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“MATHEHATICS”[2m, 3m, 4m, 5m and 6m]

DIFFERENTIATION Differentiate the following functions w.r.to 풙 from first principle method

1) 푦 = 푥 2) 푦 = √푥 3) 푦 = √푥 4) 풚 = 풆풙 5) 풚 = 풂풙 6) 풚 = 퐥퐨퐠풆 풙

7) 푦 = 푠푖푛푥 8) 푦 = 푠푖푛푎푥 9) 푦 = cos푎푥 10) 푦 = tan푚푥 11) 푦 = 풄풐풕ퟓ풙 12) 풚 = 풄풐풔풆풄 ퟔ풙

13) 풚 = 풆ퟐ풙 14) 풚 = 풄풐풔ퟐ풙. 15) 풚 = 풔풊풏ퟐ풙 16) y= logax 17) y=secax 18) y=eax+b

Differentiation of function of a function:- Differentiate the following functions w.r.to 풙.

1) 퐼푓푦 = log(푠푒푐푥 + 푡푎푛푥) ,푃푇 = 푠푒푐푥

2) 푰풇풚 = 퐥퐨퐠 풙 + √풙ퟐ + 풂ퟐ ,푷푻 풅풚풅풙

= ퟏ

풙ퟐ 풂ퟐ

3) 퐼푓푦 = 푡ℎ푒푛 푃푇 = − 푠푒푐

4) 퐼푓푦 = 푥 + √푥 + 1 ,푃푇 (푥 + 1) = 푛 푦

5) 푰풇풚 = 풍풐품 ퟏ 풄풐풔풙ퟏ 풄풐풔풙

풕풉풆풏 푷풓풐풗풆 풕풉풂풕 풅풚풅풙

= ퟐ 풄풐풔풆풄풙

6) 퐼푓푦 = 푠푖푛(log 푥) 푡ℎ푒푛 푃푟표푣푒 푡ℎ푎푡 =

7) 풚 = 퐥퐨퐠풆 풆(ퟏ 풔풊풏풉풙) 8) 푦 = tanh (푒 )

9) 퐈퐟 퐲 = 퐭퐚퐧 ퟏ 풔풊풏풙ퟏ 풄풐풔풙

,푷.푻. 풅풚풅풙

= ퟏퟐ

10) 퐈퐟 퐲 = 퐭퐚퐧 ퟏ ퟏ 풄풐풔풙ퟏ 풄풐풔풙

,푷.푻. 풅풚풅풙

= ퟏퟐ

11) If y = cos ,푃.푇. =

12) If y = tan √ √√ √

,푃.푇. =√

13) 퐈퐟 ퟐ풙ퟐ + ퟑ풙퐲 + ퟒ퐲ퟐ = ퟐퟒ,풇풊풏풅 풅풚풅풙

풂풕(ퟏ ,ퟐ )

14) 퐼푓 푦 푙표푔푥 = 푥 − 푦,푃푟표푣푒 푡ℎ푎푡 =( )

15) 푰풇풆풙 풚 = 풆풙 + 풆풚,푷풓풐풗풆 풕풉풂풕 풅풚풅풙

= −풆풚 풙

16) 푰풇풔풆풄 풙 풚풙 풚

= 풂,푷풓풐풗풆 풕풉풂풕 풅풚풅풙

= 풚풙

17) 푰풇 풔풊풏풚 = 풙 퐬퐢퐧(풂 + 풚), 푷풓풐풗풆 풕풉풂풕 풅풚풅풙

= 풔풊풏ퟐ(풂 풚)풔풊풏풂

18) 퐼푓 푥 1 + 푦 + 푦√1 + 푥 = 0 푎푛푑 푦 ≠ 푥,푃푟표푣푒 푡ℎ푎푡 =( )

19) 퐼푓 푥 = 푎휃 푎푛푑 푦 = , 푡ℎ푒푛 푠ℎ표푤 푡ℎ푎푡 + = 0

20) 푰풇 풙 = ퟏ 풕ퟐ

ퟏ 풕ퟐ 풂풏풅 풚 = ퟐ풕

ퟏ 풕ퟐ, 풕풉풆풏 풔풉풐풘 풕풉풂풕 풅풚

풅풙= − 풙

풚

21) 푰풇 풙 = 풂[풄풐풔휽 + 휽풔풊풏휽]풂풏풅 풚 = 풂[풔풊풏휽 − 휽풄풐풔휽],풑풓풐풗풆 풕풉풂풕 풅풚풅풙

= 풕풂풏휽

22) 푰풇 풙 = 풂 풄풐풔휽 + 퐥퐨퐠 (풕풂풏 휽ퟐ 풂풏풅 풚 = 풂[풔풊풏휽],풑풓풐풗풆 풕풉풂풕 풅풚

풅풙= 풕풂풏휽

23) 푰풇 풙 = 풂(휽 + 풔풊풏휽)풂풏풅 풚 = 풂(ퟏ − 풄풐풔휽),풑풓풐풗풆 풕풉풂풕 풅풚풅풙 휽 흅

ퟐ

= ퟏ

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24) Differentiate 퐬퐢퐧 ퟏ ퟐ풙ퟏ 풙ퟐ

풘. 풓. 풕풐 퐜퐨퐬 ퟏ ퟏ 풙ퟐ

ퟏ 풙ퟐ

25) Differentiate tan √ 푤. 푟. 푡표 tan 푥.

26) Differentiate w. r. to 푥: 푖)푦 = 푥 푖푖)푦 = (sin 푥) 푖푖푖) 푦 = 푥

27) 푰풇풙풎풚풏 = (풙 + 풚)풎 풏, 풕풉풆풏 풑풓풐풗풆 풕풉풂풕 풅풚풅풙

= 풚풙

28) 푰풇풙풚 = 풆풙 풚, 풕풉풆풏 풑풓풐풗풆 풕풉풂풕 풅풚풅풙

= 풍풐품풙(ퟏ 풍풐품풙)ퟐ

29) 푰풇 풚 = 풙풙풙...

풑풓풐풗풆 풕풉풂풕 풅풚풅풙

= 풚ퟐ

풙(ퟏ 풚풍풐품풙)

30) 푰풇 풚 = 풍풐품풙 + 풍풐품풙+ 풍풐품풙+ ⋯… … … ∝ find dy/dx

31) 푰풇 풚 = 풙퐬퐢퐧 ퟏ 풙 + (퐬퐢퐧 ퟏ 풙)풙 풇풊풏풅 풅풚풅풙

32) 퐼푓 푦 = (sinh 푥) , 푡ℎ푒푛 푠ℎ표푤 푡ℎ푎푡 (1 + 푥 )푦 + 푥푦 − 2 = 0 33) 퐼풇 풚 = 퐚퐜퐨퐬(퐥퐨퐠풙) + 퐛퐬퐢퐧(풍풐품풙) , 풕풉풆풏 풔풉풐풘 풕풉풂풕 풙ퟐ풚ퟐ + 풙풚ퟏ + 풚 = ퟎ

34) 푰풇 풚 = 풙 + √ퟏ + 풙ퟐퟓ

, 풕풉풆풏 풔풉풐풘 풕풉풂풕 (ퟏ+ 풙ퟐ)풚ퟐ + 풙풚ퟏ − ퟐퟓ풚 = ퟎ

35) 퐼푓 푦 = 푥푐표푠ℎ푥, 푡ℎ푒푛 푝푟표푣푒 푡ℎ푎푡 푥푦 − 2푦 − 푥푦 + 2푐표푠ℎ푥 = 0 36) 퐼푓푥 = 푠푖푛휃 푎푛푑 푦 = sin(푚휃) ,푆.푇. (1− 푥 )푦 − 푥푦 + 푚 푦 = 0

37) 푰풇풚 = 풆퐦퐬퐢퐧 ퟏ 풙, 풔풉풐풘 풕풉풂풕 (ퟏ − 풙ퟐ)풚ퟐ − 풙풚ퟏ −풎ퟐ풚 = ퟎ

38) 퐼푓푥 = 푎(휃 + 푠푖푛휃) 푎푛푑 푦 = 푎(1 − 푐표푠휃), 푝푟표푣푒 푡ℎ푎푡 =

39) 푰풇풙 = 풂(풄풐풔휽 + 휽풔풊풏휽) 풂풏풅 풚 = 풂(풔풊풏휽 − 휽풄풐풔휽),풑풓풐풗풆 풕풉풂풕 풅ퟐ풚풅풙ퟐ 휽 흅

ퟒ

= ퟖ√ퟐ풂흅

Applications of the Derivative

1) If the curves 풙ퟐ

퐀+ 퐲ퟐ

퐁= ퟏand풙

ퟐ

퐚+ 퐲ퟐ

퐛= ퟏcut each other orthogonally, then prove that A - B = a - b.

2) If curve ax2 +by2 =1 and Ax2 +By2 =1 cut each other orthoganlly,then show that − = −

3) Prove that in the curve푦 = 푒 , the subnormal varies as the square of the ordinate and sub tangent is

constant.

4) Prove that for the curve 푎푦 = (푥 + 푏) , the square of the sub tangent at any point varies as the

subnormal

5) The volume of a sphere is increasing at 4흅 c.c./sec. Find the rate of increase of the radius

and of the surface area when the volume is 288흅 c.c. and find the rate of increase of volume

after 3 seconds

6) A ladder 13 feet long is sliding on a smooth floor against a smooth wall. If the lower end of ladder

slides away at the rate of ft/sec. Find the rate at which the upper end slides down when the lower

end is 5ft from the wall.

7) A man 6ft tall moves away from a source of light 20ft above the ground level and his rate of

walking being 4miles/hr. At what rate, is the length of the shadow changing? At what rate is

the tip of shadow moving?

8) A right circular cone has a depth of 12cms and a base if 9cms. Water is poured into it at the

rate of ퟏ ퟏퟐ c.c.per second. Find the rate of rise of water level and the rate of increase of water

surface, when the depth of the water level if 4cms.

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9) Show that of all rectangles with given perimeter, the square has the largest area.

10) Product of two numbers is 16. Find the numbers when their sum is minimum.

11) The sum of two numbers is 48. Find the numbers when their Product is maximum.

12) Show that the rectangle of maximum area that can be inscribed in a circle of radius ‘a’

is a square of side √ퟐ a.

13) Find the maximum rectangle that can be inscribed in a semi circle of unit radius .

14) Among all right angled traingles of given hypotenus , show that area is greatest when it is

isoscles.

15) Show that of all rectangles of given area , the square has minimum perimeter.

INTEGRATION

1) ∫ ퟏퟏ 풔풊풏풙

풅풙

2) ∫ 푑푥

3) ∫ 푑푥

4) ∫√ퟏ + 풔풊풏ퟐ풙 풅풙

5) ∫ 풔풊풏풙 풄풐풔풙√ퟏ 풔풊풏ퟐ풙

풅풙

6) ∫ 푠푖푛2푥 푐표푠3푥 푑푥 7) ∫ 푠푖푛 푥 푑푥 8) ∫ 푐표푠 푥 푑푥

9) ∫ 푑푥

10) )∫ √ퟏ + 풔풊풏풙풅풙

11) ∫ (√푥 +√

) 푑푥

12) ∫ 푡푎푛 푥 푠푒푐 푥푑푥

13) ∫ 푑푥

14) )∫ 푒 (5푥 + 2)푑푥 15) 풙

16) )∫ 풆풔풊풏ퟐ풙 풔풊풏ퟐ풙풅풙

17) ∫ 풕풂풏ퟒ풙풅풙 18) ∫ 풔풆풄ퟒ풙풅풙 19) ∫ 푐표푠푒푐 푥푑푥

20) ∫ 푑푥

21) ∫ 푑푥

22) ∫ 푑푥

23) ∫ 풙풆 ퟏ 풆풙 ퟏ

풙풆 풆풙풅풙

24) ∫ 풔풆풄ퟐ풙 풕풂풏풙풔풆풄ퟐ풙 풕풂풏ퟐ풙

풅풙

25) ∫ 푑푥

26) ∫ 푑푥

27) ∫√

푑푥

28) ∫ 푑푥

29) ∫ 푑푥

30) ∫ ퟏퟒ풄풐풔ퟐ풙 ퟗ풔풊풏ퟐ풙

풅풙

31) ∫ 푑푥

32) ∫ 푑푥

33) ∫ 푑푥

34) ∫ 푑푥

35) ∫ 푑푥

36) ∫√

푑푥

37) )∫ ퟑ풄풐풔풙 풔풊풏풙ퟒ풄풐풔풙 ퟑ풔풊풏풙

풅풙

38) ∫ 푑푥

39) ∫ 푑푥

40) ∫ ퟏퟏퟑ ퟓ풄풐풔풙

풅풙

41) ∫ 푑푥

42) ∫ [ ] 푑푥

43) ∫ 푒 [ ] 푑푥

44) ∫ 풆풙[풔풊풏풙 ퟏ풄풐풔풙 ퟏ

] 풅

Definite Integrals And Their Application.

PROPERTIES OF DEFINITE INTEGRALS: [6 mark question in PART-D]

ퟏ)푷풓풐풗풆 풕풉풂풕 ∫ 풇(풙)풅풙 = ∫ 풇(풂 − 풙)풅풙 풂풏풅 풔풉풐풘 풕풉풂풕∫ √풄풐풔풙√풔풊풏풙 √풄풐풔풙

흅ퟐퟎ 풅풙 = 흅

ퟒ풂ퟎ

풂ퟎ

2)푃푟표푣푒 푡ℎ푎푡∫ 푓(푥)푑푥 = ∫ 푓(푎 − 푥)푑푥 푎푛푑 푠ℎ표푤 푡ℎ푎푡 ∫ ( )( )푑푥 =

4)푷풓풐풗풆 풕풉풂풕퐥퐨퐠(ퟏ + 풙)ퟏ + 풙ퟐ

ퟏ

ퟎ풅풙 = 퐥퐨퐠(ퟏ + 풕풂풏풙)

흅ퟒ

ퟎ

풅풙 =흅ퟖ 풍풐품ퟐ

5)푷풓풐풗풆 풕풉풂풕풙풔풊풏풙

ퟏ + 풄풐풔ퟐ풙

흅

ퟎ 풅풙 =

풙풕풂풏풙풔풆풄풙+ 풄풐풔풙

흅

ퟎ 풅풙 =

흅ퟐ

ퟒ

6)푃푟표푣푒 푡ℎ푎푡푥

1 + 푐표푠 푥 푑푥 =휋

2√2

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7)푷풓풐풗풆 풕풉풂풕 퐥퐨퐠(풔풊풏풙)

흅ퟐ

ퟎ

풅풙 =흅ퟐ 풍풐품

ퟏퟐ

8)푷풓풐풗풆 풕풉풂풕풙풕풂풏풙

풔풆풄풙+ 풕풂풏풙

흅

ퟎ 풅풙 =

흅ퟐ − ퟐ흅ퟐ

9)푷풓풐풗풆 풕풉풂풕풙풅풙

풂ퟐ풄풐풔ퟐ풙 + 풃ퟐ풔풊풏ퟐ풙

흅

ퟎ=

흅ퟐ

ퟐ풂풃

11)푃푟표푣푒 푡ℎ푎푡1

푠푖푛푥 + 푐표푠푥 푑푥 =1√2

푙표푔√2 + 1√2− 1

ퟏퟐ)푷풓풐풗풆 풕풉풂풕√풙+ ퟐ

√풙+ ퟐ + √ퟓ− 풙풅풙

ퟑ

ퟎ=ퟑퟐ

APPLICATION OF DEFINITE INTEGRAL AS AN AREA: (ퟑퟒ풕풉question for 5 marks)

1) Find the area enclosed between the curve 푦 = 11푥−24 − 푥 and the line 푦 = 푥.

2) Find the area enclosed between the parabolas 푦 = 4푎푥 푎푛푑 푥 = 4푎푦.

3) Find the area enclosed between the parabolas 푖)4푦 = 9푥 푎푛푑 3푥 = 16푦. 푖푖) 푦 = 6푥 푎푛푑 푥 = 6푦.

ii)y2 =x and x2 =y

5) Find the area of a circle 푥 + 푦 = 6 by integration. ( 푥 + 푦 = 3 )

6) Prove that the area of the ellipse

+ = 1 is 휋푎푏 푠푞푢푎푟푒 푢푛푖푡푠.(By the method of integration)

7) Find the area of the ellipse 4푥 + 9푦 = 36 by integration.

X. Differential Equations

1) Find the order and degree of the differential equations,

1)풙 풅풚풅풙

+ ퟑ풅풚풅풙

= 풚ퟐ

2)1 + = 7

3) 1 + =

4) ퟏ + 풅풚풅풙

ퟐퟑퟒ

= 풌 풅ퟐ풚풅풙ퟐ

2) Solve:푥 + 푦 = 0

3) Solve: =

4) Solve:푑푥 + 푥 푡푎푛푦 푑푦 = 0 5) Solve the differential equation (풚ퟐ + 풚)풅풙 + (풙ퟐ + 풙 ) 풅풚 = ퟎ

Solve: (푒 + 1)푐표푠푥 푑푥 + 푒 푠푖푛푥 푑푦 = 0 6) Solve: = 3푥 + 4푦 + 6푥푦 + 2

7) Solve:풚 − 풙 풅풚풅풙

= 풂 풚ퟐ + 풅풚풅풙

8) Solve: 3푒 푡푎푛푦 푑푥 + (1 − 푒 ) 푠푒푐 푦 푑푦 = 0. 9) Find the differential equation of family of straight lines with slope and y intercept are eqaul. 10) Find the D.E of family of circles touching y axis at origin. 11) Find the D.E. of family of parabolas with x axis as axis and (a,0) as the focus. 12) Find the D.E. of family of parabolas with axis as x axis and origin at its vertex.

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CONTRIBUTED BY SRI. BHADRASHETTY, LECTURER, SRI. SUCHEENDRA, LECTURER, AND SRI VASUDEVA. K.H, LECTURER

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