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