Calc BC Ch. 8

L'H Rule

ONLY WHEN lim= 0/0 or ∞/∞ lim x→c f(x)/g(x) = lim x→c f'(x)/g'(x), cont if needed

Choosing u

L (lnx,logx) I (inverse trig: ex. arctan) A (algebraic ex. xⁿ) T (trig ex. cos) E (exponential ex. eⁿ, 2ⁿ)

alternative form of derivative

f(x)-f(a)/x-a=f'(x) (like the first term)

indeterminate powers and L'H rule (0⁰, 1^∞, ∞⁰)

f(x)^g(x) = e^(ln[f(x)]^g(x)] = e^(g(x)ln(f(x)))

indeterminate products and L'H rule (0•∞)

f(x)•g(x) = f(x)/(1/g(x)) or g(x)/(1/f(x)) when they both go to 0 or ∞, then use L'H rule to solve

when D(x) has repeated irreducible quadratics

factor out and combine like terms, system of equations

comparison theorem

if f&g are cont a≤g(x)≤f(x) - If ∫a∞ f(x)dx converges, ∫a∞ g(x)dx will also converge - If ∫a∞ g(x)dx diverges, so will∫a∞ f(x)dx

volume by disc method

r(x)=f(x) (sometimes) A(x)=π(r(x))² V=∫A(x)dx

integration by parts: ∫udv=

uv-∫vdu

divergent

when improper integral w/ indefinite limits does not exist

convergent

when improper integral w/ indefinite limits exists

indeterminate differences and L'H rule (∞-∞)

when lim lim x→c[f(x)-g(x)] results with ∞-∞, try combining f(x)-g(x) into one function (common denominator)

definition of improper integrals with infinite integration limits

∫ a-∞ f(x)dx = lim b→∞ ∫ a-b f(x)dx

surface area

∫2πr(x)√1+(f'(x))²

definition of improper integrals with infinite discontinuities

∫ab if f is discont at a: lim t→a⁺∫ab f(x)dx if c∈(a,b) and f is discont at c: lim t→c⁻∫at f(x)dx+lim t→c⁺ ∫tb f(x)dx

improper integrals: infinite discontinuities ex.

∫₀¹ (1/x²) dx

improper integrals: indefinite limits of integration ex.

∫₁∞ (1/x²) dx

special type of improper integral

∫₁∞ dx/(x^p) = 1/(p-1) IF p>1 OR divergent if p≤1