AP Calc BC Midterm

logistic equation

f(x)= L/(1+Ce^(-kx))

euler’s method

Δy= f’(x)Δx

∫du/√(a^2-u^2)

arcsin u/a +C

∫du/(a^2+u^2)

arctan u/a +C

∫du/u√(u^2-a^2)

1/a arcsec |u|/a +C

∫tanu

\-ln |cos u| +C

∫cotu

ln |sinu| +C

∫secu

ln |secu+tanu| +C

∫cscu

\-ln |cscu+cotu| +C

disc method

V=π∫(R(x)^2)dx

Washer Method

V=π∫(R(x)^2-(r(x)^2))dx

Known Cross Section

∫A(x)dx

Arc Length

S=∫√(1+(f’(X))^2)dx

Surface Area

S= 2π∫r(X)√(1+(f’(x))^2)dx

Integration by Parts

∫udv= uv- ∫vdu

geometric series

sum ar^n, converges to a/(1-r) if r≥1, diverges if r≤1

p-series

1/n^p, 0≤p≤1 diverges, p≥1 converges

alternating series

sum (-1)^n(a(n)), converges if limit a(n) =0 and a(n+1)>a(n)

harmonic series

sum 1/n, p=1, always converges to 0

nth term test

limit of a(n) as approaches ∞, =0 converges, ≠0 diverges

ratio test

limit of |a(n+1)/a(n)|, if >1 converges, if

integral test

sum a(n) and f(X) both diverge or converge if f is positive, continuous and decreasing

limit comparison test

limit of b(n) = limit of a(n), b(n) is >a(n) or

direct comparison test

a(n)

absolute convergence

if sum |a(n)| converges, sum a(n) converges

conditional convergence

if sum |a(n)| diverges, sum a(n) converges