Applications of Differential Calculus — Formula Sheet

Normal line is perpendicular to tangent, so slope of normal = -1/f'(a)

Rolle's Theorem

If f continuous on [a,b], differentiable on (a,b), and f(a) = f(b), then ∃ c ∈ (a,b) where f'(c) = 0

Lagrange's MVT

If f continuous on [a,b] and differentiable on (a,b), then ∃ c ∈ (a,b) where f'(c) = [f(b) - f(a)]/(b - a)

Maclaurin series is Taylor series with a = 0, expands function in powers of x

Taylor series

f(x) = Σ[f⁽ⁿ⁾(a)/n!]·(x-a)ⁿ = f(a) + f'(a)(x-a) + f''(a)(x-a)²/2! + ...

Rolle's Theorem

If f is continuous on [a,b], differentiable on (a,b), and f(a) = f(b), then ∃c ∈ (a,b) where f'(c) = 0.

A function f(x) is increasing on interval I if f'(x) ≥ 0 for all x ∈ I (strictly increasing if f'(x) > 0).