# Calculus 2 Cheat Sheet by ejj1999

### 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

### 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=?

### 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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