Calculus 2 Cheat Sheet
Taylor Series
1/1-x 1+x+x +x +... ∑ x
sin(x) x -x /3!+x /5!-
+...
∑ (-1) x /‐
(2n+1)!
e 1+x+x /2!+‐
x /3!+...
∑ x /n!
cos(x) 1-x /2!+x /4!-
+...
∑ (-1) x /(2n)!
centered around 0
(1/1-x only valid for -1<x<1.)
Trig Sub's
√(x +a ) x=atan(θ)
√(a -x ) x-asin(θ)
√(x -a ) x=asec(θ)
b-ax x= √b / √a sin(θ)
ax +b x= √b / √a tan(θ)
ax -b x= √b / √a sec(θ)
Convergence|Divergence test
N 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|c‐
onverge
Ratio test r=
lim(n>∞)
|an+1/an|
r<1 converge
r>1 diverge
Alternating
series test
lim(n>∞)
an
=0 ∑ (-1) an
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
∫e dx e /f (x)+C
∫(1/x)dx ln(x)+C
∫(1/x )dx (x /n+1)+C
∫dx/√(a-x ) arcsin(x/√(a))+C
∫dx/x +a (1/√a)arctan(x/√a)+C
Important Derivatives
d/dx arctan f(x) f (x)/x+1
d/dx sec(θ) sec(θ)tan(θ)
Power Series
general form ∑ an(x-a)
an = sequence of coeff.
center x=a
radius of conver‐
gence
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) +k
Directrix y=k-(1/4a)
Focus (h,k+1/4a)
x=a(y-k) +h
Directrix x=h-(1/4a)
Focus (h+1/4a,k)
Equations for Ellipses
(x-h) /a + (y-k)
/b =1
c=√(|a -b |)
eccentricity c/(max a|b)
foci (on major
axis)
when x= center and y=
center
y= horizontal axis
x= vertical axis
Trig Identities
sec (θ) tan (θ)+1
sin (θ) 1-cos (θ)
tan (θ) sec (θ)-1
cos (θ) [1+cos(2θ)]/2
sin (θ) [1-cos(2θ)]/2
double angle cos (θ) (1+cos(2θ)/2
double angle sin (θ) (1-cos(2θ)/2
Polar Coordinates & Area
Area ∫1/2 (f(x)) 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 x +y ‐
=r
2 3n
1 3 5 n 2n+1
x2
3
n
24 n 2n
2 2
2 2
2 2
2
2
2
th
n
2
f(x) f(x)'
n n+1
2
2
' 2
n
2
2
2 2 ‐
2 2
2 2
2 2
22
2 2
2
2
2
2
2
2 2
2
