|
1/1-x
|
1+x+x2+x3+...
|
∑ xn
|
|
sin(x)
|
x1-x3/3!+x5/5!-+...
|
∑ (-1)nx2n+1/(2n+1)!
|
|
ex
|
1+x+x2/2!+x3/3!+...
|
∑ xn/n!
|
|
cos(x)
|
1-x2/2!+x4/4!-+...
|
∑ (-1)nx2n/(2n)!
|
|
√(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)
|
|
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
|
|
∫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
|
|
∫sec2(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
|
|
d/dx arctan f(x)
|
f'(x)/x2+1
|
|
d/dx sec(θ)
|
sec(θ)tan(θ)
|
|
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
|
|
Horizontal Tangents (x)
|
when dy/dx=0 t=?
|
|
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)
|
|
(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
|
|
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
|
|
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
|
Cheatography n. The art of mapping complex subjects into clear, instantly navigable reference guides.
