AP Calculus BC

arc length in rectangular form

L=∫√(1+(f’(x))²dx

∫(tanu)du

-ln|cosu|+C

∫(cotu)du

ln|sinu|+C

∫(secu)du

ln|secu+tanu|+C

∫(cscu)du

-ln|cscu+cotu|+C

∫(sec²u)du

tanu+C

∫(csc²u)du

-cotu+C

∫(secu∙tanu)du

secu+C

∫(cscu∙cotu)du

-cscu+C

∫du/√(1-x²)

arcsin(u/a)+C

∫du/(a²+u²)

(1/a)arctan(u/a)+C

integration by parts theorem

∫u∙dv=u∙v-∫v∙du

guidelines for choosing “u” in integration by parts

Logarithm, Inverse trigonometric, Algebraic, Trigonometric, Exponential (LIATE)

Pythagorean identity

sin²x+cos²x=1

logistic differential equation form

dy/dx=ky(1-y/L)

logistic growth equation form

y=L/(1+be^-kt)

At what point does a logistic growth curve increase the fastest?

when the curve reaches half its carrying capacity (L)

What is the parametric form of the derivative?

dy/dx=(dy/dt)/(dx/dt), dx/dt≠0

When does a smooth curve have a horizontal tangent line?

at t where dy/dt=0, but dx/dt≠0

When does a smooth curve have a vertical tangent line?

at t where dx/dt=0, but dy/dt≠0

What two formulas do you use to convert polar coordinates to rectangular coordinates?

x=r∙cosθ and y=r∙sinθ

What three formulas do you use to convert rectangular coordinates to polar coordinates?

r=√(x²+y²), θ=arctan(y/x) when x>0, and θ=arctan(y/x)+π when x<0

slope in polar form

dy/dx=(f(θ)cosθ+f’(θ)sinθ)/(-f(θ)sinθ+f’(θ)cosθ)

area in polar coordinates

A=1/2∫((f(θ))²-(f₂(θ))²) dθ

arc length in polar coordinates

L=∫√(r²+(dr/dθ)²) dθ

general expression for a geometric sequence

a₁∙r^n-1

sum of a convergent geometric series

S=(first term of the series)/(1-the common ratio)

limit of the n^th term of a convergent series

if the sum of a_n converges, then (lim_n→∞)(a_n)=0

n^th term for divergence

if (lim_n→∞)(a_n)≠0, then the sum of a_n diverges

integral test

if f is positive, continuous, and decreasing for x≥m where m is a positive integer greater than 1 and a_n=f(x), then ∞Σa_n and m→∞∫f(x)dx either both converge or diverge

convergence of a p-series

∞Σ1/n^p converges if p>1

divergence of a p-series

∞Σ1/n^p diverges if 0<p≤1

direct comparison test

let 0<a_n≤b_n for all n; if ∞Σb_n converges, then ∞Σa_n converges; ∞Σa_n diverges, then ∞Σb_n diverges

limit comparison test

a_n>0, b_n>0, and (lim_n→∞)(a_n/b_n)=L where L is finite and positive, then the two series ∞Σa_n and ∞Σb_n either both converge or both diverge

alternating series test

let a_n>0; the alternating series ∞Σ(-1)ⁿa_n and ∞Σ(-1)ⁿ⁺¹a_n converge if (lim_n→∞)(a_n)=0 and if a_n+1≤a_n (a_n values are nonincreasing)

definition of absolute convergence

∞Σa_n is absolutely convergent if and only if ∞Σ|a_n| converges

definition of conditional convergence

∞Σa_n is conditionally convergent if and only if ∞Σa_n converges but ∞Σ|a_n| diverges

alternating series remainder

|S-S_n|=|R_n|≤|a_n+1|

ratio test

1. Σa_n converges if (lim_n→∞)|a_n+1/a_n|<1

2. Σa_n diverges if (lim_n→∞)|a_n+1/a_n|>1 or =∞

3. inconclusive if (lim_n→∞)|a_n+1/a_n|=1

root test

1. Σa_n converges if (lim_n→∞)ⁿ√|a_n|<1

2. Σa_n diverges if (lim_n→∞)ⁿ√|a_n|>1 or =∞

3. inconclusive if (lim_n→∞)ⁿ√|a_n|=1 (and for p-series)

definition of nth Taylor Polynomial

if f has n derivatives at c, then the polynomial P_n(x)=((f⁽ⁿ⁾(c))/n!)(x-c)ⁿ is for f centered at c

error of a Taylor Polynomial

error=|R_n(x)|=|f(x)-P_n(x)|

Taylor’s inequality (error as a bound)

|R_n(x)|=|((max[f⁽ⁿ⁺¹⁾(z)])/(n+1)!)(x-c)ⁿ⁺¹|

definition of a power series

an infinite series of the form Σa_n(x-c)ⁿ centered at c

convergence of a power series

1. series converges only at c (R=0) or

2. series converges absolutely for |x-c|<R, and diverges for |x-c|>R (check endpoint convergence) or

3. series converges absolute for all x (R=∞)

speed of velocity vector

√((x'(t))²+(y'(t))²)

arc length in parametric form

L=∫√((x’(x))²+(y’(x))²dx