Math Final Rules Review

f = o(g)

f grows at a slower rate than g as x → ∞

f = O(g)

f grows at a slower or equal rate than g as x → ∞

sinhx

(e^x - e^-x)/2

coshx

(e^x + e^-x)/2

Washer formula for volume

V = π ∫ r²(x) dx

Cross-sections formula for volume

V = ∫ A(x) dx, where A can depend on the shape of the cross-section

Cylindrical shells formula for volume

V = 2π ∫ r(x) h(x) dx — often r(x) = x and h(x) = f(x)

Formula for surface area of solids formed by rotation

S = ∫ 2πy √[1 + (dy/dx)2] dx

Rules for P series

P > 1 → diverges

P = 1 → diverges

P < 1 → converges

Rules for ratio and root test

P > 1 → diverges

P = 1 → inconclusive

P < 1 → converges

LCT: if 0 < lim < ∞…

both series have the same behavior

LCT: if lim = 0…

and ∑b_n converges, a_n converges

LCT: if lim = ∞…

and b_n diverges, a_n diverges

Alternating series test

If…

• b_n is eventually decreasing

• b_n > 0

• lim as n → ∞ of b_n = 0

… then a_n converges