AP Calculus AB Differential Equations and Slope Fields Test Solutions and Scoring (PDF)

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

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This differential equations test answers 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.

Use this answer key to score the differential equations test out of 20 points. Each solution shows the work and a scoring guide that says how the points are awarded. Give partial credit for a correct setup, a correct derivative or integral, and a correct final statement with units where they apply.

AP Calculus AB · Common Core State 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

Differential equations test answers: worked solutions

Question 1 – Verify a solution [2 pts]

\(\frac{dy}{dx}=4(-3)e^{-3x}=-12e^{-3x}\) and \(-3y=-3\cdot 4e^{-3x}=-12e^{-3x}\). They are equal.

Scoring: 1 point for the derivative; 1 point for showing both sides are equal.

Question 2 – Slope field analysis [3 pts]

\(2-(-1)=3\).
Where \(x=y\), along the line \(y=x\).
At \((0,1)\) the slope is \(0-1=-1\), so the solution is decreasing there.

Slope field for dy/dx = x − y.

Scoring: 1 point per part.

Question 3 – Euler's method [2 pts]

Step 1: slope at \((0,1)\) is \(-1\), so \(y_{1}=1+0.5(-1)=0.5\) at \(x=0.5\).

Step 2: slope at \((0.5,0.5)\) is \(2(0.5)-0.5=0.5\), so \(y_{2}=0.5+0.5(0.5)=0.75\).

Final answer: \(y(1)\approx 0.75\).

Scoring: 1 point per step.

Question 4 – Separation of variables [3 pts]

Separate: \(y\,dy=2x\,dx\). Integrate: \(\frac{y^{2}}{2}=x^{2}+C\).

\(y(0)=3\) gives \(C=\frac{9}{2}\), so \(y^{2}=2x^{2}+9\).

Final answer: \(y=\sqrt{2x^{2}+9}\).

Scoring: 1 point for separating; 1 point for integrating; 1 point for the particular solution.

Question 5 – Exponential growth [3 pts]

\(P(t)=800e^{0.05t}\).
\(P(6)=800e^{0.3}\approx 1079.9\).
\(800e^{0.05t}=1600\) gives \(t=\frac{\ln 2}{0.05}\approx 13.9\) years.

Fish population in a lake.

Scoring: 1 point per part.

Question 6 – Matching a slope field [2 pts]

The slope is \(0\) along \(y=0\) and has the same sign as \(y\), and it does not change with \(x\).

Final answer: \(\frac{dy}{dx}=y\).

A slope field.

Scoring: 1 point for the match; 1 point for the explanation.

Question 7 – Cooling [3 pts]

\(\frac{du}{dt}=-0.2u\), \(u=Ce^{-0.2t}\), \(u(0)=5-25=-20\), so \(T(t)=25-20e^{-0.2t}\).
\(T(5)=25-20e^{-1}\approx 17.64\) °C.

Scoring: (a) 1 point for the substitution and equation; 1 point for the solution; (b) 1 point.

Question 8 – Half-life [2 pts]

\(e^{-8k}=\frac{1}{2}\). After 24 days: \(\left(\frac{1}{2}\right)^{3}=\frac{1}{8}\). After 12 days: \(e^{-12k}=\left(\frac{1}{2}\right)^{3/2}=\frac{1}{2\sqrt{2}}\approx 0.354\).

Scoring: 1 point for each fraction.

Total score

Total

20 points

Key facts behind this differential equations test answers

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 answers

Score your test with the point values shown for each question. Then read the worked solution of every question where you lost points and note the method to review.

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