AP Calculus AB Unit 5: Analytical Applications of Differentiation

Unit 5 covers the analytical applications of differentiation: increasing and decreasing intervals, extrema, concavity, the Mean Value Theorem and optimisation. It is the largest AB unit in the bank, and it is where derivative rules become reasoning.

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160 questions3 CED topics100% with a figure
How this unit behaves

What 160 questions on it look like

This is the largest unit in AP Calculus AB by practice volume — 160 questions against the subject's other units. Weight your revision accordingly.

With only 3 topics, this unit is narrow enough to master completely — an unusual opportunity in a course this size.

100% of the questions come with a figure — a graph, diagram or data display. Reading the figure correctly is most of the work before any content knowledge applies.

Determining Intervals on Which a Function Is Increasing or Decreasing100
Determining Concavity of Functions over Their Domains50
Sketching Graphs of Functions and Their Derivatives10

Questions per CED topic.

From the bank

Three real analytical applications of differentiation questions

Drawn from the practice pool, not from the mock papers.

Determining Intervals on Which a Function Is Increasing or Decreasing · Connecting Representations · with figure

Use Figure 1. On which interval is f increasing, and where does f have a local maximum?

  1. Af is increasing on (-3, -1); local maximum at x=-1.correct
  2. Bf is increasing on (-infinity, -3); local maximum at x=-3.
  3. Cf is increasing on (-1, infinity); local maximum at x=-3.
  4. Df is decreasing on (-3, -1); local minimum at x=-1.
Why A is correct

The graph rises on (-3, -1). At x=-1, f changes from increasing to decreasing, so f has a local maximum there.

Determining Concavity of Functions over Their Domains · Connecting Representations · with figure

Use Figure 1. On which interval is f concave down?

  1. A(-2.5, -1.5)correct
  2. B(-infinity, -2.5) only
  3. C(-1.5, infinity) only
  4. D(-infinity, -2.5) union (-1.5, infinity)
Why A is correct

The graph bends downward between the marked inflection points, so f is concave down on (-2.5, -1.5).

Determining Intervals on Which a Function Is Increasing or Decreasing · Connecting Representations · with figure

Use Figure 1. On which interval is f increasing, and where does f have a local maximum?

  1. Af is increasing on (-infinity, -2.5); local maximum at x=-2.5.
  2. Bf is increasing on (-2.5, -0.5); local maximum at x=-0.5.correct
  3. Cf is increasing on (-0.5, infinity); local maximum at x=-2.5.
  4. Df is decreasing on (-2.5, -0.5); local minimum at x=-0.5.
Why B is correct

The graph rises on (-2.5, -0.5). At x=-0.5, f changes from increasing to decreasing, so f has a local maximum there.

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FAQ

Analytical Applications of Differentiation questions

What does the first derivative tell me?

Where the function increases or decreases, and where extrema can occur. The second derivative gives concavity and confirms which kind of extremum it is.

How do optimisation questions work?

Write the quantity to optimise, use the constraint to reduce it to one variable, differentiate, and justify the answer with a sign analysis.
Last reviewed 2026-08-28. Unit and topic names follow the College Board course framework. Question counts describe the PrepScore practice bank, not the exam.
Same course
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Practise analytical applications of differentiation until the reasoning is automatic.

Real AP questions with a full explanation on every answer, and a mistake bank that only clears when you get it right.

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