Worksheets · AP Calculus AB · Unit 2

AP Calculus AB Unit 2 FRQ Practice: Derivatives

Four exam-style free-response questions on the definition of the derivative and the basic derivative rules: one with a graphing calculator, three without. Each is scored out of 9, with a scoring guide and a printable PDF with room to write, free to use in class. Good practice before the Unit 2 Progress Check in AP Classroom. Multiple choice for this unit: Unit 2 worksheet.

The exam rarely builds a whole free-response question on Unit 2, so this set is practice with its skills in the free-response format. The mixed set is the closest to a real exam.

Sigma Prep · sigmaprep.io/worksheets/ap-calculus-ab

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AP Calculus AB Unit 2 Free Response: Derivatives

Date Version A

Directions: Show your work in the space under each part: the setup, the values you use and your reasoning. A justification needs a mathematical reason, with the conditions of any theorem or test you use checked. An answer with no supporting work may earn no credit. Give decimal answers correct to three decimal places unless a part says otherwise. Each question is worth 9 points. Unless a question says otherwise, the domain of a function f is every real number x for which f(x) is a real number.

This set is practice with Unit 2 skills in the free-response format: the exam almost never builds a whole question on Unit 2 alone. The mixed free-response set is the closest to a real exam.

Part A: A graphing calculator is required.

Question 1

Set your calculator to radian mode. When you use it to solve an equation, find a derivative at a point or evaluate a definite integral, write down the equation, function or integral you used.

  1. 1.
    A new app is downloaded at a rate modeled by r(t)=4+7te−t/2 thousand downloads per day, where t is the number of days since the app was released, for 0≤t≤6.
    1. (a)
      Find the average rate of change of r(t) over the interval 0≤t≤2. Indicate units of measure. Show the setup for your calculations.
    2. (b)
      Evaluate r′(4), and interpret the value in the context of the problem, using correct units.
    1. (c)
      Must there be a time t, for 0<t<2, at which the rate at which the app is downloaded is 7 thousand downloads per day? Justify your answer.
    2. (d)
      According to the model, at what times t, for 0<t<6, is the rate at which the app is downloaded equal to 7 thousand downloads per day? Show the setup for your calculations.

Part B: No calculator is allowed.

Questions 2–4
  1. 2.
    The functions f and g are differentiable for all real numbers. The table gives values of f, g and their derivatives at selected values of x.
    x01235f(x)−11−4−6−8f′(x)4−6−3−11g(x)021−1−2g′(x)1−2−23−3
    1. (a)
      Let w be the function defined below, where a and b are constants. If w is continuous at x=1, write an equation that a and b must satisfy. Show the work that leads to your answer.
      w(x)={f(x)for x≤1a g(x)+bfor x>1
    2. (b)
      Find the values of a and b for which w is differentiable at x=1. Show the work that leads to your answer.
    1. (c)
      Let h be the function defined by h(x)=f(x)+k g(x), where k is a constant. The line tangent to the graph of h at x=3 is parallel to the line y=−10x. Find the value of k. Show the work that leads to your answer.
    2. (d)
      Let q be the function defined by q(x)=f(x)g(x). Find q′(5). Show the work that leads to your answer.
  2. 3.
    For all real numbers x, the function f is given by f(x)=(x2−7x+11)ex.
    1. (a)
      Let ℓ be the line tangent to the graph of f at x=0. Write an equation for ℓ. Show the work that leads to your answer.
    2. (b)
      Find the x-coordinate of each point at which the line tangent to the graph of f is horizontal. Show the work that leads to your answer.
    1. (c)
      Let r be the greater of the two x-coordinates found in part (b). A differentiable function g satisfies g(r)=4 and g′(r)=−1. The function k is defined by k(x)=f(x)g(x). Find k′(r). Show the work that leads to your answer.
    2. (d)
      Let h be the function defined below. Is h differentiable at x=0? Justify your answer.
      h(x)={f(x)for x≤05x+11for x>0
  3. 4.
    Runner A runs along a straight path. Her velocity, in meters per second, is given by a differentiable function vA of time t, in seconds, for 0≤t≤32. Selected values of vA(t) are given in the table. Runner B runs along the same path in the same direction with velocity vB(t)=512t−1120t2 meters per second for 0≤t≤32.t (seconds) 04101732vA(t) (meters per second) 25676
    1. (a)
      From the data in the table, approximate vA′(23), and show the computations that lead to your answer. What does this value tell you in the context of the problem? Include units of measure.
    2. (b)
      Find the acceleration of runner B at time t=8. Indicate units of measure. Is the velocity of runner B increasing or decreasing at time t=8? Give a reason for your answer.
    1. (c)
      Must there be a time t, for 0<t<10, at which the velocity of runner A is 3 meters per second? Justify your answer.
    2. (d)
      Runner B's pace, the number of seconds she takes per meter, is P(t)=1vB(t) seconds per meter. Find P′(30). Show the work that leads to your answer. Indicate units of measure.

      End of Version A

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.

Scoring guide · Version A

Question 1Rates of change from a table or a rate model · calculator · 9 points

(a) 2 points Average rate of change · Topic 2.1

  • 1 point: r(2)−r(0)2
  • 1 point: 2.575 thousand downloads per day per day

Solution

  1. On the calculator, r(0)=4.000 and r(2)=9.150.
  2. The average rate of change is the change in r divided by the change in t: 2.575 thousand downloads per day per day.

(b) 2 points Derivative value with units interpretation · Topic 2.1

  • 1 point: r′(4)=−0.947
  • 1 point: interpretation with the time, the rate and units: at t=4 days, the rate at which the app is downloaded is decreasing at 0.947 thousand downloads per day per day

Solution

  1. Use the calculator's derivative of r at t=4: r′(4)=−0.947.
  2. Its sign says the rate at which the app is downloaded is decreasing at t=4, by 0.947 thousand downloads per day per day.

Model answer. r′(4)=−0.947 (calculator). At time t=4 days, the rate at which the app is downloaded is decreasing at a rate of 0.947 thousand downloads per day per day.

(c) 2 points IVT justification · Topic 1.16

  • 1 point: r(0)=4 and r(2)=9.150, so 7 is between them, with r continuous on 0≤t≤2
  • 1 point: yes, by the Intermediate Value Theorem (eligible only with the first point)

Solution

  1. On the calculator, r(0)=4 and r(2)=9.150, so 7 is between them.
  2. r is a sum and product of continuous functions, so the Intermediate Value Theorem applies on 0≤t≤2.

Model answer. Yes. r is continuous on 0≤t≤2, because it is built from continuous functions, and r(0)=4<7<9.150=r(2). Therefore, by the Intermediate Value Theorem, there must be a time t, for 0<t<2, at which r(t)=7.

(d) 3 points Times when a model reaches a value · Topic 1.16

  • 1 point: r(t)=7
  • 1 point: t=0.570
  • 1 point: t=4.854

Solution

  1. Graph y=r(t) and y=7 on 0<t<6.
  2. They meet twice: at t=0.570, while the rate rises, and at t=4.854, after it peaks.

Question 2Derivative rules from a table of values · no calculator · 9 points

(a) 2 points Value for continuity · Topic 1.13

  • 1 point: lim⁡x→1−w(x)=f(1)=1 and lim⁡x→1+w(x)=a g(1)+b=2a+b
  • 1 point: 2a+b=1

Solution

  1. w(1)=f(1)=1 is also the limit from the left; g is differentiable, hence continuous, so the limit from the right is a g(1)+b.
  2. Setting them equal, with g(1)=2 from the table: 2a+b=1.

(b) 3 points Constants for differentiability · Topic 2.4

  • 1 point: equal one-sided derivatives: f′(1)=a g′(1), that is, −6=−2a
  • 1 point: a=3
  • 1 point: b=−5, using the equation from part (a)

Solution

  1. Differentiability at x=1 needs continuity, 2a+b=1, and equal slopes from the two sides: f′(1)=a g′(1), that is, −6=−2a.
  2. So a=3, and the equation from part (a) gives b=−5.

(c) 2 points Horizontal, vertical or given-slope tangent · Topic 2.2

  • 1 point: h′(3)=f′(3)+k g′(3)=−10: −1+k(3)=−10
  • 1 point: k=−3

Solution

  1. Parallel lines have equal slopes, so the slope of the tangent line is −10, and the slope of the tangent line at x=3 is h′(3)=f′(3)+k g′(3).
  2. From the table, −1+k(3)=−10, so k=−3.

(d) 2 points Quotient rule · Topic 2.9

  • 1 point: the quotient rule with the table values:
    q′(5)=f′(5)g(5)−f(5)g′(5)(g(5))2=(1)(−2)−(−8)(−3)(−2)2
  • 1 point: q′(5)=−132

Solution

  1. The quotient rule gives q′(x)=f′(x)g(x)−f(x)g′(x)(g(x))2.
  2. From the table, f(5)=−8, f′(5)=1, g(5)=−2 and g′(5)=−3, so q′(5)=−132.

Question 3Derivatives and tangent lines of a given function · no calculator · 9 points

(a) 3 points Tangent line or tangent-line approximation · Topic 2.2

  • 1 point: product rule:
    f′(x)=(2x−7)ex+(x2−7x+11)ex
  • 1 point: slope f′(0)=4
  • 1 point: the tangent line y=11+4x

Solution

  1. Use the product rule:
    f′(x)=(2x−7)ex+(x2−7x+11)ex; simplified, f′(x)=(x2−5x+4)ex.
  2. f(0)=11 and f′(0)=4.
  3. The tangent line is y=11+4x.

(b) 2 points Horizontal, vertical or given-slope tangent · Topic 2.2

  • 1 point: sets f′(x)=0
  • 1 point: x=1 and x=4

Solution

  1. A horizontal tangent has slope 0, so solve f′(x)=0, with f′ from part (a).
  2. The quadratic factor is (x−1)(x−4), which is 0 at x=1 and x=4.

(c) 2 points Product rule · Topic 2.8

  • 1 point: the product rule with f′(r)=0 from part (b):
    k′(4)=f′(4)g(4)+f(4)g′(4)=0⋅(4)+(−e4)(−1)
  • 1 point: k′(r)=e4

Solution

  1. r is the greater of the two x-coordinates found in part (b), so r=4; there the tangent line is horizontal, so f′(4)=0.
  2. The product rule gives k′(x)=f′(x)g(x)+f(x)g′(x); with f(4)=−e4, g(4)=4 and g′(4)=−1, k′(4)=e4.

(d) 2 points Differentiability at a point · Topic 2.4

  • 1 point: lim⁡x→0−h(x)=f(0)=11 and lim⁡x→0+h(x)=11, so h is continuous at x=0
  • 1 point: no: the slope from the left, f′(0)=4 (part (a)), is not equal to the slope from the right, 5

Solution

  1. Check continuity at x=0: both one-sided limits equal 11=h(0).
  2. Compare the one-sided derivatives: f′(0)=4 from the left and 5, the slope of the line, from the right.
  3. They differ, so h is not differentiable at x=0 (the graph has a corner there).

Model answer. lim⁡x→0−h(x)=f(0)=11 and lim⁡x→0+h(x)=11, so h is continuous at x=0. For x<0, h′(x)=f′(x), and f′(0)=4, the slope of the tangent line in part (a); for x>0, h′(x)=5. Since 4≠5, the one-sided derivatives are not equal, so h is not differentiable at x=0.

Question 4A rate or a velocity from a table, without a calculator · no calculator · 9 points

(a) 2 points Table estimate of a derivative · Topic 2.3

  • 1 point: approximation vA(32)−vA(17)32−17=−115
  • 1 point: interpretation with the time, the rate and units: at t=23 seconds, the velocity of runner A is decreasing at about 115 meters per second per second

Solution

  1. vA′(23) is the acceleration of runner A at t=23. The table times on either side of it are t=17 and t=32; the average rate of change of vA between them approximates vA′(23).
  2. 6−715=−115 meters per second per second: her velocity is decreasing at about 115 meters per second per second.

Model answer. vA′(23)≈vA(32)−vA(17)32−17=6−715=−115 meters per second per second. At time t=23 seconds, the velocity of runner A is decreasing at a rate of approximately 115 meters per second per second.

(b) 2 points Increasing or decreasing at a time · Topic 2.5

  • 1 point: vB′(8)=1760 meters per second per second
  • 1 point: increasing, because vB′(8)>0

Solution

  1. Acceleration is the derivative of velocity; by the power rule, vB′(t)=512−160t.
  2. At t=8:
    vB′(8)=1760 meters per second per second. Its sign decides whether vB is increasing or decreasing there.

Model answer. The acceleration of runner B is vB′(t)=512−160t, so at t=8 it is vB′(8)=1760 meters per second per second. Since vB′(8)>0, the velocity of runner B is increasing at time t=8.

(c) 2 points IVT justification · Topic 1.16

  • 1 point: vA is differentiable, so it is continuous on 0≤t≤10, and vA(0)=2<3<6=vA(10)

    Also acceptable: The same argument on 0≤t≤4: vA(0)=2<3<5=vA(4).

  • 1 point: yes, by the Intermediate Value Theorem (eligible only with the first point)

Solution

  1. From the table, vA(0)=2<3<6=vA(10).
  2. A differentiable function is continuous, so the Intermediate Value Theorem applies to vA on the closed interval.

(d) 3 points Quotient rule · Topic 2.9

  • 1 point: P′(t)=−vB′(t)(vB(t))2
  • 1 point: P′(30)=1300
  • 1 point: units:
    P′(30)=1300 seconds per meter per second

Solution

  1. Write P=1vB and use the quotient rule: the numerator 1 has derivative 0, so P′(t)=−vB′(t)(vB(t))2.
  2. From the formula, vB(30)=5 and vB′(30)=512−160(30)=−112.
  3. P′(30)=− ⁣(−112)⋅125=1300 seconds per meter per second.

What does the AP Calculus AB free response ask on Unit 2?

A fair share: Unit 2 is about 8% of the free-response points since 2012.

This set has 4 questions in that mix, 1 with a calculator (about 26% of Unit 2 free-response points are in the calculator section), each scored out of 9 with a scoring guide.

Counted from the official AP Calculus AB free-response questions, 2012–2026. No official question is reproduced here.

AP Calculus AB Unit 2 free-response settings by share of exam points

Free-response settings on this set and their share of official AP Calculus AB free-response points, 2012–2026
SettingQuestions on this setShare of Unit 2 free-response pointsShare of all free-response pointsLast on the exam
Rates of change from a table or a rate modelCalculator1 of 420%–2026
Derivative rules from a table of valuesNo calculator1 of 426%–2026
Derivatives and tangent lines of a given functionNo calculator1 of 420%–2021
A rate or a velocity from a table, without a calculatorNo calculator1 of 415%–2025

A part counts toward a setting when its primary skill is in Unit 2, in the same section, and the setting asks that skill (in the part’s own setting where there is one), on the main exams 2012–2026, by its points; a part that fits two settings is shared between them.

Multiple-choice practice for this unit: AP Calculus AB Unit 2 Worksheet: Derivatives.

Why these kinds of question: AP Calculus AB Unit 2 Review: The Most-Tested Derivative Questions.

Questions about this free-response set

Unit 2, Differentiation: Definition and Fundamental Properties, has 10 topics in the College Board's course description: average and instantaneous rates of change; the derivative as a limit and derivative notation, including tangent lines; estimating a derivative from a table or graph; where a function is and is not differentiable; the power rule; the constant, sum and difference rules; the derivatives of sin x, cos x, eˣ and ln x; the product and quotient rules; and the derivatives of tan x, cot x, sec x and csc x. The course description weights the unit at 10–15% of the multiple-choice section.

Each question is worth 9 points, spread over its three or four parts. Points are earned for specific pieces of work: a setup, an antiderivative, a value with its units, a conclusion with its reason. The scoring guide lists every point and what earns it, then a worked solution for each part. On the exam the six free-response questions are half of the score.

Only where the exam allows one. On this set, question 1 is Part A, where a graphing calculator is required, and questions 2–4 are Part B, without one. A calculator question has parts that need one (a value to three decimal places, an equation solved on the calculator); its other parts can be done by hand, as on the exam.

We tagged every free-response part from 2012 to 2026 by the skill it tests and the setting of its question. The Unit 2 settings the exam leans on most get the most questions here, each in its own context with its own parts.

No. Every question here is original, written by Sigma Prep in the style of the exam: the settings, functions, numbers and wording are our own, and no official question or scoring guideline is copied or reworded. Every value and statement in the scoring guide was checked by an independent computer-algebra check before the set was published.

Yes. Each version has its own: open it under the questions, or download the PDF, which prints the questions with room to write and then the scoring guide on its own pages. There is also a PDF without the scoring guide to hand out.

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