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Calculus 2 Cheat Sheet by

math

Taylor Series

1/1-x
1+x+x­2+­x+...
∑ xn
sin(x)
x1-x­3­/3!­+x­5/­5!-+...
∑ (-1)x­2n+­1/­(2n+1)!
ex
1+x+x­2/­2!+­x/3­!+...
∑ xn/n!
cos(x)
1-x2­/2!­+x­4/­4!-+...
∑ (-1)x­2n­/(2n)!
centered around 0
(1/1-x only valid for -1<­x<1.)

Trig Sub's

√(x2­+a2)
x=atan(θ)
√(a2­-x2)
x-asin(θ)
√(x2­-a2)
x=asec(θ)
b-ax2
x= √b / √a sin(θ)
ax2+b
x= √b / √a tan(θ)
ax2-b
x= √b / √a sec(θ)

Conver­gen­ce|­Div­ergence test

Nth term test for divergence
lim(n>∞) an
≠0 ∑an diverges
P-Test
converge p>1
diverge p≤1
Limit Comparison
L= lim(n>∞) (an/bn)
L≠0 series both diverg­e|c­onverge
Ratio test
r= lim(n>∞) |an+1/an|
r<1 converge r>1 diverge
Altern­ating series test
lim(n>∞) an
=0 ∑ (-1)nan converges
 

Common Integrals

∫sin(x)dx
-cos(x)+C
∫cos(x)dx
sin(x)+C
∫tan(x)dx
-ln(co­s(x))+C
∫sec(x)dx
ln(sec­(x)­+ta­n(x))+C
∫csc(x)dx
-ln(cs­c(x­)+c­ot(­x))+C
∫cot(x)dx
ln(sin­(x))+C
∫sec(x)dx
tan(x)+C
∫ef(x)dx
ef(x)­/f­'­(x)+C
∫(1/x)dx
ln(x)+C
∫(1/x­n)dx
(xn+1­/n­+1)+C
∫dx/√(­a-x2)
arcsin­(x/­√(a))+C
∫dx/x2+a
(1/√a)­arc­tan­(x/­√a)+C

Important Deriva­tives

d/dx arctan f(x)
f'(x­)/x­2+1
d/dx sec(θ)
sec(θ)­tan(θ)

Power Series

general form
∑ an(x-a)n
an = sequence of coeff.
center
x=a
radius of conver­gence
R=lim(­n>∞) |an/an+1|
endpoints
x=a+R and x=a-R in series

Parametric Curves

Horizontal Tangents (x)
when dy/dx=0 t=?
 

Equations for Parabola

y=a(x-­h)2+k
Directrix
y=k-(1/4a)
Focus
(h,k+1/4a)
x=a(y-­k)2+h
Directrix
x=h-(1/4a)
Focus
(h+1/4a,k)

Equations for Ellipses

(x-h)­2/a2 + (y-k)­2/b2 =1
c=√(|a­2­-b2|)
eccent­ricity
c/(max a|b)
foci (on major axis)
when x= center and y= center
y= horizontal axis
x= vertical axis

Trig Identities

sec2(θ)
tan2­(θ)+1
sin2(θ)
1-cos­2(θ)
tan2(θ)
sec2­(θ)-1
cos2(θ)
[1+cos­(2θ)]/2
sin2(θ)
[1-cos­(2θ)]/2
double angle cos2(θ)
(1+cos­(2θ)/2
double angle sin2(θ)
(1-cos­(2θ)/2

Polar Coordi­nates & Area

Area
∫1/2 (f(x))2 dx
One petal of r=sin(nθ)
interval [0,π/n]
One petal of r=cos(nθ)
[-π/2n­,π/2n]
Polar > Cartesian
x=rcos(θ) y=rsin(θ)
Cartesian > Polar
tan(θ)=y/x x2+y­2­=r2

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