AP Calculus AB

Advanced Placement Calculus AB covering limits, derivatives, and integrals.

Understanding LimitsIntroduction to DerivativesFundamentals of Integrals
Techniques of DifferentiationApplications of DerivativesDefinite Integrals and the Fundamental Theorem
Calculus in Physics and EngineeringCalculus in Economics and BiologyCalculus and Everyday Problems
Mastering Multiple-Choice QuestionsShowing Work and Earning Free Response Points
Understanding LimitsIntroduction to Deri...Fundamentals of Inte...Techniques of Differ...Applications of Deri...Definite Integrals a...Calculus in Physics ...Calculus in Economic...Calculus and Everyda...Mastering Multiple-C...Showing Work and Ear...
Advanced Topics

Applications of Derivatives

Why Take Derivatives?

Derivatives aren’t just for math class—they help us solve real-world problems like finding maximum profit, safe speeds, and even predicting population growth.

Optimization

Use derivatives to find maximum or minimum values of functions—critical for business, engineering, and science.

Related Rates

If two (or more) things are changing together, derivatives help us relate their rates of change.

Motion and Physics

In physics, velocity is the derivative of position, and acceleration is the derivative of velocity.

Steps for Solving Application Problems

  1. Set up your variables.
  2. Write equations relating them.
  3. Use derivatives to find rates or optimize.
  4. Interpret your answer in context.

Examples

  • Finding the dimensions of a box with maximum volume given a fixed surface area.

  • Determining how fast a shadow grows as a person walks away from a light.

In a Nutshell

Derivatives help us optimize, relate changing quantities, and model motion.

Next
Definite Integrals and the Fundamental Theorem