AP Calculus AB Differential Equations and Slope Fields Test with Point Values (PDF)

AP Calculus AB math test: Differential Equations and Slope Fields – Maths PDF

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This differential equations test for AP Calculus AB covers differential equations and slope fields (Common Core AP CALC AB 7.1, AP CALC AB 7.2, AP CALC AB 7.3, AP CALC AB 7.4, AP CALC AB 7.5, AP CALC AB 7.6, AP CALC AB 7.7), with a free PDF to print.

This test checks differential equations, slope fields, Euler’s method, separation of variables and exponential growth and decay (AP Calculus AB, 7.1 to 7.7). It has 8 questions worth 20 points in total. Allow 45 minutes, work without notes, and show all work, since justification earns points on the AP exam.

Time and points

About this test

Grade: AP Calculus AB
Chapter: Differential Equations and Slope Fields
Standards: AP CALC AB 7.1, AP CALC AB 7.2, AP CALC AB 7.3, AP CALC AB 7.4, AP CALC AB 7.5, AP CALC AB 7.6, AP CALC AB 7.7
Time: 45 minutes
Total: 20 points

Differential equations test: questions

Question 1 – Verify a solution [2 pts]

Show that \(y=4e^{-3x}\) is a solution of \(\frac{dy}{dx}=-3y\).

Question 2 – Slope field analysis [3 pts]

Slope field for dy/dx = x − y.

The slope field is for \(\frac{dy}{dx}=x-y\). (a) Find the slope at \((2,-1)\). (1 point) (b) Where are the segments horizontal? (1 point) (c) Describe the solution through \((0,1)\) near that point (rising or falling). (1 point)

Question 3 – Euler's method [2 pts]

Use Euler’s method with \(h=0.5\) to estimate \(y(1)\) for \(\frac{dy}{dx}=2x-y\), \(y(0)=1\).

Question 4 – Separation of variables [3 pts]

Solve \(\frac{dy}{dx}=\frac{2x}{y}\) with \(y(0)=3\).

Question 5 – Exponential growth [3 pts]

Fish population in a lake.

A fish population satisfies \(\frac{dP}{dt}=0.05P\) with \(P(0)=800\) and \(t\) in years. (a) Find \(P(t)\). (1 point) (b) Find \(P(6)\). (1 point) (c) When does the population reach 1600? (1 point)

Question 6 – Matching a slope field [2 pts]

A slope field.

The slope field depends only on \(y\): horizontal segments at \(y=0\), positive slopes for \(y > 0\) and negative slopes for \(y < 0\). Which equation matches: \(\frac{dy}{dx}=x\), \(\frac{dy}{dx}=y\) or \(\frac{dy}{dx}=x+y\)? Explain.

Question 7 – Cooling [3 pts]

A cold drink at 5 °C is placed in a 25 °C room and warms according to \(\frac{dT}{dt}=-0.2(T-25)\) with \(t\) in minutes. (a) Let \(u=T-25\) and solve for \(T(t)\). (2 points) (b) Find \(T(5)\). (1 point)

Question 8 – Half-life [2 pts]

A radioactive sample decays by \(\frac{dN}{dt}=-kN\) with a half-life of 8 days. What fraction of the sample remains after 24 days? After 12 days (exact)?

Key facts behind this differential equations test

Key facts

Verify a solution by substituting y and y′ into the equation.
A slope field shows dy/dx at grid points; solution curves follow the segments.
Euler: y_new = y_old + h · (dy/dx at the old point), x_new = x_old + h.
dy/dt = ky has the general solution y = Ce^(kt); growth if k is positive, decay if k is negative.

How to use this differential equations test

Time yourself: the test is designed for an average student in 45 minutes. Answer the questions you are sure of first, then come back to the longer ones and check your units.

All AP Calculus AB chapters: AP Calculus AB math.

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