DOUBLE INTEGRALS BY CHANGE OF VARIABLE IN POLAR COORDINATES:
â Procedure for evaluating double integrals by changing variables form cartesian to polar
coordinates:
(1). Substitute x = r cos Ξ and y = r sin Ξ.
(2). Find Jacobian J = r.
(3). Replace dx dy by | J |dr dΞ.
(4). Find ð« and ð in terms of x and y.
(5). Draw the region. Hence, we get
⬠f(x, y)dx dy = ⬠f(r cos Ξ, r sin Ξ) r dr dΞ.
METHOD â 14: D.I. BY CHANGE OF VARIABLE IN POLAR COORDINATES
C 1 By changing into polar co â ordinates, evaluate â« â«dx dy
1
0
1
0
.
ðð§ð¬ð°ðð«: ð.
DARSHAN INSTITUTE OF ENGINEERING & TECHNOLOGY Mathematics-I (3110014)
U N I T - 5 [ 1 2 5 ]
H 2 By changing into polar coordinates, evaluate â« â« eâ(x
2+y2) dx dy.
â
0
â
0
Also
sketch the regions.
ðð§ð¬ð°ðð«: ð/ð.
C 3 Evaluate the integralâ« â«
1
(1 + x2 + y2)2 dx dy.
â
0
â
0
ðð§ð¬ð°ðð«: ð/ð.
S-19 (3)
T 4
Evaluateâ« â« (x2 + y2) dy dx
â4âx2
0
2
0
by changing into polar coordinates.
ðð§ð¬ð°ðð«: ðð.
W-18 (4)
T 5
Change into polar co â ordinates, evalute â« â« eâ(x2+y2) dy dx
âa2âx2
0
a
0
.
ðð§ð¬ð°ðð«: (ð â ðâðð)ð/ð.
H 6
Evaluate the intrgralâ« â« y2â(x2 + y2)
âa2ây2
0
dy dx
a
0
by changing into polar
co â ordinates.
ðð§ð¬ð°ðð«: ððð/ðð.
C 7 t By changing into polar coâordinates, evaluate the integral
â« â« (x2 + y2) dy dx
â2axâx2
0
2a
0
.
ðð§ð¬ð°ðð«: ðððð/ð.
H 8
By change into polar coordinates, evaluate â« â« dy dx
âa2âx2
ââa2âx2
a
âa
.
ðð§ð¬ð°ðð«: ððð.
DARSHAN INSTITUTE OF ENGINEERING & TECHNOLOGY Mathematics-I (3110014)
U N I T - 5 [ 1 2 6 ]
H 9 Evaluate the integral by change into the polar coordinates.
â« â«âx2+y2
x
0
1
0
dy dx.
ðð§ð¬ð°ðð«: {ð
âð + ð¥ðšð (ð + âð )} /ð.
T 10 By transforming into polar coordinates, evaluateâ« â«
x
x2 + y2 dxdy
a
y
a
0
.
ðð§ð¬ð°ðð«: ðð/ð.
C 11 Integrate f(x , y) =
ln (x2 + y2)
âx2 + y2 over the region 1 †x2 + y2 †e by
changing into polar coordinates.
ðð§ð¬ð°ðð«: ðð( ð â âð ).
H 12 By transforming into polar co â ordinates, findâ¬
x2y2
x2 + y2 dx dy over
the annular region between the circles x2 + y2 = a2 & x2 + y2 = b2(a < b).
ðð§ð¬ð°ðð«: ð(ðð â ðð)/ðð.
H 13 Evaluate â¬(x2 + y2) dA,
R
where R is the annular region between the two
circles x2 + y2 = 1 and x2 + y2 = 5, by changing into polar coordinate.
ðð§ð¬ð°ðð«: ððð.
T 14
Evaluate the integral â« â« x2 â y2
x2 + y2
y
y2
4a
dx dy
4a
0
by transforming into polar
co â ordinates.
ðð§ð¬ð°ðð«: ððð{(ð/ð) â (ð/ð)}.
DARSHAN INSTITUTE OF ENGINEERING & TECHNOLOGY Mathematics-I (3110014)
U N I T - 5 [ 1 2 7 ]
â TRIPLE INTEGRALS BY CHANGE OF VARIABLE OF INTEGRATION:
â Procedure for evaluating triple integrals by change of variable:
(1). Find ð±, ð², and ð³ in terms of u, v and w by transformation.
(2). Find Jacobian J(u, v, w).
(3). Find limits for u, v and w.
(4). The new integral form is
âf(x, y, z)dx dy dz =âf(g1, g2, g3)|J| du dv dw,
where g1 = g1(u, v, w), g2 = g2(u, v, w), g3 = g3(u, v, w).
(5). Calculate the example as the example of triple integrals.
METHOD â 15: T.I. BY CHANGE OF VARIABLE OF INTEGRATION
H 1 Change into spherical polar coordinate and hence evaluate
â« â« â«1
âa2 â x2 â y2 â z2
âa2âx2ây2
0
âa2âx2
0
dz dy dx.
a
0
ðð§ð¬ð°ðð«: ðð ðð.
C 2
Evaluate â« â« â« (x2 + y2) dz dy dx
2
âx2+y2
â4âx2
ââ4âx2
2
â2
.
ðð§ð¬ð°ðð«: ðð/ð.
â â â â â â â â â â
DARSHAN INSTITUTE OF ENGINEERING & TECHNOLOGY Mathematics-I (3110014)
U N I T - 6 [ 1 2 9 ]
UNIT 6
â ELEMENTARY ROW TRANSFORMATION/OPERATION ON A MATRIX:
â Rij or Ri â Rj means interchange of ith and jth rows.
â k â Ri means multiplication of all the elements of ith row by non-zero scalar k.
â Rij(k) or Rj + k â Ri means multiplication of all the elements of ith row by nonzero scalar
k and add it into jth row.
â ROW ECHELON FORM OF MATRIX:
â Procedure to convert the matrix into Row Echelon Form is as follow:
(1). Every zero row of the matrix occurs below the non-zero rows.
(2). Arrange all the rows in decreasing order.
(3). Make all the entries zero below the leading (first non-zero entry of the row)
element of 1st row.
(4). Repeat step-3 for each row.
â REDUCED ROW ECHELON FORM OF MATRIX:
â Procedure to convert the matrix into Reduced Row Echelon Form is as follow:
(1). Convert the given matrix into Row Echelon Form.
(2). Make all the leading elements 1(one).
(3). Make all the entries zero above the leading element 1(one) of each row.
â RANK OF A MATRIX:
â Let A be an mà n matrix. Then the positive integer r is said to be the rank of A if
(1). There is at least one minor of order r which is non-zero.
(2). Every (r + 1)th order minor of A is zero.
â Rank of matrix A is denoted by Ï(A) ðð Rank(A).
â NOTES:
â Ï(AB) †Ï(A) ðð Ï(AB) †Ï(B).
DARSHAN INSTITUTE OF ENGINEERING & TECHNOLOGY Mathematics-I (3110014)
U N I T - 6 [ 1 3 0 ]
â Ï(A) = Ï(AT).
â For a matrix A of order mà n, Ï(A) †min{m, n}.
â Procedure to find the Rank of given matrix is as follow:
(1). Reduced the given matrix in Row Echelon Form.
(2). Number of non-zero raw in Row-Echelon Form is the Rank of given matrix.
METHOD â 1: ECHELON FORM AND RANK OF MATRIX
C 1 Determine whether the following matrices are in Row-Echelon Form,
Reduced Row-Echelon Form, both or none.
(ð). [3] (ð). [1 2] (ð). [0 00 0
] (ð). [1 1 00 1 00 0 0
] (ð). [1 0 00 1 00 2 0
]
(ð). [1 2 30 0 00 0 1
] (ð). [1 0 0 40 1 0 70 0 1 â1
] (ð). [1 4 â3 70 1 6 20 0 1 5
]
(ð). [000
100
210
6â10
001] (ðð). [
0 1 â2 0 10 0 0 1 300
00
0 0 00 0 0
].
ðð§ð¬ð°ðð«: (ð). ððð¡ðð¥ðšð§ ð ðšð«ðŠ (ð). & (ð). ððšðð¡ (ð). ððð¡ðð¥ðšð§ ð ðšð«ðŠ (ð). ððšð§ð
(ð). ððšð§ð (ð). ððšðð¡ (ð). & (ð). ððð¡ðð¥ðšð§ ð ðšð«ðŠ (ðð). ððšðð¡.
H 2 Is the matrix [
1 0 4 60 1 5 70 0 0 1
] in Row Echelon Form or Reduced Row Echelon
Form?
ðð§ð¬ð°ðð«: ððð¡ðð¥ðšð§ ð ðšð«ðŠ.
C 3
Find Reduced Row Echelon Form of A = [
2 4 3 41 2 1 313
37
2 24 8
].
ðð§ð¬ð°ðð«: [
ð ð ð ðð ð ð ððð
ðð
ð âðð ð
].
DARSHAN INSTITUTE OF ENGINEERING & TECHNOLOGY Mathematics-I (3110014)
U N I T - 6 [ 1 3 1 ]
H 4 Find Reduced Row Echelon Form of A = [
0 0 â2 0 7 122 4 â10 6 12 282 4 â5 6 â5 â1
].
ðð§ð¬ð°ðð«: [ð ð ð ð ð ðð ð ð ð ð ðð ð ð ð ð ð
].
T 5
Convert [
1202
3606
â2â550
0â2108
2404
0â31518
0â156
] into its equivalent Reduced Row
Echelon Form.
ðð§ð¬ð°ðð«: [
ðððð
ðððð
ðððð
ðððð
ðððð
ðððð
ððð ðâð
].
C 6 Reduce matrix A = [
1 5 3 â22 0 4 14 8 9 â1
] to Row Echelon Form and find itâs Rank.
ðð§ð¬ð°ðð«: [ð ð ð âðð âðð âð ðð ð âð/ð ð
] & ð(ð) = ð.
W-18 (3)
H 7 Find the Rank of the following matrices:
[2 4 61 2 34 8 12
] , [
1 2 3 42 4 6 834
68
9 1212 16
],
[1 2 32 3 43 5 7
] , [
1 2 3 â1â2 â1 â3 â110
01
1 11 â1
] , [1 2 â1 42 4 3 5â1 â2 6 â7
],
[5 3 14 40 1 2 11 â1 2 0
] , [1 â1 2 03 1 0 0â1 2 4 0
] & [
1 2 3 02 4 3 236
28
1 37 5
].
ðð§ð¬ð°ðð«: ð, ð, ð, ð, ð, ð, ð & ð.
T 8 Define the rank of a matrix and find the rank of the matrix
A = [2 â1 04 5 â31 â4 7
].
ðð§ð¬ð°ðð«: ð(ð) = ð.
S-19 (3)
DARSHAN INSTITUTE OF ENGINEERING & TECHNOLOGY Mathematics-I (3110014)
U N I T - 6 [ 1 3 2 ]
C 9 Convert the following matrix in to Reduced Row Echelon Form and hence
find the Rank of a matrix A = [1 2 31 4 22 6 5
].
ðð§ð¬ð°ðð«: [ð ð ðð ð âð/ðð ð ð
] & ð(ð) = ð.
H 10 Find the Rank of the matrix A = [
1 2â3 1â2 3
4 05 29 2
].
ðð§ð¬ð°ðð«: ð(ð) = ð.
H 11 Find the Rank of the matrix A = [
1 2â1
413
532
202].
ðð§ð¬ð°ðð«: ð(ð) = ð.
H 12 Find the Rank of the matrix A = [
1 2 34 5 67 8 9
].
ðð§ð¬ð°ðð«: ð(ð) = ð.
T 13
Find the Rank of the matrix A = [
â1 3
24
2â7â5â9
0222
4 0 4â4
5 1 6â4
â3
417
].
ðð§ð¬ð°ðð«: ð(ð) = ð.
C 14 h
Find the Rank of the matrix A =
[ 3451015
4561116
5671217
6781318
7891419]
.
ðð§ð¬ð°ðð«: ð(ð) = ð.
H 15
Obtain the Reduced Row Echelon Form of the matrix A = [
1 31 2
2 21 3
2 43 7
3 44 8
] and
hence find the Rank of the matrix A.
ðð§ð¬ð°ðð«: ð(ð) = ð.
C 16 Find the Rank of A = [
1 4 7
258
369
000 ] in terms of determinants.
ðð§ð¬ð°ðð«: ð(ð) = ð.
DARSHAN INSTITUTE OF ENGINEERING & TECHNOLOGY Mathematics-I (3110014)
U N I T - 6 [ 1 3 3 ]
H 17 Find the Rank of the matrix A = [
13â1
â112
204
000] in terms of determinants.
ðð§ð¬ð°ðð«: ð(ð) = ð.
â MATRIX EQUATION:
â Let us consider system of m linear equations with n unknowns(variables):
a11x1 + a12x2 + a13x3 + â¯+ a1nxn = b1
a21x1 + a22x2 + a23x3 + â¯+ a2nxn = b2
⊠âŠâŠâŠâŠâŠâŠâŠâŠâŠâŠâŠâŠâŠâŠâŠâŠâŠâŠâŠ
am1x1 + am2x2 + am3x3 + â¯+ amnxn = bm
â The matrix representation of above system is AX = B, where
A =
[ a11 a12 a13a21 a22 a23a31 a32 a33
â¯
a1na2na3n
â® â± â®am1 am2 am3 ⯠amn]
mxn
,B =
[ b1b2b3â®bm]
mx1
and X =
[ x1x2x3â®xn]
nx1
.
â The matrix A is called the co-efficient matrix of above system.
â The matrix X is called the variable matrix of above system.
â The matrix B is called the constant matrix of above system.
â AUGMENTED MATRIX:
â The augmented matrix of the above linear system is defined as follow:
[A|B] ðð [A ⶠB] =
[ a11 a12 a13a21 a22 a23a31 a32 a33
â¯
a1n â® b1a2n â® b2a3n â® b3
â® â± â®am1 am2 am3 ⯠amn â® bm]
mx(n+1)
.
â CONSISTENT AND INCONSISTENT SYSTEM:
â If the system of linear equations AX = B has solution (unique or infinitely many), then
the system is called consistent system and if system has no solution, then the system is
called inconsistent system.
DARSHAN INSTITUTE OF ENGINEERING & TECHNOLOGY Mathematics-I (3110014)
U N I T - 6 [ 1 3 4 ]
â In Gauss Elimination and Gauss-Jordan method if Ï(A) = Ï(A ⶠB) then system is
consistent otherwise it is inconsistent.
â HOMOGENEOUS AND NON-HOMOGENEOUS SYSTEM
â A linear system AX = B is called Homogeneous linear system if matrix ð = ð (null or zero
column matrix). Otherwise it is called Non-homogeneous linear system.
â Non-homogeneous linear system (i. e. B â ð) has three types of solutions.
(1). No solution, (2). Unique solution & (3). Infinitely many solutions.
â Homogeneous linear system ( i. e. B = ð) has two types of solutions.
(1). Unique solution (trivial solution: (0, 0,âŠ,0))
(2). Infinitely many solutions (non-trivial solution)
â For the linear system in which no. of equations = no. of unknown (square system),
No solution ⹠[0 0 ⊠0 : #]
Unique solution â¹ [0 0 ⊠# : 0] or [0 0 ⊠# : â]
Infinite solutions ⹠[0 0 ⊠0 : 0]
â Solution by rank:
Ï(A) = Ï(A ⶠB) = n â Unique solution,
Ï(A) = Ï(A ⶠB) < n â Infinitely many solution,
Ï(A) â Ï(A ⶠB) â No solution,where n = no. of unknowns.
â NOTES:
â In the case of infinitely many solutions:
⢠The number of free variables = n â Ï(A),where n = no. of unknowns.
â If the Non-Homogeneous system of equations AX = B is square system (no. of unknown
= no. of equations) then find the determinant of co-efficient matrix A.
⢠If det(A) â 0, then the system of linear equations has unique solution.
⢠If det(A) = 0, then the system of linear equations has either no solution or infinitely
many solutions.
DARSHAN INSTITUTE OF ENGINEERING & TECHNOLOGY Mathematics-I (3110014)
U N I T - 6 [ 1 3 5 ]
â If the homogeneous system of equations AX = ð is square system (no. of unknown = no.
of equations) then find the determinant of co-efficient matrix A.
⢠If det(A) â 0, then the system of linear equations has unique solution It means trivial
solution.
⢠If det(A) = 0, then the system of linear equations has infinitely many solutions.
â NOTE:
â Since the homogeneous system AX = ð contains trivial solution, this system is always
consistent.
â GAUSS ELIMINATION METHOD TO SOLVE SYSTEM OF LINEAR EQUATIONS:
â Procedure to solve the given system of linear equations using Gauss Elimination Method:
(1). Start with augmented matrix [A ⶠB].
(2). Convert [A ⶠB] into Row Echelon Form with leading element of each row is one
(1).
(3). Apply back substitution for getting equations.
(4). Solve the equations and find the unknown variables (i.e. solution).
METHOD â 2: GAUSS ELIMINATION METHOD
C 1 Using Gauss Elimination method solve the following system
âx + 3y + 4z = 30, 3x + 2y â z = 9, 2x â y + 2z = 10.
ðð§ð¬ð°ðð«: (ð, ð, ð).
W-19 (7)
H 2 Solve by using Gauss Elimination Method:
(ð). x1 + x2 + 2x3 = 8, âx1 â 2x2 + 3x3 = 1, 3x1 â 7x2 + 4x3 = 10.
(ð). x1 + 2x2 + 3x3 = 4, 2x1 + 5x2 + 3x3 = 5, x1 + 8x3 = 9.
ðð§ð¬ð°ðð«: (ð). (ð, ð, ð) & (ð). (ð, ð, ð).
H 3 Solve the given system of linear equations by using Gauss Elimination
Method: 2x + y â z = 4, x â y + 2z = â2, â x + 2y â z = 2.
ðð§ð¬ð°ðð«: (ð, ð,âð).
DARSHAN INSTITUTE OF ENGINEERING & TECHNOLOGY Mathematics-I (3110014)
U N I T - 6 [ 1 3 6 ]
H 4 Solve system of linear equations by using Gauss Elimination method, if
solution exists.
x + y + 2z = 9, 2x + 4y â 3z = 1, 3x + 6y â 5z = 0.
ðð§ð¬ð°ðð«: (ð, ð, ð).
W-18 (4)
C 5 Solve by using Gauss Elimination Method:
â1
x +3
y +4
z = 30,
3
x +2
y â1
z = 9,
2
x â1
y +2
z = 10.
ðð§ð¬ð°ðð«: (ð ðâ , ð ðâ , ð ðâ ).
T 6 Solve the given system of linear equations by using Gauss Elimination
Method:
x + y + z = 3, x + 2y â z = 4, x + 3y + 2z = 4.
ðð§ð¬ð°ðð«: (ðð/ð, ð/ð,âð/ð).
T 7 Solve the given system of linear equations using Gauss Elimination Method:
2x1 + x2 + 2x3 + x4 = 6, 6x1 â x2 + 6x3 + 12x4 = 36,
4x1 + 3x2 + 3x3 â 3x4 = 1, 2x1 + 2x2 â x3 + x4 = 10.
ðð§ð¬ð°ðð«: (ð/ð, ð/ð,âð/ð, ð/ð).
C 8 Solve by using Gauss Elimination Method:
â2b + 3c = 1, 3a + 6b â 3c = â2, 6a + 6b + 3c = 5.
ðð§ð¬ð°ðð«: ððš ð¬ðšð¥ð®ðð¢ðšð§.
H 9 Solve by using Gauss Elimination Method:
(1). â2y + 3z = 1, 3x + 6y â 3z = â2, 6x + 6y + 3z = 5.
(2). 3x + y â 3z = 13, 2x â 3y + 7z = 5, 2x + 19y â 47z = 32.
ðð§ð¬ð°ðð«: ððš ð¬ðšð¥ð®ðð¢ðšð§ ððšð« ðððð¡ ð¬ð²ð¬ðððŠ.
C 10 Solve the given system of linear equations by using Gauss Elimination
Method:
x1â2x2 + 3x3 = â2, âx1 + x2 â 2x3 = 3, 2x1 â x2 + 3x3 = â7.
ðð§ð¬ð°ðð«: {(âð â ð, ð â ð, ð)/ð â â}.
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U N I T - 6 [ 1 3 7 ]
C 11 Solve by using Gauss Elimination Method:
x1 + 3x2 â 2x3 + 2x5 = 0, 2x1 + 6x2 â 5x3 â 2x4 + 4x5 â 3x6 = â1,
5x3 + 10x4 + 15x6 = 5, 2x1 + 6x2 + 8x4 + 4x5 + 18x6 = 6.
ðð§ð¬ð°ðð«: ð±ð = âðð« â ðð¬ â ðð, ð±ð = ð«, ð±ð = âðð¬, ð±ð = ð¬, ð±ð = ð,
ð±ð = ð ðâ ; ð°ð¡ðð«ð ð«, ð¬, ð â â.
H 12 Solve by using Gauss Elimination Method:
x1 â 2x2 â x3 + 3x4 = 1, 2x1 â 4x2 + x3 = 5, x1 â 2x2 + 2x3 â 3x4 = 4.
ðð§ð¬ð°ðð«: ð±ð = ð + ððð â ðð, ð±ð = ðð, ð±ð = ð + ððð, ð±ð = ðð, ðð, ðð â â.
T 13 Solve by using Gauss Elimination Method:
(1). 5x + 3y + 7z = 4, 3x + 26y + 2z = 9, 7x + 2y + 10z = 5.
(2). 2x1 + 2x2 + 2x3 = 0, â2x1 + 5x2 + 2x3 = 1, 8x1 + x2 + 4x3 = â1.
ðð§ð¬ð°ðð«: (ð). {((ð â ððð) ððâ , (ð + ð) ððâ , ð )/ð â â} &
(ð). {ð±ð = (âð â ðð) ðâ , ð±ð = (ð â ðð) ðâ , ð±ð = ð;ð°ð¡ðð«ð ð ð â}.
C 14 The augmented matrix of a linear system has the form [
â210
315
1â1â1
:::
abc].
Determine when the linear system is consistent.
Determine when the linear system is inconsistent.
Does the linear system have a unique solution or infinitely many solutions?
ðð§ð¬ð°ðð«: ð + ðð â ð = ð, ð + ðð â ð â ð & ðð§ðð¢ð§ððð¥ð² ðŠðð§ð² ð¬ðšð¥ð®ðð¢ðšð§ð¬.
H 15 Determine when the given augmented matrix represents a consistent linear
system. [121
01â1
251
:::
abc].
ðð§ð¬ð°ðð«: ð + ð â ðð = ð.
H 16 For what choices of parameter λ the following system is consistent?
x1 + x2 + 2x3 + x4 = 1, x1 + 2x3 = 0, 2x1+2x2 + 3x3 = λ,
x2 + x3 + 3x4 = 2λ.
ðð§ð¬ð°ðð«: ð ðšð« ð = ð ð ð¢ð¯ðð§ ð¬ð²ð¬ðððŠ ð¢ð¬ ððšð§ð¬ð¢ð¬ððð§ð.
DARSHAN INSTITUTE OF ENGINEERING & TECHNOLOGY Mathematics-I (3110014)
U N I T - 6 [ 1 3 8 ]
T 17 Determine the values of k, for which the equations 3x â y + 2z = 1, â4x +
2y â 3z = k, 2x + z = k2 possesses solution. Find solutions in each case.
ðð§ð¬ð°ðð«: ð€ = ð â {((ð â ð)/ð, (ðð + ð)/ð, ð)/ð â â} &
ð€ = âð â {((ð â ð)/ð, (ð + ð)/ð, ð)/ð â â}.
C 18 For which value of λ and k the following system has (i) No solution, (ii)
Unique solution, (iii) An infinite no. of solution.
x + y + z = 6, x + 2y + 3z = 10, x + 2y + λz = k.
ðð§ð¬ð°ðð«: (ð¢) ð = ð, ð€ â ðð â ð§ðš ð¬ðšð¥ð®ðð¢ðšð§,
(ð¢ð¢) ð â ð, ð€ â â â ð®ð§ð¢ðªð®ð ð¬ðšð¥ð®ðð¢ðšð§ &
(ð¢ð¢ð¢) ð = ð, ð€ = ðð â ð¢ð§ðð¢ð§ð¢ðð ð¬ðšð¥ð®ðð¢ðšð§ð¬.
H 19 For which values of âaâ will the following system has (i) no solution, (ii)
unique solution, (iii) Infinitely many solutions.
x + 2y â 3z = 4, 3x â y + 5z = 2, 4x + y + (a2 â 14)z = a + 2.
ðð§ð¬ð°ðð«: (ð¢) ð = âð â ð§ðš ð¬ðšð¥ð®ðð¢ðšð§,
(ð¢ð¢) ð â â â { âð, ð } â ð®ð§ð¢ðªð®ð ð¬ðšð¥ð®ðð¢ðšð§ &
(ð¢ð¢ð¢) ð = ð â ð¢ð§ðð¢ð§ð¢ðð ð¬ðšð¥ð®ðð¢ðšð§ð¬.
T 20 Determine the value of k so that the system of homogeneous equations
2x + y + 2z = 0, x + y + 3z = 0, 4x + 3y + kz = 0 has
(a) Trivial solution.
(b) Non-trivial solution. Also find non-trivial solution.
ðð§ð¬ð°ðð«: (ð)ðð ð€ â ð â ðð«ð¢ð¯ð¢ðð¥ ð¬ðšð¥ð®ðð¢ðšð§. ð¢. ð. (ð, ð, ð) &
(ð) ðð ð€ = ð â ð± = ð€; ð² = âðð€; ð³ = ð€,ð°ð¡ðð«ð ð€ ð¢ð¬ ð ð©ðð«ððŠðððð«.
C 21 Find real value of λ for which the equations
(1 â λ)x â y + z = 0, 2x + (1 â λ)y = 0, 2y â (1 + λ)z = 0 have solution
other than x = y = z = 0. Also find the solution for each value of λ.
ðð§ð¬ð°ðð«: ð = ðâ¹ ð± = ð; ð² = ð€; ð³ = ð€,ð°ð¡ðð«ð ð€ â â ð¢ð¬ ðð«ðð¢ðð«ðð«ð².
â GAUSS-JORDAN ELIMINATION METHOD TO SOLVE THE SYSTEM OF LINEAR EQUATIONS:
â Procedure to solve the system of linear equations by Gauss-Jordan Elimination Method:
(1). Start with the augmented matrix [A ⶠB].
DARSHAN INSTITUTE OF ENGINEERING & TECHNOLOGY Mathematics-I (3110014)
U N I T - 6 [ 1 3 9 ]
(2). Convert [A ⶠB] into reduced row echelon form.
(3). Find the unknowns, which is required solutions.
METHOD â 3: GAUSS - JORDAN METHOD
C 1 Solve the following system of linear equations by Gauss-Jordan Method.
x + y + 2z = 8, â x â 2y + 3z = 1, 3x â 7y + 4z = 10.
ðð§ð¬ð°ðð«: (ð, ð, ð).
H 2 Solve by using Gauss-Jordan Method:
(1). x + y + z = 6, x + 2y + 3z = 14, 2x + 4y + 7z = 30.
(2). x1 + 2x2 + 3x3 = 1, 2x1 + 5x2 + 3x3 = 6, x1 + 8x3 = â6.
(3). x1 + 2x2 + 3x3 = 4, 2x1 + 5x2 + 3x3 = 5, x1 + 8x3 = 9.
ðð§ð¬ð°ðð«: (ð). (ð, ð, ð) (ð). (ð, ð, âð) (ð). (ð, ð, ð).
H 3 Solve the system of linear equations by using Gauss-Jordan Elimination
Method.
x + 4y â 3z = 0, âx â 3y + 5z = â3, 2x + 8y â 5z = 1.
ðð§ð¬ð°ðð«: (ðð,âð, ð).
H 4 Solve by using Gauss-Jordan Method:
(1). 2x â y â 3z = 0, â x + 2y â 3z = 0, x + y + 4z = 0.
(2). 2x1 + x2 + 3x3 = 0, x1 + 2x2 = 0, x2 + x3 = 0.
ðð§ð¬ð°ðð«: ðð«ð¢ð¯ð¢ðð¥ ð¬ðšð¥ð®ðð¢ðšð§ (ð, ð, ð) ððšð« ððšðð¡ ð¬ð²ð¬ðððŠ.
T 5 Solve by using Gauss-Jordan Method,
where 0 †α †2Ï, 0 †β †2Ï, 0 †γ †Ï.
2 sin α â cosβ + 3 tan γ = 3, 4 sin α + 2 cosβ â 2 tan γ = 2,
6 sin α â 3 cosβ + tan γ = 9.
ðð§ð¬ð°ðð«: (ð,âð, ð).
C 6 Solve by using Gauss-Jordan Method:
â2y + 3z = 1, 3x + 6y â 3z = â2, 6x + 6y + 3z = 5.
ðð§ð¬ð°ðð«: ðð§ððšð§ð¬ð¢ð¬ððð§ð(ð§ðš ð¬ðšð¥ð®ðð¢ðšð§).
DARSHAN INSTITUTE OF ENGINEERING & TECHNOLOGY Mathematics-I (3110014)
U N I T - 6 [ 1 4 0 ]
C 7 Use Gauss-Jordan algorithm to solve the system of linear equations:
2x1 + 2x2 â x3 + x5 = 0, â x1 â x2 + 2x3 â 3x4 + x5 = 0,
x1 + x2 â 2x3 â x5 = 0, x3 + x4 + x5 = 0.
ðð§ð¬ð°ðð«: {(âð¬ â ð, ð¬,âð, ð, ð)/ ð, ð¬ â â}.
S-19 (4)
H 8 Solve the system of linear equations v + 3w â 2x = 0, 2u + v â 4w+ 3x
= 0, 2u + 3v + 2wâ x = 0,â4u â 3v + 5wâ 4x = 0.
ðð§ð¬ð°ðð«: ð® = (ðð¬ â ðð)/ð,ð¯ = ðð â ðð¬,ð° = ð¬, ð± = ð,ð°ð¡ðð«ð ð¬, ð â â.
H 9 Solve x1 + x2 + 2x3 â 5x4 = 3, 2x1 + 5x2 â x3 â 9x4 = â3,
2x1 + x2 â x3 + 3x4 = â11, x1 â 3x2 + 2x3 + 7x4 = â5 using Gauss-Jordan
Method.
ðð§ð¬ð°ðð«: {(âð â ðð, ð + ðð, ð + ðð, ð)/ð â â}.
T 10 Solve by using Gauss-Jordan Method:
(1). x1 + 3x2 + x4 = 0, x1 + 4x2 + 2x3 = 0, â2x2 â 2x3 â x4 = 0.
(2). 2x1 â 4x2 + x3 + x4 = 0, x1 â 2x2 â x3 + x4 = 0.
ðð§ð¬ð°ðð«: (ð). {(ðð,âðð, ðð/ð, ð)/ð â â},
(ð). {((ððð â ððð)/ð, ðð, ðð/ð, ðð)/ðð & ðð â â}.
T 11 Solve using by Gauss-Jordan Method:
(i). 3x â y â z = 0, x + y + 2z = 0, 5x + y + 3z = 0 &
(ii). x + y â z +w = 0, x â y + 2z â w = 0, 3x + y + w = 0.
ðð§ð¬ð°ðð«: (i). {(ð± = â ð
ðð, ð² = â
ð
ðð, ð³ = ð,ð°ð¡ðð«ð ð â â)},
(ii). {(ð± = âð
ð ðð, ð² =
ð
ððð â ðð, ð³ = ðð, ð° = ðð ; ðð, ðð â â)}.
â INVERSE BY GAUSS - JORDAN METHOD:
â Procedure to find the inverse of Matrix by Gauss-Jordan Method is as follow:
(1). Start with augmented matrix [A ⶠI], where I is the identity matrix of the same size
of matrix A.
(2). Convert [A ⶠI] into reduced row echelon form.
(3). The inverse of the matrix A is I with respective changes. i.e. [I ⶠAâ1].
DARSHAN INSTITUTE OF ENGINEERING & TECHNOLOGY Mathematics-I (3110014)
U N I T - 6 [ 1 4 1 ]
METHOD â 4: INVERSE BY GAUSS - JORDAN METHOD
C 1 Find the inverse of the given matrix by Gauss-Jordan Method
A = [1 â23 â1
].
ðð§ð¬ð°ðð«: ðâð =ð
ð [âð ðâð ð
].
H 2 If possible then find the inverse of the given matrix by Gauss-Jordan
Method A = [6 â4â3 2
].
ðð§ð¬ð°ðð«: ððšð ð©ðšð¬ð¬ð¢ðð¥ð ððððð®ð¬ð ððð(ð) = ð.
C 3 Find the inverse of matrix A = [
2 1 33 1 21 2 3
] by Gauss-Jordan Method.
ðð§ð¬ð°ðð«: ðâð = ð
ð[âð ð âðâð ð ðð âð âð
].
H 4 If possible then find the inverse of matrix by Gauss-Jordan Method
A = [1 0 1â1 1 â10 1 0
].
ðð§ð¬ð°ðð«: ððšð ð©ðšð¬ð¬ð¢ðð¥ð ððððð®ð¬ð ððð(ð) = ð.
H 5 Find the inverse of the given matrix by Gauss-Jordan Method
A = [2 6 62 7 62 7 7
].
ðð§ð¬ð°ðð«: ðâð = [ð/ð ð âðâð ð ðð âð ð
].
T 6 Find the inverse of matrix A = [
1 0 1â1 1 10 1 0
] using Gauss-Jordan Method.
ðð§ð¬ð°ðð«: [ð/ð âð/ð ð/ðð ð ðð/ð ð/ð âð/ð
].
T 7 Find the inverse of matrix A=[
0 1 21 2 33 1 1
] by Gauss-Jordan Method.
ðð§ð¬ð°ðð«: ðâð =ð
ð [ð âð ðâð ð âðð âð ð
].
DARSHAN INSTITUTE OF ENGINEERING & TECHNOLOGY Mathematics-I (3110014)
U N I T - 6 [ 1 4 2 ]
C 8 Using Gauss â Jordan method find Aâ1 for A = [
1 2 32 5 31 0 8
].
ðð§ð¬ð°ðð«: ðâð = [âðð ðð ððð âð âðð âð âð
].
W-19 (7)
H 9 Find the inverse of matrix A = [
1 3 31 4 31 3 4
] using row operations.
ðð§ð¬ð°ðð«: ðâð = [ð âð âðâð ð ðâð ð ð
].
H 10 Find Aâ1 for A = [
0 1 â13 1 11 2 â1
], if exists.
ðð§ð¬ð°ðð«: ðâð = [ð ð âðâð âð ðâð âð ð
].
H 11 Find the inverse of A = [
3 â3 42 â3 40 â1 1
] using row operations.
ðð§ð¬ð°ðð«: ðâð = [ð âð ðâð ð âðâð ð âð
].
T 12 Find the inverse of following matrices by Gauss-Jordan Method
[
1111
0333
0055
0007
] & [
1023
â111â2
0121
2â116
].
ðð§ð¬ð°ðð«: [
ðâð/ððð
ðð/ðâð/ðð
ððð/ðâð/ð
ðððð/ð
] & [
ðâððâð
âðâððâð
ððâðð
âðððð
].
T 13
Find the inverse of [
1121
2341
3331
1231
] using Gauss-Jordan Method.
ðð§ð¬ð°ðð«: [
ðððâð
âðâððð
ððâðâð
ðâððð
].
DARSHAN INSTITUTE OF ENGINEERING & TECHNOLOGY Mathematics-I (3110014)
U N I T - 6 [ 1 4 3 ]
â EIGENVALUES, EIGENVECTORS & EIGEN SPACE:
â Let A be an n à n matrix, then the scalar λ is called an Eigen Value of A, if there exist a
nonzero vector X such that AX = λX.
â The non-zero vector X is called an Eigen Vector of A corresponding to λ.
â The solution space of system [A â λI]X = O is called Eigen Space of matrix A
corresponding to λ.
â NOTES:
â Let A be an n à n matrix then
(1). An Eigen Value of A is a scalar λ such that det(A â λI) = 0.
(2). The Eigen Vectors of A corresponding to λ are the non-zero solutions of
(A â λI) X = 0. Here det(A â λI) = 0 is called the Characteristic Equation of A.
â Eigen value is also known as characteristic root/latent value/proper roots and Eigen
Vector is also known as characteristic vector/latent vector/proper vector corresponding
to the eigen value λ of the matrix.
â The set of all eigenvalues of matrix A is called Spectrum of A.
â The characteristic equation of 2 Ã 2 matrix A is ðð â ððð + ðð = ð.
Where, S1=Sum of the principal diagonal elements (Trace(A)) and S2= Determinant of A.
â The characteristic equation of 3 Ã 3 matrix A is ðð â ðððð + ððð â ðð = ð.
Where S1= Sum of the principal diagonal elements or Trace(A),
S2= Sum of the minors of the principal diagonal elements &
S3= Determinant of A.
â Procedure for finding eigen values, eigen vectors and eigen spaces for 3 Ã 3 matrix A.
(1). Find the characteristic equation det(A â λI) = 0. It will be a polynomial equation
of degree 3 in the variable λ.
(2). Find the roots of the characteristic equation. These are the Eigen Values of A,
say λ1, λ2, λ3.
DARSHAN INSTITUTE OF ENGINEERING & TECHNOLOGY Mathematics-I (3110014)
U N I T - 6 [ 1 4 4 ]
(3). For each Eigen Value λi, find the Eigen Vectors corresponding to λi by solving the
Homogeneous system [A â λiI] â X = O. (Here the solution space of this system is
Eigen Space corresponding to λi).
â The procedure remains same for the matrices of order 2.
â PROPERTIES OF EIGEN VALUES AND EIGEN VECTORS:
â Trace(A)=Sum of the eigen values of matrix A.
â Det(A)=Product of the eigen values of matrix A.
â The eigen values of triangular matrix are the elements on its principal diagonal.
â If one of the eigen value is zero then det(A)=0, hence Aâ1 does not exist.
â The square matrix A and AT have the same eigen values but eigen vectors need not to be
same.
â If λ is an eigen value of a non-singular matrix A and X is an eigen vector correspond to λ
then 1
λ is an eigen value of Aâ1 and X is an eigen vector correspond to
1
λ .
â If λ is an eigen value of matrix A and X is an eigen vector correspond to λ then λk is an
eigen value of Ak and X is an eigen vector.
â If λ is an eigen value of matrix A then λ ± k is an eigen value of A ± kI.
â If λ an is eigen value of matrix A then kλ is an eigen value of kA.
METHOD â 5: EIGEN VALUES, EIGEN VECTORS AND EIGEN SPACE
C 1 Define the eigen value, eigen vector and eigen space.
C 2 Let A = [3 08 â1
] then find the eigen values of A, A25 , 3A, Aâ1, AT, A +
2I & A3 â 5I.
ðð§ð¬ð°ðð«: â ð, ð â ð, ð25 â ð, ð â ð, ð/ð â ð, ð ð, ð & â ð, ðð.
H 3 Find the eigen values of A and Aâ1, where A = [1 20 4
].
ðð§ð¬ð°ðð«: ð, ð & ð, ð/ð.
W-18 (2)
DARSHAN INSTITUTE OF ENGINEERING & TECHNOLOGY Mathematics-I (3110014)
U N I T - 6 [ 1 4 5 ]
C 4 Find the eigen values of A = [
â5 4 340 0 40 0 4
]. Is A invertible?
ðð§ð¬ð°ðð«: â ð, ð, ð & ð§ðš.
H 5 If A = [
1 3 40 0 50 0 9
], then find the eigen values of the matrix AT. Is A an
invertible?
ðð§ð¬ð°ðð«: ð, ð, ð & ð§ðš.
H 6 If A = [
1 2 â30 2 30 0 2
] then find the eigen values of AT and 5A.
ðð§ð¬ð°ðð«: ð, ð, ð & ð, ðð, ðð.
H 7 Find the eigen values of A = [
1 0 05 2 012 15 3
] and hence deduce eigen values of
A5 and Aâ1.
ðð§ð¬ð°ðð«: ð, ð, ð ð, ðð, ðð & ð, ð/ð,ð/ð.
H 8
Find the eigen values of A9 for A = [
1000
30.500
7300
11842
].
ðð§ð¬ð°ðð«: ð, ð. ðð, ð, ðð.
T 9 Find the eigen values of the following matrices:
(ð). [1 00 2
] (ð). [5 31 3
] (ð). [10 â94 â2
] (ð). [1 2 â30 2 30 0 2
] (ð).[4 0 1â2 1 0â2 0 1
]
(ð). [3 0 â51/5 â1 01 1 â2
].
ðð§ð¬ð°ðð«: (ð). ð, ð (ð). ð, ð (ð). ð, ð (ð). ð, ð, ð (ð). ð, ð, ð (ð). ð,ââð,âð.
C 10 Let A = [0 11 0
], then find the eigen values, eigen vectors corresponding to
each eigen values of A. Also write the eigen vector space for each eigen
values.
ðð§ð¬ð°ðð«: ð & â ð, [ðð]& [
âðð] , ðð = { ð [
ðð] /ð â â }, ðâð = { ð [
âðð] /ð â â }.
DARSHAN INSTITUTE OF ENGINEERING & TECHNOLOGY Mathematics-I (3110014)
U N I T - 6 [ 1 4 6 ]
H 11 Find the eigen values and eigen vectors of the matrix A = [1 â2â5 4
].
ðð§ð¬ð°ðð«: ð, âð & (ð, ð)T, (âð/ð, ð)T.
T 12 Show that if 0 < Ξ < Ï , then A = [cosΞ âsinΞsinΞ cosΞ
] has no real eigen values
and consequently no eigen vectors.
C 13 Find the eigen values and eigen vectors of the matrix A = [
3 1 40 2 60 0 5
].
ðð§ð¬ð°ðð«: ð, ð, ð & (âð, ð, ð)T, (ð, ð, ð)T, (ð, ð, ð)T.
H 14 Find the eigen values and basis for eigen space for the matrix
A = [3 â1 0â1 2 â10 â1 3
].
ðð§ð¬ð°ðð«: ð, ð, ð & { (ð, ð, ð)T, (ð,âð, ð)T , (âð, ð, ð)T }.
T 15 Let A = [
1 0 â11 2 12 2 3
], then find eigen values, eigen vectors corresponding to
each eigen values of A & write eigen space for each eigen values.
ðð§ð¬ð°ðð«: ð, ð, ð & [âððð] , [
âðð
ð ð
] ,
[ â
ð
ð ð
ð ð ]
& ðð = { ð [â1ðð] /ð â â },
ðð = { ð [âðð/ðð
]/ð â â }, ðð = { ð [âð/ðð/ðð
]/ð â â }.
C 16 Find the eigen values and bases for the eigen space of the matrix
A=[0 1 11 0 11 1 0
].
ðð§ð¬ð°ðð«: â ð,âð, ð & {(âð, ð, ð)T, (âð, ð, ð)T, (ð, ð, ð)T}.
H 17 Find the eigen values and corresponding eigen vectors for the matrix
A = [1 2 20 2 1â1 2 2
].
ðð§ð¬ð°ðð«: ð, ð, ð & (âð,âð, ð)ð, (ð, ð, ð)ð.
W-18 (7)
DARSHAN INSTITUTE OF ENGINEERING & TECHNOLOGY Mathematics-I (3110014)
U N I T - 6 [ 1 4 7 ]
H 18 Find the eigen values and bases for the eigen space of the matrix
A=[1 0 01 2 0â3 5 2
].
ðð§ð¬ð°ðð«: ð, ð, ð & {(ð,âð, ð)T, (ð, ð, ð)T}.
T 19 Find the eigen values and eigen vectors of A=[
2 1 20 2 â10 1 0
].
ðð§ð¬ð°ðð«: ð, ð, ð & (âð, ð , ð )T, (ð, ð, ð)T.
C 20 Find the eigen values and corresponding eigen vectors of the matrix
A=[5 0 11 1 0â7 1 0
].
ðð§ð¬ð°ðð«: ð, ð, ð & (âð/ð,âð/ð, ð)T.
H 21 Find the eigen values and eigen vectors of the matrix [
0 1 00 0 11 â3 3
].
ðð§ð¬ð°ðð«: ð, ð, ð & (ð, ð, ð)ð.
W-19 (4)
â ALGEBRAIC AND GEOMETRIC MULTIPLICITY OF AN EIGENVALUE:
â If eigenvalue of matrix A is repeated n-times, then n is called the Algebraic Multiplicity
(A.M.) of λ.
Example: If λ = 2, 2, 3 then A.M. of λ = 2 is 2 and A.M. of λ = 3 is 1.
â The number of linearly independent eigenvectors corresponding to λ is called the
Geometric Multiplicity (G.M.) of λ. i.e., G.M.= No. of variablesâRank (A â λI).
â G.M.†A.M. for each eigen value.
â Procedure for finding A.M. and G.M. is as follow:
(1). Find eigen values of given matrix.
(2). Find A.M. using definition.
(3). Find eigen vectors for each eigen values.
(4). Find G.M. using definition.
DARSHAN INSTITUTE OF ENGINEERING & TECHNOLOGY Mathematics-I (3110014)
U N I T - 6 [ 1 4 8 ]
METHOD â 6: ALGEBRAIC AND GEOMETRIC MULTIPLICITY
C 1 Explain the algebraic multiplicity and geometric multiplicity with example.
H 2 Find the geometric and algebraic multiplicity of each eigen values of A =
[2 01 2
].
ðð§ð¬ð°ðð«: ð.ð. ðšð ð = ð & ð.ð. ðšð ð = ð.
C 3 Find the algebraic and geometric multiplicity of each of eigen value of
[0 1 11 0 11 1 0
].
ðð§ð¬ð°ðð«: ð.ð. (âð & ð) = ð & ð & ð.ð. (âð & ð) = ð, ð.
C 4 Find A.M. and G.M. for each eigen values of A = [
â3 â7 â52 4 31 2 2
].
ðð§ð¬ð°ðð«: ð.ð. (ð) = ð & ð.ð. (ð) = ð.
H 5 Find the eigen values and eigen vectors of [
2 2 11 3 11 2 2
]. Also find algebraic and
geometric multiplicity of each eigen values.
ðð§ð¬ð°ðð«: ð, ð, ð, [ððð] , [
ðâðð] , [
ððâð] , ð.ð.= ð, ð & ð.ð. = ð, ð.
T 6 C Find the eigen values and eigen vectors of [
8 â6 2â6 7 â42 â4 3
]. Also find
algebraic and geometric multiplicity of each eigen values.
ðð§ð¬ð°ðð«: ð, ð, ðð, [ð/ððð
] , [âðâð/ðð
] , [ðâðð] , ð.ð.= ð, ð, ð & ð.ð. = ð, ð, ð.
â DIAGONALIZATION:
â A matrix A is said to be diagonalizable if it is similar to diagonal matrix.
â It means a matrix A is diagonalizable if there exists an invertible matrix P such that
Pâ1AP = D, where D is a diagonal matrix known as spectral matrix and P is known as
modal matrix.
DARSHAN INSTITUTE OF ENGINEERING & TECHNOLOGY Mathematics-I (3110014)
U N I T - 6 [ 1 4 9 ]
â An n à n matrix is diagonalizable if and only if it has n linearly independent eigenvectors.
â If the eigenvalues of an n à n matrix are all distinct then the matrix is always
diagonalizable.
â The necessary and sufficient condition for a square matrix to be diagonalizable is that the
geometric multiplicity of each eigenvalues is same as its algebraic multiplicity.
â Procedure to check whether the given matrix is diagonalizable or not:
(1). Find the eigen values of a given matrix.
(2). If all the eigen values are distinct then the given matrix is diagonalizable,
otherwise follow step (3).
(3). Find A.M. and G.M. for each eigen value.
(4). If both the multiplicities are same for all eigen value then the given matrix is
diagonalizable.
â Procedure for finding a matrix P and Pâ1AP is as follow:
(1). Find the eigen values and eigen vectors of a given matrix.
(2). Construct matrix P by writing all the eigen vectors as its column vectors.
(3). Find Pâ1.
(4). Find Pâ1AP and verify that Pâ1AP is diagonal matrix or not. The diagonal matrix
D = Pâ1AP will have the eigenvalues on its main diagonal.
â Note that the order of the eigen vectors used to form P will determine the order in which
eigen values appear on the main diagonal of D.
â Pâ1AP = D â¹ A = PDPâ1.
â Let A be a diagonalizable matrix and P is an invertible matrix such that A = PDPâ1 then
An = PDnPâ1, ân â â.
â If D = Diag[a b c] then Dn = Diag[an bn cn].
â Every diagonal matrix is diagonalizable.
DARSHAN INSTITUTE OF ENGINEERING & TECHNOLOGY Mathematics-I (3110014)
U N I T - 6 [ 1 5 0 ]
METHOD â 7: DIAGONALIZATION
C 1 Check whether the following matrices are diagonalizable or not:
[10 â94 â2
], [0 34 0
], [1 0 01 2 0â3 5 2
] & [4 0 1â2 1 0â2 0 1
].
ðð§ð¬ð°ðð«: ððš, ð²ðð¬, ð§ðš & ð²ðð¬.
H 2 Check whether the following matrices are diagonalizable or not:
[0 00 0
], [1 00 1
], [1 2 20 2 1â1 2 2
] & [0 0 â21 2 11 0 3
].
ðð§ð¬ð°ðð«: ððð¬, ð²ðð¬, ð§ðš & ð²ðð¬.
T 3 Check whether the following matrices are diagonalizable or not:
[3 0 â51/5 â1 01 1 â2
], [â2 0 1â6 â2 019 5 â4
], [â1 0 1â1 3 0â4 13 â1
] & [5 0 11 1 0â7 1 0
].
ðð§ð¬ð°ðð«: ððð¬, ð²ðð¬, ð²ðð¬ & ð§ðš.
T 4 Check whether the following matrices are diagonalizable or not:
[
0100
0010
21â20
0001
] & [
â1/3000
0â1/300
0010
0001/2
].
ðð§ð¬ð°ðð«: ððð¬ & ð²ðð¬.
T 5 If A = [
0 1 11 0 11 1 0
] and B = [â1 1 00 â1 10 0 2
] then show that A is diagonalizable
but B is not diagonalizable.
C 6 Let A = [3 08 â1
]. Find a matrix P such that Pâ1AP is diagonal matrix.
ðð§ð¬ð°ðð«: ð = [ð ð/ðð ð
].
H 7 Find the matrix P that diagonalize A = [1 06 â1
]. Also determine Pâ1AP.
ðð§ð¬ð°ðð«: ð = [ð/ð ðð ð
] & ðâððð = [ð ðð âð
].
DARSHAN INSTITUTE OF ENGINEERING & TECHNOLOGY Mathematics-I (3110014)
U N I T - 6 [ 1 5 1 ]
T 8 For the matrix A = [â4 â63 5
], find the nonsingular matrix P and the
diagonal matrix D such that D = Pâ1AP.
ðð§ð¬ð°ð«: ð = [âð âðð ð
] & ð = [ð ðð âð
].
C 9 Let A = [
0 1 11 0 11 1 0
]. Find an orthonormal matrix P such that Pâ1AP is
diagonal matrix.
ðð§ð¬ð°ðð«: ð =
[ ð
âð â
ð
âð â
ð
âð ð
âð
ð
âð ð
ð
âð ð
ð
âð ]
.
H 10 c Let A = [
1 0 00 1 10 1 1
]. Find an orthogonal matrix P such that Pâ1AP is diagonal
matrix.
ðð§ð¬ð°ðð«: ð = [ð ð ðâð ð ðð ð ð
].
H 11 Find a matrix P that diagonalize A = [
2 0 â20 3 00 0 3
]. Also determine Pâ1AP.
ðð§ð¬ð°ðð«: ð = [ð ð âðð ð ðð ð ð
] & ðâððð = [ð ð ðð ð ðð ð ð
].
T 12 Find a matrix P that diagonalize A = [
â1 4 â2â3 4 0â3 1 3
]. Also determine Pâ1AP.
ðð§ð¬ð°ðð«: ð = [ð ð/ð ð/ðð ð ð/ðð ð ð
] & ðâððð = [ð ð ðð ð ðð ð ð
].
C 13 Prove that A = [
0 0 â21 2 11 0 3
] is diagonalizable and use it to find A13.
ðð§ð¬ð°ðð«: ð = [âð âð ðð ð ðð ð ð
] & ððð = [âðððð ð âððððððððð ðððð ðððððððð ð ððððð
].
S-19 (7)
DARSHAN INSTITUTE OF ENGINEERING & TECHNOLOGY Mathematics-I (3110014)
U N I T - 6 [ 1 5 2 ]
T 14 Let A = [
6 â2 2â2 3 â12 â1 3
]. Find matrix P such that Pâ1AP is diagonal matrix
and find A5.
ðð§ð¬ð°ðð«: ð = [âð/ð ð/ð ðð ð âðð ð ð
] & ðð = [ððððð âððððð ðððððâððððð ðððð âððððððððð âðððð ðððð
].
â THE CAYLEY-HAMILTON THEOREM:
â Every square matrix satisfies its own characteristic equation. i.e., if A is an n à n matrix,
and PA(λ) = 0 is its charateristic equation, then PA(A) = ðn.
â That means the characteristic equation is satisfied by A.
â By using the Cayley-Hamilton theorem we can find Aâ1 and integer power of A. i.e., An .
â Procedure to verify Cayley-Hamilton theorem for 2 Ã 2 matrix is as follow:
(1). Find the characteristic equation of given matrix A.
(2). Put λ = A in the characteristic equation and give it equation (1).
(3). Find A2 using given matrix.
(4). Put the values of A2, A, I in equation (1) and verify it.
(5). If you want to find Aâ1 using Cayley-Hamilton theorem then multiply equation (1)
both sides by Aâ1 and substitute the values of A and I.
â Procedure to verify Cayley-Hamilton theorem for 3 Ã 3 matrix is as follow:
(1). Find the characteristic equation of given matrix A.
(2). Put λ = A in the characteristic equation and give it equation (1).
(3). Find A3 & A2 using given matrix.
(4). Put the values of A3, A2, A, I in equation (1) and verify it.
(5). If you want to find Aâ1 using Cayley-Hamilton theorem then multiply equation (1)
both sides by Aâ1 and substitute the values of A2, A and I. By the similar way we
can find Aâ2, A2, A4 etcâŠ
DARSHAN INSTITUTE OF ENGINEERING & TECHNOLOGY Mathematics-I (3110014)
U N I T - 6 [ 1 5 3 ]
METHOD â 8: THE CAYLEY - HAMILTON THEOREM
C 1 State Cayley-Hamilton theorem. W-18 (1)
C 2 Using Cayley-Hamilton theorem find Aâ1 for A = [1 â12 3
].
ðð§ð¬ð°ðð«: ð
ð [ð ðâð ð
].
W-19 (4)
T 3 Using Cayley-Hamilton theorem, find Aâ1, Aâ2 for A = [1 21 1
].
ðð§ð¬ð°ðð«: ðâð = [âð ðð âð
] & ðâð = [ð âðâð ð
].
C 4 Verify Cayley-Hamilton theorem and find inverse for [
2 â1 1â1 2 â11 â1 2
].
ðð§ð¬ð°ðð«: ð
ð [ð ð âðð ð ðâð ð ð
].
H 5 State Cayley-Hamilton theorem and verify it for the matrix A = [
4 0 1â2 1 0â2 0 1
]. S-19 (7)
T 6 Using Cayley-Hamilton theorem, find inverse of [
1 3 74 2 31 2 1
].
ðð§ð¬ð°ðð«: ð
ðð[âð ðð âðâð âð ððð ð âðð
].
C 7 Verify Cayley-Hamilton theorem for A = [
1 1 23 1 12 3 1
] and hence find A4.
ðð§ð¬ð°ðð«: [ððð ððð ðððððð ððð ðððððð ððð ððð
].
H 8 Verify Cayley-Hamilton theorem for A = [
6 â1 1â2 5 â12 1 7
] and hence find A4.
ðð§ð¬ð°ðð«: [ðððð âððð ððððâðððð ððð âðððððððð ððð ðððð
].
DARSHAN INSTITUTE OF ENGINEERING & TECHNOLOGY Mathematics-I (3110014)
U N I T - 6 [ 1 5 4 ]
C 9 Determine Aâ1 by using Cayley-Hamilton theorem for matrix A =
[1 3 20 â1 4â2 1 5
]. Hence find the matrix represented by A8 â 5A7 â A6 +
37A5 + A4 â 5A3 â 3A2 + 41A+ 3I.
ðð§ð¬ð°ðð«: ðâð =ð
ðð [ð ðð âððð âð ðð ð ð
] & [ðð ð âðððð âðð âðððð ð âðð
].
T 10 Verify Cayley-Hamilton theorem for the matrix A = [
2 1 10 1 01 1 2
] and hence
find the matrix represented by A8 â 5A7 + 7A6 â 3A5 + A4 â 5A3 + 8A2 â
2A + I.
ðð§ð¬ð°ðð«: [ð ð ðð ð ðð ð ð
].
â â â â â â â â â â
DARSHAN INSTITUTE OF ENGINEERING & TECHNOLOGY Mathematics-I (3110014)
L i s t o f F o r m u l a e , R e s u l t s & S y m b o l s [ 1 5 5 ]
FORMULAE, RESULTS & SYMBOLS
STANDARD SYMBOLS:
α alpha Ο xu Ï sigma
β beta η eta Ï tau
γ gamma ζ zeta Ï, Ω omega
Ύ delta λ lamda Πcapital gamma
ε epsilon â del Î capital delta
Ξ, Πtheta Ό mu Σ capital sigma
Ï, Ί phi Ï pi â infinity
Ï psi Ï rho â capital del
BASIC FORMULAE:
(1). (a + b)2 = a2 + 2ab + b2 & (a â b)2 = a2 â 2ab + b2.
(2). (a + b)3 = a3 + 3a2b + 3ab2 + b3 = a3 + b3 + 3ab(a + b).
(3). (a â b)3 = a3 â 3a2b + 3ab2 â b3 = a3 â b3 â 3ab(a â b).
(4). a3 + b3 = (a + b)(a2 â ab + b2) & a3 â b3 = (a â b)(a2 + ab + b2).
(5). (a + b + c)2 = a2 + b2 + c2 + 2ab + 2bc + 2ca.
(6). a3 + b3 + c3 â 3abc = (a + b + c)(a2 + b2 + c2 â ab â bc â ac).
STANDARD SERIES:
(1). 1 + 2 + 3 + â¯+ n = n(n + 1)
2= â k
n
k = 1
.
(2). 12 + 22 + 32 +â¯+ n2 = n(n + 1)(2n + 1)
6= â k2
n
k = 1
.
(3). 13 + 23 + 33 +â¯+ n3 = [ n(n + 1)
2]
2
= â k3n
k = 1
.
DARSHAN INSTITUTE OF ENGINEERING & TECHNOLOGY Mathematics-I (3110014)
L i s t o f F o r m u l a e , R e s u l t s & S y m b o l s [ 1 5 6 ]
(4). 1
12+1
22+1
32+â¯â = â
1
k2
n
k = 1
= Ï2
6.
TRIGONOMETRIC IDENTITIES AND FORMULAE:
(1). cos2 Ξ + sin2 Ξ = 1
(2). 1 + tan2 Ξ = sec2 Ξ (3). 1 + cot2 Ξ = cosec2Ξ
(4). sin 2Ξ = 2 sin Ξ cosΞ =2 tan Ξ
1 + tan2 Ξ (5). tan 2Ξ =
2 tan Ξ
1 â tan2 Ξ
(6). cos 2Ξ = cos2 Ξ â sin2 Ξ (7). cos 2Ξ =1 â tan2 Ξ
1 + tan2 Ξ
(8). cos2 Ξ = 1 + cos 2Ξ
2 (9). sin2 Ξ =
1 â cos2Ξ
2
(10). tan2 Ξ = 1 â cos2Ξ
1 + cos2Ξ (11). cot2 Ξ =
1 + cos 2Ξ
1 â cos 2Ξ
(12). sin(α + β) = sin α cosβ + cos α sin β (13). cos(α + β) = cosα cos β â sin α sin β
(14). sin(α â β) = sin α cosβ â cos α sin β (15). cos(α â β) = cosα cos β + sin α sin β
(16). tan(α + β) =tan α + tan β
1 â tan α tan β (17). cot(α + β) =
cot α cot β â 1
cot α + cot β
(18). tan(α â β) =tan α â tan β
1 + tan α tan β (19). cot(α â β) =
cot α cot β + 1
cot α â cot β
(20). sin C + sin D = 2 sin ( C+D
2) cos (
CâD
2) (21). cosC + cosD = 2 cos (
C+D
2) cos (
CâD
2)
(22). sin C â sin D = 2 cos ( C+D
2) sin (
CâD
2) (23).
cosC â cosD =
â2 sin ( C+D
2) sin (
CâD
2)
(24). 2 sin α cosβ = sin(α + β) + sin(α â β) (25). 2 sin α sin β = cos(α â β)âcos(α + β)
(26). 2 cosα sin β = sin(α + β) â sin(α â β) (27). 2 cosα cos β
= cos(α + β) + cos(α â β)
DARSHAN INSTITUTE OF ENGINEERING & TECHNOLOGY Mathematics-I (3110014)
L i s t o f F o r m u l a e , R e s u l t s & S y m b o l s [ 1 5 7 ]
VALUES OF CIRCULAR FUNCTIONS:
(1). sin nÏ = 0, â n â †(2). cosnÏ = (â1)n, â n â â€
(3). sin(2n â 1) Ï
2= (â1)n+1, â n â â (4). cos (2n â 1)
Ï
2 = 0, â n â â
(5). If sin Ξ = 0 then Ξ = nÏ, â n â †(6). If cos Ξ = 1 then Ξ = 2nÏ, â n â â€
(7). If sin Ξ = 1 then Ξ = (4n + 1)Ï
2 , â n â â€+
(8). If cos Ξ = 0 then Ξ = (2n + 1)Ï
2 , â n â â€
INVERSE TRIGONOMETRIC FUNCTIONS:
(1). sinâ1x = cosecâ11
x = cosâ1 (â1 â x2) = tanâ1 (
x
â1 â x2 )
(2). cosâ1x = secâ11
x = sinâ1 (â1 â x2) = tanâ1 (
â1 â x2
x)
(3). tanâ1x = cotâ11
x = sinâ1 (
x
â1 â x2 ) = cosâ1 (
1
â1 â x2 )
(4). sinâ1x + cosâ1 x = Ï
2 (5). secâ1x + cosecâ1x =
Ï
2
(6). tanâ1x + cotâ1x = Ï
2 (7).
tanâ1x + tanâ1y = tanâ1 ( Ï
2) ; xy
= 1
(8). sinâ1(âx) = âsinâ1x (9). cosâ1(âx) = Ï â cosâ1x
(10). tanâ1(âx) = âtanâ1x (11). cotâ1(âx) = Ï â cotâ1x
(12). secâ1(âx) = Ï â secâ1x (13). cosecâ1(âx) = âcosecâ1x
(14). sinâ1x + sinâ1y = sinâ1 (xâ1 â y2 + yâ1 â x2)
DARSHAN INSTITUTE OF ENGINEERING & TECHNOLOGY Mathematics-I (3110014)
L i s t o f F o r m u l a e , R e s u l t s & S y m b o l s [ 1 5 8 ]
(15). cosâ1x + cosâ1y = cosâ1 (xy â â1 â y2â1 â x2)
DIFFERENTIATION:
(1). d
dx xn = nxnâ1 (2).
d
dx ax = ax log a, a â â+ â {1}
(3). d
dx eax = a eax (4).
d
dx log x =
1
x , x â 0
(5). d
dx sin x = cos x (6).
d
dx cos x = â sin x
(7). d
dx tan x = sec2x (8).
d
dx cot x = âcosec2x
(9). d
dx sec x = sec x tan x (10).
d
dx cosec x = â cosec x cot x
(11). d
dx sinâ1x =
1
â1 â x2 (12).
d
dx cosâ1x =
â1
â1 â x2
(13). d
dx tanâ1x =
1
1 + x2 (14).
d
dx cotâ1x =
â1
1 + x2
(15). d
dx secâ1x =
1
|x| âx2 â 1 , |x| > 1 (16).
d
dx cosecâ1x =
â1
|x| âx2 â 1 , |x|
> 1
(17). d
dx (u â v) = u
dv
dx + v
du
dx (18). d
dx ( u
v) =
v du dx
â u dv dx
v2
INTEGRATION:
(1). â«xn dx = xn+1
n + 1 , n â â1 (2). â«
1
x dx = log | x |
(3). â«(ax + b)n dx = (ax + b)n+1
a(n + 1) , n â â1
DARSHAN INSTITUTE OF ENGINEERING & TECHNOLOGY Mathematics-I (3110014)
L i s t o f F o r m u l a e , R e s u l t s & S y m b o l s [ 1 5 9 ]
(4). â«dx
(ax + b) =
1
a log | ax + b | (5). â«
dx
x(ax + b) =
1
b log (
x
ax + b )
(6). â«â(ax + b)n dx = 2
a
â(ax + b)n+2
n + 2, n â â2
(7). â« âax + b
x dx = 2âax + b + bâ«
dx
xâax + b
(8). â«dx
(a2 + x2) dx =
1
a tanâ1 (
x
a) (9). â«
dx
(a2 â x2) =
1
2a log |
x + a
x â a |
(10). â«dx
(a2 + x2)2 =
x
2a2(a2 + x2) +
1
2a3 tanâ1 (
x
a)
(11). â«dx
(a2 â x2)2 =
x
2a2(a2 â x2) +
1
4a3 log |
x + a
x â a |
(12). â«âa2 + x2 dx = x
2 âa2 + x2 +
a2
2 sinhâ1 (
x
a)
(13). â«âx2 â a2 dx = x
2âx2 â a2 â
a2
2 log | x + âx2 â a2 |
(14). â«âa2 â x2 dx = x
2âa2 â x2 +
a2
2sinâ1 (
x
a)
(15). â«dx
âa2 + x2 = sinhâ1 (
x
a) = log (x + âa2 + x2)
(16). â«x2âa2 + x2 dx = x(a2 + 2x2)âa2 + x2
8â a4
8 sinhâ1 (
x
a)
(17). â«x2
âa2 + x2 dx = â
a2
2 sinhâ1 (
x
a) +
xâa2 + x2
2
(18). â«dx
x âa2 + x2 = â
1
a log(
a + âa2 + x2
x)
(19). â«dx
âa2 â x2 = sinâ1
x
a
(20). â«âa2 â x2 dx = x
2âa2 â x2 +
a2
2sinâ1 (
x
a)
(21). â«x2
âa2 â x2 dx =
a2
2sinâ1 (
x
a) â
x
2âa2 â x2
DARSHAN INSTITUTE OF ENGINEERING & TECHNOLOGY Mathematics-I (3110014)
L i s t o f F o r m u l a e , R e s u l t s & S y m b o l s [ 1 6 0 ]
(22). â«dx
x âa2 â x2 = â
1
a log |
a + âa2 â x2
x |
(23). â«dx
âx2 â a2 = coshâ1 (
x
a) = log | x + âx2 â a2 |
(24). â«x2
âx2 â a2 dx =
a2
2coshâ1 (
x
a) +
x
2âx2 â a2
(25). â«dx
xâx2 â a2 = 1
asecâ1 (
x
a) =
1
a cosâ1 (
a
x )
(26). â«dx
â2ax â x2 = sinâ1 (
x â a
a)
(27). â«â2ax â x2 dx = x â a
2â2ax â x2 +
a2
2sinâ1 (
x â a
a)
(28). â«x
â2ax â x2 dx = asinâ1 (
x â a
a) â â2ax â x2
(29). â«sin ax dx = â1
a cos ax (30). â«cos ax dx =
1
a sin ax
(31). â«sinn ax dx = â (sinnâ1 ax) cos ax
na+ n â 1
nâ« sinnâ2 ax dx
(32). â«cosn ax dx = cosnâ1 ax sin ax
na+ n â 1
nâ«cosnâ2 ax dx
(33). â«sin ax cos bx dx = â cos(a + b) x
2(a + b)â cos(a â b) x
2(a â b), a2 â b2
(34). â«sin ax sin bx dx = sin(a â b) x
2(a â b)â sin(a + b) x
2(a + b), a2 â b2
(35). â«cos ax cosbx dx = sin(a â b) x
2(a â b)+ sin(a + b) x
2(a + b), a2 â b2
(36). â«sin ax cos ax dx = â cos 2ax
4a (37). â«cot ax dx =
1
a log(sin ax)
(38). â«sinn ax cos ax dx = sinn+1 ax
(n + 1)a, n â â1
(39). â«cosn ax sin ax dx = â cosn+1 ax
(n + 1)a, n â â1
DARSHAN INSTITUTE OF ENGINEERING & TECHNOLOGY Mathematics-I (3110014)
L i s t o f F o r m u l a e , R e s u l t s & S y m b o l s [ 1 6 1 ]
(40). â« tan ax dx = â1
a log(cos ax) =
1
a log(sec ax)
(41). â«sinn ax cosm ax dx = â
sinnâ1 ax cosm+1 ax
a(m + n)+ n â 1
m + n â« sinnâ2 ax cosm ax dx, n
â âm
(42). â«dx
b + c sin ax = â
2
aâb2 â c2 tanâ1 [â
b â c
b + c tan (
Ï
4â ax
2)] , b2 > c2
= â1
aâc2 â b2 log [
c + b sin ax + âc2 â b2 cos ax
b + c sin ax] , b2 < c2
(43). â«dx
1 + sin ax = â
1
a tan (
Ï
4â ax
2) (44). â«
dx
1 â sin ax = 1
a tan (
Ï
4+ ax
2)
(45). â«dx
1 + cos ax = 1
a tan (
ax
2) (46). â«
dx
1 â cos ax = â
1
a cot (
ax
2)
(47). â« tan2 ax dx =1
a tan ax â x (48). â«cot2 ax dx = â
1
a cot ax â x
(49). â« tann ax dx = tannâ1 ax
a(n â 1)â â« tannâ2 ax dx, n â 1
(50). â«cotn ax dx = â cotnâ1 ax
a(n â 1)ââ« cotnâ2 ax dx, n â 1
(51). â«sec ax dx =1
a log| sec ax + tan ax |
(52). â« cosec ax dx = â1
a log| cosec ax â cot ax |
(53). â«sec2 ax dx =1
a tan ax (54). â«cosec2 ax dx = â
1
a cot ax
(55). â«secn ax dx = secnâ2 ax tan ax
a(n â 1)+ n â 2
n â 1 â« secnâ2 ax dx, n â 1
(56). â« cosecn ax dx = â cosecnâ2 ax cot ax
a(n â 1)+ n â 2
n â 1 â« cosecnâ2 ax dx, n â 1
(57). â«secn ax tan ax dx = secn ax
na (58). â«cosecn ax cot ax dx = â
cosecn ax
na
(59). â«sinâ1 ax dx = x sinâ1 ax +1
a â1 â a2x2
DARSHAN INSTITUTE OF ENGINEERING & TECHNOLOGY Mathematics-I (3110014)
L i s t o f F o r m u l a e , R e s u l t s & S y m b o l s [ 1 6 2 ]
(60). â«cosâ1 ax dx = x cosâ1 ax â1
a â1 â a2x2
(61). â« tanâ1 ax dx = x tanâ1 ax â1
2a log| 1 + a2x2 |
(62). â«eax dx = eax
a (63). â«bax dx =
1
a bax
logb , b > 0, b â 1
(64). â«eax sin bx dx =eax
a2 + b2 (a sin bx â b cosbx)
(65). â«eax cos bx dx = eax
a2 + b2 (a cos bx + b sin bx)
(66). â«xn log ax dx =xn+1
n + 1 log ax â
xn+1
(n + 1)2 , n â 1
(67). â« logax dx = x log ax â x
(68). â«uv dx = uâ«v dx â â«{ du
dx â â« v dx } dx
(69).
When u is a polynomial function then it is better to use following rule:
â«uv dx = (u)(v1) â (uâ²)(v2) + (u
â²â²)(v3) â (uâ²â²â²)(v4) +â¯
where dashes of u denotes the derivatives and the subscripts of v denotes integrations.
(70). In = â« sinn x
Ï 2
0
dx = â« cosn x
Ï 2
0
dx = n â 1
n Inâ2, where I0 =
Ï
2, I1 = 1
(71). â« f(x) dx = 2â« f(x) dx
a 2
0
if f(a â x) = f(x)
a
0
= 0 if f(a â x) = âf(x)
(72). â« f(x) dx = 2â« f(x) dx
a
0
if f(x) is an even function
a
âa
= 0 if f(x) is an odd function
(73). â«[f(x)] n f â²(x) dx =
[f(x)] n+1
n + 1+ c, n â â1
(74). â« f â²(x)
f(x) dx = log|f(x)| + c (75). â« f â²(x) ef(x) dx = ef(x)
DARSHAN INSTITUTE OF ENGINEERING & TECHNOLOGY Mathematics-I (3110014)
L i s t o f F o r m u l a e , R e s u l t s & S y m b o l s [ 1 6 3 ]
RELATION WITH CARTESIAN COORDINATES:
(1).
Polar coordinates (r, Ξ) ⶠx = r cosΞ , y = r sin Ξ
ⶠr = âx2 + y2 , Ξ = tanâ1y
x
(2). Cylindrical coordinates (r, Ξ, z): x = r cosΞ , y = r sin Ξ , z = z
(3). Spherical coordinates (r, Ξ, â ): x = r sin Ξ cosâ , y = r sin Ξ sin â , z = r cosΞ
HYPERBOLIC FUNCTIONS:
(1). sinh x = ex â eâx
2 (2). cosh x =
ex + eâx
2
(3). sech x =1
cosh x =
2
ex + eâx (4). cosech x =
1
sin h x =
2
ex â eâx
(5). tanh x = sinh x
cosh x= ex â eâx
ex + eâx (6). coth x =
cosh x
sinh x= ex + eâx
ex â eâx
(7). sinh(âx) = âsinh x (8). cosh(âx) = cosh x
(9). cosh2 x â sinh2 x = 1
FREQUENTLY USED LIMIT:
(1). limx â 0
sin x
x= 1 (2). lim
x â 0
tan x
x= 1
(3). limx â 0
ax â 1
x= loge a (4).
limx â a
xn â an
x â a
= nanâ1; n is rational no.
(5). limn â â
1
n = 0 (6). lim
n â â
1
ân = 0
(7). limn â â
(n) 1 n = 1 (8). lim
n â â
logn
n= 0
(9). limn â â
(1 +x
n )n
= ex (10). limn â â
(x) 1 n = 1, x > 0
DARSHAN INSTITUTE OF ENGINEERING & TECHNOLOGY Mathematics-I (3110014)
L i s t o f F o r m u l a e , R e s u l t s & S y m b o l s [ 1 6 4 ]
(11). limn â â
xn
n!= 0, x > 0 (12). lim
n â âxn = 0, | x | < 1
LOGARITHM RULES:
(1). log1 = 0 (2). log e = 1
(3). log(xy) = log(x) + log(y) (4). log ( x
y) = log(x) â log(y)
(5). log(xy) = y log x (6). logy x = logz x
logz y
AREA:
Sr. No. Type Perimeter Area Remarks
(1). Square 4a a2 Side âaâ
(2). Rectangle 2a + 2b a b Sides âaâ & âbâ
(3). Triangle a + b + c (b h)/2 Altitude âhâ & Base âbâ
(4). Circle 2Ïr Ï r2 Radius ârâ
(5). Sector of Circle --- (r2 Ξ)/2 Radius ârâ & Angle âΞâ
VOLUME:
Sr. No. Type Volume Surface Area Remarks
(1). Cube a3 6a2 Side âaâ
(2). Cuboid a b c 2(ab + bc + ca) Sides âaâ, âbâ & âcâ
(3). Right circular Cylinder Ï r2 h 2 Ï r h Altitude âhâ & Radius ârâ
(4). Right circular Cone (Ï r2 h)/3 Ï r âr2 + h2 Altitude âhâ & Radius ârâ
(5). Sphere (4 Ï r3)/3 4 Ï r2 Radius ârâ
DARSHAN INSTITUTE OF ENGINEERING & TECHNOLOGY Mathematics-I (3110014)
L i s t o f F o r m u l a e , R e s u l t s & S y m b o l s [ 1 6 5 ]
VALUE OF SOME CONSTANTS:
(1). e = 2.71828 (2). Ï = 3.14159
(3). Log 1 = 0 (any base â 1) (4). ÏR = 180°
(5). 1° = (Ï
180 )R
= 0.017453R (6). 1R = ( 180
Ï)â
(7). 1° = 60â (8). 1â = 60â²â²
Here ° = degree, R = radian, â² = minute and â²â² = second
TRIGONOMETRIC TABLE:
ð° ð° ðð° ðð° ðð° ðð° ððð° ððð° ððð°
ðð ð ð
ð
ð
ð
ð
ð
ð
ð ð
ðð
ð ðð
ð¬ð¢ð§ð ð ð
ð
ð
âð âð
ð ð ð âð ð
ððšð¬ ð ð âð
ð
ð
âð
ð
ð ð âð ð ð
ððð§ð ð ð
âð ð âð â ð ââ ð
ððšð¬ðð ð â ð âð ð
âð ð â âð â
ð¬ððð ð ð
âð âð ð â âð â ð
ððšð ð â âð ð ð
âð ð â ð â
â â â â â â â â â â
DARSHAN INSTITUTE OF ENGINEERING & TECHNOLOGY Mathematics-I (3110014)
L i s t o f A s s i g n m e n t s [ 1 6 6 ]
LIST OF ASSIGNMENTS
ASSIGNMENT UNIT METHOD-NO. (EXAMPLE NO.)
1 1
M-1 (Ex â 4, 6, 8, 11, 13, 14)
M-2 (Ex â 3, 6)
M-4 (Ex â 3, 7, 8)
M-5 (Ex â 4, 8, 10, 12, 14)
M-8 (Ex â 4, 7, 11)
M-9 (Ex â 4, 6, 7)
2 2
M-4 (Ex â 3, 4, 7, 8, 10)
M-9 (Ex â 2, 5)
M-10 (Ex â 5, 7, 9, 12, 20, 21)
M-11 (Ex â 4, 6, 7, 12, 16)
M-13 (Ex â 3, 4, 7, 10)
M-17 (Ex â 3, 4, 9, 12, 15, 16)
M-19 (Ex â 3, 4, 7, 9)
M-20 (Ex â 5, 6, 7, 9, 11, 14, 18)
3 3
M-1 (Ex â 4, 5, 10, 12)
M-2 (Ex â 2, 5, 6, 9, 13, 16, 19)
M-3 (Ex â 3, 7, 8)
M-4 (Ex â 3, 4, 6)
DARSHAN INSTITUTE OF ENGINEERING & TECHNOLOGY Mathematics-I (3110014)
L i s t o f A s s i g n m e n t s [ 1 6 7 ]
4 4
M-2 (Ex â 3, 4, 9, 10)
M-3 (Ex â 2, 6, 8, 11, 14, 16, 17, 18, 20)
M-4 (Ex â 2, 5, 8, 9, 12, 14)
M-6 (Ex â 3, 5, 6, 7)
M-7 (Ex â 5, 6, 7)
M-8 (Ex â 3, 4, 7, 10, 13)
M-10 (Ex â 5, 8, 9, 11, 13)
5 5
M-1 (Ex â 2, 7, 8, 13, 14)
M-2 (Ex â 4, 5, 7, 10, 11, 13, 14, 16,19)
M-3 (Ex â 2, 6, 7, 10, 12, 15, 18, 22)
M-5 (Ex â 1, 5, 10, 12, 14, 17, 18)
6 6
M-1 (Ex â 5, 8, 10, 13, 16, 17)
M-2 (Ex â 3, 6, 7, 8, 13, 17, 19)
M-3 (Ex â 3, 5, 9)
M-4 (Ex â 2, 6, 7, 13)
M-5 (Ex â 3, 8, 11, 12, 15, 19, 21)
M-7 (Ex â 2, 4, 5, 8, 12, 14)
M-8 (Ex â 1, 3, 6, 10)
GUJARAT TECHNOLOGICAL UNIVERSITY Bachelor of Engineering
Subject Code: 3110014
Page 1 of 2
w.e.f. AY 2018-19
SUBJECT NAME: Mathematics-1
1st Year
Type of course: Basic Science Course
Prerequisite: Algebra, Trigonometry, Geometry
Rationale: The study of rate of changes, understanding to compute area, volume and express the
function in terms of series, to apply matrix algebra.
Teaching and Examination Scheme:
Teaching Scheme Credits Examination Marks Total
Marks L T P C Theory Marks Practical Marks
ESE (E) PA (M) ESE (V) PA (I)
3 2 0 5 70 30 0 0 100
Content:
Sr. No. Content Total
Hrs
% Weightage
01
Indeterminate Forms and LâHÃŽspitalâs Rule. 01
15 %
Improper Integrals, Convergence and divergence of the integrals, Beta
and Gamma functions and their properties. 03
Applications of definite integral, Volume using cross-sections, Length of
plane curves, Areas of Surfaces of Revolution 03
02
Convergence and divergence of sequences, The Sandwich Theorem for
Sequences, The Continuous Function Theorem for Sequences, Bounded
Monotonic Sequences, Convergence and divergence of an infinite series,
geometric series, telescoping series, ððð term test for divergent series,
Combining series, Harmonic Series, Integral test, The p - series, The
Comparison test, The Limit Comparison test, Ratio test, Raabeâs Test,
Root test, Alternating series test, Absolute and Conditional convergence,
Power series, Radius of convergence of a power series, Taylor and
Maclaurin series.
08 20 %
03
Fourier Series of 2ð periodic functions, Dirichletâs conditions for
representation by a Fourier series, Orthogonality of the trigonometric
system, Fourier Series of a function of period 2ð, Fourier Series of
even and odd functions, Half range expansions.
04 10 %
04
Functions of several variables, Limits and continuity, Test for non
existence of a limit, Partial differentiation, Mixed derivative theorem,
differentiability, Chain rule, Implicit differentiation, Gradient,
Directional derivative, tangent plane and normal line, total
differentiation, Local extreme values, Method of Lagrange Multipliers.
08 20 %
05
Multiple integral, Double integral over Rectangles and general regions,
double integrals as volumes, Change of order of integration, double
integration in polar coordinates, Area by double integration, Triple
integrals in rectangular, cylindrical and spherical coordinates, Jacobian,
multiple integral by substitution.
08 20 %
06 Elementary row operations in Matrix, Row echelon and Reduced row
echelon forms, Rank by echelon forms, Inverse by Gauss-Jordan method, 07 15%
GUJARAT TECHNOLOGICAL UNIVERSITY Bachelor of Engineering
Subject Code: 3110014
Page 2 of 2
w.e.f. AY 2018-19
Solution of system of linear equations by Gauss elimination and Gauss-
Jordan methods. Eigen values and eigen vectors, Cayley-Hamilton
theorem, Diagonalization of a matrix.
Suggested Specification table with Marks (Theory):
Distribution of Theory Marks
R Level U Level A Level N Level E Level C Level
10 25 35 0 0 0
Legends: R: Remembrance; U: Understanding; A: Application, N: Analyze and E: Evaluate C: Create and
above Levels (Revised Bloomâs Taxonomy).
Reference Books:
(1) Maurice D. Weir, Joel Hass, Thomas' Calculus, Early Transcendentals, 13e, Pearson, 2014.
(2) Howard Anton, Irl Bivens, Stephens Davis, Calculus, 10e, Wiley, 2016.
(3) James Stewart, Calculus: Early Transcendentals with Course Mate, 7e, Cengage, 2012.
(4) Anton and Rorres, Elementary Linear Algebra, Applications version,, Wiley India Edition.
(5) T. M. Apostol, Calculus, Volumes 1 & 2,, Wiley Eastern.
(6) Erwin Kreyszig, Advanced Engineering Mathematics, Wiley India Edition.
(7) Peter O'Neill, Advanced Engineering Mathematics, 7th Edition, Cengage.
Course Outcomes
The objective of this course is to familiarize the prospective engineers with techniques in calculus,
multivariate analysis and matrices. It aims to equip the students with standard concepts and tools at an
intermediate to advanced level that will serve them well towards tackling more advanced level of
mathematics and applications that they would find useful in their disciplines.
Sr.
No.
Course Outcomes Weightage in %
1
To apply differential and integral calculus to improper integrals and to
determine applications of definite integral. Apart from some other
applications they will have a basic understanding of indeterminate forms,
Beta and Gamma functions.
15
2
To apply the various tests of convergence to sequence, series and the tool of
power series and fourier series for learning advanced Engineering
Mathematics.
30
3 To compute directional derivative, maximum or minimum rate of change
and optimum value of functions of several variables. 20
4 To compute the areas and volumes using multiple integral techniques.
20
5 To perform matrix computation in a comprehensive manner. 15
List of Open Source Software/learning website:
Scilab, MIT Opencourseware.
1
Seat No.: ________ Enrolment No.___________
GUJARAT TECHNOLOGICAL UNIVERSITY BE âSEMESTER 1&2(NEW SYLLABUS)EXAMINATION- WINTER 2018
Subject Code: 3110014 Date: 07-01-2019
Subject Name: Mathematics - I
Time: 10:30 am to 01:30 pm Total Marks: 70 Instructions:
1. Attempt all questions.
2. Make suitable assumptions wherever necessary.
3. Figures to the right indicate full marks.
Marks
Q.1 (a) State Cayleyâ Hamilton theorem. Find eigen values of A and 1A , where
1 2
0 4A
03
(b) State Lâ Hospitalâs Rule. Use it to evaluate
2 20
1 1lim
sinx x x
04
(c) Investigate convergence of the following integrals:
(i)
35 2
5
1
xdx
x
(ii)
10 5
270
1
1
x xdx
x
07
Q.2
(a) Test the convergence of series 1
4 5
6
n n
nn
03
(b) State the p-series test. Discuss the convergence of the series
32
1
3 2n
n
n n
04
(c) State DâAlembertâs ratio test and Cauchyâs root test. Discuss the
convergence of the following series:
(i)
11
4 1 !n
nn
n
n
(ii) 2 log
nn
n
n
07
OR
(c) Test the convergence of the series 2 31
1 2 3 4 5 6 7 8 9 10 11 12
x x x
;
0x
07
Q.3
(a) Reduce matrix
1 5 3 2
2 0 4 1
4 8 9 1
A
to row echelon form and find its rank. 03
(b) Derive half range sine series of f x x , 0 x 04
(c) Find the eigen values and corresponding eigen vectors for the matrix A
where
1 2 2
0 2 1
1 2 2
A
07
2
OR
Q.3 (a) Expand exsin(x) in power of x up to the terms containing x6. 03
(b) Solve system of linear equation by Gauss Elimination method, if solution
exists.
2 9; 2 4 3 1; 3 6 5 0x y z x y z x y z
04
(c) Find Fourier series of
, 1 0
2, 0 1
x xf x
x
07
Q.4 (a) Discuss the continuity of the function f defined as
3 3
2 2; , 0,0
,
0 , 0,0
x yx y
x yf x y
x y
03
(b) Define gradient of a function. Use it to find directional derivative of
3 2, ,f x y z x xy z at 1,1,0P in the direction of ËË Ë2 3 6a i j k .
04
(c) Find the shortest and largest distance from the point 1,2, 1 to the sphere
2 2 2 24x y z .
07
OR
Q.4 (a) Find the extreme values of 3 2 2 23 3 3 4x xy x y 03
(b) Evaluate
22 42 2
0 0
x
x y dydx
by changing into polar coordinates. 04
(c) (i) If 2 2 2u x y y z z x then find out
u u u
x y z
(ii) If 3 3 6x y xy then find dy
dx and
2
2
d y
dx
07
Q.5
(a) Evaluate sinR
y xy dA , where R is the region bounded by 1x , 2x ,
0y and 2
y
.
03
(b) By changing the order of integration, evaluate
3 3
2 2
0 y
xdxdy
x y 04
(c) Find the volume below the surface 2 2z x y , above the plane 0z , and
inside the cylinder 2 2 2x y y .
07
OR
Q.5
(a) Evaluate integral 2 2
R
rdrd
a r
over the region R which is one loop of
2 2 cos 2r a
03
(b) Evaluate the integral log 2 2
1 1 1
xe y ex y dzdxdy . 04
(c) Find the volume of the solid obtained by rotating the region R enclosed by
the curves y x and 2y x about the line 2y .
07
1
Seat No.: ________ Enrolment No.___________
GUJARAT TECHNOLOGICAL UNIVERSITY BE - SEMESTERâI &II (NEW) EXAMINATION â SUMMER-2019
Subject Code: 3110014 Date: 06/06/2019 Subject Name: Mathematics â I Time: 10:30 AM TO 01:30 PM Total Marks: 70 Instructions:
1. Attempt all questions. 2. Make suitable assumptions wherever necessary. 3. Figures to the right indicate full marks.
Marks
Q.1 (a) Use LâHospitalâs rule to find the limit of limð¥â1
(ð¥
ð¥â1â
1
ððð¥). 03
(b) Define Gamma function and evaluate â« ð-ð¥2
ðð¥â
0. 04
(c) Evaluate â« â« ððŠ3ððŠðð¥
1âð¥
3
3
0.
07
Q.2 (a) Define the convergence of a sequence (ðð) and verify whether
the sequence whose ðð¡â term is ðð = (ð+1
ðâ1)
ðconverges or not.
03
(b) Sketch the region of integration and evaluate the integral
⬠(ðŠ â 2ð¥2)ððŽð
where ð is the region inside the square |ð¥| +
|ðŠ| = 1.
04
(c) (i) Find the sum of the series â1
4ððâ¥2 and â4
(4ðâ3)(4ð+1)ðâ¥1 .
(ii) Use Taylorâs series to estimate ð ðð38°.
07
OR
(c) Evaluate the integrals â« â«1
(1+ð¥2+ðŠ2)2ðð¥ððŠ
â
0
â
0 and
â« â« â«ðð2ð¥ð ððððŠ2
ðŠ2ðð¥ððŠðð§
ðð3
0
1
âð§3 .
1
0
07
Q.3 (a) If an electrostatic field ðž acts on a liquid or a gaseous polar
dielectric, the net dipode moment ð per unit volume is ð(ðž) =ððž+ðâðž
ððžâðâðžâ
1
ðž. Show that lim
ðžâ0+ð(ðž) = 0.
03
(b) For what values of the constant ð does the second derivative
test guarantee that ð(ð¥, ðŠ) = ð¥2 + ðð¥ðŠ + ðŠ2 will have a
saddle point at (0,0)? A local minimum at (0,0)?
04
(c) Find the series radius and interval of convergence for
â(3ð¥â2)ð
ðâð=0 . For what values of ð¥ does the series converge
absolutely?
07
OR
Q.3 (a) Determine whether the integral â«ðð¥
ð¥â1
3
0 converges or diverges. 03
(b) Find the volume of the solid generated by revolving the region
bounded by ðŠ = âð¥ and the lines ðŠ = 1, ð¥ = 4 about the line
ðŠ = 1.
04
2
(c) Check the convergence of the series
â(ððð)3
ð3âð=1 and â (â1)ð(âð + âðâ
ð=0 â âð) .
07
Q.4 (a) Show that the function ð(ð¥, ðŠ) =
2ð¥2ðŠ
ð¥4+ðŠ2 has no limit as (ð¥, ðŠ)
approaches to (0,0).
03
(b) Suppose ð is a differentiable function of ð¥ and ðŠ and ð(ð¢, ð£) =ð(ðð¢ + ð ððð£, ðð¢ + ððð ð£). Use the following table to calculate
ðð¢(0,0), ðð£(0,0), ðð¢(1,2) and ðð£(1,2).
ð ð ðð¥ ððŠ
(0,0) 3 6 4 8
(1,2) 6 3 2 5
04
(c) Find the Fourier series of 2ð âperiodic function ð(ð¥) =
ð¥2, 0 < ð¥ < 2ð and hence deduce that ð2
6= â
1
ð2âð=0 .
07
OR
Q.4 (a) Verify that the function ð¢ = ðâðŒ2ð2ð¡ â ð ðððð¥ is a solution f the
heat conduction euation ð¢ð¡ = ðŒ2ð¢ð¥ð¥.
03
(b) Find the half-range cosine series of the function
ð(ð¥) = {2, â2 < ð¥ < 00, 0 < ð¥ < 2
.
04
(c) Find the points on the sphere ð¥2 + ðŠ2 + ð§2 = 4 that are
closest to and farthest from the point (3,1, â1). 07
Q.5 (a) Find the directional derivative ð·ð¢ð(ð¥, ðŠ) if ð(ð¥, ðŠ) = ð¥3 â
3ð¥ðŠ + 4ðŠ2and ð¢ is the unit vector given by angle ð =ð
6. What
is ð·ð¢ð(1,2)?
03
(b) Find the area of the region bounded y the curves ðŠ = ð ððð¥, ðŠ =
ððð ð¥ and the lines ð¥ = 0 and ð¥ =ð
4.
04
(c) Prove that ðŽ = [0 0 â21 2 11 0 3
] is diagonalizable and use it to
find ðŽ13.
07
OR
Q.5 (a) Define the rank of a matrix and find the rank of the matrix ðŽ =
[2 â1 04 5 â31 â4 7
].
03
(b) Use Gauss-Jordan algorithm to solve the system of linear
equations 2ð¥1 + 2ð¥2 â ð¥3 + ð¥5 = 0 âð¥1 â ð¥2 + 2ð¥3 â 3ð¥4 + ð¥5 = 0
ð¥1 + ð¥2 â 2ð¥3 â ð¥5 = 0
ð¥3 + ð¥4 + ð¥5 = 0
04
(c) State Cayley-Hamilton theorem and verify if for the matrix
ðŽ = [4 0 1
â2 1 0â2 0 1
].
07
Seat No.: ________ Enrolment No.___________
GUJARAT TECHNOLOGICAL UNIVERSITY BE - SEMESTERâ I & II (NEW) EXAMINATION â WINTER 2019
Subject Code: 3110014 Date: 17/01/2020 Subject Name: Mathematics â I
Time: 10:30 AM TO 01:30 PM Total Marks: 70 Instructions:
1. Attempt all questions. 2. Make suitable assumptions wherever necessary. 3. Figures to the right indicate full marks.
MARKS
Q.1 (a) Find the equations of the tenagent plane and normal line to the surface
ð¥2 + ðŠ2 + ð§2 = 3 at the point (1,1, .1)
03
(b) Evaluate limð¥â0
ð¥ðð¥âððð(1+ð¥)
ð¥2 . 04
(c) Using Gauss Elimination method solve the following sy .stem
-x+3y+4z =30
3x+2y-z =9
2x-y+2z =10
07
Q.2 (a) Test the convergence of the series
1
3+ (
2
5)
2
+ (3
7)
3
+âââ + (ð
2ð+1)
ð
+ ââ .â
03
(b) Discuss the Maxima and Minima of the function 3ð¥2 â ðŠ2 + .ð¥3 04
(c) Find the fourier series of ð(ð¥) =(ðâð¥)
2 in the interval (0,2Ï .) 07
OR
(c) Change the order of integration and evaluate â« â« ð ðððŠ2 ððŠ .ðð¥1
ð¥
1
0 07
Q.3 (a) Find the value of ðœ (7
2,
5
2.) 03
(b) Obtain the fourier cosine series of the function f(x) = ðð¥ in the range (0,ð .) 04
(c) Find the maximum and minimum distance fom the point (1,2,2) to the
sphere ð¥2 + ðŠ2 + ð§2 = .36
07
OR
Q.3 (a) Test the convergence of the series .......4
1
3
1
2
1
1
12222
03
(b) Evaluate â¬(ð¥2 â ðŠ2) ðð¥ ððŠ over the triangle with the vertices (0,1),
(1,1), (1,2 .)
04
(c) Find the volume of the solid generated by rotating the plane region
bounded by ðŠ =1
ð¥, x=1 and x=3 about the X axis.
07
Q.4 (a) Evaluate â« â« ð ðð .ððsin ð
0
ð
0 03
(b) Express f(x) = 2ð¥3 + 3ð¥2 â 8ð¥ + 7 in terms of (x- .2) 04
(c) Using Gauss-Jordan method find 1A for
801
352
321
A 07
OR
Q.4 (a) Using Cayley-Hamilton Theorem find 1A for
32
11A 03
(b) Evaluate â«
ðð¥
ð¥2+1
â
0 04
(c) Test the convergence of the series
ð¥
1â2+
ð¥2
3â4+
ð¥3
5â6+
ð¥4
7â8+âââ
07
Q.5 (a) Evaluate â« â« ð¥ðŠ ððŠ .ðð¥2
1
1
0 03
(b) Find the eigen values and eigenvectors of the matrix [
0 1 00 0 11 â3 3
] 04
(c) If u = f (x-y, y-z, z-x) then show that ðð¢
ðð¥+
ðð¢
ððŠ+
ðð¢
ðð§= .0 07
OR
Q.5 (a) Find the directional derivatives of f = ð¥ðŠ2 + ðŠð§2 at the point (2,-1,1) in
the direction of i+2j+2k.
03
(b) Test the convergence of the series ââð
ð2+1âð=1 04
(c) Evaluate â ð¥ðŠð§ ðð¥ ððŠ ðð§ over the positive octant of the sphere ð¥2 +
ðŠ2 + ð§2 = .4. 07
***********