AP Calculus AB, AP exams, 10–15%

Unit 4: Contextual Applications of Differentiation

Unit 4 puts derivatives to work in applied settings. Motion along a line, related rates, linear approximation, and L'Hospital's rule are the core techniques, and interpreting a derivative with correct units and meaning is a persistent requirement. Related rates problems appear on the free-response section regularly.

Position: 4 of 8 units
Published weighting: 10–15% in the official framework
Topics covered: 6 topics in this guide

What is covered

Topics inside this unit.

These topics describe the scope students most often review for this part of AP Calculus AB. Treat them as a study checklist rather than an exhaustive list of what a single exam form can sample.

Interpreting the derivative in context: Explaining what a rate of change means in a situation, with correct units.
Straight-line motion: Position, velocity, acceleration, speed, and determining when an object changes direction.
Rates of change in applied contexts: Modeling change in non-motion situations such as volume, temperature, or population.
Related rates: Relating changing quantities through an equation and differentiating with respect to time.
Linear approximation: Using the tangent line to approximate function values, and determining whether the estimate is over or under.
L'Hospital's rule: Evaluating indeterminate quotient limits after verifying the hypotheses.

Study tips

How to practice Contextual Applications.

Section-specific guidance that follows from how this material is usually tested.

Write units in every contextual answer; the free-response rubrics award points for correct interpretation with units.
For related rates, set up the geometric relationship before differentiating, and label which rate is given and which is wanted.
Distinguish speed from velocity carefully; questions about speeding up and slowing down turn on the sign comparison.

Where this fits

Unit 4: Contextual Applications of Differentiation is 4 of 8 units in the AP Calculus AB guide, and it carries a published weighting of 10–15% in the official framework. Use the full guide to balance it against everything else you have to review.

Back to the AP Calculus AB guide Official exam information

Sample Contextual Applications questions

Practice questions, not yet expert-reviewed

Original practice questions written to the published framework for this section. No expert has reviewed them, and they are not taken from any official exam. Try each one before you open the answer.

Question 1Foundational
A particle moves along a line so that its position at time t seconds is s(t) = t^3 - 6t^2 + 9t meters. What is the particle's velocity at t = 2 seconds?
1. A-3 meters per second
2. B3 meters per second
3. C-12 meters per second
4. D2 meters per second
Show answer
Answer: A. -3 meters per second
Velocity is the derivative of position: v(t) = 3t^2 - 12t + 9, so v(2) = 12 - 24 + 9 = -3. The negative sign says the particle is moving in the negative direction, and dropping it reports speed rather than velocity.
Key insight: Velocity carries a sign that reports direction; speed is its absolute value.
Question 2Foundational
What is the value of lim x→0 of (e^x - 1)/x?
1. A0
2. B1
3. Ce
4. DThe limit does not exist
Show answer
Answer: B. 1
Substitution gives 0/0, so L'Hospital's rule applies: differentiate the numerator and denominator separately to get e^x/1, which approaches 1 as x → 0. Equivalently, this quotient is the difference quotient for the derivative of e^x at x = 0.
Key insight: Check the indeterminate form first; L'Hospital's rule applies only to 0/0 or ∞/∞.
Question 3Intermediate
The radius of a circle increases at a constant rate of 3 centimeters per second. How fast is the area of the circle increasing at the instant the radius is 5 centimeters?
1. A6π square centimeters per second
2. B10π square centimeters per second
3. C25π square centimeters per second
4. D30π square centimeters per second
Show answer
Answer: D. 30π square centimeters per second
From A = πr^2, differentiating with respect to time gives dA/dt = 2πr(dr/dt) = 2π(5)(3) = 30π. Answering 10π uses 2πr alone and forgets dr/dt, and 6π multiplies only the constants 2π and 3.
Key insight: Every variable in a related rates equation is differentiated with respect to time, producing its own rate factor.
Question 4Intermediate
Let f(x) = √x. Using the line tangent to the graph of f at x = 16, what is the linear approximation of √16.4?
1. A4.025
2. B4.05
3. C4.2
4. D4.4
Show answer
Answer: B. 4.05
Since f(16) = 4 and f'(x) = 1/(2√x), the slope at x = 16 is 1/8. The tangent line gives 4 + (1/8)(0.4) = 4.05. A slope of 1/16 instead of 1/8 produces 4.025, and a slope of 1/2 produces 4.2.
Key insight: A linear approximation is f(a) + f'(a)(x - a); the slope must be evaluated at the anchor point a.
Question 5Advanced
A particle moves along a line with velocity v(t) = t^2 - 4t + 3 meters per second. At t = 1.5 seconds, is the particle speeding up or slowing down?
1. ASpeeding up
2. BSlowing down
3. CNeither, because the speed is momentarily constant
4. DIt cannot be determined without the position function
Show answer
Answer: A. Speeding up
At t = 1.5, v(1.5) = 2.25 - 6 + 3 = -0.75 and a(t) = v'(t) = 2t - 4 gives a(1.5) = -1. Velocity and acceleration have the same sign, so the speed is increasing and the particle is speeding up. A negative velocity alone does not mean slowing down.
Key insight: Speeding up means velocity and acceleration share a sign, regardless of what that sign is.

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Unit 4: Contextual Applications of Differentiation