Insurance and Annuity — Formula Sheet

Equation of tangent

y - y₁ = m(x - x₁)

Chain rule is essential

dy/dt = (dy/dx) × (dx/dt)

Rolle's Theorem

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

LMVT

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

At any point P(x₁, y₁) on curve y = f(x), the slope of tangent = dy/dx|(x₁,y₁) and slope of normal = -1/(dy/dx)|(x₁,y₁).

For a differentiable function f(x), the approximate value at x = a + h is given by f(a + h) ≈ f(a) + h × f'(a).

If function f is continuous on [a,b], differentiable on (a,b), and f(a) = f(b), then there exists at least one point c in (a,b) where f'(c) = 0.

If function f is continuous on [a,b] and differentiable on (a,b), then there exists at least one point c in (a,b) where f'(c) = [f(b) - f(a)]/(b - a).