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Differential Equations Test – AP Calculus AB Math (Printable PDF)

Last updated: October 22, 2026 · Written by the Math Made Easy math teachers

AP Calculus AB Math Tests – Mathbot, the Math Made Easy robot coach, with free printable tests for AP Calculus AB

This 45-minute test follows the AP Calculus AB style: two multiple-choice items and free-response questions requiring a full solution. Show every step. The total is 20 points, and no calculator is needed except for numerical answers.

Standards: AP Calculus AB Unit 7 · Level: Grade 12 AP Calculus AB

Time: 45 minutes · 8 questions · Total: 20 points · Calculator: as allowed by your teacher

Question 1 Solution with an initial condition (2 points)

Which function is the solution of dydx = 3y − 6 with y(0) = 7?

  1. 2 + 5e3x
  2. 2 + 7e3x
  3. 7e3x
  4. 7 + e3x

Question 2 Matching a slope field (2 points)

Which differential equation matches the slope field shown?

−4−3−2−11234−3−2−11234xy
Slope field.
  1. dydx = x + y
  2. dydx = x − y
  3. dydx = y − x
  4. dydx = xy

Question 3 Separable equation (3 points)

Consider dydx = 6x2y with y(1) = 3.

  1. Separate the variables and integrate both sides. (2 points)
  2. Find the particular solution. (1 point)

Question 4 Radioactive decay (3 points)

A radioactive sample of mass m (grams) satisfies dm/dt = −0.04m with m(0) = 250, t in years.

  1. Write m(t). (1 point)
  2. Find the half-life. (1 point)
  3. When will 40 grams remain? (1 point)

Question 5 Verifying a solution (3 points)

Let y = e2x + x.

  1. Show that y satisfies y′ − 2y = 1 − 2x. (2 points)
  2. Show that y satisfies the initial condition y(0) = 1. (1 point)

Question 6 Slope field reasoning (3 points)

The slope field for dydx = y(4 − y) is shown.

  1. Name the equilibrium solutions. (1 point)
  2. For the solution with y(0) = 1, find the limit of y as x → ∞. (1 point)
  3. Is the solution concave up or concave down where 0 < y < 2? Justify. (1 point)
−112345−112345xy
Slope field.

Question 7 Cooling model (2 points)

A drink at 80°F is placed in a 20°F refrigerator. Its temperature T satisfies dT/dt = −0.1(T − 20), t in minutes.

  1. Find T(5). (1 point)
  2. What does T approach as t → ∞? (1 point)

Question 8 Tangent line approximation (2 points)

The function y = f(x) satisfies dydx = x2 − y with f(1) = 2.

  1. Write the equation of the tangent line at x = 1. (1 point)
  2. Use it to approximate f(1.1). (1 point)

View the Differential Equations test answer key with scoring

Differential Equations lessonDifferential Equations worksheetDifferential Equations quiz

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