• 59 questions

AP Calculus Unit 7 Practice Test

Unofficial practice for Unit 7 of AP Calculus AB and BC: slope fields, separation of variables, exponential and logistic models.

What is a differential equation?

Pick an answer, then press Check answer. Then you see Correct or Not quite, with the reason. Keys: A to D pick, Enter checks.

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Unofficial practice questions, not affiliated with the College Board. Bank checked October 14, 2026.

Next step AP Calculus integrals: Unit 6AP Calculus BC Unit 9: parametric and polar

All 59 questions

Topics 7.1 and 7.2: Writing and verifying differential equations 12

  • A cup of coffee cools in a 20 °C room. The rate of change of its temperature T is proportional to the difference between T and the room temperature. Which equation says this? Wikipedia, "Newton's law of cooling"; course topic 7.1
  • For which value of k is y = e^(kx) a solution of dy/dx + 5y = 0? Wikipedia, "Differential equation"; course topic 7.2
  • How many functions are solutions of dy/dx = 2x? Wikipedia, "Differential equation"; course topic 7.2
  • The rate at which a population P grows is proportional to the population. Which differential equation says this? Wikipedia, "Differential equation"; course topic 7.1
  • The rate of change of y with respect to x is inversely proportional to y. Which equation says this? Wikipedia, "Differential equation"; course topic 7.1
  • The velocity v of a falling object changes at a rate equal to 10 minus one fifth of the velocity. Which equation says this? Wikipedia, "Differential equation"; course topic 7.1
  • Water enters a tank at 6 liters per minute and leaks out at a rate proportional to the square root of the volume V in the tank. With k > 0, which equation models V? Wikipedia, "Differential equation"; course topic 7.1
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Topics 7.1 and 7.2: Writing and verifying differential equations, continued

  • What is a differential equation? Wikipedia, "Differential equation"; course topic 7.1
  • Which differential equation has y = xe^x as a solution? Wikipedia, "Differential equation"; course topic 7.2
  • Which function is a solution of dy/dx = 3y? Wikipedia, "Differential equation"; course topic 7.2
  • Which function is a solution of y″ + 4y = 0? Wikipedia, "Differential equation"; course topic 7.2
  • Which function satisfies x · dy/dx = 2y for x > 0? Wikipedia, "Differential equation"; course topic 7.2

Topics 7.3 and 7.4: Slope fields 9

  • For dy/dx = y − 2, a solution curve passes through the point (0, 5). What does the slope field show about it as x increases? Wikipedia, "Slope field"; course topic 7.4
  • For dy/dx = y − 2, which solution is a horizontal line? Wikipedia, "Slope field"; course topic 7.4
  • In a slope field, the segments along each horizontal line are parallel to one another, but they change from one horizontal line to the next. Which equation fits? Wikipedia, "Slope field"; course topic 7.3
  • In the slope field for dy/dx = x + y, along which line are all the segments horizontal? Wikipedia, "Slope field"; course topic 7.3
  • In the slope field for dy/dx = x − y, what is the slope of the segment at the point (3, 1)? Wikipedia, "Slope field"; course topic 7.3
  • In the slope field for dy/dx = x/y, where is the slope undefined? Wikipedia, "Slope field"; course topic 7.3
  • In the slope field for dy/dx = xy, in which quadrants do the segments have negative slope? Wikipedia, "Slope field"; course topic 7.3
  • The slope field of dy/dx = −x/y has slope −3/4 at the point (3, 4). Which family of curves are its solutions? Wikipedia, "Slope field"; course topic 7.4
  • What does a slope field for dy/dx = F(x, y) show? Wikipedia, "Slope field"; course topic 7.3

Topic 7.5: Euler's method (BC only) 7

  • Euler's method is used on dy/dx = 2y with y(0) = 3. What does one step of size 0.1 give for y(0.1)? Wikipedia, "Euler method"; course topic 7.5
  • Euler's method is used on dy/dx = x + y with y(0) = 1 and two steps of size 0.5. What is the approximation of y(1)? Wikipedia, "Euler method"; course topic 7.5
  • Euler's method is used on dy/dx = x − y with y(0) = 2 and two steps of size 0.5. What is the approximation of y(1)? Wikipedia, "Euler method"; course topic 7.5
  • Euler's method is used on dy/dx = x² with y(1) = 2 and two steps of size 0.5. What is the approximation of y(2)? Wikipedia, "Euler method"; course topic 7.5
  • Euler's method with a positive step is used on dy/dx = y with y(0) = 1 to approximate y(1). The solution curve is concave up. How does the approximation compare with the true value? Wikipedia, "Euler method"; course topic 7.5
  • One step of Euler's method starts at (x₀, y₀) with step size h for dy/dx = F(x, y). What is the new y-value? Wikipedia, "Euler method"; course topic 7.5
  • What is the usual effect of using a smaller step size in Euler's method? Wikipedia, "Euler method"; course topic 7.5

Topics 7.6 and 7.7: Separation of variables 16

  • A particle on a line has dv/dt = 12t² and velocity v(1) = 7. What is v(t)? Wikipedia, "Separation of variables"; course topic 7.7
  • dy/dx = sin(x²) has no antiderivative in elementary functions. Which expression is still the particular solution with y(1) = 5? Wikipedia, "Fundamental theorem of calculus"; course topic 7.7
  • How many solutions of dy/dx = 2x pass through the point (1, 4)? Wikipedia, "Initial value problem"; course topic 7.7
  • The solution of dy/dx = xy² with y(0) = 1 is y = 2/(2 − x²). On which interval is it valid? Wikipedia, "Initial value problem"; course topic 7.7
  • What is the general solution of dy/dt = −2(y − 10)? Wikipedia, "Separation of variables"; course topic 7.6
  • What is the general solution of dy/dx = 2xe^(−y)? Wikipedia, "Separation of variables"; course topic 7.6
  • What is the general solution of dy/dx = 3x²y? Wikipedia, "Separation of variables"; course topic 7.6
  • What is the general solution of dy/dx = x/y? Wikipedia, "Separation of variables"; course topic 7.6
  • What is the general solution of dy/dx = y², for y not equal to 0? Wikipedia, "Separation of variables"; course topic 7.6
  • What is the particular solution of dy/dx = 2x(y − 1) with y(0) = 3? Wikipedia, "Separation of variables"; course topic 7.7
  • What is the particular solution of dy/dx = 2x/y with y(0) = 3? Wikipedia, "Separation of variables"; course topic 7.7
  • What is the particular solution of dy/dx = xy² with y(0) = 1? Wikipedia, "Separation of variables"; course topic 7.7
  • What is the particular solution of dy/dx = y cos x with y(0) = 4? Wikipedia, "Separation of variables"; course topic 7.7
  • What is the particular solution of dy/dx = −x/y with y(3) = −4? Wikipedia, "Separation of variables"; course topic 7.7
  • Which of these equations cannot be solved by separating the variables? Wikipedia, "Separation of variables"; course topic 7.6
  • Which step correctly separates the variables in dy/dx = (x + 1)/y²? Wikipedia, "Separation of variables"; course topic 7.6

Topic 7.8: Exponential models 8

  • A colony of bacteria follows dN/dt = 0.4N, with t in days. How fast is the colony growing at the moment N = 250? Wikipedia, "Exponential growth"; course topic 7.8
  • A culture grows according to dy/dt = ky and doubles every 5 hours. What is k? Wikipedia, "Exponential growth"; course topic 7.8
  • A population of 500 grows according to dP/dt = kP and reaches 1,500 after 4 years. What is the population after 8 years? Wikipedia, "Exponential growth"; course topic 7.8
  • A population satisfies dP/dt = 0.03P with P(0) = 200. What is P(t)? Wikipedia, "Exponential growth"; course topic 7.8
  • A quantity decays exponentially with a half-life of 6 hours. What fraction of the starting amount is left after 30 hours? Wikipedia, "Half-life"; course topic 7.8
  • A substance decays according to dy/dt = −0.1y, with t in years. What is its half-life? Wikipedia, "Half-life"; course topic 7.8
  • For y = 80e^(−0.2t), which differential equation and starting value does y satisfy? Wikipedia, "Exponential decay"; course topic 7.8
  • y satisfies dy/dt = ky with k < 0 and y(0) > 0. What happens to y as t grows without bound? Wikipedia, "Exponential decay"; course topic 7.8

Topic 7.9: Logistic models (BC only) 7

  • A logistic solution starts at a positive value below half its carrying capacity a. What is the shape of its graph? Wikipedia, "Logistic function"; course topic 7.9
  • A population follows dP/dt = 0.002P(500 − P) and starts at P(0) = 800. What happens to P over time? Wikipedia, "Logistic function"; course topic 7.9
  • A population follows dP/dt = 0.002P(500 − P) with P(0) = 50. At what population is it growing fastest? Wikipedia, "Logistic function"; course topic 7.9
  • A population follows dP/dt = 0.002P(500 − P). What is its carrying capacity? Wikipedia, "Logistic function"; course topic 7.9
  • A population follows dP/dt = 3P − P²/20 with P(0) = 10. What is the limit of P(t) as t grows without bound? Wikipedia, "Logistic function"; course topic 7.9
  • A rumor spreads through a school of 900 students at a rate jointly proportional to the number y who have heard it and the number who have not. Which equation models y? Wikipedia, "Logistic function"; course topic 7.9
  • For dP/dt = 0.002P(500 − P), what is the largest value the growth rate dP/dt can have? Wikipedia, "Logistic function"; course topic 7.9

How this list was compiled: Our own questions, written to Unit 7 (Differential Equations; topics 7.1 to 7.9; topics 7.5 and 7.9 are BC only) of the College Board's AP Calculus AB and BC Course and Exam Description, https://apcentral.collegeboard.org/media/pdf/ap-calculus-ab-and-bc-course-and-exam-description.pdf, read 2026-10-14. Every answer is mathematics worked from what the question prints, and the computed ones were worked out a second time with computer algebra on 2026-10-14. The rule, theorem or definition behind each answer was read on 2026-10-14 in the Wikipedia article named under the question. Licence: Own work.

How it works

This AP Calculus Unit 7 practice test has 59 questions of our own on Differential Equations, from slope fields to exponential and logistic models. It is unofficial and is not affiliated with or endorsed by the College Board, which owns AP.

Read the full guide

What Unit 7 covers

The College Board's course description for AP Calculus AB and BC, read October 14, 2026, gives Unit 7 nine topics. Six sets follow them: writing and verifying differential equations (12 questions), slope fields (9), Euler's method (7), separation of variables (16), exponential models (8) and logistic models (7). The description gives the topics no weights of their own, so the spread is ours.

AB or BC

Euler's method and logistic models each have their own set here, so an AB student can skip those two. The other four sets hold 45 questions.

  • Separate before you integrate: every y with dy on one side, every x with dx on the other.
  • Write the constant as soon as you integrate, then use the starting value to find it.
  • A logistic model grows fastest when the quantity is half the carrying capacity.

Questions and answers

What is Unit 7 in AP Calculus?

Differential Equations. Its topics cover writing and checking a differential equation, slope fields, Euler's method, separation of variables, and exponential and logistic models.

How much of the AP Calculus exam is Unit 7?

From 5 to 10% of the multiple-choice section on both the AB and the BC exam, by the College Board's course pages.

Is Euler's method on the AP Calculus AB exam?

No. The course description marks Euler's method (topic 7.5) and logistic models (topic 7.9) as BC only. The other seven topics of Unit 7 belong to both courses.

Is this the AP Calculus Unit 7 progress check?

No. The Progress Check for Unit 7 is the College Board's: about 15 multiple-choice questions in AB or about 20 in BC, and 3 free-response questions, assigned in AP Classroom. The questions on this page are ours.

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