BA

x

y

relative maximumf '(x)=0

zerof (x)=0

relative minimumf '(x)=0

absolute maximum

inflection pointf ''(x)=0

a b c d e

k

bf '(x)

d e+ - +

cf ''(x)

e+-

Area = ∫ f (x) dxa

b

Domain:( - ∞, e ]-∞ < x ≤ e

Range:( - ∞, k ]-∞ < y ≤ k

Sample Function f(x)BA

x

y

relative maximumf '(x)=0

zerof (x)=0

relative minimumf '(x)=0

absolute maximum

inflection pointf ''(x)=0

a b c d e

k

bf '(x)

d e+ - +

cf ''(x)

e+-

Area = ∫ f (x) dxa

b

Domain:( - ∞, e ]-∞ < x ≤ e

Range:( - ∞, k ]-∞ < y ≤ k

Sample Function f(x)BA

x

y

relative maximumf '(x)=0

zerof (x)=0

relative minimumf '(x)=0

absolute maximum

inflection pointf ''(x)=0

a b c d e

k

bf '(x)

d e+ - +

cf ''(x)

e+-

Area = ∫ f (x) dxa

b

Domain:( - ∞, e ]-∞ < x ≤ e

Range:( - ∞, k ]-∞ < y ≤ k

Sample Function f(x)

sin ² x + cos ² x = 11 + tan ² x = sec ² x1 + cot ² x = csc ² x

cos (a+b) = cos a cos b - sin a sin bsin (a+b) = sin a cos b + cos a sin b

cos (a - b) = cos a cos b + sin a sin bsin (a - b) = sin a cos b - cos a sin b

transcendental integrals

Double Angle Identitiessin 2x = 2 sin x cos x

cos 2x = cos ² x - sin ² xcos ² x = 1+ cos 2x

2sin ² x = 1- cos 2x

2

Odd/Even Identitiessin (- x) = - sin xcos (- x) = cos xtan (- x) = - tan xcot (- x) = - cot xsec (- x) = sec x

csc (- x) = - csc x

C

B

A

ac

bx

sin x =a/c= opposite/hypotenusecos x = b/c = adjacent/hypotenusesec x = c/b = hypotenuse/adjacentcsc x = c/a = hypotenuse/oppositetan x = a/b = sin x/cos x = opposite/adjacentcot x = b/a = cos x/sin x = adjacent/opposite

trig in a nutshell

logs in a nutshell

personal notes

ln (xy) = ln x + ln yln (x/y) = lnx - lnyln x = n ln xln e = e = xln 1 = 0 ln e = 1

ln x

limn → ∞

n= e(1+ )1

nif: a = x

log x = b

b

a

e10log x = log x

log x = ln x

3 step test for continuity:1. f(c) exists2. lim exists x->c

3. lim = f(c) x->c

derivatives

integration by parts

velocity & motion

volumes & areas

disc & shell methods

partial fractions trig substitutions

transcendental derivatives

Product Rulef ' (u v) = udv + vdu

Chain Rule (f o g)' = f ' g '

Quotient Rule

f ' ( ) =v du - u dv

v ²uv

(Lo D Hi minus Hi D Lo over Lo Lo)

f (x + h) - f (x)h

f ' (x) = limh -> 0

definition of the derivative:

Power Rulef ' ( x ) = c x

Addition Rulef ' (u + v) = f ' u + f ' v

c c -1

Mean Value Theorem

f (b) - f (a)b - a

= f ' (c)

l’Hôpital’s RuleWhen

lim f (x)g(x)x → a

lim f ' (x)g ' (x)x → a=

00

∞∞

lim f (x)g(x)x → a = OR

trig derivatives

Standard Trig(d/dx)(csc u) = - csc u cot u(d/dx)(sec u) = sec u tan u

(d/dx)(cot u) = - csc ² u(d/dx)(tan u) = sec ² u(d/dx)(cos u) = - sin u(d/dx)(sin u) = cos u

ddx

1 duu dxln u =

ddx

dudx

e = eu u

ddx

dudx

a = a ln au u

1

1

f (x) = ax

f (x) = log xa

(Use h(b1+b2)/2 for

trapezoids of

different height)

∫ f (x) dx = F(a) - F(b)

where F is theantiderivative of f

a

b

integrals

Power Rule:∫x dx = x +C

a+1x ≠ - 1

a a+1Average Value

f (x) dxAvg. (f (x)) =a

b 1 . b - a∫

LRAM RRAM MRAM

Rectangular Approximation Methods (RAM)

(use b x h for approximation)

Trapezoidal Rule

T = (y + 2y + 2y + ... + 2y + y )

a & b = bounds

n = number of intervals

b - a2n 210 n - 1 n

First FundamentalTheorem of Calculus

Second FundamentalTheorem of Calculus

(Leibniz's Rule)

dvdx

dudx

= f (v) - f (u)

ddx

f (t) dt∫u (x)

v (x)

= sec | |+ C-1du

u u² - a²1a

ua∫

trigonometric integrals

∫ sin x dx = - cos x + C

∫ cos x dx = sinx + C

∫ sec² x dx = tan x + C

∫ csc² x dx = - cot x + C

∫ sec x tan x dx = sec x + C

∫ csc x cot x dx = - csc x + C

∫ tan x dx = - ln |cos x| + C

∫ cot x dx = ln |sin x| + C

∫ sin² x dx = x - sin 2x + C

2 4

∫ cos² x dx = x + sin 2x + C

2 4

Standard Trig

Inverse Trig

= tan + C-1du

a² + u²

1a

ua∫= sin + C

-1du

a² - u²∫ua

u u∫ e du = e + C ∫ a du = + C

uu

aln a

duu∫ = ln |u| + C ∫ ln x dx = x ln x - x + C

∫a

b2r dx

∫c

d2

r dy ∫a

b

r h dx2π

∫c

d

r h dy2π

∫c

d22(R - r )dyππ

∫a

b2 2

(R - r )dxππ

Volume

X-Axis

Y-Axis

Disc(no hole)

Discw/ Hole

Shell

r = radius R = Outside radiusr = inside radius

r = radiush = height

s(t) or x(t)

v(t)

a(t)position

velocity

the velocity equation

acceleration

dd

∫∫

s(t) = ½ g t ² + v t + s

g = - 32 ft / s ², - 9.8 m / s ²O O

SA(sphere) = 4 π r ²

V(sphere) = π r ³43

V(cone) = π r ² h13s² 3

4A =

∫udv = uv - ∫vduPriority:

Logarithmic

Inverse Trig

Algebraic

Trigonometric

Euler's Constant (e)

Tabular Integration

∫ [(algebraic)(trigonometric/e)]dxex: ∫ 2x³cosx dx

2x³6x²12x120

cosxsinx-cosx-sinxcosx

+-

+-

Alt

ern

ati

ng

Sig

ns

d ∫

2x³sinx + 6x²cosx - 12x sin x - 12 x cos x +C

BA

x

y

relative maximumf '(x)=0

zerof (x)=0

relative minimumf '(x)=0

absolute maximum

inflection pointf ''(x)=0

a b c d e

k

bf '(x)

d e+ - +

cf ''(x)

e+-

Area = ∫ f (x) dxa

b

Domain:( - ∞, e ]-∞ < x ≤ e

Range:( - ∞, k ]-∞ < y ≤ k

Sample Function f(x)

sin ² x + cos ² x = 11 + tan ² x = sec ² x1 + cot ² x = csc ² x

cos (a+b) = cos a cos b - sin a sin bsin (a+b) = sin a cos b + cos a sin b

cos (a - b) = cos a cos b + sin a sin bsin (a - b) = sin a cos b - cos a sin b

transcendental integrals

Double Angle Identitiessin 2x = 2 sin x cos x

cos 2x = cos ² x - sin ² xcos ² x = 1+ cos 2x

2sin ² x = 1- cos 2x

2

Odd/Even Identitiessin (- x) = - sin xcos (- x) = cos xtan (- x) = - tan xcot (- x) = - cot xsec (- x) = sec x

csc (- x) = - csc x

C

B

A

ac

bx

sin x =a/c= opposite/hypotenusecos x = b/c = adjacent/hypotenusesec x = c/b = hypotenuse/adjacentcsc x = c/a = hypotenuse/oppositetan x = a/b = sin x/cos x = opposite/adjacentcot x = b/a = cos x/sin x = adjacent/opposite

trig in a nutshell

logs in a nutshell

personal notes

ln (xy) = ln x + ln yln (x/y) = lnx - lnyln x = n ln xln e = e = xln 1 = 0 ln e = 1

ln x

limn → ∞

n= e(1+ )1

nif: a = x

log x = b

b

a

e10log x = log x

log x = ln x

3 step test for continuity:1. f(c) exists2. lim exists x->c

3. lim = f(c) x->c

derivatives

integration by parts

velocity & motion

volumes & areas

disc & shell methods

partial fractions trig substitutions

transcendental derivatives

Product Rulef ' (u v) = udv + vdu

Chain Rule (f o g)' = f ' g '

Quotient Rule

f ' ( ) =v du - u dv

v ²uv

(Lo D Hi minus Hi D Lo over Lo Lo)

f (x + h) - f (x)h

f ' (x) = limh -> 0

definition of the derivative:

Power Rulef ' ( x ) = c x

Addition Rulef ' (u + v) = f ' u + f ' v

c c -1

Mean Value Theorem

f (b) - f (a)b - a

= f ' (c)

l’Hôpital’s RuleWhen

lim f (x)g(x)x → a

lim f ' (x)g ' (x)x → a=

00

∞∞

lim f (x)g(x)x → a = OR

trig derivatives

Standard Trig(d/dx)(csc u) = - csc u cot u(d/dx)(sec u) = sec u tan u

(d/dx)(cot u) = - csc ² u(d/dx)(tan u) = sec ² u(d/dx)(cos u) = - sin u(d/dx)(sin u) = cos u

ddx

1 duu dxln u =

ddx

dudx

e = eu u

ddx

dudx

a = a ln au u

1

1

f (x) = ax

f (x) = log xa

(Use h(b1+b2)/2 for

trapezoids of

different height)

∫ f (x) dx = F(a) - F(b)

where F is theantiderivative of f

a

b

integrals

Power Rule:∫x dx = x +C

a+1x ≠ - 1

a a+1Average Value

f (x) dxAvg. (f (x)) =a

b 1 . b - a∫

LRAM RRAM MRAM

Rectangular Approximation Methods (RAM)

(use b x h for approximation)

Trapezoidal Rule

T = (y + 2y + 2y + ... + 2y + y )

a & b = bounds

n = number of intervals

b - a2n 210 n - 1 n

First FundamentalTheorem of Calculus

Second FundamentalTheorem of Calculus

(Leibniz's Rule)

dvdx

dudx

= f (v) - f (u)

ddx

f (t) dt∫u (x)

v (x)

= sec | |+ C-1du

u u² - a²1a

ua∫

trigonometric integrals

∫ sin x dx = - cos x + C

∫ cos x dx = sinx + C

∫ sec² x dx = tan x + C

∫ csc² x dx = - cot x + C

∫ sec x tan x dx = sec x + C

∫ csc x cot x dx = - csc x + C

∫ tan x dx = - ln |cos x| + C

∫ cot x dx = ln |sin x| + C

∫ sin² x dx = x - sin 2x + C

2 4

∫ cos² x dx = x + sin 2x + C

2 4

Standard Trig

Inverse Trig

= tan + C-1du

a² + u²

1a

ua∫= sin + C

-1du

a² - u²∫ua

u u∫ e du = e + C ∫ a du = + C

uu

aln a

duu∫ = ln |u| + C ∫ ln x dx = x ln x - x + C

∫a

b2r dx

∫c

d2

r dy ∫a

b

r h dx2π

∫c

d

r h dy2π

∫c

d22(R - r )dyππ

∫a

b2 2

(R - r )dxππ

Volume

X-Axis

Y-Axis

Disc(no hole)

Discw/ Hole

Shell

r = radius R = Outside radiusr = inside radius

r = radiush = height

s(t) or x(t)

v(t)

a(t)position

velocity

the velocity equation

acceleration

dd

∫∫

s(t) = ½ g t ² + v t + s

g = - 32 ft / s ², - 9.8 m / s ²O O

SA(sphere) = 4 π r ²

V(sphere) = π r ³43

V(cone) = π r ² h13s² 3

4A =

∫udv = uv - ∫vduPriority:

Logarithmic

Inverse Trig

Algebraic

Trigonometric

Euler's Constant (e)

Tabular Integration

∫ [(algebraic)(trigonometric/e)]dxex: ∫ 2x³cosx dx

2x³6x²12x120

cosxsinx-cosx-sinxcosx

+-

+-

Alt

ern

ati

ng

Sig

ns

d ∫

2x³sinx + 6x²cosx - 12x sin x - 12 x cos x +C

BA

x

y

relative maximumf '(x)=0

zerof (x)=0

relative minimumf '(x)=0

absolute maximum

inflection pointf ''(x)=0

a b c d e

k

bf '(x)

d e+ - +

cf ''(x)

e+-

Area = ∫ f (x) dxa

b

Domain:( - ∞, e ]-∞ < x ≤ e

Range:( - ∞, k ]-∞ < y ≤ k

Sample Function f(x)

sin ² x + cos ² x = 11 + tan ² x = sec ² x1 + cot ² x = csc ² x

cos (a+b) = cos a cos b - sin a sin bsin (a+b) = sin a cos b + cos a sin b

cos (a - b) = cos a cos b + sin a sin bsin (a - b) = sin a cos b - cos a sin b

transcendental integrals

Double Angle Identitiessin 2x = 2 sin x cos x

cos 2x = cos ² x - sin ² xcos ² x = 1+ cos 2x

2sin ² x = 1- cos 2x

2

Odd/Even Identitiessin (- x) = - sin xcos (- x) = cos xtan (- x) = - tan xcot (- x) = - cot xsec (- x) = sec x

csc (- x) = - csc x

C

B

A

ac

bx

sin x =a/c= opposite/hypotenusecos x = b/c = adjacent/hypotenusesec x = c/b = hypotenuse/adjacentcsc x = c/a = hypotenuse/oppositetan x = a/b = sin x/cos x = opposite/adjacentcot x = b/a = cos x/sin x = adjacent/opposite

trig in a nutshell

logs in a nutshell

personal notes

ln (xy) = ln x + ln yln (x/y) = lnx - lnyln x = n ln xln e = e = xln 1 = 0 ln e = 1

ln x

limn → ∞

n= e(1+ )1

nif: a = x

log x = b

b

a

e10log x = log x

log x = ln x

3 step test for continuity:1. f(c) exists2. lim exists x->c

3. lim = f(c) x->c

derivatives

integration by parts

velocity & motion

volumes & areas

disc & shell methods

partial fractions trig substitutions

transcendental derivatives

Product Rulef ' (u v) = udv + vdu

Chain Rule (f o g)' = f ' g '

Quotient Rule

f ' ( ) =v du - u dv

v ²uv

(Lo D Hi minus Hi D Lo over Lo Lo)

f (x + h) - f (x)h

f ' (x) = limh -> 0

definition of the derivative:

Power Rulef ' ( x ) = c x

Addition Rulef ' (u + v) = f ' u + f ' v

c c -1

Mean Value Theorem

f (b) - f (a)b - a

= f ' (c)

l’Hôpital’s RuleWhen

lim f (x)g(x)x → a

lim f ' (x)g ' (x)x → a=

00

∞∞

lim f (x)g(x)x → a = OR

trig derivatives

Standard Trig(d/dx)(csc u) = - csc u cot u(d/dx)(sec u) = sec u tan u

(d/dx)(cot u) = - csc ² u(d/dx)(tan u) = sec ² u(d/dx)(cos u) = - sin u(d/dx)(sin u) = cos u

ddx

1 duu dxln u =

ddx

dudx

e = eu u

ddx

dudx

a = a ln au u

1

1

f (x) = ax

f (x) = log xa

(Use h(b1+b2)/2 for

trapezoids of

different height)

∫ f (x) dx = F(a) - F(b)

where F is theantiderivative of f

a

b

integrals

Power Rule:∫x dx = x +C

a+1x ≠ - 1

a a+1Average Value

f (x) dxAvg. (f (x)) =a

b 1 . b - a∫

LRAM RRAM MRAM

Rectangular Approximation Methods (RAM)

(use b x h for approximation)

Trapezoidal Rule

T = (y + 2y + 2y + ... + 2y + y )

a & b = bounds

n = number of intervals

b - a2n 210 n - 1 n

First FundamentalTheorem of Calculus

Second FundamentalTheorem of Calculus

(Leibniz's Rule)

dvdx

dudx

= f (v) - f (u)

ddx

f (t) dt∫u (x)

v (x)

= sec | |+ C-1du

u u² - a²1a

ua∫

trigonometric integrals

∫ sin x dx = - cos x + C

∫ cos x dx = sinx + C

∫ sec² x dx = tan x + C

∫ csc² x dx = - cot x + C

∫ sec x tan x dx = sec x + C

∫ csc x cot x dx = - csc x + C

∫ tan x dx = - ln |cos x| + C

∫ cot x dx = ln |sin x| + C

∫ sin² x dx = x - sin 2x + C

2 4

∫ cos² x dx = x + sin 2x + C

2 4

Standard Trig

Inverse Trig

= tan + C-1du

a² + u²

1a

ua∫= sin + C

-1du

a² - u²∫ua

u u∫ e du = e + C ∫ a du = + C

uu

aln a

duu∫ = ln |u| + C ∫ ln x dx = x ln x - x + C

∫a

b2r dx

∫c

d2

r dy ∫a

b

r h dx2π

∫c

d

r h dy2π

∫c

d22(R - r )dyππ

∫a

b2 2

(R - r )dxππ

Volume

X-Axis

Y-Axis

Disc(no hole)

Discw/ Hole

Shell

r = radius R = Outside radiusr = inside radius

r = radiush = height

s(t) or x(t)

v(t)

a(t)position

velocity

the velocity equation

acceleration

dd

∫∫

s(t) = ½ g t ² + v t + s

g = - 32 ft / s ², - 9.8 m / s ²O O

SA(sphere) = 4 π r ²

V(sphere) = π r ³43

V(cone) = π r ² h13s² 3

4A =

∫udv = uv - ∫vduPriority:

Logarithmic

Inverse Trig

Algebraic

Trigonometric

Euler's Constant (e)

Tabular Integration

∫ [(algebraic)(trigonometric/e)]dxex: ∫ 2x³cosx dx

2x³6x²12x120

cosxsinx-cosx-sinxcosx

+-

+-

Alt

ern

ati

ng

Sig

ns

d ∫

2x³sinx + 6x²cosx - 12x sin x - 12 x cos x +C

BA

x

y

relative maximumf '(x)=0

zerof (x)=0

relative minimumf '(x)=0

absolute maximum

inflection pointf ''(x)=0

a b c d e

k

bf '(x)

d e+ - +

cf ''(x)

e+-

Area = ∫ f (x) dxa

b

Domain:( - ∞, e ]-∞ < x ≤ e

Range:( - ∞, k ]-∞ < y ≤ k

Sample Function f(x)

sin ² x + cos ² x = 11 + tan ² x = sec ² x1 + cot ² x = csc ² x

cos (a+b) = cos a cos b - sin a sin bsin (a+b) = sin a cos b + cos a sin b

cos (a - b) = cos a cos b + sin a sin bsin (a - b) = sin a cos b - cos a sin b

transcendental integrals

Double Angle Identitiessin 2x = 2 sin x cos x

cos 2x = cos ² x - sin ² xcos ² x = 1+ cos 2x

2sin ² x = 1- cos 2x

2

Odd/Even Identitiessin (- x) = - sin xcos (- x) = cos xtan (- x) = - tan xcot (- x) = - cot xsec (- x) = sec x

csc (- x) = - csc x

C

B

A

ac

bx

sin x =a/c= opposite/hypotenusecos x = b/c = adjacent/hypotenusesec x = c/b = hypotenuse/adjacentcsc x = c/a = hypotenuse/oppositetan x = a/b = sin x/cos x = opposite/adjacentcot x = b/a = cos x/sin x = adjacent/opposite

trig in a nutshell

logs in a nutshell

personal notes

ln (xy) = ln x + ln yln (x/y) = lnx - lnyln x = n ln xln e = e = xln 1 = 0 ln e = 1

ln x

limn → ∞

n= e(1+ )1

nif: a = x

log x = b

b

a

e10log x = log x

log x = ln x

3 step test for continuity:1. f(c) exists2. lim exists x->c

3. lim = f(c) x->c

derivatives

integration by parts

velocity & motion

volumes & areas

disc & shell methods

partial fractions trig substitutions

transcendental derivatives

Product Rulef ' (u v) = udv + vdu

Chain Rule (f o g)' = f ' g '

Quotient Rule

f ' ( ) =v du - u dv

v ²uv

(Lo D Hi minus Hi D Lo over Lo Lo)

f (x + h) - f (x)h

f ' (x) = limh -> 0

definition of the derivative:

Power Rulef ' ( x ) = c x

Addition Rulef ' (u + v) = f ' u + f ' v

c c -1

Mean Value Theorem

f (b) - f (a)b - a

= f ' (c)

l’Hôpital’s RuleWhen

lim f (x)g(x)x → a

lim f ' (x)g ' (x)x → a=

00

∞∞

lim f (x)g(x)x → a = OR

trig derivatives

Standard Trig(d/dx)(csc u) = - csc u cot u(d/dx)(sec u) = sec u tan u

(d/dx)(cot u) = - csc ² u(d/dx)(tan u) = sec ² u(d/dx)(cos u) = - sin u(d/dx)(sin u) = cos u

ddx

1 duu dxln u =

ddx

dudx

e = eu u

ddx

dudx

a = a ln au u

1

1

f (x) = ax

f (x) = log xa

(Use h(b1+b2)/2 for

trapezoids of

different height)

∫ f (x) dx = F(a) - F(b)

where F is theantiderivative of f

a

b

integrals

Power Rule:∫x dx = x +C

a+1x ≠ - 1

a a+1Average Value

f (x) dxAvg. (f (x)) =a

b 1 . b - a∫

LRAM RRAM MRAM

Rectangular Approximation Methods (RAM)

(use b x h for approximation)

Trapezoidal Rule

T = (y + 2y + 2y + ... + 2y + y )

a & b = bounds

n = number of intervals

b - a2n 210 n - 1 n

First FundamentalTheorem of Calculus

Second FundamentalTheorem of Calculus

(Leibniz's Rule)

dvdx

dudx

= f (v) - f (u)

ddx

f (t) dt∫u (x)

v (x)

= sec | |+ C-1du

u u² - a²1a

ua∫

trigonometric integrals

∫ sin x dx = - cos x + C

∫ cos x dx = sinx + C

∫ sec² x dx = tan x + C

∫ csc² x dx = - cot x + C

∫ sec x tan x dx = sec x + C

∫ csc x cot x dx = - csc x + C

∫ tan x dx = - ln |cos x| + C

∫ cot x dx = ln |sin x| + C

∫ sin² x dx = x - sin 2x + C

2 4

∫ cos² x dx = x + sin 2x + C

2 4

Standard Trig

Inverse Trig

= tan + C-1du

a² + u²

1a

ua∫= sin + C

-1du

a² - u²∫ua

u u∫ e du = e + C ∫ a du = + C

uu

aln a

duu∫ = ln |u| + C ∫ ln x dx = x ln x - x + C

∫a

b2r dx

∫c

d2

r dy ∫a

b

r h dx2π

∫c

d

r h dy2π

∫c

d22(R - r )dyππ

∫a

b2 2

(R - r )dxππ

Volume

X-Axis

Y-Axis

Disc(no hole)

Discw/ Hole

Shell

r = radius R = Outside radiusr = inside radius

r = radiush = height

s(t) or x(t)

v(t)

a(t)position

velocity

the velocity equation

acceleration

dd

∫∫

s(t) = ½ g t ² + v t + s

g = - 32 ft / s ², - 9.8 m / s ²O O

SA(sphere) = 4 π r ²

V(sphere) = π r ³43

V(cone) = π r ² h13s² 3

4A =

∫udv = uv - ∫vduPriority:

Logarithmic

Inverse Trig

Algebraic

Trigonometric

Euler's Constant (e)

Tabular Integration

∫ [(algebraic)(trigonometric/e)]dxex: ∫ 2x³cosx dx

2x³6x²12x120

cosxsinx-cosx-sinxcosx

+-

+-

Alt

ern

ati

ng

Sig

ns

d ∫

2x³sinx + 6x²cosx - 12x sin x - 12 x cos x +C

BA

x

y

relative maximumf '(x)=0

zerof (x)=0

relative minimumf '(x)=0

absolute maximum

inflection pointf ''(x)=0

a b c d e

k

bf '(x)

d e+ - +

cf ''(x)

e+-

Area = ∫ f (x) dxa

b

Domain:( - ∞, e ]-∞ < x ≤ e

Range:( - ∞, k ]-∞ < y ≤ k

Sample Function f(x)

sin ² x + cos ² x = 11 + tan ² x = sec ² x1 + cot ² x = csc ² x

cos (a+b) = cos a cos b - sin a sin bsin (a+b) = sin a cos b + cos a sin b

cos (a - b) = cos a cos b + sin a sin bsin (a - b) = sin a cos b - cos a sin b

transcendental integrals

Double Angle Identitiessin 2x = 2 sin x cos x

cos 2x = cos ² x - sin ² xcos ² x = 1+ cos 2x

2sin ² x = 1- cos 2x

2

Odd/Even Identitiessin (- x) = - sin xcos (- x) = cos xtan (- x) = - tan xcot (- x) = - cot xsec (- x) = sec x

csc (- x) = - csc x

C

B

A

ac

bx

sin x =a/c= opposite/hypotenusecos x = b/c = adjacent/hypotenusesec x = c/b = hypotenuse/adjacentcsc x = c/a = hypotenuse/oppositetan x = a/b = sin x/cos x = opposite/adjacentcot x = b/a = cos x/sin x = adjacent/opposite

trig in a nutshell

logs in a nutshell

personal notes

ln (xy) = ln x + ln yln (x/y) = lnx - lnyln x = n ln xln e = e = xln 1 = 0 ln e = 1

ln x

limn → ∞

n= e(1+ )1

nif: a = x

log x = b

b

a

e10log x = log x

log x = ln x

3 step test for continuity:1. f(c) exists2. lim exists x->c

3. lim = f(c) x->c

derivatives

integration by parts

velocity & motion

volumes & areas

disc & shell methods

partial fractions trig substitutions

transcendental derivatives

Product Rulef ' (u v) = udv + vdu

Chain Rule (f o g)' = f ' g '

Quotient Rule

f ' ( ) =v du - u dv

v ²uv

(Lo D Hi minus Hi D Lo over Lo Lo)

f (x + h) - f (x)h

f ' (x) = limh -> 0

definition of the derivative:

Power Rulef ' ( x ) = c x

Addition Rulef ' (u + v) = f ' u + f ' v

c c -1

Mean Value Theorem

f (b) - f (a)b - a

= f ' (c)

l’Hôpital’s RuleWhen

lim f (x)g(x)x → a

lim f ' (x)g ' (x)x → a=

00

∞∞

lim f (x)g(x)x → a = OR

trig derivatives

Standard Trig(d/dx)(csc u) = - csc u cot u(d/dx)(sec u) = sec u tan u

(d/dx)(cot u) = - csc ² u(d/dx)(tan u) = sec ² u(d/dx)(cos u) = - sin u(d/dx)(sin u) = cos u

ddx

1 duu dxln u =

ddx

dudx

e = eu u

ddx

dudx

a = a ln au u

1

1

f (x) = ax

f (x) = log xa

(Use h(b1+b2)/2 for

trapezoids of

different height)

∫ f (x) dx = F(a) - F(b)

where F is theantiderivative of f

a

b

integrals

Power Rule:∫x dx = x +C

a+1x ≠ - 1

a a+1Average Value

f (x) dxAvg. (f (x)) =a

b 1 . b - a∫

LRAM RRAM MRAM

Rectangular Approximation Methods (RAM)

(use b x h for approximation)

Trapezoidal Rule

T = (y + 2y + 2y + ... + 2y + y )

a & b = bounds

n = number of intervals

b - a2n 210 n - 1 n

First FundamentalTheorem of Calculus

Second FundamentalTheorem of Calculus

(Leibniz's Rule)

dvdx

dudx

= f (v) - f (u)

ddx

f (t) dt∫u (x)

v (x)

= sec | |+ C-1du

u u² - a²1a

ua∫

trigonometric integrals

∫ sin x dx = - cos x + C

∫ cos x dx = sinx + C

∫ sec² x dx = tan x + C

∫ csc² x dx = - cot x + C

∫ sec x tan x dx = sec x + C

∫ csc x cot x dx = - csc x + C

∫ tan x dx = - ln |cos x| + C

∫ cot x dx = ln |sin x| + C

∫ sin² x dx = x - sin 2x + C

2 4

∫ cos² x dx = x + sin 2x + C

2 4

Standard Trig

Inverse Trig

= tan + C-1du

a² + u²

1a

ua∫= sin + C

-1du

a² - u²∫ua

u u∫ e du = e + C ∫ a du = + C

uu

aln a

duu∫ = ln |u| + C ∫ ln x dx = x ln x - x + C

∫a

b2r dx

∫c

d2

r dy ∫a

b

r h dx2π

∫c

d

r h dy2π

∫c

d22(R - r )dyππ

∫a

b2 2

(R - r )dxππ

Volume

X-Axis

Y-Axis

Disc(no hole)

Discw/ Hole

Shell

r = radius R = Outside radiusr = inside radius

r = radiush = height

s(t) or x(t)

v(t)

a(t)position

velocity

the velocity equation

acceleration

dd

∫∫

s(t) = ½ g t ² + v t + s

g = - 32 ft / s ², - 9.8 m / s ²O O

SA(sphere) = 4 π r ²

V(sphere) = π r ³43

V(cone) = π r ² h13s² 3

4A =

∫udv = uv - ∫vduPriority:

Logarithmic

Inverse Trig

Algebraic

Trigonometric

Euler's Constant (e)

Tabular Integration

∫ [(algebraic)(trigonometric/e)]dxex: ∫ 2x³cosx dx

2x³6x²12x120

cosxsinx-cosx-sinxcosx

+-

+-

Alt

ern

ati

ng

Sig

ns

d ∫

2x³sinx + 6x²cosx - 12x sin x - 12 x cos x +C

BA

x

y

relative maximumf '(x)=0

zerof (x)=0

relative minimumf '(x)=0

absolute maximum

inflection pointf ''(x)=0

a b c d e

k

bf '(x)

d e+ - +

cf ''(x)

e+-

Area = ∫ f (x) dxa

b

Domain:( - ∞, e ]-∞ < x ≤ e

Range:( - ∞, k ]-∞ < y ≤ k

Sample Function f(x)

sin ² x + cos ² x = 11 + tan ² x = sec ² x1 + cot ² x = csc ² x

cos (a+b) = cos a cos b - sin a sin bsin (a+b) = sin a cos b + cos a sin b

cos (a - b) = cos a cos b + sin a sin bsin (a - b) = sin a cos b - cos a sin b

transcendental integrals

Double Angle Identitiessin 2x = 2 sin x cos x

cos 2x = cos ² x - sin ² xcos ² x = 1+ cos 2x

2sin ² x = 1- cos 2x

2

Odd/Even Identitiessin (- x) = - sin xcos (- x) = cos xtan (- x) = - tan xcot (- x) = - cot xsec (- x) = sec x

csc (- x) = - csc x

C

B

A

ac

bx

sin x =a/c= opposite/hypotenusecos x = b/c = adjacent/hypotenusesec x = c/b = hypotenuse/adjacentcsc x = c/a = hypotenuse/oppositetan x = a/b = sin x/cos x = opposite/adjacentcot x = b/a = cos x/sin x = adjacent/opposite

trig in a nutshell

logs in a nutshell

personal notes

ln (xy) = ln x + ln yln (x/y) = lnx - lnyln x = n ln xln e = e = xln 1 = 0 ln e = 1

ln x

limn → ∞

n= e(1+ )1

nif: a = x

log x = b

b

a

e10log x = log x

log x = ln x

3 step test for continuity:1. f(c) exists2. lim exists x->c

3. lim = f(c) x->c

derivatives

integration by parts

velocity & motion

volumes & areas

disc & shell methods

partial fractions trig substitutions

transcendental derivatives

Product Rulef ' (u v) = udv + vdu

Chain Rule (f o g)' = f ' g '

Quotient Rule

f ' ( ) =v du - u dv

v ²uv

(Lo D Hi minus Hi D Lo over Lo Lo)

f (x + h) - f (x)h

f ' (x) = limh -> 0

definition of the derivative:

Power Rulef ' ( x ) = c x

Addition Rulef ' (u + v) = f ' u + f ' v

c c -1

Mean Value Theorem

f (b) - f (a)b - a

= f ' (c)

l’Hôpital’s RuleWhen

lim f (x)g(x)x → a

lim f ' (x)g ' (x)x → a=

00

∞∞

lim f (x)g(x)x → a = OR

trig derivatives

Standard Trig(d/dx)(csc u) = - csc u cot u(d/dx)(sec u) = sec u tan u

(d/dx)(cot u) = - csc ² u(d/dx)(tan u) = sec ² u(d/dx)(cos u) = - sin u(d/dx)(sin u) = cos u

ddx

1 duu dxln u =

ddx

dudx

e = eu u

ddx

dudx

a = a ln au u

1

1

f (x) = ax

f (x) = log xa

(Use h(b1+b2)/2 for

trapezoids of

different height)

∫ f (x) dx = F(a) - F(b)

where F is theantiderivative of f

a

b

integrals

Power Rule:∫x dx = x +C

a+1x ≠ - 1

a a+1Average Value

f (x) dxAvg. (f (x)) =a

b 1 . b - a∫

LRAM RRAM MRAM

Rectangular Approximation Methods (RAM)

(use b x h for approximation)

Trapezoidal Rule

T = (y + 2y + 2y + ... + 2y + y )

a & b = bounds

n = number of intervals

b - a2n 210 n - 1 n

First FundamentalTheorem of Calculus

Second FundamentalTheorem of Calculus

(Leibniz's Rule)

dvdx

dudx

= f (v) - f (u)

ddx

f (t) dt∫u (x)

v (x)

= sec | |+ C-1du

u u² - a²1a

ua∫

trigonometric integrals

∫ sin x dx = - cos x + C

∫ cos x dx = sinx + C

∫ sec² x dx = tan x + C

∫ csc² x dx = - cot x + C

∫ sec x tan x dx = sec x + C

∫ csc x cot x dx = - csc x + C

∫ tan x dx = - ln |cos x| + C

∫ cot x dx = ln |sin x| + C

∫ sin² x dx = x - sin 2x + C

2 4

∫ cos² x dx = x + sin 2x + C

2 4

Standard Trig

Inverse Trig

= tan + C-1du

a² + u²

1a

ua∫= sin + C

-1du

a² - u²∫ua

u u∫ e du = e + C ∫ a du = + C

uu

aln a

duu∫ = ln |u| + C ∫ ln x dx = x ln x - x + C

∫a

b2r dx

∫c

d2

r dy ∫a

b

r h dx2π

∫c

d

r h dy2π

∫c

d22(R - r )dyππ

∫a

b2 2

(R - r )dxππ

Volume

X-Axis

Y-Axis

Disc(no hole)

Discw/ Hole

Shell

r = radius R = Outside radiusr = inside radius

r = radiush = height

s(t) or x(t)

v(t)

a(t)position

velocity

the velocity equation

acceleration

dd

∫∫

s(t) = ½ g t ² + v t + s

g = - 32 ft / s ², - 9.8 m / s ²O O

SA(sphere) = 4 π r ²

V(sphere) = π r ³43

V(cone) = π r ² h13s² 3

4A =

∫udv = uv - ∫vduPriority:

Logarithmic

Inverse Trig

Algebraic

Trigonometric

Euler's Constant (e)

Tabular Integration

∫ [(algebraic)(trigonometric/e)]dxex: ∫ 2x³cosx dx

2x³6x²12x120

cosxsinx-cosx-sinxcosx

+-

+-

Alt

ern

ati

ng

Sig

ns

d ∫

2x³sinx + 6x²cosx - 12x sin x - 12 x cos x +C

BA

x

y

relative maximumf '(x)=0

zerof (x)=0

relative minimumf '(x)=0

absolute maximum

inflection pointf ''(x)=0

a b c d e

k

bf '(x)

d e+ - +

cf ''(x)

e+-

Area = ∫ f (x) dxa

b

Domain:( - ∞, e ]-∞ < x ≤ e

Range:( - ∞, k ]-∞ < y ≤ k

Sample Function f(x)

sin ² x + cos ² x = 11 + tan ² x = sec ² x1 + cot ² x = csc ² x

cos (a+b) = cos a cos b - sin a sin bsin (a+b) = sin a cos b + cos a sin b

cos (a - b) = cos a cos b + sin a sin bsin (a - b) = sin a cos b - cos a sin b

transcendental integrals

Double Angle Identitiessin 2x = 2 sin x cos x

cos 2x = cos ² x - sin ² xcos ² x = 1+ cos 2x

2sin ² x = 1- cos 2x

2

Odd/Even Identitiessin (- x) = - sin xcos (- x) = cos xtan (- x) = - tan xcot (- x) = - cot xsec (- x) = sec x

csc (- x) = - csc x

C

B

A

ac

bx

sin x =a/c= opposite/hypotenusecos x = b/c = adjacent/hypotenusesec x = c/b = hypotenuse/adjacentcsc x = c/a = hypotenuse/oppositetan x = a/b = sin x/cos x = opposite/adjacentcot x = b/a = cos x/sin x = adjacent/opposite

trig in a nutshell

logs in a nutshell

personal notes

ln (xy) = ln x + ln yln (x/y) = lnx - lnyln x = n ln xln e = e = xln 1 = 0 ln e = 1

ln x

limn → ∞

n= e(1+ )1

nif: a = x

log x = b

b

a

e10log x = log x

log x = ln x

3 step test for continuity:1. f(c) exists2. lim exists x->c

3. lim = f(c) x->c

derivatives

integration by parts

velocity & motion

volumes & areas

disc & shell methods

partial fractions trig substitutions

transcendental derivatives

Product Rulef ' (u v) = udv + vdu

Chain Rule (f o g)' = f ' g '

Quotient Rule

f ' ( ) =v du - u dv

v ²uv

(Lo D Hi minus Hi D Lo over Lo Lo)

f (x + h) - f (x)h

f ' (x) = limh -> 0

definition of the derivative:

Power Rulef ' ( x ) = c x

Addition Rulef ' (u + v) = f ' u + f ' v

c c -1

Mean Value Theorem

f (b) - f (a)b - a

= f ' (c)

l’Hôpital’s RuleWhen

lim f (x)g(x)x → a

lim f ' (x)g ' (x)x → a=

00

∞∞

lim f (x)g(x)x → a = OR

trig derivatives

Standard Trig(d/dx)(csc u) = - csc u cot u(d/dx)(sec u) = sec u tan u

(d/dx)(cot u) = - csc ² u(d/dx)(tan u) = sec ² u(d/dx)(cos u) = - sin u(d/dx)(sin u) = cos u

ddx

1 duu dxln u =

ddx

dudx

e = eu u

ddx

dudx

a = a ln au u

1

1

f (x) = ax

f (x) = log xa

(Use h(b1+b2)/2 for

trapezoids of

different height)

∫ f (x) dx = F(a) - F(b)

where F is theantiderivative of f

a

b

integrals

Power Rule:∫x dx = x +C

a+1x ≠ - 1

a a+1Average Value

f (x) dxAvg. (f (x)) =a

b 1 . b - a∫

LRAM RRAM MRAM

Rectangular Approximation Methods (RAM)

(use b x h for approximation)

Trapezoidal Rule

T = (y + 2y + 2y + ... + 2y + y )

a & b = bounds

n = number of intervals

b - a2n 210 n - 1 n

First FundamentalTheorem of Calculus

Second FundamentalTheorem of Calculus

(Leibniz's Rule)

dvdx

dudx

= f (v) - f (u)

ddx

f (t) dt∫u (x)

v (x)

= sec | |+ C-1du

u u² - a²1a

ua∫

trigonometric integrals

∫ sin x dx = - cos x + C

∫ cos x dx = sinx + C

∫ sec² x dx = tan x + C

∫ csc² x dx = - cot x + C

∫ sec x tan x dx = sec x + C

∫ csc x cot x dx = - csc x + C

∫ tan x dx = - ln |cos x| + C

∫ cot x dx = ln |sin x| + C

∫ sin² x dx = x - sin 2x + C

2 4

∫ cos² x dx = x + sin 2x + C

2 4

Standard Trig

Inverse Trig

= tan + C-1du

a² + u²

1a

ua∫= sin + C

-1du

a² - u²∫ua

u u∫ e du = e + C ∫ a du = + C

uu

aln a

duu∫ = ln |u| + C ∫ ln x dx = x ln x - x + C

∫a

b2r dx

∫c

d2

r dy ∫a

b

r h dx2π

∫c

d

r h dy2π

∫c

d22(R - r )dyππ

∫a

b2 2

(R - r )dxππ

Volume

X-Axis

Y-Axis

Disc(no hole)

Discw/ Hole

Shell

r = radius R = Outside radiusr = inside radius

r = radiush = height

s(t) or x(t)

v(t)

a(t)position

velocity

the velocity equation

acceleration

dd

∫∫

s(t) = ½ g t ² + v t + s

g = - 32 ft / s ², - 9.8 m / s ²O O

SA(sphere) = 4 π r ²

V(sphere) = π r ³43

V(cone) = π r ² h13s² 3

4A =

∫udv = uv - ∫vduPriority:

Logarithmic

Inverse Trig

Algebraic

Trigonometric

Euler's Constant (e)

Tabular Integration

∫ [(algebraic)(trigonometric/e)]dxex: ∫ 2x³cosx dx

2x³6x²12x120

cosxsinx-cosx-sinxcosx

+-

+-

Alt

ern

ati

ng

Sig

ns

d ∫

2x³sinx + 6x²cosx - 12x sin x - 12 x cos x +C

BA

x

y

relative maximumf '(x)=0

zerof (x)=0

relative minimumf '(x)=0

absolute maximum

inflection pointf ''(x)=0

a b c d e

k

bf '(x)

d e+ - +

cf ''(x)

e+-

Area = ∫ f (x) dxa

b

Domain:( - ∞, e ]-∞ < x ≤ e

Range:( - ∞, k ]-∞ < y ≤ k

Sample Function f(x)

sin ² x + cos ² x = 11 + tan ² x = sec ² x1 + cot ² x = csc ² x

cos (a+b) = cos a cos b - sin a sin bsin (a+b) = sin a cos b + cos a sin b

cos (a - b) = cos a cos b + sin a sin bsin (a - b) = sin a cos b - cos a sin b

transcendental integrals

Double Angle Identitiessin 2x = 2 sin x cos x

cos 2x = cos ² x - sin ² xcos ² x = 1+ cos 2x

2sin ² x = 1- cos 2x

2

Odd/Even Identitiessin (- x) = - sin xcos (- x) = cos xtan (- x) = - tan xcot (- x) = - cot xsec (- x) = sec x

csc (- x) = - csc x

C

B

A

ac

bx

sin x =a/c= opposite/hypotenusecos x = b/c = adjacent/hypotenusesec x = c/b = hypotenuse/adjacentcsc x = c/a = hypotenuse/oppositetan x = a/b = sin x/cos x = opposite/adjacentcot x = b/a = cos x/sin x = adjacent/opposite

trig in a nutshell

logs in a nutshell

personal notes

ln (xy) = ln x + ln yln (x/y) = lnx - lnyln x = n ln xln e = e = xln 1 = 0 ln e = 1

ln x

limn → ∞

n= e(1+ )1

nif: a = x

log x = b

b

a

e10log x = log x

log x = ln x

3 step test for continuity:1. f(c) exists2. lim exists x->c

3. lim = f(c) x->c

derivatives

integration by parts

velocity & motion

volumes & areas

disc & shell methods

partial fractions trig substitutions

transcendental derivatives

Product Rulef ' (u v) = udv + vdu

Chain Rule (f o g)' = f ' g '

Quotient Rule

f ' ( ) =v du - u dv

v ²uv

(Lo D Hi minus Hi D Lo over Lo Lo)

f (x + h) - f (x)h

f ' (x) = limh -> 0

definition of the derivative:

Power Rulef ' ( x ) = c x

Addition Rulef ' (u + v) = f ' u + f ' v

c c -1

Mean Value Theorem

f (b) - f (a)b - a

= f ' (c)

l’Hôpital’s RuleWhen

lim f (x)g(x)x → a

lim f ' (x)g ' (x)x → a=

00

∞∞

lim f (x)g(x)x → a = OR

trig derivatives

Standard Trig(d/dx)(csc u) = - csc u cot u(d/dx)(sec u) = sec u tan u

(d/dx)(cot u) = - csc ² u(d/dx)(tan u) = sec ² u(d/dx)(cos u) = - sin u(d/dx)(sin u) = cos u

ddx

1 duu dxln u =

ddx

dudx

e = eu u

ddx

dudx

a = a ln au u

1

1

f (x) = ax

f (x) = log xa

(Use h(b1+b2)/2 for

trapezoids of

different height)

∫ f (x) dx = F(a) - F(b)

where F is theantiderivative of f

a

b

integrals

Power Rule:∫x dx = x +C

a+1x ≠ - 1

a a+1Average Value

f (x) dxAvg. (f (x)) =a

b 1 . b - a∫

LRAM RRAM MRAM

Rectangular Approximation Methods (RAM)

(use b x h for approximation)

Trapezoidal Rule

T = (y + 2y + 2y + ... + 2y + y )

a & b = bounds

n = number of intervals

b - a2n 210 n - 1 n

First FundamentalTheorem of Calculus

Second FundamentalTheorem of Calculus

(Leibniz's Rule)

dvdx

dudx

= f (v) - f (u)

ddx

f (t) dt∫u (x)

v (x)

= sec | |+ C-1du

u u² - a²1a

ua∫

trigonometric integrals

∫ sin x dx = - cos x + C

∫ cos x dx = sinx + C

∫ sec² x dx = tan x + C

∫ csc² x dx = - cot x + C

∫ sec x tan x dx = sec x + C

∫ csc x cot x dx = - csc x + C

∫ tan x dx = - ln |cos x| + C

∫ cot x dx = ln |sin x| + C

∫ sin² x dx = x - sin 2x + C

2 4

∫ cos² x dx = x + sin 2x + C

2 4

Standard Trig

Inverse Trig

= tan + C-1du

a² + u²

1a

ua∫= sin + C

-1du

a² - u²∫ua

u u∫ e du = e + C ∫ a du = + C

uu

aln a

duu∫ = ln |u| + C ∫ ln x dx = x ln x - x + C

∫a

b2r dx

∫c

d2

r dy ∫a

b

r h dx2π

∫c

d

r h dy2π

∫c

d22(R - r )dyππ

∫a

b2 2

(R - r )dxππ

Volume

X-Axis

Y-Axis

Disc(no hole)

Discw/ Hole

Shell

r = radius R = Outside radiusr = inside radius

r = radiush = height

s(t) or x(t)

v(t)

a(t)position

velocity

the velocity equation

acceleration

dd

∫∫

s(t) = ½ g t ² + v t + s

g = - 32 ft / s ², - 9.8 m / s ²O O

SA(sphere) = 4 π r ²

V(sphere) = π r ³43

V(cone) = π r ² h13s² 3

4A =

∫udv = uv - ∫vduPriority:

Logarithmic

Inverse Trig

Algebraic

Trigonometric

Euler's Constant (e)

Tabular Integration

∫ [(algebraic)(trigonometric/e)]dxex: ∫ 2x³cosx dx

2x³6x²12x120

cosxsinx-cosx-sinxcosx

+-

+-

Alt

ern

ati

ng

Sig

ns

d ∫

2x³sinx + 6x²cosx - 12x sin x - 12 x cos x +C

px + q = A + B (x+a)(x+b) (x+a) (x+b)

If you see:2 2

a + x2 2a - x2 2x - a

Use:

x = a tan θ

x = a sin θ

x = a sec θ

px + q = A + B2 2(x+a) (x+a) (x+a)

2 px - qx + r = A + Bx + C2 2(x+a)(x +bx+c) (x+a) (x +bx+c)

A BBCcalculus

a definitive sheetby chad valencia, ucla mathematics major

version 2.0.2000, rev 1

Inverse Trig

sin u =-1

1

1 - u²d

dx

tan u =-1 1

1 + u²d

dx

sec u =-1 1

|u| u² - 1d

dx

dudx

dudx

dudx

solved through trig substitution:∫sec u du = ln |sec u + tan u| + C

A BBC

BC

calculus

a definitive sheetby chad valencia, ucla mathematics major

version 2.0.2000, rev 1

Σn = 0

an

∞last number

sequence

first number

Greek lettersigma(sum of)

improper integrals

sequences

taylor/maclaurin series

hyperbolic trig functions

series

Polar

parametric

vectors

extraneous bc concepts

personal notes

dydx

=

dydt

dxdt

First Derivative:

=dxdt

d²ydx²

Second Derivative:

dydx( )d

a

b dydt

dxdt( ) ( )+

2 2

Arc Length:

dt

Surface Area (x-axis):

a

b dydt

dxdt( ) ( )+

2 22 y dt

Surface Area (y-axis):

a

b dydt

dxdt( ) ( )+

2 22 x dt

Vector Valued Functionsr(t) = x(t)i + y(t)j

r'(t) = x'(t)i + y'(t)j∫r(t)dt = (∫x(t)dt + c)i + (∫y(t)dt + c)j

Velocity Equation for Vectors:(-½ g t² + S )j + V t0 0

where v = #( cos t i + sin t j)0

# = initial velocity/muzzle speed

Notation

v = ai + bj<a,b>

Vector Length

(magnitude/norm):

||v|| = a² + b²

Unit Vector:v

||v||

T(t) = r ' (t) ||r ' (t)||

If T = u1i + u2j N = -u2i + u1j

Unit Tangent andUnit Normal Vectors dy

dtdxdt( ) ( )+

2 2

Speed

Limits of Common Sequences:

Convergence/Divergence:

Let L be a finite number

limn → ∞

an= LConvergent

limn → ∞

an≠ LDivergent

limn → ∞

limn → ∞nn

1nn= = 1

limn → ∞

n

= e(1+ )xnxn

x

limn → ∞

= 0nx

n!

limn → ∞

ln nn = 0

limn → ∞

x = 0n |x|<1

(fraction)

1

∞1xp

Converges if p > 1Diverges if 0 < p < 1

P Series Test: Comparison Test

if f(x) < convergent function,f(x) is convergent

if f(x) > divergent function,f(x) is divergent

Limit Comparison Test:

limx → ∞

f(x)g(x)= L

Let f(x) be a known convergent or divergent function:

0 < L < ∞

f(x) & g(x) both converge or both diverge

limn → ∞

n= e(1+ )1

n

Surface Area:

a

b dydx( )+

2dx2π y 1

WorkWork = Force x Distance

= Density x Volume x Distance= Density x Sum of Areas x Distance

Hooke’s Law for SpringsForce = f(x) = kx

(k is a constant, x is the distance)Work = ò f(x)dx, from position A to position B

Newton’s Law of Cooling

T - T = (T - T )eS 0 S

- kt

T = temperature of object at a given timeTs = temperature of surroundings

To = temperature at time zerot = time

Arc Length (y-axis):

c

d dxdy( )+

2dy1

Arc Length (x-axis):

a

b dydx( )+

2dx1

Circlesr = a cos θr = a sin θ

Basic Shapes(pink = cosine, blue = sine)

Lemniscatesr² = a cos θr² = a sin θ

Spiral ofArchimedes

r = aθ

Limaconsw/ Inner Loopr = a + b cos θr = a + b sin θ

|a|<|b|

Limaconsw/ Dimple

r = a + b cos θr = a + b sin θ

|a|>|b|

Cardioidsr = a + b cos θr = a + b sin θ

|a|=|b|

Rosesr = a cos bθr = a sin bθ

If b is odd, b = number of petalsIf b is even, 2b = number of petals

Polar Conversion (x,y) <-> (r,θ)

x² + y² = r²x = r cos θy = r sin θ

tan θ =yx

Polar Surface Area (y-axis):

α

β 22π r cos θ r +( )dr

dθ

2dθ

Polar Sloper ' sin θ + r cos θr ' cos θ - r sin θ

Polar Surface Area (x-axis):

α

β 2

2π r sin θ r +( )drdθ

2dθ

Polar Arc Length:

α

β2

r+( )drdθ

2dθ

Polar Area:

α

β 2r dθ

12

Geometric Series

Finite Series: Infinite Series:∞

n = 1

a r =n-1 a1-r

|r| <1if |r| > 1, series diverges

=Sn

a(1-r )(1-r)

n

Nth Term Test for Divergence

limn → ∞

∞

n = k

an an ≠ 0 series is divergent

Given: If:

Integral Test∞

n = k

an

∞

k∫ ax dx

If integral converges, series convergesIf integral diverges, series diverges

Comparison Test

if Σan < convergent series,

Σan is convergentif Σan > divergent series,

Σan is divergent

Limit Comparison Test

Let Σbn be a knownconvergent or divergent series:

limn → ∞

Σan

Σbn

= ρ 0 < ρ < ∞

Σan & Σbn

both converge or both diverge

∞

n = 1

an(-1)n-1

limn → ∞

3. an = 0

Alternating Series Test (AST)

Converges if:1. All terms are positive

2. an > an+1 for all n.

Power Series1. Use ratio test on theabsolute value of the series.

2. Set ρ < 1 to find the interval

3. To check bounds, pluginto original equation4. Take ½ of the interval to find radius of convergence

Error (Alternating Series/Taylor)Error = Actual - ApproximateError < First Unused Term

Root Testn

limn → ∞ anρ =

Ratio Test

limn → ∞ρ =

an+1

an Converges if ρ < 1No conclusion if ρ = 1Divergent if ρ > 1

2 3 n nP(x) = f (a) + f ' (a)(x - a) + f ''(a)(x - a) + f ''' (a)(x - a) + … + f (a)(x - a) 2! 3! n!

Taylor Polynomial

Σn = 0

∞ n 2n( - 1) x(2n)!

2 4 6cos x = 1- x + x - x + …= 2! 4! 6!

Common MacLaurin Series

Σn = 0

∞ n 2n+1( - 1) x(2n+1)!

3 5 7sin x = x - x + x - x + … = 3! 5! 7!

Σn = 0

∞ n nf ( a)(x - a) n!

Σn = 0

∞

n n(-1) x 1 = 1 - x + x² - x³+... =1+x

Σn = 0

∞

nx 1 = 1 + x + x² + x³+... =1- x

Σn = 0

∞ nxn!

xe = 1 + x + x² + x³+... = 2! 3!

y(x,y)

Rectangular

x

Polar(r,θ)

rθ

x xcosh x = e + e 2

x xsinh x = e - e 2

x xtanhx = e - e x -xe + e

sech x = 2 x xe + ecsch x = 2 x xe - e

x xcothx = e + e x -xe - e

Standard Hyperbolic Trig Derivatives

(d/dx)(csch u) = - csch u coth u (du/dx)

(d/dx)(sech u) = - sech u tanh u (du/dx)

(d/dx)(coth u) = - csch ² u (du/dx)

(d/dx)(tanh u) = sech ² u (du/dx)

(d/dx)(cosh u) = sinh u (du/dx)

(d/dx)(sinh u) = cosh u (du/dx)

sinh 2x = 2 sinh x cosh xcosh 2x = cosh ² x - sinh ² x

cosh ² x = cosh 2x+1 2

sinh ² x = cosh 2x - 1 2

cosh² x - sinh² x = 1tanh² x + sech ² x = 1

coth² x - csch² x = 1

Inverse Hyperbolic Trig Derivatives

sech u = , 0<u<1-1 1

u u² - 1d

dx

dudx

csch u = , u ≠0-1 1

|u| u² - 1d

dx

dudx

tanh u = , |u|<1-1 1

1 - u²d

dxdudx

coth u = , |u|>1-1 1

1 - u²d

dxdudx

sinh u =-1 1

u² + 1d

dx

dudx

cosh u = , u > 1-1 1

u² - 1d

dx

dudx

Every function splitsinto Even and Odd parts:

f(x) = + f(x) + f(-x)

2 f(x) - f(-x)

2even odd

Hyperbolic Trig Integrals

∫ sinh u du = cosh u + C

∫ cosh u du = sinh u + C

∫ sech² u du = tanh u + C

∫ csch² u du = - coth u + C

∫ sech x tanh u du =- sech u + C

∫ csch x coth u du = - csch u + C

A BBC

BC

calculus

a definitive sheetby chad valencia, ucla mathematics major

version 2.0.2000, rev 1

Σn = 0

an

∞last number

sequence

first number

Greek lettersigma(sum of)

improper integrals

sequences

taylor/maclaurin series

hyperbolic trig functions

series

Polar

parametric

vectors

extraneous bc concepts

personal notes

dydx

=

dydt

dxdt

First Derivative:

=dxdt

d²ydx²

Second Derivative:

dydx( )d

a

b dydt

dxdt( ) ( )+

2 2

Arc Length:

dt

Surface Area (x-axis):

a

b dydt

dxdt( ) ( )+

2 22 y dt

Surface Area (y-axis):

a

b dydt

dxdt( ) ( )+

2 22 x dt

Vector Valued Functionsr(t) = x(t)i + y(t)j

r'(t) = x'(t)i + y'(t)j∫r(t)dt = (∫x(t)dt + c)i + (∫y(t)dt + c)j

Velocity Equation for Vectors:(-½ g t² + S )j + V t0 0

where v = #( cos t i + sin t j)0

# = initial velocity/muzzle speed

Notation

v = ai + bj<a,b>

Vector Length

(magnitude/norm):

||v|| = a² + b²

Unit Vector:v

||v||

T(t) = r ' (t) ||r ' (t)||

If T = u1i + u2j N = -u2i + u1j

Unit Tangent andUnit Normal Vectors dy

dtdxdt( ) ( )+

2 2

Speed

Limits of Common Sequences:

Convergence/Divergence:

Let L be a finite number

limn → ∞

an= LConvergent

limn → ∞

an≠ LDivergent

limn → ∞

limn → ∞nn

1nn= = 1

limn → ∞

n

= e(1+ )xnxn

x

limn → ∞

= 0nx

n!

limn → ∞

ln nn = 0

limn → ∞

x = 0n |x|<1

(fraction)

1

∞1xp

Converges if p > 1Diverges if 0 < p < 1

P Series Test: Comparison Test

if f(x) < convergent function,f(x) is convergent

if f(x) > divergent function,f(x) is divergent

Limit Comparison Test:

limx → ∞

f(x)g(x)= L

Let f(x) be a known convergent or divergent function:

0 < L < ∞

f(x) & g(x) both converge or both diverge

limn → ∞

n= e(1+ )1

n

Surface Area:

a

b dydx( )+

2dx2π y 1

WorkWork = Force x Distance

= Density x Volume x Distance= Density x Sum of Areas x Distance

Hooke’s Law for SpringsForce = f(x) = kx

(k is a constant, x is the distance)Work = ò f(x)dx, from position A to position B

Newton’s Law of Cooling

T - T = (T - T )eS 0 S

- kt

T = temperature of object at a given timeTs = temperature of surroundings

To = temperature at time zerot = time

Arc Length (y-axis):

c

d dxdy( )+

2dy1

Arc Length (x-axis):

a

b dydx( )+

2dx1

Circlesr = a cos θr = a sin θ

Basic Shapes(pink = cosine, blue = sine)

Lemniscatesr² = a cos θr² = a sin θ

Spiral ofArchimedes

r = aθ

Limaconsw/ Inner Loopr = a + b cos θr = a + b sin θ

|a|<|b|

Limaconsw/ Dimple

r = a + b cos θr = a + b sin θ

|a|>|b|

Cardioidsr = a + b cos θr = a + b sin θ

|a|=|b|

Rosesr = a cos bθr = a sin bθ

If b is odd, b = number of petalsIf b is even, 2b = number of petals

Polar Conversion (x,y) <-> (r,θ)

x² + y² = r²x = r cos θy = r sin θ

tan θ =yx

Polar Surface Area (y-axis):

α

β 22π r cos θ r +( )dr

dθ

2dθ

Polar Sloper ' sin θ + r cos θr ' cos θ - r sin θ

Polar Surface Area (x-axis):

α

β 2

2π r sin θ r +( )drdθ

2dθ

Polar Arc Length:

α

β2

r+( )drdθ

2dθ

Polar Area:

α

β 2r dθ

12

Geometric Series

Finite Series: Infinite Series:∞

n = 1

a r =n-1 a1-r

|r| <1if |r| > 1, series diverges

=Sn

a(1-r )(1-r)

n

Nth Term Test for Divergence

limn → ∞

∞

n = k

an an ≠ 0 series is divergent

Given: If:

Integral Test∞

n = k

an

∞

k∫ ax dx

If integral converges, series convergesIf integral diverges, series diverges

Comparison Test

if Σan < convergent series,

Σan is convergentif Σan > divergent series,

Σan is divergent

Limit Comparison Test

Let Σbn be a knownconvergent or divergent series:

limn → ∞

Σan

Σbn

= ρ 0 < ρ < ∞

Σan & Σbn

both converge or both diverge

∞

n = 1

an(-1)n-1

limn → ∞

3. an = 0

Alternating Series Test (AST)

Converges if:1. All terms are positive

2. an > an+1 for all n.

Power Series1. Use ratio test on theabsolute value of the series.

2. Set ρ < 1 to find the interval

3. To check bounds, pluginto original equation4. Take ½ of the interval to find radius of convergence

Error (Alternating Series/Taylor)Error = Actual - ApproximateError < First Unused Term

Root Testn

limn → ∞ anρ =

Ratio Test

limn → ∞ρ =

an+1

an Converges if ρ < 1No conclusion if ρ = 1Divergent if ρ > 1

2 3 n nP(x) = f (a) + f ' (a)(x - a) + f ''(a)(x - a) + f ''' (a)(x - a) + … + f (a)(x - a) 2! 3! n!

Taylor Polynomial

Σn = 0

∞ n 2n( - 1) x(2n)!

2 4 6cos x = 1- x + x - x + …= 2! 4! 6!

Common MacLaurin Series

Σn = 0

∞ n 2n+1( - 1) x(2n+1)!

3 5 7sin x = x - x + x - x + … = 3! 5! 7!

Σn = 0

∞ n nf ( a)(x - a) n!

Σn = 0

∞

n n(-1) x 1 = 1 - x + x² - x³+... =1+x

Σn = 0

∞

nx 1 = 1 + x + x² + x³+... =1- x

Σn = 0

∞ nxn!

xe = 1 + x + x² + x³+... = 2! 3!

y(x,y)

Rectangular

x

Polar(r,θ)

rθ

x xcosh x = e + e 2

x xsinh x = e - e 2

x xtanhx = e - e x -xe + e

sech x = 2 x xe + ecsch x = 2 x xe - e

x xcothx = e + e x -xe - e

Standard Hyperbolic Trig Derivatives

(d/dx)(csch u) = - csch u coth u (du/dx)

(d/dx)(sech u) = - sech u tanh u (du/dx)

(d/dx)(coth u) = - csch ² u (du/dx)

(d/dx)(tanh u) = sech ² u (du/dx)

(d/dx)(cosh u) = sinh u (du/dx)

(d/dx)(sinh u) = cosh u (du/dx)

sinh 2x = 2 sinh x cosh xcosh 2x = cosh ² x - sinh ² x

cosh ² x = cosh 2x+1 2

sinh ² x = cosh 2x - 1 2

cosh² x - sinh² x = 1tanh² x + sech ² x = 1

coth² x - csch² x = 1

Inverse Hyperbolic Trig Derivatives

sech u = , 0<u<1-1 1

u u² - 1d

dx

dudx

csch u = , u ≠0-1 1

|u| u² - 1d

dx

dudx

tanh u = , |u|<1-1 1

1 - u²d

dxdudx

coth u = , |u|>1-1 1

1 - u²d

dxdudx

sinh u =-1 1

u² + 1d

dx

dudx

cosh u = , u > 1-1 1

u² - 1d

dx

dudx

Every function splitsinto Even and Odd parts:

f(x) = + f(x) + f(-x)

2 f(x) - f(-x)

2even odd

Hyperbolic Trig Integrals

∫ sinh u du = cosh u + C

∫ cosh u du = sinh u + C

∫ sech² u du = tanh u + C

∫ csch² u du = - coth u + C

∫ sech x tanh u du =- sech u + C

∫ csch x coth u du = - csch u + C

A BBC

BC

calculus

a definitive sheetby chad valencia, ucla mathematics major

version 2.0.2000, rev 1

Σn = 0

an

∞last number

sequence

first number

Greek lettersigma(sum of)

improper integrals

sequences

taylor/maclaurin series

hyperbolic trig functions

series

Polar

parametric

vectors

extraneous bc concepts

personal notes

dydx

=

dydt

dxdt

First Derivative:

=dxdt

d²ydx²

Second Derivative:

dydx( )d

a

b dydt

dxdt( ) ( )+

2 2

Arc Length:

dt

Surface Area (x-axis):

a

b dydt

dxdt( ) ( )+

2 22 y dt

Surface Area (y-axis):

a

b dydt

dxdt( ) ( )+

2 22 x dt

Vector Valued Functionsr(t) = x(t)i + y(t)j

r'(t) = x'(t)i + y'(t)j∫r(t)dt = (∫x(t)dt + c)i + (∫y(t)dt + c)j

Velocity Equation for Vectors:(-½ g t² + S )j + V t0 0

where v = #( cos t i + sin t j)0

# = initial velocity/muzzle speed

Notation

v = ai + bj<a,b>

Vector Length

(magnitude/norm):

||v|| = a² + b²

Unit Vector:v

||v||

T(t) = r ' (t) ||r ' (t)||

If T = u1i + u2j N = -u2i + u1j

Unit Tangent andUnit Normal Vectors dy

dtdxdt( ) ( )+

2 2

Speed

Limits of Common Sequences:

Convergence/Divergence:

Let L be a finite number

limn → ∞

an= LConvergent

limn → ∞

an≠ LDivergent

limn → ∞

limn → ∞nn

1nn= = 1

limn → ∞

n

= e(1+ )xnxn

x

limn → ∞

= 0nx

n!

limn → ∞

ln nn = 0

limn → ∞

x = 0n |x|<1

(fraction)

1

∞1xp

Converges if p > 1Diverges if 0 < p < 1

P Series Test: Comparison Test

if f(x) < convergent function,f(x) is convergent

if f(x) > divergent function,f(x) is divergent

Limit Comparison Test:

limx → ∞

f(x)g(x)= L

Let f(x) be a known convergent or divergent function:

0 < L < ∞

f(x) & g(x) both converge or both diverge

limn → ∞

n= e(1+ )1

n

Surface Area:

a

b dydx( )+

2dx2π y 1

WorkWork = Force x Distance

= Density x Volume x Distance= Density x Sum of Areas x Distance

Hooke’s Law for SpringsForce = f(x) = kx

(k is a constant, x is the distance)Work = ò f(x)dx, from position A to position B

Newton’s Law of Cooling

T - T = (T - T )eS 0 S

- kt

T = temperature of object at a given timeTs = temperature of surroundings

To = temperature at time zerot = time

Arc Length (y-axis):

c

d dxdy( )+

2dy1

Arc Length (x-axis):

a

b dydx( )+

2dx1

Circlesr = a cos θr = a sin θ

Basic Shapes(pink = cosine, blue = sine)

Lemniscatesr² = a cos θr² = a sin θ

Spiral ofArchimedes

r = aθ

Limaconsw/ Inner Loopr = a + b cos θr = a + b sin θ

|a|<|b|

Limaconsw/ Dimple

r = a + b cos θr = a + b sin θ

|a|>|b|

Cardioidsr = a + b cos θr = a + b sin θ

|a|=|b|

Rosesr = a cos bθr = a sin bθ

If b is odd, b = number of petalsIf b is even, 2b = number of petals

Polar Conversion (x,y) <-> (r,θ)

x² + y² = r²x = r cos θy = r sin θ

tan θ =yx

Polar Surface Area (y-axis):

α

β 22π r cos θ r +( )dr

dθ

2dθ

Polar Sloper ' sin θ + r cos θr ' cos θ - r sin θ

Polar Surface Area (x-axis):

α

β 2

2π r sin θ r +( )drdθ

2dθ

Polar Arc Length:

α

β2

r+( )drdθ

2dθ

Polar Area:

α

β 2r dθ

12

Geometric Series

Finite Series: Infinite Series:∞

n = 1

a r =n-1 a1-r

|r| <1if |r| > 1, series diverges

=Sn

a(1-r )(1-r)

n

Nth Term Test for Divergence

limn → ∞

∞

n = k

an an ≠ 0 series is divergent

Given: If:

Integral Test∞

n = k

an

∞

k∫ ax dx

If integral converges, series convergesIf integral diverges, series diverges

Comparison Test

if Σan < convergent series,

Σan is convergentif Σan > divergent series,

Σan is divergent

Limit Comparison Test

Let Σbn be a knownconvergent or divergent series:

limn → ∞

Σan

Σbn

= ρ 0 < ρ < ∞

Σan & Σbn

both converge or both diverge

∞

n = 1

an(-1)n-1

limn → ∞

3. an = 0

Alternating Series Test (AST)

Converges if:1. All terms are positive

2. an > an+1 for all n.

Power Series1. Use ratio test on theabsolute value of the series.

2. Set ρ < 1 to find the interval

3. To check bounds, pluginto original equation4. Take ½ of the interval to find radius of convergence

Error (Alternating Series/Taylor)Error = Actual - ApproximateError < First Unused Term

Root Testn

limn → ∞ anρ =

Ratio Test

limn → ∞ρ =

an+1

an Converges if ρ < 1No conclusion if ρ = 1Divergent if ρ > 1

2 3 n nP(x) = f (a) + f ' (a)(x - a) + f ''(a)(x - a) + f ''' (a)(x - a) + … + f (a)(x - a) 2! 3! n!

Taylor Polynomial

Σn = 0

∞ n 2n( - 1) x(2n)!

2 4 6cos x = 1- x + x - x + …= 2! 4! 6!

Common MacLaurin Series

Σn = 0

∞ n 2n+1( - 1) x(2n+1)!

3 5 7sin x = x - x + x - x + … = 3! 5! 7!

Σn = 0

∞ n nf ( a)(x - a) n!

Σn = 0

∞

n n(-1) x 1 = 1 - x + x² - x³+... =1+x

Σn = 0

∞

nx 1 = 1 + x + x² + x³+... =1- x

Σn = 0

∞ nxn!

xe = 1 + x + x² + x³+... = 2! 3!

y(x,y)

Rectangular

x

Polar(r,θ)

rθ

x xcosh x = e + e 2

x xsinh x = e - e 2

x xtanhx = e - e x -xe + e

sech x = 2 x xe + ecsch x = 2 x xe - e

x xcothx = e + e x -xe - e

Standard Hyperbolic Trig Derivatives

(d/dx)(csch u) = - csch u coth u (du/dx)

(d/dx)(sech u) = - sech u tanh u (du/dx)

(d/dx)(coth u) = - csch ² u (du/dx)

(d/dx)(tanh u) = sech ² u (du/dx)

(d/dx)(cosh u) = sinh u (du/dx)

(d/dx)(sinh u) = cosh u (du/dx)

sinh 2x = 2 sinh x cosh xcosh 2x = cosh ² x - sinh ² x

cosh ² x = cosh 2x+1 2

sinh ² x = cosh 2x - 1 2

cosh² x - sinh² x = 1tanh² x + sech ² x = 1

coth² x - csch² x = 1

Inverse Hyperbolic Trig Derivatives

sech u = , 0<u<1-1 1

u u² - 1d

dx

dudx

csch u = , u ≠0-1 1

|u| u² - 1d

dx

dudx

tanh u = , |u|<1-1 1

1 - u²d

dxdudx

coth u = , |u|>1-1 1

1 - u²d

dxdudx

sinh u =-1 1

u² + 1d

dx

dudx

cosh u = , u > 1-1 1

u² - 1d

dx

dudx

Every function splitsinto Even and Odd parts:

f(x) = + f(x) + f(-x)

2 f(x) - f(-x)

2even odd

Hyperbolic Trig Integrals

∫ sinh u du = cosh u + C

∫ cosh u du = sinh u + C

∫ sech² u du = tanh u + C

∫ csch² u du = - coth u + C

∫ sech x tanh u du =- sech u + C

∫ csch x coth u du = - csch u + C

A BBC

BC

calculus

a definitive sheetby chad valencia, ucla mathematics major

version 2.0.2000, rev 1

Σn = 0

an

∞last number

sequence

first number

Greek lettersigma(sum of)

improper integrals

sequences

taylor/maclaurin series

hyperbolic trig functions

series

Polar

parametric

vectors

extraneous bc concepts

personal notes

dydx

=

dydt

dxdt

First Derivative:

=dxdt

d²ydx²

Second Derivative:

dydx( )d

a

b dydt

dxdt( ) ( )+

2 2

Arc Length:

dt

Surface Area (x-axis):

a

b dydt

dxdt( ) ( )+

2 22 y dt

Surface Area (y-axis):

a

b dydt

dxdt( ) ( )+

2 22 x dt

Vector Valued Functionsr(t) = x(t)i + y(t)j

r'(t) = x'(t)i + y'(t)j∫r(t)dt = (∫x(t)dt + c)i + (∫y(t)dt + c)j

Velocity Equation for Vectors:(-½ g t² + S )j + V t0 0

where v = #( cos t i + sin t j)0

# = initial velocity/muzzle speed

Notation

v = ai + bj<a,b>

Vector Length

(magnitude/norm):

||v|| = a² + b²

Unit Vector:v

||v||

T(t) = r ' (t) ||r ' (t)||

If T = u1i + u2j N = -u2i + u1j

Unit Tangent andUnit Normal Vectors dy

dtdxdt( ) ( )+

2 2

Speed

Limits of Common Sequences:

Convergence/Divergence:

Let L be a finite number

limn → ∞

an= LConvergent

limn → ∞

an≠ LDivergent

limn → ∞

limn → ∞nn

1nn= = 1

limn → ∞

n

= e(1+ )xnxn

x

limn → ∞

= 0nx

n!

limn → ∞

ln nn = 0

limn → ∞

x = 0n |x|<1

(fraction)

1

∞1xp

Converges if p > 1Diverges if 0 < p < 1

P Series Test: Comparison Test

if f(x) < convergent function,f(x) is convergent

if f(x) > divergent function,f(x) is divergent

Limit Comparison Test:

limx → ∞

f(x)g(x)= L

Let f(x) be a known convergent or divergent function:

0 < L < ∞

f(x) & g(x) both converge or both diverge

limn → ∞

n= e(1+ )1

n

Surface Area:

a

b dydx( )+

2dx2π y 1

WorkWork = Force x Distance

= Density x Volume x Distance= Density x Sum of Areas x Distance

Hooke’s Law for SpringsForce = f(x) = kx

(k is a constant, x is the distance)Work = ò f(x)dx, from position A to position B

Newton’s Law of Cooling

T - T = (T - T )eS 0 S

- kt

T = temperature of object at a given timeTs = temperature of surroundings

To = temperature at time zerot = time

Arc Length (y-axis):

c

d dxdy( )+

2dy1

Arc Length (x-axis):

a

b dydx( )+

2dx1

Circlesr = a cos θr = a sin θ

Basic Shapes(pink = cosine, blue = sine)

Lemniscatesr² = a cos θr² = a sin θ

Spiral ofArchimedes

r = aθ

Limaconsw/ Inner Loopr = a + b cos θr = a + b sin θ

|a|<|b|

Limaconsw/ Dimple

r = a + b cos θr = a + b sin θ

|a|>|b|

Cardioidsr = a + b cos θr = a + b sin θ

|a|=|b|

Rosesr = a cos bθr = a sin bθ

If b is odd, b = number of petalsIf b is even, 2b = number of petals

Polar Conversion (x,y) <-> (r,θ)

x² + y² = r²x = r cos θy = r sin θ

tan θ =yx

Polar Surface Area (y-axis):

α

β 22π r cos θ r +( )dr

dθ

2dθ

Polar Sloper ' sin θ + r cos θr ' cos θ - r sin θ

Polar Surface Area (x-axis):

α

β 2

2π r sin θ r +( )drdθ

2dθ

Polar Arc Length:

α

β2

r+( )drdθ

2dθ

Polar Area:

α

β 2r dθ

12

Geometric Series

Finite Series: Infinite Series:∞

n = 1

a r =n-1 a1-r

|r| <1if |r| > 1, series diverges

=Sn

a(1-r )(1-r)

n

Nth Term Test for Divergence

limn → ∞

∞

n = k

an an ≠ 0 series is divergent

Given: If:

Integral Test∞

n = k

an

∞

k∫ ax dx

If integral converges, series convergesIf integral diverges, series diverges

Comparison Test

if Σan < convergent series,

Σan is convergentif Σan > divergent series,

Σan is divergent

Limit Comparison Test

Let Σbn be a knownconvergent or divergent series:

limn → ∞

Σan

Σbn

= ρ 0 < ρ < ∞

Σan & Σbn

both converge or both diverge

∞

n = 1

an(-1)n-1

limn → ∞

3. an = 0

Alternating Series Test (AST)

Converges if:1. All terms are positive

2. an > an+1 for all n.

Power Series1. Use ratio test on theabsolute value of the series.

2. Set ρ < 1 to find the interval

3. To check bounds, pluginto original equation4. Take ½ of the interval to find radius of convergence

Error (Alternating Series/Taylor)Error = Actual - ApproximateError < First Unused Term

Root Testn

limn → ∞ anρ =

Ratio Test

limn → ∞ρ =

an+1

an Converges if ρ < 1No conclusion if ρ = 1Divergent if ρ > 1

2 3 n nP(x) = f (a) + f ' (a)(x - a) + f ''(a)(x - a) + f ''' (a)(x - a) + … + f (a)(x - a) 2! 3! n!

Taylor Polynomial

Σn = 0

∞ n 2n( - 1) x(2n)!

2 4 6cos x = 1- x + x - x + …= 2! 4! 6!

Common MacLaurin Series

Σn = 0

∞ n 2n+1( - 1) x(2n+1)!

3 5 7sin x = x - x + x - x + … = 3! 5! 7!

Σn = 0

∞ n nf ( a)(x - a) n!

Σn = 0

∞

n n(-1) x 1 = 1 - x + x² - x³+... =1+x

Σn = 0

∞

nx 1 = 1 + x + x² + x³+... =1- x

Σn = 0

∞ nxn!

xe = 1 + x + x² + x³+... = 2! 3!

y(x,y)

Rectangular

x

Polar(r,θ)

rθ

x xcosh x = e + e 2

x xsinh x = e - e 2

x xtanhx = e - e x -xe + e

sech x = 2 x xe + ecsch x = 2 x xe - e

x xcothx = e + e x -xe - e

Standard Hyperbolic Trig Derivatives

(d/dx)(csch u) = - csch u coth u (du/dx)

(d/dx)(sech u) = - sech u tanh u (du/dx)

(d/dx)(coth u) = - csch ² u (du/dx)

(d/dx)(tanh u) = sech ² u (du/dx)

(d/dx)(cosh u) = sinh u (du/dx)

(d/dx)(sinh u) = cosh u (du/dx)

sinh 2x = 2 sinh x cosh xcosh 2x = cosh ² x - sinh ² x

cosh ² x = cosh 2x+1 2

sinh ² x = cosh 2x - 1 2

cosh² x - sinh² x = 1tanh² x + sech ² x = 1

coth² x - csch² x = 1

Inverse Hyperbolic Trig Derivatives

sech u = , 0<u<1-1 1

u u² - 1d

dx

dudx

csch u = , u ≠0-1 1

|u| u² - 1d

dx

dudx

tanh u = , |u|<1-1 1

1 - u²d

dxdudx

coth u = , |u|>1-1 1

1 - u²d

dxdudx

sinh u =-1 1

u² + 1d

dx

dudx

cosh u = , u > 1-1 1

u² - 1d

dx

dudx

Every function splitsinto Even and Odd parts:

f(x) = + f(x) + f(-x)

2 f(x) - f(-x)

2even odd

Hyperbolic Trig Integrals

∫ sinh u du = cosh u + C

∫ cosh u du = sinh u + C

∫ sech² u du = tanh u + C

∫ csch² u du = - coth u + C

∫ sech x tanh u du =- sech u + C

∫ csch x coth u du = - csch u + C

A BBC

BC

calculus

a definitive sheetby chad valencia, ucla mathematics major

version 2.0.2000, rev 1

Σn = 0

an

∞last number

sequence

first number

Greek lettersigma(sum of)

improper integrals

sequences

taylor/maclaurin series

hyperbolic trig functions

series

Polar

parametric

vectors

extraneous bc concepts

personal notes

dydx

=

dydt

dxdt

First Derivative:

=dxdt

d²ydx²

Second Derivative:

dydx( )d

a

b dydt

dxdt( ) ( )+

2 2

Arc Length:

dt

Surface Area (x-axis):

a

b dydt

dxdt( ) ( )+

2 22 y dt

Surface Area (y-axis):

a

b dydt

dxdt( ) ( )+

2 22 x dt

Vector Valued Functionsr(t) = x(t)i + y(t)j

r'(t) = x'(t)i + y'(t)j∫r(t)dt = (∫x(t)dt + c)i + (∫y(t)dt + c)j

Velocity Equation for Vectors:(-½ g t² + S )j + V t0 0

where v = #( cos t i + sin t j)0

# = initial velocity/muzzle speed

Notation

v = ai + bj<a,b>

Vector Length

(magnitude/norm):

||v|| = a² + b²

Unit Vector:v

||v||

T(t) = r ' (t) ||r ' (t)||

If T = u1i + u2j N = -u2i + u1j

Unit Tangent andUnit Normal Vectors dy

dtdxdt( ) ( )+

2 2

Speed

Limits of Common Sequences:

Convergence/Divergence:

Let L be a finite number

limn → ∞

an= LConvergent

limn → ∞

an≠ LDivergent

limn → ∞

limn → ∞nn

1nn= = 1

limn → ∞

n

= e(1+ )xnxn

x

limn → ∞

= 0nx

n!

limn → ∞

ln nn = 0

limn → ∞

x = 0n |x|<1

(fraction)

1

∞1xp

Converges if p > 1Diverges if 0 < p < 1

P Series Test: Comparison Test

if f(x) < convergent function,f(x) is convergent

if f(x) > divergent function,f(x) is divergent

Limit Comparison Test:

limx → ∞

f(x)g(x)= L

Let f(x) be a known convergent or divergent function:

0 < L < ∞

f(x) & g(x) both converge or both diverge

limn → ∞

n= e(1+ )1

n

Surface Area:

a

b dydx( )+

2dx2π y 1

WorkWork = Force x Distance

= Density x Volume x Distance= Density x Sum of Areas x Distance

Hooke’s Law for SpringsForce = f(x) = kx

(k is a constant, x is the distance)Work = ò f(x)dx, from position A to position B

Newton’s Law of Cooling

T - T = (T - T )eS 0 S

- kt

T = temperature of object at a given timeTs = temperature of surroundings

To = temperature at time zerot = time

Arc Length (y-axis):

c

d dxdy( )+

2dy1

Arc Length (x-axis):

a

b dydx( )+

2dx1

Circlesr = a cos θr = a sin θ

Basic Shapes(pink = cosine, blue = sine)

Lemniscatesr² = a cos θr² = a sin θ

Spiral ofArchimedes

r = aθ

Limaconsw/ Inner Loopr = a + b cos θr = a + b sin θ

|a|<|b|

Limaconsw/ Dimple

r = a + b cos θr = a + b sin θ

|a|>|b|

Cardioidsr = a + b cos θr = a + b sin θ

|a|=|b|

Rosesr = a cos bθr = a sin bθ

If b is odd, b = number of petalsIf b is even, 2b = number of petals

Polar Conversion (x,y) <-> (r,θ)

x² + y² = r²x = r cos θy = r sin θ

tan θ =yx

Polar Surface Area (y-axis):

α

β 22π r cos θ r +( )dr

dθ

2dθ

Polar Sloper ' sin θ + r cos θr ' cos θ - r sin θ

Polar Surface Area (x-axis):

α

β 2

2π r sin θ r +( )drdθ

2dθ

Polar Arc Length:

α

β2

r+( )drdθ

2dθ

Polar Area:

α

β 2r dθ

12

Geometric Series

Finite Series: Infinite Series:∞

n = 1

a r =n-1 a1-r

|r| <1if |r| > 1, series diverges

=Sn

a(1-r )(1-r)

n

Nth Term Test for Divergence

limn → ∞

∞

n = k

an an ≠ 0 series is divergent

Given: If:

Integral Test∞

n = k

an

∞

k∫ ax dx

If integral converges, series convergesIf integral diverges, series diverges

Comparison Test

if Σan < convergent series,

Σan is convergentif Σan > divergent series,

Σan is divergent

Limit Comparison Test

Let Σbn be a knownconvergent or divergent series:

limn → ∞

Σan

Σbn

= ρ 0 < ρ < ∞

Σan & Σbn

both converge or both diverge

∞

n = 1

an(-1)n-1

limn → ∞

3. an = 0

Alternating Series Test (AST)

Converges if:1. All terms are positive

2. an > an+1 for all n.

Power Series1. Use ratio test on theabsolute value of the series.

2. Set ρ < 1 to find the interval

3. To check bounds, pluginto original equation4. Take ½ of the interval to find radius of convergence

Error (Alternating Series/Taylor)Error = Actual - ApproximateError < First Unused Term

Root Testn

limn → ∞ anρ =

Ratio Test

limn → ∞ρ =

an+1

an Converges if ρ < 1No conclusion if ρ = 1Divergent if ρ > 1

2 3 n nP(x) = f (a) + f ' (a)(x - a) + f ''(a)(x - a) + f ''' (a)(x - a) + … + f (a)(x - a) 2! 3! n!

Taylor Polynomial

Σn = 0

∞ n 2n( - 1) x(2n)!

2 4 6cos x = 1- x + x - x + …= 2! 4! 6!

Common MacLaurin Series

Σn = 0

∞ n 2n+1( - 1) x(2n+1)!

3 5 7sin x = x - x + x - x + … = 3! 5! 7!

Σn = 0

∞ n nf ( a)(x - a) n!

Σn = 0

∞

n n(-1) x 1 = 1 - x + x² - x³+... =1+x

Σn = 0

∞

nx 1 = 1 + x + x² + x³+... =1- x

Σn = 0

∞ nxn!

xe = 1 + x + x² + x³+... = 2! 3!

y(x,y)

Rectangular

x

Polar(r,θ)

rθ

x xcosh x = e + e 2

x xsinh x = e - e 2

x xtanhx = e - e x -xe + e

sech x = 2 x xe + ecsch x = 2 x xe - e

x xcothx = e + e x -xe - e

Standard Hyperbolic Trig Derivatives

(d/dx)(csch u) = - csch u coth u (du/dx)

(d/dx)(sech u) = - sech u tanh u (du/dx)

(d/dx)(coth u) = - csch ² u (du/dx)

(d/dx)(tanh u) = sech ² u (du/dx)

(d/dx)(cosh u) = sinh u (du/dx)

(d/dx)(sinh u) = cosh u (du/dx)

sinh 2x = 2 sinh x cosh xcosh 2x = cosh ² x - sinh ² x

cosh ² x = cosh 2x+1 2

sinh ² x = cosh 2x - 1 2

cosh² x - sinh² x = 1tanh² x + sech ² x = 1

coth² x - csch² x = 1

Inverse Hyperbolic Trig Derivatives

sech u = , 0<u<1-1 1

u u² - 1d

dx

dudx

csch u = , u ≠0-1 1

|u| u² - 1d

dx

dudx

tanh u = , |u|<1-1 1

1 - u²d

dxdudx

coth u = , |u|>1-1 1

1 - u²d

dxdudx

sinh u =-1 1

u² + 1d

dx

dudx

cosh u = , u > 1-1 1

u² - 1d

dx

dudx

Every function splitsinto Even and Odd parts:

f(x) = + f(x) + f(-x)

2 f(x) - f(-x)

2even odd

Hyperbolic Trig Integrals

∫ sinh u du = cosh u + C

∫ cosh u du = sinh u + C

∫ sech² u du = tanh u + C

∫ csch² u du = - coth u + C

∫ sech x tanh u du =- sech u + C

∫ csch x coth u du = - csch u + C

A BBC

BC

calculus

a definitive sheetby chad valencia, ucla mathematics major

version 2.0.2000, rev 1

Σn = 0

an

∞last number

sequence

first number

Greek lettersigma(sum of)

improper integrals

sequences

taylor/maclaurin series

hyperbolic trig functions

series

Polar

parametric

vectors

extraneous bc concepts

personal notes

dydx

=

dydt

dxdt

First Derivative:

=dxdt

d²ydx²

Second Derivative:

dydx( )d

a

b dydt

dxdt( ) ( )+

2 2

Arc Length:

dt

Surface Area (x-axis):

a

b dydt

dxdt( ) ( )+

2 22 y dt

Surface Area (y-axis):

a

b dydt

dxdt( ) ( )+

2 22 x dt

Vector Valued Functionsr(t) = x(t)i + y(t)j

r'(t) = x'(t)i + y'(t)j∫r(t)dt = (∫x(t)dt + c)i + (∫y(t)dt + c)j

Velocity Equation for Vectors:(-½ g t² + S )j + V t0 0

where v = #( cos t i + sin t j)0

# = initial velocity/muzzle speed

Notation

v = ai + bj<a,b>

Vector Length

(magnitude/norm):

||v|| = a² + b²

Unit Vector:v

||v||

T(t) = r ' (t) ||r ' (t)||

If T = u1i + u2j N = -u2i + u1j

Unit Tangent andUnit Normal Vectors dy

dtdxdt( ) ( )+

2 2

Speed

Limits of Common Sequences:

Convergence/Divergence:

Let L be a finite number

limn → ∞

an= LConvergent

limn → ∞

an≠ LDivergent

limn → ∞

limn → ∞nn

1nn= = 1

limn → ∞

n

= e(1+ )xnxn

x

limn → ∞

= 0nx

n!

limn → ∞

ln nn = 0

limn → ∞

x = 0n |x|<1

(fraction)

1

∞1xp

Converges if p > 1Diverges if 0 < p < 1

P Series Test: Comparison Test

if f(x) < convergent function,f(x) is convergent

if f(x) > divergent function,f(x) is divergent

Limit Comparison Test:

limx → ∞

f(x)g(x)= L

Let f(x) be a known convergent or divergent function:

0 < L < ∞

f(x) & g(x) both converge or both diverge

limn → ∞

n= e(1+ )1

n

Surface Area:

a

b dydx( )+

2dx2π y 1

WorkWork = Force x Distance

= Density x Volume x Distance= Density x Sum of Areas x Distance

Hooke’s Law for SpringsForce = f(x) = kx

(k is a constant, x is the distance)Work = ò f(x)dx, from position A to position B

Newton’s Law of Cooling

T - T = (T - T )eS 0 S

- kt

T = temperature of object at a given timeTs = temperature of surroundings

To = temperature at time zerot = time

Arc Length (y-axis):

c

d dxdy( )+

2dy1

Arc Length (x-axis):

a

b dydx( )+

2dx1

Circlesr = a cos θr = a sin θ

Basic Shapes(pink = cosine, blue = sine)

Lemniscatesr² = a cos θr² = a sin θ

Spiral ofArchimedes

r = aθ

Limaconsw/ Inner Loopr = a + b cos θr = a + b sin θ

|a|<|b|

Limaconsw/ Dimple

r = a + b cos θr = a + b sin θ

|a|>|b|

Cardioidsr = a + b cos θr = a + b sin θ

|a|=|b|

Rosesr = a cos bθr = a sin bθ

If b is odd, b = number of petalsIf b is even, 2b = number of petals

Polar Conversion (x,y) <-> (r,θ)

x² + y² = r²x = r cos θy = r sin θ

tan θ =yx

Polar Surface Area (y-axis):

α

β 22π r cos θ r +( )dr

dθ

2dθ

Polar Sloper ' sin θ + r cos θr ' cos θ - r sin θ

Polar Surface Area (x-axis):

α

β 2

2π r sin θ r +( )drdθ

2dθ

Polar Arc Length:

α

β2

r+( )drdθ

2dθ

Polar Area:

α

β 2r dθ

12

Geometric Series

Finite Series: Infinite Series:∞

n = 1

a r =n-1 a1-r

|r| <1if |r| > 1, series diverges

=Sn

a(1-r )(1-r)

n

Nth Term Test for Divergence

limn → ∞

∞

n = k

an an ≠ 0 series is divergent

Given: If:

Integral Test∞

n = k

an

∞

k∫ ax dx

If integral converges, series convergesIf integral diverges, series diverges

Comparison Test

if Σan < convergent series,

Σan is convergentif Σan > divergent series,

Σan is divergent

Limit Comparison Test

Let Σbn be a knownconvergent or divergent series:

limn → ∞

Σan

Σbn

= ρ 0 < ρ < ∞

Σan & Σbn

both converge or both diverge

∞

n = 1

an(-1)n-1

limn → ∞

3. an = 0

Alternating Series Test (AST)

Converges if:1. All terms are positive

2. an > an+1 for all n.

Power Series1. Use ratio test on theabsolute value of the series.

2. Set ρ < 1 to find the interval

3. To check bounds, pluginto original equation4. Take ½ of the interval to find radius of convergence

Error (Alternating Series/Taylor)Error = Actual - ApproximateError < First Unused Term

Root Testn

limn → ∞ anρ =

Ratio Test

limn → ∞ρ =

an+1

an Converges if ρ < 1No conclusion if ρ = 1Divergent if ρ > 1

2 3 n nP(x) = f (a) + f ' (a)(x - a) + f ''(a)(x - a) + f ''' (a)(x - a) + … + f (a)(x - a) 2! 3! n!

Taylor Polynomial

Σn = 0

∞ n 2n( - 1) x(2n)!

2 4 6cos x = 1- x + x - x + …= 2! 4! 6!

Common MacLaurin Series

Σn = 0

∞ n 2n+1( - 1) x(2n+1)!

3 5 7sin x = x - x + x - x + … = 3! 5! 7!

Σn = 0

∞ n nf ( a)(x - a) n!

Σn = 0

∞

n n(-1) x 1 = 1 - x + x² - x³+... =1+x

Σn = 0

∞

nx 1 = 1 + x + x² + x³+... =1- x

Σn = 0

∞ nxn!

xe = 1 + x + x² + x³+... = 2! 3!

y(x,y)

Rectangular

x

Polar(r,θ)

rθ

x xcosh x = e + e 2

x xsinh x = e - e 2

x xtanhx = e - e x -xe + e

sech x = 2 x xe + ecsch x = 2 x xe - e

x xcothx = e + e x -xe - e

Standard Hyperbolic Trig Derivatives

(d/dx)(csch u) = - csch u coth u (du/dx)

(d/dx)(sech u) = - sech u tanh u (du/dx)

(d/dx)(coth u) = - csch ² u (du/dx)

(d/dx)(tanh u) = sech ² u (du/dx)

(d/dx)(cosh u) = sinh u (du/dx)

(d/dx)(sinh u) = cosh u (du/dx)

sinh 2x = 2 sinh x cosh xcosh 2x = cosh ² x - sinh ² x

cosh ² x = cosh 2x+1 2

sinh ² x = cosh 2x - 1 2

cosh² x - sinh² x = 1tanh² x + sech ² x = 1

coth² x - csch² x = 1

Inverse Hyperbolic Trig Derivatives

sech u = , 0<u<1-1 1

u u² - 1d

dx

dudx

csch u = , u ≠0-1 1

|u| u² - 1d

dx

dudx

tanh u = , |u|<1-1 1

1 - u²d

dxdudx

coth u = , |u|>1-1 1

1 - u²d

dxdudx

sinh u =-1 1

u² + 1d

dx

dudx

cosh u = , u > 1-1 1

u² - 1d

dx

dudx

Every function splitsinto Even and Odd parts:

f(x) = + f(x) + f(-x)

2 f(x) - f(-x)

2even odd

Hyperbolic Trig Integrals

∫ sinh u du = cosh u + C

∫ cosh u du = sinh u + C

∫ sech² u du = tanh u + C

∫ csch² u du = - coth u + C

∫ sech x tanh u du =- sech u + C

∫ csch x coth u du = - csch u + C

A BBC

BC

calculus

a definitive sheetby chad valencia, ucla mathematics major

version 2.0.2000, rev 1

Σn = 0

an

∞last number

sequence

first number

Greek lettersigma(sum of)

improper integrals

sequences

taylor/maclaurin series

hyperbolic trig functions

series

Polar

parametric

vectors

extraneous bc concepts

personal notes

dydx

=

dydt

dxdt

First Derivative:

=dxdt

d²ydx²

Second Derivative:

dydx( )d

a

b dydt

dxdt( ) ( )+

2 2

Arc Length:

dt

Surface Area (x-axis):

a

b dydt

dxdt( ) ( )+

2 22 y dt

Surface Area (y-axis):

a

b dydt

dxdt( ) ( )+

2 22 x dt

Vector Valued Functionsr(t) = x(t)i + y(t)j

r'(t) = x'(t)i + y'(t)j∫r(t)dt = (∫x(t)dt + c)i + (∫y(t)dt + c)j

Velocity Equation for Vectors:(-½ g t² + S )j + V t0 0

where v = #( cos t i + sin t j)0

# = initial velocity/muzzle speed

Notation

v = ai + bj<a,b>

Vector Length

(magnitude/norm):

||v|| = a² + b²

Unit Vector:v

||v||

T(t) = r ' (t) ||r ' (t)||

If T = u1i + u2j N = -u2i + u1j

Unit Tangent andUnit Normal Vectors dy

dtdxdt( ) ( )+

2 2

Speed

Limits of Common Sequences:

Convergence/Divergence:

Let L be a finite number

limn → ∞

an= LConvergent

limn → ∞

an≠ LDivergent

limn → ∞

limn → ∞nn

1nn= = 1

limn → ∞

n

= e(1+ )xnxn

x

limn → ∞

= 0nx

n!

limn → ∞

ln nn = 0

limn → ∞

x = 0n |x|<1

(fraction)

1

∞1xp

Converges if p > 1Diverges if 0 < p < 1

P Series Test: Comparison Test

if f(x) < convergent function,f(x) is convergent

if f(x) > divergent function,f(x) is divergent

Limit Comparison Test:

limx → ∞

f(x)g(x)= L

Let f(x) be a known convergent or divergent function:

0 < L < ∞

f(x) & g(x) both converge or both diverge

limn → ∞

n= e(1+ )1

n

Surface Area:

a

b dydx( )+

2dx2π y 1

WorkWork = Force x Distance

= Density x Volume x Distance= Density x Sum of Areas x Distance

Hooke’s Law for SpringsForce = f(x) = kx

(k is a constant, x is the distance)Work = ò f(x)dx, from position A to position B

Newton’s Law of Cooling

T - T = (T - T )eS 0 S

- kt

T = temperature of object at a given timeTs = temperature of surroundings

To = temperature at time zerot = time

Arc Length (y-axis):

c

d dxdy( )+

2dy1

Arc Length (x-axis):

a

b dydx( )+

2dx1

Circlesr = a cos θr = a sin θ

Basic Shapes(pink = cosine, blue = sine)

Lemniscatesr² = a cos θr² = a sin θ

Spiral ofArchimedes

r = aθ

Limaconsw/ Inner Loopr = a + b cos θr = a + b sin θ

|a|<|b|

Limaconsw/ Dimple

r = a + b cos θr = a + b sin θ

|a|>|b|

Cardioidsr = a + b cos θr = a + b sin θ

|a|=|b|

Rosesr = a cos bθr = a sin bθ

If b is odd, b = number of petalsIf b is even, 2b = number of petals

Polar Conversion (x,y) <-> (r,θ)

x² + y² = r²x = r cos θy = r sin θ

tan θ =yx

Polar Surface Area (y-axis):

α

β 22π r cos θ r +( )dr

dθ

2dθ

Polar Sloper ' sin θ + r cos θr ' cos θ - r sin θ

Polar Surface Area (x-axis):

α

β 2

2π r sin θ r +( )drdθ

2dθ

Polar Arc Length:

α

β2

r+( )drdθ

2dθ

Polar Area:

α

β 2r dθ

12

Geometric Series

Finite Series: Infinite Series:∞

n = 1

a r =n-1 a1-r

|r| <1if |r| > 1, series diverges

=Sn

a(1-r )(1-r)

n

Nth Term Test for Divergence

limn → ∞

∞

n = k

an an ≠ 0 series is divergent

Given: If:

Integral Test∞

n = k

an

∞

k∫ ax dx

If integral converges, series convergesIf integral diverges, series diverges

Comparison Test

if Σan < convergent series,

Σan is convergentif Σan > divergent series,

Σan is divergent

Limit Comparison Test

Let Σbn be a knownconvergent or divergent series:

limn → ∞

Σan

Σbn

= ρ 0 < ρ < ∞

Σan & Σbn

both converge or both diverge

∞

n = 1

an(-1)n-1

limn → ∞

3. an = 0

Alternating Series Test (AST)

Converges if:1. All terms are positive

2. an > an+1 for all n.

Power Series1. Use ratio test on theabsolute value of the series.

2. Set ρ < 1 to find the interval

3. To check bounds, pluginto original equation4. Take ½ of the interval to find radius of convergence

Error (Alternating Series/Taylor)Error = Actual - ApproximateError < First Unused Term

Root Testn

limn → ∞ anρ =

Ratio Test

limn → ∞ρ =

an+1

an Converges if ρ < 1No conclusion if ρ = 1Divergent if ρ > 1

2 3 n nP(x) = f (a) + f ' (a)(x - a) + f ''(a)(x - a) + f ''' (a)(x - a) + … + f (a)(x - a) 2! 3! n!

Taylor Polynomial

Σn = 0

∞ n 2n( - 1) x(2n)!

2 4 6cos x = 1- x + x - x + …= 2! 4! 6!

Common MacLaurin Series

Σn = 0

∞ n 2n+1( - 1) x(2n+1)!

3 5 7sin x = x - x + x - x + … = 3! 5! 7!

Σn = 0

∞ n nf ( a)(x - a) n!

Σn = 0

∞

n n(-1) x 1 = 1 - x + x² - x³+... =1+x

Σn = 0

∞

nx 1 = 1 + x + x² + x³+... =1- x

Σn = 0

∞ nxn!

xe = 1 + x + x² + x³+... = 2! 3!

y(x,y)

Rectangular

x

Polar(r,θ)

rθ

x xcosh x = e + e 2

x xsinh x = e - e 2

x xtanhx = e - e x -xe + e

sech x = 2 x xe + ecsch x = 2 x xe - e

x xcothx = e + e x -xe - e

Standard Hyperbolic Trig Derivatives

(d/dx)(csch u) = - csch u coth u (du/dx)

(d/dx)(sech u) = - sech u tanh u (du/dx)

(d/dx)(coth u) = - csch ² u (du/dx)

(d/dx)(tanh u) = sech ² u (du/dx)

(d/dx)(cosh u) = sinh u (du/dx)

(d/dx)(sinh u) = cosh u (du/dx)

sinh 2x = 2 sinh x cosh xcosh 2x = cosh ² x - sinh ² x

cosh ² x = cosh 2x+1 2

sinh ² x = cosh 2x - 1 2

cosh² x - sinh² x = 1tanh² x + sech ² x = 1

coth² x - csch² x = 1

Inverse Hyperbolic Trig Derivatives

sech u = , 0<u<1-1 1

u u² - 1d

dx

dudx

csch u = , u ≠0-1 1

|u| u² - 1d

dx

dudx

tanh u = , |u|<1-1 1

1 - u²d

dxdudx

coth u = , |u|>1-1 1

1 - u²d

dxdudx

sinh u =-1 1

u² + 1d

dx

dudx

cosh u = , u > 1-1 1

u² - 1d

dx

dudx

Every function splitsinto Even and Odd parts:

f(x) = + f(x) + f(-x)

2 f(x) - f(-x)

2even odd

Hyperbolic Trig Integrals

∫ sinh u du = cosh u + C

∫ cosh u du = sinh u + C

∫ sech² u du = tanh u + C

∫ csch² u du = - coth u + C

∫ sech x tanh u du =- sech u + C

∫ csch x coth u du = - csch u + C

A BBC

BC

calculus

a definitive sheetby chad valencia, ucla mathematics major

version 2.0.2000, rev 1

Σn = 0

an

∞last number

sequence

first number

Greek lettersigma(sum of)

improper integrals

sequences

taylor/maclaurin series

hyperbolic trig functions

series

Polar

parametric

vectors

extraneous bc concepts

personal notes

dydx

=

dydt

dxdt

First Derivative:

=dxdt

d²ydx²

Second Derivative:

dydx( )d

a

b dydt

dxdt( ) ( )+

2 2

Arc Length:

dt

Surface Area (x-axis):

a

b dydt

dxdt( ) ( )+

2 22 y dt

Surface Area (y-axis):

a

b dydt

dxdt( ) ( )+

2 22 x dt

Vector Valued Functionsr(t) = x(t)i + y(t)j

r'(t) = x'(t)i + y'(t)j∫r(t)dt = (∫x(t)dt + c)i + (∫y(t)dt + c)j

Velocity Equation for Vectors:(-½ g t² + S )j + V t0 0

where v = #( cos t i + sin t j)0

# = initial velocity/muzzle speed

Notation

v = ai + bj<a,b>

Vector Length

(magnitude/norm):

||v|| = a² + b²

Unit Vector:v

||v||

T(t) = r ' (t) ||r ' (t)||

If T = u1i + u2j N = -u2i + u1j

Unit Tangent andUnit Normal Vectors dy

dtdxdt( ) ( )+

2 2

Speed

Limits of Common Sequences:

Convergence/Divergence:

Let L be a finite number

limn → ∞

an= LConvergent

limn → ∞

an≠ LDivergent

limn → ∞

limn → ∞nn

1nn= = 1

limn → ∞

n

= e(1+ )xnxn

x

limn → ∞

= 0nx

n!

limn → ∞

ln nn = 0

limn → ∞

x = 0n |x|<1

(fraction)

1

∞1xp

Converges if p > 1Diverges if 0 < p < 1

P Series Test: Comparison Test

if f(x) < convergent function,f(x) is convergent

if f(x) > divergent function,f(x) is divergent

Limit Comparison Test:

limx → ∞

f(x)g(x)= L

Let f(x) be a known convergent or divergent function:

0 < L < ∞

f(x) & g(x) both converge or both diverge

limn → ∞

n= e(1+ )1

n

Surface Area:

a

b dydx( )+

2dx2π y 1

WorkWork = Force x Distance

= Density x Volume x Distance= Density x Sum of Areas x Distance

Hooke’s Law for SpringsForce = f(x) = kx

(k is a constant, x is the distance)Work = ò f(x)dx, from position A to position B

Newton’s Law of Cooling

T - T = (T - T )eS 0 S

- kt

T = temperature of object at a given timeTs = temperature of surroundings

To = temperature at time zerot = time

Arc Length (y-axis):

c

d dxdy( )+

2dy1

Arc Length (x-axis):

a

b dydx( )+

2dx1

Circlesr = a cos θr = a sin θ

Basic Shapes(pink = cosine, blue = sine)

Lemniscatesr² = a cos θr² = a sin θ

Spiral ofArchimedes

r = aθ

Limaconsw/ Inner Loopr = a + b cos θr = a + b sin θ

|a|<|b|

Limaconsw/ Dimple

r = a + b cos θr = a + b sin θ

|a|>|b|

Cardioidsr = a + b cos θr = a + b sin θ

|a|=|b|

Rosesr = a cos bθr = a sin bθ

If b is odd, b = number of petalsIf b is even, 2b = number of petals

Polar Conversion (x,y) <-> (r,θ)

x² + y² = r²x = r cos θy = r sin θ

tan θ =yx

Polar Surface Area (y-axis):

α

β 22π r cos θ r +( )dr

dθ

2dθ

Polar Sloper ' sin θ + r cos θr ' cos θ - r sin θ

Polar Surface Area (x-axis):

α

β 2

2π r sin θ r +( )drdθ

2dθ

Polar Arc Length:

α

β2

r+( )drdθ

2dθ

Polar Area:

α

β 2r dθ

12

Geometric Series

Finite Series: Infinite Series:∞

n = 1

a r =n-1 a1-r

|r| <1if |r| > 1, series diverges

=Sn

a(1-r )(1-r)

n

Nth Term Test for Divergence

limn → ∞

∞

n = k

an an ≠ 0 series is divergent

Given: If:

Integral Test∞

n = k

an

∞

k∫ ax dx

If integral converges, series convergesIf integral diverges, series diverges

Comparison Test

if Σan < convergent series,

Σan is convergentif Σan > divergent series,

Σan is divergent

Limit Comparison Test

Let Σbn be a knownconvergent or divergent series:

limn → ∞

Σan

Σbn

= ρ 0 < ρ < ∞

Σan & Σbn

both converge or both diverge

∞

n = 1

an(-1)n-1

limn → ∞

3. an = 0

Alternating Series Test (AST)

Converges if:1. All terms are positive

2. an > an+1 for all n.

Power Series1. Use ratio test on theabsolute value of the series.

2. Set ρ < 1 to find the interval

3. To check bounds, pluginto original equation4. Take ½ of the interval to find radius of convergence

Error (Alternating Series/Taylor)Error = Actual - ApproximateError < First Unused Term

Root Testn

limn → ∞ anρ =

Ratio Test

limn → ∞ρ =

an+1

an Converges if ρ < 1No conclusion if ρ = 1Divergent if ρ > 1

2 3 n nP(x) = f (a) + f ' (a)(x - a) + f ''(a)(x - a) + f ''' (a)(x - a) + … + f (a)(x - a) 2! 3! n!

Taylor Polynomial

Σn = 0

∞ n 2n( - 1) x(2n)!

2 4 6cos x = 1- x + x - x + …= 2! 4! 6!

Common MacLaurin Series

Σn = 0

∞ n 2n+1( - 1) x(2n+1)!

3 5 7sin x = x - x + x - x + … = 3! 5! 7!

Σn = 0

∞ n nf ( a)(x - a) n!

Σn = 0

∞

n n(-1) x 1 = 1 - x + x² - x³+... =1+x

Σn = 0

∞

nx 1 = 1 + x + x² + x³+... =1- x

Σn = 0

∞ nxn!

xe = 1 + x + x² + x³+... = 2! 3!

y(x,y)

Rectangular

x

Polar(r,θ)

rθ

x xcosh x = e + e 2

x xsinh x = e - e 2

x xtanhx = e - e x -xe + e

sech x = 2 x xe + ecsch x = 2 x xe - e

x xcothx = e + e x -xe - e

Standard Hyperbolic Trig Derivatives

(d/dx)(csch u) = - csch u coth u (du/dx)

(d/dx)(sech u) = - sech u tanh u (du/dx)

(d/dx)(coth u) = - csch ² u (du/dx)

(d/dx)(tanh u) = sech ² u (du/dx)

(d/dx)(cosh u) = sinh u (du/dx)

(d/dx)(sinh u) = cosh u (du/dx)

sinh 2x = 2 sinh x cosh xcosh 2x = cosh ² x - sinh ² x

cosh ² x = cosh 2x+1 2

sinh ² x = cosh 2x - 1 2

cosh² x - sinh² x = 1tanh² x + sech ² x = 1

coth² x - csch² x = 1

Inverse Hyperbolic Trig Derivatives

sech u = , 0<u<1-1 1

u u² - 1d

dx

dudx

csch u = , u ≠0-1 1

|u| u² - 1d

dx

dudx

tanh u = , |u|<1-1 1

1 - u²d

dxdudx

coth u = , |u|>1-1 1

1 - u²d

dxdudx

sinh u =-1 1

u² + 1d

dx

dudx

cosh u = , u > 1-1 1

u² - 1d

dx

dudx

Every function splitsinto Even and Odd parts:

f(x) = + f(x) + f(-x)

2 f(x) - f(-x)

2even odd

Hyperbolic Trig Integrals

∫ sinh u du = cosh u + C

∫ cosh u du = sinh u + C

∫ sech² u du = tanh u + C

∫ csch² u du = - coth u + C

∫ sech x tanh u du =- sech u + C

∫ csch x coth u du = - csch u + C

© 2000 Chad A. Valencia. All Rights Reserved.