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Taylor Series1/1-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 testNth 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 Derivativesd/dx arctan f(x) | f'(x)/x2+1 | d/dx sec(θ) | sec(θ)tan(θ) |
Power Seriesgeneral 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 CurvesHorizontal Tangents (x) | when dy/dx=0 t=? |
| | Equations for Parabolay=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 Identitiessec2(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 & AreaArea | ∫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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