Tables and Formulas

8/3/2019 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, )



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,



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)



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



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,



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 ,



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



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

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