AP CALCULUS · MODULE 02B

Applications of Derivatives
Extrema, Optimization, Related Rates & More

Using derivatives to analyze functions, solve real-world problems, and understand motion.

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This module covers the many applications of derivatives: curve analysis, optimization, related rates, L'Hôpital's rule, and more.

Increasing/Decreasing & First Derivative Test

Where $f'(x) > 0$ means increasing, $f'(x) < 0$ means decreasing. First derivative test for local extrema.

curve analysis → start

Concavity & Second Derivative Test

$f''(x) > 0$ means concave up, $f''(x) < 0$ means concave down. Inflection points and second derivative test for extrema.

curve analysis → start

Optimization (Max/Min Problems)

Finding maximum and minimum values in real-world contexts: area, volume, cost, profit, etc.

applications → start

Problems involving rates of change that are related through an equation (e.g., expanding balloon, sliding ladder).

applications → start

Linearization (Tangent Line Approximation)

Approximating functions using tangent lines: $L(x) = f(a) + f'(a)(x-a)$.

approximation → start

L'Hôpital's Rule

Evaluating limits of the form $\frac{0}{0}$ or $\frac{\infty}{\infty}$ using derivatives.

limits → start

Mean Value Theorem (MVT)

If $f$ is continuous on $[a,b]$ and differentiable on $(a,b)$, then $f'(c) = \frac{f(b)-f(a)}{b-a}$ for some $c$.

theorem → start

Practice Problems

20 solved problems covering all application topics.

practice → start

20-question interactive quiz with instant feedback.

assessment → start