Worksheets · AP Calculus AB

AP Calculus AB Unit 7 Worksheet: Differential Equations

20 exam-style multiple-choice questions on separation of variables, particular solutions, slope fields and exponential growth and decay, free to use in class. Good practice before the Unit 7 Progress Check in AP Classroom. Free response for this unit: Unit 7 FRQ set.

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

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AP Calculus AB Unit 7: Differential Equations

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 1–18
  1. For the differential equation dydx=x+y, the points (x,y) at which the slope field has slope 3 lie on which of the following?
    1. (A)y=−x+3
    2. (B)y=−x−3
    3. (C)y=x+3
    4. (D)y=−x
  2. Which of the following is the general solution of dydx=−3x2y, where C is an arbitrary constant?
    1. (A)y=Cex3
    2. (B)y=e−x3
    3. (C)y=e−x3+C
    4. (D)y=Ce−x3
  3. The differential equation y′′+16y=0 has which of the following as a solution?
    1. (A)y=cos⁡(4x)+3
    2. (B)y=e−4x
    3. (C)y=cos⁡(16x)
    4. (D)y=−3sin⁡(4x)
  4. −3−2−10123−3−2−10123xy
    The slope field drawn above belongs to which differential equation?
    1. (A)dydx=(x−1)2
    2. (B)dydx=y2
    3. (C)dydx=x2−y
    4. (D)dydx=x2
  5. The balance B of a savings account increases with time t, and at each moment its rate of change is proportional to the balance at that moment. Which of the following differential equations expresses this, with k a positive constant?
    1. (A)dBdt=kB
    2. (B)dBdt=kB2
    3. (C)dBdt=kB
    4. (D)dBdt=kt
  6. For t≥0, K(t) is the temperature, in degrees Fahrenheit, of a cast-iron skillet t minutes after it is taken off a stove. It changes according to dKdt=−16(K−67), and K(0)=257. Which expression below is K(t)?
    1. (A)257−953t
    2. (B)67+257e−t6
    3. (C)67−e−t6
    4. (D)67+190e−t6
  7. Which of the following lists every horizontal line y=c that is a solution curve of dydx=(y−2)(x−1)?
    1. (A)y=1 and y=2
    2. (B)y=2 only
    3. (C)y=−2 only
    4. (D)y=1 only
  8. A curve in the xy-plane passes through (2,0), and its slope at each of its points (x,y) is 1−yx2. Which equation describes the curve?
    1. (A)y=1−e1x−12
    2. (B)y=e1x−12−1
    3. (C)y=1−e12−1x
    4. (D)y=12−1x
  9. Every graph below shows the slope field of dydx=x−y together with a curve. In which graph could the curve be the solution with y(−1)=2?
    (A)
    −3−2−10123−3−2−10123xy
    (B)
    −3−2−10123−3−2−10123xy
    (C)
    −3−2−10123−3−2−10123xy
    (D)
    −3−2−10123−3−2−10123xy
    1. (A)Graph (A)
    2. (B)Graph (B)
    3. (C)Graph (C)
    4. (D)Graph (D)
  10. For the solution of dydx=xy2 whose graph passes through (1,−2), what is y when x=3?
    1. (A)−15
    2. (B)−29
    3. (C)27
    4. (D)−217
  11. For t≥0, S(t) is the mass of salt, in kilograms, dissolved in a mixing tank t minutes after brine starts to flow in. It changes according to dSdt=13(45−S), and S(0)=0. How many minutes after brine starts to flow in does S reach 30?
    1. (A)13ln⁡3
    2. (B)3ln⁡ ⁣(32)
    3. (C)2
    4. (D)3ln⁡3
  12. A solution curve of dydx=y2x+1 passes through the point (0,−1). At x=1, the y-coordinate of this curve is
    1. (A)3−2
    2. (B)−3
    3. (C)−33
    4. (D)3
  13. Below is a student's attempt to find the solution of dydx=3xy that has y=4 at x=2. Which step is the first one that contains an error?
    Step 1:  ∫1y dy=∫3x dxStep 2:  ln⁡∣y∣=3x22+CStep 3:  ∣y∣=Ke3x22Step 4:  Since y=4 when x=2, 4=Ke6, so K=4.
    Step 5:  y=4e3x22
    1. (A)Step 2
    2. (B)Step 3
    3. (C)Step 4
    4. (D)Step 5
  14. The particular solution of dydx=13xy3 with y(1)=1 is defined on an interval around x=1. Which of the following is the largest such open interval?
    1. (A)x>−2
    2. (B)all real numbers x
    3. (C)−2<x<2
    4. (D)x<2
  15. Which of the following functions satisfies both dydx=3(3−y)cos⁡x and y(0)=0?
    1. (A)y=1−e−3sin⁡x
    2. (B)y=9sin⁡x
    3. (C)y=3−3e−3sin⁡x
    4. (D)y=3−e−3sin⁡x
  16. A student's work for solving dydx=3x2y, with y=2 when x=0, is shown below. In which step does the first error appear?
    Step 1:  ∫1y dy=∫3x2 dxStep 2:  ln⁡∣y∣=x3+CStep 3:  ∣y∣=ex3+CStep 4:  Since y=2 when x=0, 2=e0+C, so C=1.
    Step 5:  y=ex3+1
    1. (A)Step 2
    2. (B)Step 3
    3. (C)Step 4
    4. (D)Step 5
  17. For t≥0, Q(t) is the charge, in microcoulombs, on a capacitor t milliseconds after the circuit is switched on. It changes according to dQdt=14(60−Q), and Q(0)=0. According to this model, what is Q(2)?
    1. (A)30
    2. (B)60−e−12
    3. (C)60−60e−12
    4. (D)60+60e−12
  18. If y=f(x) solves dydx=ysec⁡2x and its graph contains the point (0,−1), what is f(1)?
    1. (A)etan⁡1−1
    2. (B)etan⁡1
    3. (C)−etan⁡1
    4. (D)−e− ⁣tan⁡1

Part B: A graphing calculator is allowed.

Questions 19–20
  1. The number of trees in a young forest follows dNdt=kN, where t is in years, and the number of trees doubles in 14 years. To three decimal places, what is k?
    1. (A)0.050
    2. (B)0.022
    3. (C)0.071
    4. (D)−0.050
  2. A sourdough starter has a mass of 80 grams when it is fed and 150 grams 2 hours later. Its mass grows at a rate proportional to its mass. According to this model, what is its mass 5 hours after it is fed?
    1. (A)255.000 grams
    2. (B)385.117 grams
    3. (C)158.306 grams
    4. (D)713.032 grams

    End of Version A

Answer key · Version A

12345678910
ADDDADBAAB
7.37.67.27.37.17.77.47.77.37.7
11121314151617181920
DBCCCBCCAB
7.77.77.77.77.77.77.77.77.87.8

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 7?

Very little of the multiple choice: Unit 7 was only 14 of the 315 official multiple-choice questions on the seven 2012–2018 practice exams (4.4%), but a separable differential equation was on the free response of 12 of the 15 exams from 2012 to 2026.

This worksheet has 20 questions in that mix: Part A (18) without a calculator and Part B (2) with one; the exam put 16% of its official Unit 7 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 7 question types: how often the exam asks each

Official AP Calculus AB multiple-choice questions by question type, Unit 7 (Differential Equations), 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
Write a differential equation from wordsTopic 7.11331
Verify a solution of a differential equationTopic 7.21221
Slope fieldsTopic 7.32664
Sketch a slope field or a solution curve on it (FRQ)Topic 7.31Asked on 17 free-response questions since 1998, most recently in 2024.
Reason about solutions from a differential equation or slope fieldTopic 7.41No official multiple-choice question is tagged to this topic in the 17 official exams (1969–2018); reasoning about solutions straight from the equation (where they rise, level off or bend) is in the course framework (7.4) and in free-response parts on concavity, limits and extrema.
General solution by separation of variablesTopic 7.61No official multiple-choice question is tagged to this topic alone in the 17 official exams (1969–2018); the general solution is the first half of every separable-equation question, and the course framework lists it as its own topic (7.6).
Separable DE with an initial conditionTopic 7.711886
Exponential growth and decay (dy/dt = ky)Topic 7.82662

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.

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

Why these kinds of question: AP Calculus AB Unit 7 Review: The Most-Tested Differential Equation Questions.

Questions about this worksheet

Unit 7, Differential Equations, has 7 AB topics in the College Board's course description: modeling situations with differential equations; verifying solutions; sketching slope fields; reasoning using slope fields; finding general solutions by separation of variables; finding particular solutions from an initial condition; and exponential models, dy/dt = ky. Topics 7.5 and 7.9 (Euler's method and logistic models) are BC only. The course description weights the unit at 5–10% of the multiple-choice section, and a separable differential equation was on the free-response section of 12 of the 15 exams from 2012 to 2026.

Only in Part B, as on the exam. Questions 1–18 are Part A, with no calculator; questions 19–20 are Part B, where the exam allows a graphing calculator (on the digital exam, the Desmos calculator built into Bluebook is available there). Both Unit 7 Part B questions need one: exponential growth and decay, such as the constant k from a doubling time or a half-life, or an amount after a time from a half-life or from two measurements. 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 7 questions into 8 kinds, and weighted each kind by how often and how recently the exam asks it. Every kind the exam actually uses gets at least one question, and the most-tested get more (11 on separable DE with an initial condition, 2 on slope fields, 2 on exponential growth and decay (dy/dt = ky)).

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 7 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.