Applications of Differential Calculus

The derivative f'(x) = dy/dx represents

The derivative f'(x) = dy/dx represents two interconnected ideas: (1) It is the slope of the tangent line to the curve at a point, telling us the dire

The tangent line at point (a

The tangent line at point (a, b) on curve y = f(x) has equation: y - b = f'(a)(x - a). This line touches the curve at exactly one point and has slope

Related rates problems involve situations where

Related rates problems involve situations where multiple quantities change with time and are related by an equation. The strategy is to: (1) Establish

Rolle's Theorem

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. Geometrically, there's a hori

A function f(x) is increasing on

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