tables and formulas
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Trigonometry Formulas
1. Definitions and Fundamental Identities
Sine:
Cosine:
Tangent:
2. Identities
cos s A - Bd = cos A cos B + sin A sin B
cos s A + Bd = cos A cos B - sin A sin B
sin s A - Bd = sin A cos B - cos A sin B
sin s A + Bd = sin A cos B + cos A sin B
cos2 u =
1 + cos 2u2
, sin2 u =
1 - cos 2u2
sin 2u = 2 sin u cos u, cos 2u = cos2 u - sin2 u
sin2 u + cos2
u = 1, sec2 u = 1 + tan2
u, csc2 u = 1 + cot2 u
sin s -ud = -sin u, cos s -ud = cos u
tan u = y x = 1cot u
cos u =xr =
1sec u
sin u =yr =
1csc u r
0 x
y
P( x, y)
y
x
cos A - cos B = -2 sin 12
s A + Bd sin 12
s A - Bd
cos A + cos B = 2 cos 12 s A + Bd cos
12 s A - Bd
sin A - sin B = 2 cos 12
s A + Bd sin 12
s A - Bd
sin A + sin B = 2 sin 12
s A + Bd cos 12
s A - Bd
sin A cos B =12
sin s A - Bd +12
sin s A + Bd
cos A cos B =12
cos s A - Bd +12
cos s A + Bd
sin A sin B =12
cos s A - Bd -12
cos s A + Bd
sin a A +p
2b = cos A, cos a A +
p
2b = -sin A
sin a A -p
2b = -cos A, cos a A -
p
2b = sin A
tan s A - Bd =tan A - tan B
1 + tan A tan B
tan s A + Bd =tan A + tan B
1 - tan A tan B
Trigonometric Functions
Radian Measure
s
r
1θ
C i r c l e o f r a d i u s
r
U n i t c i r c
l e
180° = p radians.
sr =
u
1= u or u =
sr,
2
45
45 90
1
1
1
1 1
1
2
4
3
2
6
4
2 2
30
9060
Degrees Radians
2
33
The angles of two common triangles, in
degrees and radians.
x
y
y cos x
Domain: (–, )
Range: [–1, 1]
0– 2–
2
23
2
y sin x
x
y
0– 2–
2
23
2
y sin x
Domain: (–, )
Range: [–1, 1]
y
x
y tan x
3
2– – –
20
2 3
2
Domain: All real numbers except odd
integer multiples of /2
Domain: All real numbers except odd
integer multiples of /2
Range: (–, )
x
y
y csc x
0
1
– 2–
2
23
2
Domain: x 0,,2, . . .Range: (–, –1]h [1, )
y
x
y cot x
0
1
– 2–
2
23
2
Domain: x 0,,2, . . .Range: (–, )
x
y
y sec x
3
2– – –
20
1
2 3
2
Range: (–, –1]h [1, )
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Taylor Series
Binomial Series
where am
k b =
msm - 1d Á sm - k + 1d
k ! for k Ú 3.am
1b = m, am
2b =
msm - 1d
2! ,
= 1 +
a
q
k = 1 am
k b x k , ƒ x ƒ 6 1,
+msm - 1dsm - 2d Á sm - k + 1d x k
k !+ Ás1 + xdm = 1 + mx +
msm - 1d x2
2!+
msm - 1dsm - 2d x3
3!+ Á
ƒ x ƒ … 1tan-1 x = x -
x3
3+
x5
5- Á + s -1dn
x2n + 1
2n + 1+ Á = a
q
n = 0
s -1dn x2n + 1
2n + 1 ,
= 2aq
n = 0
x2n + 1
2n + 1 , ƒ x ƒ 6 1ln
1 + x1 - x
= 2 tanh-1 x = 2 a x +x3
3+
x5
5+ Á +
x2n + 1
2n + 1+ Á b
-1 6 x … 1lns1 + xd = x -x2
2
+x3
3
- Á + s -1dn - 1 x n
n
+ Á = aq
n = 1
s -1dn - 1 x n
n
,
cos x = 1 -x2
2!+
x4
4!- Á + s -1dn
x2n
s2nd!+ Á = a
q
n = 0
s -1dn x2n
s2nd! , ƒ x ƒ 6 q
ƒ x ƒ 6 qsin x = x -x3
3!+
x5
5!- Á + s -1dn
x2n + 1
s2n + 1d!+ Á = a
q
n = 0
s -1dn x2n + 1
s2n + 1d! ,
e x = 1 + x +x2
2!+ Á +
xn
n!+ Á = a
q
n = 0
x n
n! , ƒ x ƒ 6 q
11 + x
= 1 - x + x2 - Á + s - xdn + Á = aq
n = 0
s -1dn x n, ƒ x ƒ 6 1
11 - x
= 1 + x + x2 + Á + xn + Á = aq
n = 0 x
n, ƒ x ƒ 6 1
SERIES
Tests for Convergence of Infinite Series
1. The nth-Term Test: Unless the series diverges.2. Geometric series: converges if otherwise it
diverges.
3. p-series: converges if otherwise it diverges.
4. Series with nonnegative terms: Try the Integral Test, Ratio
Test, or Root Test. Try comparing to a known series with the
Comparison Test or the Limit Comparison Test.
5. Series with some negative terms: Does convergeyes, so does since absolute convergence implies c
vergence.
6. Alternating series: converges if the series satisfies
conditions of the Alternating Series Test.
gan
gan
gƒ an ƒ
p 7 1;g1>n p
ƒ r ƒ 6 1;gar n an:
0,
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Vector Triple Products
u * sv * wd = su # wdv - su # vdw
su * vd # w = sv * wd # u = sw * ud # v
Formulas for Grad, Div, Curl, and the Laplacian
The Fundamental Theorem of Line Integrals
1. Let be a vector field whose components are
continuous throughout an open connected region D in space. Then
there exists a differentiable function ƒ such that
if and only if for all points A and B in D the value of is inde-
pendent of the path joining A to B in D.
2. If the integral is independent of the path from A to B, its value is
L B
A
F # d r = ƒs Bd - ƒs Ad .
1 B
A F # d r
F = §ƒ = 0ƒ0 x
i + 0ƒ0 y
j + 0ƒ0 z
k
F = M i + N j + Pk
Green’s Theorem and Its Generalization to Three Dimensions
Normal form of Green’s Theorem:
Divergence Theorem:
Tangential form of Green’s Theorem:
Stokes’Theorem: FC
F # d r = 6S
§ * F # n d s
FC
F # d r = 6 R
§ * F # k dA
6S
F # n d s = 9 D
§ # F dV
FC F # n ds =
6 R § # F dA
Gradient
Divergence
Curl
Laplacian §2ƒ =02ƒ
0 x2+
02ƒ
0 y 2+
02ƒ
0 z2
§ * F = 4i j k
00 x
00 y
00 z
M N P 4
§ # F =0 M
0 x+
0 N
0 y+
0 P0 z
§ƒ =0ƒ
0 x i +
0ƒ
0 y j +
0ƒ
0 z k
Cartesian ( x, y, z)
i, j, and k are unit vectors
in the directions of
increasing x, y, and z.
and are thescalar components of
F( x, y, z) in these
directions.
P M , N ,
F1 * s§ * F2d + F2 * s§ * F1d
§sF1# F2d = sF1
# §dF2 + sF2# §dF1 +
§ * saF1 + bF2d = a§ * F1 + b§ * F2
§ # saF1 + bF2d = a§ # F1 + b§ # F2
§ * s g Fd = g § * F + § g * F
§ # s g Fd = g § # F + § g # F
§sƒ g d = ƒ§ g + g §ƒ
§ * s§ƒd = 0
s§ * Fd * F = sF # §dF -12
§sF # Fd
§ * s§ * Fd = §s§ # Fd - s§ # §dF = §s§ # Fd - §2F
s§ # F2dF1 - s§ # F1dF2
§ * sF1 * F2d = sF2# §dF1 - sF1
# §dF2 +
§ # sF1 * F2d = F2# § * F1 - F1
# § * F2
Vector Identities
In the identities here, ƒ and g are differentiable scalar functions, F, and are differentiable vector fields, and a and b are real
constants.
F2F1 ,
VECTOR OPERATOR FORMULAS (CARTESIAN FORM)
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BASIC ALGEBRA FORMULAS
Arithmetic Operations
Laws of Signs
Zero Division by zero is not defined.
If
For any number a:
Laws of Exponents
If
The Binomial Theorem For any positive integer n,
For instance,
Factoring the Difference of Like Integer Powers,
For instance,
Completing the Square If
The Quadratic Formula If and then
x =-b ; 2 b2 - 4ac
2a.
ax2 + bx + c = 0,a Z 0
ax2 + bx + c = au 2 + C au = x + sb>2ad, C = c -b2
4ab
a Z 0,
a4 - b4 = sa - bdsa3 + a2b + ab2 + b3d .
a3 - b3 = sa - bdsa2 + ab + b2d,
a2 - b2 = sa - bdsa + bd,
an - bn = sa - bdsan- 1 + an- 2b + an - 3b2 + Á + abn- 2 + bn- 1d
n>1
sa + bd3 = a3 + 3a2b + 3ab2 + b3, sa - bd3 = a3 - 3a2b + 3ab2 - b3.
sa + bd2 = a2 + 2ab + b2, sa - bd2 = a2 - 2ab + b2
+nsn - 1dsn - 2d
1 # 2 # 3 an- 3b3 + Á + nabn- 1 + bn .
sa + bdn = an + nan- 1b +nsn - 1d
1 # 2 an- 2b2
am
an = am - n, a0 = 1, a-m =1
am .
a Z 0,
am>n = 2 n
am = A2 n
a B maman = am + n, sabdm = ambm, samdn = amn,
a # 0 = 0 # a = 0
0a = 0, a0 = 1, 0a = 0a Z 0:
-s -ad = a, -ab
= -ab
=a
-b
ab
+cd
=ad + bc
bd , a>b
c>d =
ab
# d c
asb + cd = ab + ac, a
b# c
d =
ac
bd
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GEOMETRY FORMULAS
Triangle Similar Triangles Pythagorean Theorem
Parallelogram Trapezoid Circle
Any Cylinder or Prism with Parallel Bases Right Circular Cylinder
Any Cone or Pyramid Right Circular Cone Sphere
V r 3, S 4r 243
h
s
r
V r 2h1
3
S rs Area of side
h
h
V Bh1
3 B
B
V r 2h
S 2rh Area of side
h
r
hh
V Bh B
B
A r 2,
C 2r r
a
b
h
A (a b)h1
2
h
b
A bh
a
bc
a2 b
2 c
2
b
cc' a'
b'
a
a'
a
b'
b
c'
c
b
h
A bh1
2
V = volume
S = lateral area or surface area,circumference, B = area of base, C = A = area,
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LIMITS
General Laws
If L, M , c, and k are real numbers and
Sum Rule:
Difference Rule:
Product Rule:
Constant Multiple Rule:
Quotient Rule:
The Sandwich Theorem
If in an open interval containing c, except possibly at and if
then
Inequalities
If in an open interval containing c, except possibly
at and both limits exist, then
Continuity
If g is continuous at L and then
lim x:c
g (ƒs xdd = g s Ld .
lim x:c ƒs xd = L ,
lim x:c
ƒs xd … lim x:c
g s xd .
x = c ,
ƒs xd … g s xd
lim x:c ƒs xd = L .
lim x:c
g s xd = lim x:c
hs xd = L ,
x = c , g s xd … ƒs xd … hs xd
lim x:c
ƒs xd
g s xd=
L M
, M Z 0
lim x:c
sk # ƒs xdd = k # L
lim x:c
sƒs xd # g s xdd = L # M
lim x:c
sƒs xd - g s xdd = L - M
lim x:
c
sƒs xd + g s xdd = L + M
lim x:c
ƒs xd = L and lim x:c
g s xd = M , then
Specific Formulas
If then
If P( x) and Q( x) are polynomials and then
If ƒ( x) is continuous at then
L’Hôpital’s Rule
If both and exist in an open interv
containing a, and on I if then
assuming the limit on the right side exists.
lim x:a
ƒs xd
g s xd= lim
x:a ƒ¿s xd
g ¿s xd,
x Z a , g ¿s xd Z 0
g ¿ƒ¿ƒsad = g sad = 0,
lim x:0
sin x x = 1 and lim
x:0 1 - cos x
x = 0
lim x:c
ƒs xd = ƒscd .
x = c ,
lim x:c
Ps xd
Qs xd=
Pscd
Qscd.
Qscd Z 0,
lim x:c
Ps xd = Pscd = an cn + an - 1 cn - 1 + Á + a0 .
Ps xd = an xn + an - 1 xn- 1 + Á + a0 ,
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DIFFERENTIATION RULES
General Formulas
Assume u and y are differentiable functions of x.
Trigonometric Functions
Exponential and Logarithmic Functions
d dx
a x = a x ln a d
dx sloga xd =
1 x ln a
d dx
e x = e x d dx
ln x =1 x
d dx
scot xd = -csc2 x d
dx scsc xd = -csc x cot x
d dx
stan xd = sec2 x d
dx ssec xd = sec x tan x
d dx
ssin xd = cos x d dx
scos xd = -sin x
d dx
sƒs g s xdd = ƒ¿s g s xdd # g ¿s xdChain Rule: d dx xn = nxn
-1 Power:
d dx
auyb =
y dudx
- u d ydx
y2
Quotient: d dx
suyd = u d ydx
+ y dudx
Product: d dx
scud = c dudx
Constant Multiple: d dx
su - yd =dudx
-d ydx
Difference: d dx
su + yd =dudx
+d ydx
Sum: d
dx scd = 0Constant: Inverse Trigonometric Functions
Hyperbolic Functions
Inverse Hyperbolic Functions
Parametric Equations
If and are differentiable, then
. y¿ =dy
dx=
dy>dt
dx>dt and d 2 y
dx2=
dy¿>dt
dx>dt
y = g st d x = ƒst d
d dx
scoth-1 xd =
1
1 - x2 d dx
scsch-1 xd = -
1
ƒ x ƒ 2 1 + x2
d dx
stanh-1 xd =
1
1 - x2 d dx
ssech-1 xd = -
1
x2 1 - x2
d dx
ssinh-1 xd =
1
2 1 + x2
d dx
scosh-1 xd =
1
2 x2 - 1
d dx
scoth xd = -csch2 x d dx
scsch xd = -csch x coth x
d dx
stanh xd = sech2 x d dx
ssech xd = -sech x tanh x
d dx
ssinh xd = cosh x d dx
scosh xd = sinh x
d dx
scot-1 xd = -
1
1 + x2 d dx
scsc-1 xd = -
1
ƒ x ƒ2 x2 - 1
d dx
stan-1 xd =
1
1 + x2 d dx
ssec-1 xd =
1
ƒ x ƒ2 x2 - 1
d dx
ssin-1 xd =
1
2 1 - x2 d
dx scos-1
xd = -1
2 1 - x2
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The Fundamental Theorem of Calculus
Part 1 If ƒ is continuous on [a, b], then is continuous on
[a, b] and differentiable on (a, b) and its derivative is ƒ( x);
Part 2 If ƒ is continuous at every point of [a, b] and F is any antiderivative of ƒ
on [a, b], then
Lb
a
ƒs xd dx = F sbd - F sad.
F ¿( x) =d dxL
x
a
ƒst d dt = ƒs xd.
F s xd =
1
x
aƒst d dt
General Formulas
Zero:
Order of Integration:
Constant Multiples:
Sums and Differences:
Additivity:
Max-Min Inequality: If max ƒ and min ƒ are the maximum and minimum values of ƒ on [a, b], then
Domination:
ƒs xd Ú 0 on [a, b] implies Lb
a
ƒs xd dx Ú 0
ƒs xd Ú g s xd on [a, b] implies Lb
a
ƒs xd dx Ú Lb
a
g s xd dx
min ƒ # sb - ad … Lb
a
ƒs xd dx … max ƒ # sb - ad .
Lb
a
ƒs xd dx + Lc
b
ƒs xd dx = Lc
a
ƒs xd dx
Lb
a
sƒs xd ; g s xdd dx = Lb
a
ƒs xd dx ; Lb
a
g s xd dx
Lb
a
-ƒs xd dx = -Lb
a
ƒs xd dx sk = -1d
Lb
a
k ƒs xd dx = k Lb
a
ƒs xd dx sAny number k d
La
b
ƒs xd dx = -Lb
a
ƒs xd dx
La
a
ƒs xd dx = 0
Substitution in Definite Integrals
Lb
a
ƒs g s xdd # g ¿s xd dx = L g sbd
g sadƒsud du
Integration by Parts
Lb
a
ƒs xd g ¿s xd dx = ƒs xd g s xd Da
b- L
b
a
ƒ¿s xd g s xd dx
INTEGRATION RULES