Cheatography
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Taylor Series
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)! |
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 |
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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 |
∫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 |
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=? |
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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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