Worksheets · AP Calculus AB

AP Calculus AB Unit 5 Worksheet: Analyzing Functions

20 exam-style multiple-choice questions on extrema, concavity, the Mean Value Theorem and the graphs of f, f′ and f″, free to use in class. Good practice before the Unit 5 Progress Check in AP Classroom. Free response for this unit: Unit 5 FRQ set.

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AP Calculus AB Unit 5: Analyzing Functions

Date Version A

Directions: Work each problem in the space below it, then choose the best of the four answer choices. Unless a question says otherwise, the domain of a function f is every real number x for which f(x) is a real number.

Part A: No calculator is allowed.

Questions 1–12
  1. t010203040H(t)2321066The water level in a canal lock is H(t) meters at time t minutes after its gates close, and H is differentiable. The table above lists H(t) at five times. Is there necessarily a time c, with 30<c<40, at which H′(c)=0? Which of the following gives a correct answer together with a correct justification?
    1. (A)Yes. H is differentiable, so it is continuous on [30,40], and H(40)−H(30)40−30=0; by the Mean Value Theorem, H′(c)=0 for some c in (30,40).
    2. (B)No. The table shows no time at which H′(t)=0.
    3. (C)No. H is not given by a formula, so the Mean Value Theorem cannot be applied.
    4. (D)Yes. H(30)=H(40), so by the Intermediate Value Theorem, H′(c)=0 for some c in (30,40).
  2. Pictured is y=f′(x) on the interval −3≤x≤4, where f is differentiable; the picture is built from line segments and a semicircle. Suppose f(−3)=3. What is the least value that f takes on [−3,4]?
    −4−3−2−1012345−4−3−2−1012y = f′(x)
    1. (A)π2−92
    2. (B)4
    3. (C)π2−152
    4. (D)3
  3. A curve is defined implicitly by x2+4y2=13. Evaluate d2ydx2 where the curve passes through (3,−1).
    1. (A)14
    2. (B)1316
    3. (C)34
    4. (D)−1316
  4. The Mean Value Theorem is applied to f(x)=3x2−2x+2 over [−3,0]. Which number c in the open interval (−3,0) does it produce?
    1. (A)−316
    2. (B)13
    3. (C)−32
    4. (D)−11
  5. The derivative of a function g is positive everywhere. For h(x)=g(x3−12x2), which choice gives the intervals on which h is increasing?
    1. (A)(−∞,∞)
    2. (B)(−∞,−8)∪(0,∞)
    3. (C)(−∞,0)∪(8,∞)
    4. (D)(12,∞)
  6. Consider the curve given by x2+2x+y3−6y2+9y=3. At which of the following points of the curve is the line tangent to the curve vertical?
    1. (A)(47−1,−3)
    2. (B)(1,3)
    3. (C)(1,0)
    4. (D)(−1,4)
  7. As x runs over the interval [−6,1], what is the least value taken by y=−2x3−15x2+4?
    1. (A)−104
    2. (B)−5
    3. (C)4
    4. (D)−121
  8. Positive numbers x and y satisfy xy=100. What is the least possible value of x+25y?
    1. (A)20
    2. (B)100
    3. (C)50
    4. (D)260
  9. A rectangle in the first quadrant has two of its sides on the coordinate axes and its upper right corner at a point (x,y) of the curve y=8−x2. Which of the following gives the area A of the rectangle as a function of x?
    1. (A)2x+2 ⁣(8−x2)
    2. (B)2x(8−x2)
    3. (C)x(8−x2)
    4. (D)8x
  10. Choose the constant k so that x=52 is a critical point of g(x)=x2ekx. What is k?
    1. (A)−45
    2. (B)45
    3. (C)−25
    4. (D)No value of k gives a critical point there.
  11. Suppose k is a constant and f(x)=x+kx. If f has a relative minimum at x=3, what is k?
    1. (A)9
    2. (B)−9
    3. (C)19
    4. (D)3
  12. A function f is differentiable for all real numbers, and f′(x)=−2x3−12x2. At which x-values does the graph of f change concavity?
    1. (A)x=−6 and x=0
    2. (B)x=−4 only
    3. (C)x=−4 and x=0
    4. (D)x=0 and x=4

Part B: A graphing calculator is allowed.

Questions 13–20
  1. The derivative of f is f′(x)=sin⁡ ⁣(x2), and f is considered only for 0<x<3.1. Exactly one x in this interval gives a relative minimum of f. Which value is it?
    1. (A)1.772
    2. (B)2.802
    3. (C)1.253
    4. (D)2.507
  2. For −1.5<x<1.5, a function f has derivative f′(x)=(x2+2)sin⁡(3x−2). Over which intervals is the graph of f concave up?
    1. (A)−0.976<x<0.161 and 1.267<x<1.500
    2. (B)−1.428<x<−0.381 and 0.667<x<1.500
    3. (C)0.161<x<1.267
    4. (D)−1.500<x<−0.976 and 0.161<x<1.267
  3. The top figure is a graph of the derivative y=f′(x) on the interval [0,4]. One of graphs (A) through (D) could be the graph of y=f(x) itself. Which one?
    Graph of f′
    −1012345−4−3−2−101234y = f′(x)
    (A)
    −1012345−4−3−2−101234
    (B)
    −1012345−4−3−2−101234
    (C)
    −1012345−4−3−2−101234
    (D)
    −1012345−4−3−2−101234
    1. (A)Graph (A)
    2. (B)Graph (B)
    3. (C)Graph (C)
    4. (D)Graph (D)
  4. A twice-differentiable function f has a graph that is concave down for 2<x<4. Its tangent line at x=3 is y=2x−5. What can be concluded about the derivative f′ near x=3?
    1. (A)f′(x)≤2 for 3<x<3.5.
    2. (B)f′(x)≥2 for 3<x<3.5.
    3. (C)f′(x) is constant for 2.5<x<3.5.
    4. (D)f′(x)<0 for 2.5<x<3.5.
  5. A function f is differentiable on [−2,5]. Its derivative f′ is graphed in the figure. Find every x where f has a relative maximum.
    −3−2−10123456−3−2−101234y = f′(x)
    1. (A)x=−1 only
    2. (B)x=3 only
    3. (C)x=−1 and x=3
    4. (D)x=1 only
  6. Pictured is y=f′(x) on the interval −3≤x≤5, where f is differentiable. Considering only points with −3<x<5, how many relative minima, relative maxima, and inflection points does the graph of f have?
    −4−3−2−10123456−2−1012y = f′(x)
    1. (A)relative minima: 1; relative maxima: 0; inflection points: 2
    2. (B)relative minima: 1; relative maxima: 1; inflection points: 2
    3. (C)relative minima: 0; relative maxima: 1; inflection points: 1
    4. (D)relative minima: 0; relative maxima: 1; inflection points: 2
  7. A function g has no breaks on [0,5] (it is continuous there), with g(0)=5 and g(5)=15. Adding which one of the following facts would make it certain that some c strictly between 0 and 5 has g′(c)=2?
    1. (A)g increases throughout [0,5].
    2. (B)Somewhere between x=0 and x=5, the graph of g has a horizontal tangent line.
    3. (C)g has no break at any x with 0<x<5.
    4. (D)g′(x) exists at every x with 0<x<5.
  8. For every real x, the first and second derivatives of F are positive. Of the tables shown, which one could hold values of F?
    1. (A)x2345F(x)−1−4−8−13
    2. (B)x2345F(x)−1258
    3. (C)x2345F(x)−141017
    4. (D)x2345F(x)−1478

    End of Version A

Answer key · Version A

12345678910
AABCCBDBCA
5.15.55.125.15.35.125.55.115.105.2
11121314151617181920
ACDDBABDDC
5.45.65.45.65.85.95.45.65.15.9

The second row is the correct letter; the third is the course topic each question tests.

Free AP Calculus AB worksheets at sigmaprep.io/worksheets/ap-calculus-abOriginal questions written by Sigma Prep. AP is a registered trademark of the College Board, which is not affiliated with this sheet.

What does the AP Calculus AB exam ask on Unit 5?

The most of any unit: Unit 5 was 65 of the 315 official multiple-choice questions on the seven 2012–2018 practice exams (20.6%).

This worksheet has 20 questions in that mix: Part A (12) without a calculator and Part B (8) with one; the exam put 43% of its official Unit 5 multiple choice in the calculator section.

Counted from the official AP Calculus AB exams: multiple choice from 17 exams, 1969–2018, free response 2012–2026. No official question is reproduced here.

AP Calculus AB Unit 5 question types: how often the exam asks each

Official AP Calculus AB multiple-choice questions by question type, Unit 5 (Analytical Applications of Differentiation), and how many are on this worksheet
Question typeOn this worksheetOfficial questions, 1969–2018Exams it was on (of 17)On the 2012–2018 practice exams
Mean Value Theorem: find c or bound valuesTopic 5.11665
Which theorem applies (MVT, Rolle, EVT, IVT)?Topic 5.1113105
Justify with the MVT or Rolle's theorem (FRQ)Topic 5.11Asked on 19 free-response questions since 1998, most recently in 2023.
Critical pointsTopic 5.21442
Increasing and decreasing intervalsTopic 5.3115126
Increasing intervals and extrema from the graph of f′Topic 5.41862
Relative extrema from a formula for f′Topic 5.4111107
Relative extrema from a formula for fTopic 5.41650
Absolute extrema from the graph of f′ and areasTopic 5.51332
Absolute extrema on an interval (candidates test)Topic 5.511094
Concavity and inflection from the graph of f′ or f″Topic 5.61866
Points of inflection from a formulaTopic 5.6114125
Concavity intervals from a formulaTopic 5.6112106
Match graphs of f and f′Topic 5.8111105
Connect signs of f, f′, f″Topic 5.911075
Which table fits f′ and f″ signs?Topic 5.91553
Set up an optimization problemTopic 5.101No official multiple-choice question is tagged to this topic in the 17 official exams (1969–2018); writing the quantity to optimize in one variable, with its domain, is the first step of every optimization question and is its own topic in the course framework (5.10).
OptimizationTopic 5.111551
Second derivative of an implicit relationTopic 5.121221
Horizontal/vertical tangents on an implicit curveTopic 5.121Asked on 10 free-response questions since 1998, most recently in 2025.

Counts are of official multiple-choice questions whose main topic is that kind, on 17 College Board exams 1969–2018 (7 released 1969–1998, 2003 and 2008 released, the 2008 practice exam and the seven 2012–2018 practice exams). No official multiple-choice questions after 2018 are public. Left out: sketch the graph of f or f′ (FRQ); second derivative test.

Free-response practice for this unit: 4 FRQs with a scoring guide.

Why these kinds of question: AP Calculus AB Unit 5 Review: The Most-Tested Extrema and Concavity Questions.

Questions about this worksheet

Unit 5, Analytical Applications of Differentiation, has 12 topics in the College Board's course description: the Mean Value Theorem; the Extreme Value Theorem, critical points, and global versus local extrema; intervals where a function is increasing or decreasing; relative extrema by the first derivative test; absolute extrema by the candidates test; concavity and points of inflection; the second derivative test; sketching the graphs of a function and its derivatives; connecting f, f′ and f″; setting up and solving optimization problems; and the behavior of implicit relations, such as horizontal and vertical tangents and the second derivative. The course description weights the unit at 15–20% of the multiple-choice section.

Only in Part B, as on the exam. Questions 1–12 are Part A, with no calculator; questions 13–20 are Part B, where the exam allows a graphing calculator (on the digital exam, the Desmos calculator built into Bluebook is available there). Most of these Unit 5 Part B questions do not need one: the graph of f′, matching the graphs of f and f′, tables that fit the signs of f′ and f″, and which theorem applies. Two in each version do: relative extrema and concavity from a derivative such as cos(x²) or (x² + 1)sin(3x − 2), worked on the calculator. A numerical answer is given to three decimal places, as on the exam. From May 2027 the exam's multiple-choice section is 29 questions without a calculator and 13 with one.

We tagged 760 official multiple-choice questions from 17 College Board exams (1969–2018) and the free-response questions from 1969 to 2026, sorted the Unit 5 questions into 22 kinds, and weighted each kind by how often and how recently the exam asks it. There are more kinds than questions. Nineteen of the 20 questions go to the most-tested kinds, one each, and one goes to setting up an optimization problem (topic 5.10), the first step of every optimization question, which the exam has never made the main topic of a multiple-choice question. It takes the place of the second derivative test, which has no official multiple-choice question and is asked on the free-response section. Left out: sketch the graph of f or f′ (FRQ); second derivative test.

The progress check is College Board's own end-of-unit quiz in AP Classroom, assigned by your teacher. Its questions are not public, and we do not have them: Sigma Prep is not affiliated with College Board. This worksheet is exam-style practice on the same unit: 20 original multiple-choice questions, weighted by how often the official exams ask each kind of question, with a no-calculator Part A and a calculator Part B. Time yourself, check the answer key, then redo the kinds you missed. For the free-response part, use the Unit 5 FRQ set, which has a scoring guide.

No. Every question here is original, written by Sigma Prep in the style of the exam. We studied which kinds of questions the official exams ask and how often; no official question is copied or reworded. Each answer was checked by an independent computer-algebra check before the sheet was published.

Yes. It is shown with the sheet, and the PDF prints it on its own last page. There is also a PDF without the key to hand out. Worked solutions are not included.

AP is a registered trademark of the College Board, which was not involved in making this worksheet and does not endorse it. All questions are original to Sigma Prep. Frozen 2026-10-24.