Differential Equations: AP Calculus AB Math Test – PDF Download

AP Calculus Differential Equations Test: questions 1–5

Question 1: Verify a solution (2 points)

Consider \(\frac{dy}{dx}=-3(y-1)\).

1. Show that \(y=2e^{-3x}+1\) is a solution. (1 point)
2. Find \(y(0)\) for this solution. (1 point)

Question 2: Slope field reading (3 points)

The slope field shown is for \(\frac{dy}{dx}=x-y\). Let \(y=f(x)\) be the solution with \(f(0)=1\).

1. Find the slope of the field at \((2,1)\) and at \((-1,2)\). (1 point)
2. Write the tangent line to the graph of \(f\) at \(x=0\) and use it to estimate \(f(0.5)\). (1 point)
3. Find \(\frac{d^{2}y}{dx^{2}}\) in terms of \(x\) and \(y\), and decide whether your estimate in part (b) is too large or too small. (1 point)

Question 3: Separable equation (3 points)

Solve \(\frac{dy}{dx}=\frac{2x}{y^{2}}\) with \(y(0)=2\). Write \(y\) as a function of \(x\).

Question 4: Bacteria model (3 points)

A bacteria culture grows so that \(\frac{dP}{dt}=kP\), where \(t\) is in hours. There are 400 bacteria at \(t=0\) and 1,600 bacteria at \(t=3\).

1. Write \(P(t)\) in terms of \(k\). (1 point)
2. Find the exact value of \(k\). (1 point)
3. Find \(P(5)\) to the nearest whole bacterium. (1 point)

Question 5: Write a differential equation (2 points)

The temperature \(T\) of a pie, in \(^{\circ}\text{F}\), changes at a rate proportional to the difference between \(T\) and the room temperature of \(72^{\circ}\text{F}\). The pie comes out of the oven at \(T(0)=350\).

1. Write a differential equation for \(T\). (1 point)
2. State the sign of the constant of proportionality and justify it. (1 point)

AP Calculus Differential Equations Test: questions 6–8

Question 6: Solution and domain (3 points)

Let \(y=f(x)\) be the solution of \(\frac{dy}{dx}=xy^{2}\) with \(f(0)=1\).

1. Find \(f(x)\). (2 points)
2. State the largest open interval containing \(x=0\) on which \(f\) is defined. (1 point)

Question 7: Radioactive decay (2 points)

An 80 g sample of an isotope decays so that \(\frac{dm}{dt}=km\), with a half-life of 30 years. Find the number of years until 10 g remain.

Question 8: Identify an equation (2 points)

A slope field is shown. Which of these could be its differential equation: \(\frac{dy}{dx}=x\), \(\frac{dy}{dx}=\frac{y}{2}\) or \(\frac{dy}{dx}=-\frac{y}{2}\)? Justify, then state \(\lim_{x\to\infty}f(x)\) for the solution with \(f(0)=1\).

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AP Calculus AB Differential Equations test

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