AP Calculus Differential Equations Quiz – Free PDF

Contents

This ap calculus differential equations quiz asks 10 multiple-choice questions on differential equations, and each answer comes with a short explanation.

Ten multiple-choice questions check slope fields, verifying solutions, separation of variables and exponential models. Choose one answer for each question, then read the explanation to see the typical mistake.

AP Calculus ABCommon Core: AP Calculus AB Unit 7

🎯 Differential Equations – AP Calculus AB math quiz

Score: 0 / 10

Answered: 0 / 10

Question 1

Which function is a solution of \(\frac{dy}{dx}=3y\) with \(y(0)=5\)?

A \(y=5e^{3x}\)

B \(y=3e^{5x}\)

C \(y=15x\)

D \(y=5e^{x}+3\)

Question 2

Find the slope given by \(\frac{dy}{dx}=xy-1\) at the point \((2,3)\).

A \(5\)

B \(6\)

C \(4\)

D \(-1\)

Question 3

What is the first step in separating \(\frac{dy}{dx}=\frac{x}{y^{2}}\)?

A \(dy=\frac{x}{y^{2}}\,dx\)

B \(y^{2}\,dy=x\,dx\)

C \(\frac{dy}{y^{2}}=x\,dx\)

D \(y\,dy=x^{2}\,dx\)

Question 4

A slope field.

Which differential equation produces this slope field?

A \(\frac{dy}{dx}=y\)

B \(\frac{dy}{dx}=x\)

C \(\frac{dy}{dx}=-y\)

D \(\frac{dy}{dx}=-x\)

Question 5

If \(\frac{dy}{dx}=-2y\) and \(y(0)=4\), what is \(y(\ln 2)\)?

A \(2\)

B \(16\)

C \(\frac{1}{4}\)

D \(1\)

Question 6

The population doubles every 2 hours.

A population satisfies \(\frac{dP}{dt}=kP\) and doubles every 2 hours from 100. What is \(P(7)\), rounded to the nearest whole number?

A 700

B 1,131

C 800

D 1,600

Question 7

A substance has a half-life of 10 years and obeys \(\frac{dy}{dt}=ky\). What is \(k\)?

A \(\frac{\ln 2}{10}\)

B \(-\frac{10}{\ln 2}\)

C \(-\frac{1}{20}\)

D \(-\frac{\ln 2}{10}\)

Question 8

A slope field.

Which constant function is a solution of the differential equation that has this slope field?

A \(y=0\)

B \(y=2\)

C \(y=1\)

D \(y=-1\)

Question 9

Which function is a solution of \(\frac{dy}{dx}=2xy\) with \(y(0)=3\)?

A \(y=e^{x^{2}}+2\)

B \(y=3e^{2x}\)

C \(y=3e^{x^{2}}\)

D \(y=x^{2}+3\)

Question 10

The temperature of a cooling object satisfies \(\frac{dT}{dt}=-0.1(T-70)\) with \(T(0)=190\). What is \(\lim_{t\to\infty}T(t)\)?

A \(0\)

B \(190\)

C \(-70\)

D \(70\)

🎉 Quiz complete!

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How does this quiz work?

Read each question, choose one answer among A, B, C and D, then check it: a short explanation appears after every answer. Your score out of 10 is shown at the end, and you can restart the quiz as often as you like.

What should you review before the quiz?

Reread the key facts of the lesson and the worked example. If a question is hard, go back to the matching part of the lesson, then try the quiz again. Write your work on scrap paper: most mistakes come from skipping a step, not from the idea itself.

Remember:

Verify a solution by differentiating and substituting.
A slope field shows \(\frac{dy}{dx}\) at grid points; solutions follow the segments.
Separable: \(g(y)\,dy=h(x)\,dx\); integrate both sides, add \(C\) once, then use the initial condition.
\(\frac{dy}{dt}=ky\) has solution \(y=y_{0}e^{kt}\).