AP®MathsCollege BoardCalculus ABExam QuestionsUnit 7: Differential EquationsUnit 7 Overview

Unit 7 Overview (College Board AP® Calculus AB): Exam Questions

1 hour8 questions
1a
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1 mark

A bottle of milk is taken out of a refrigerator and placed in a pan of hot water to be warmed. The increasing function M models the temperature of the milk at time t, where M(t) is measured in degrees Celsius (°C) and t is the number of minutes since the bottle was placed in the pan. M satisfies the differential equation \displaystyle \frac{\text{d} M}{\text{d} t} = \frac{1}{4} (40 - M). At time t = 0, the temperature of the milk is 5°C. It can be shown that M(t) < 40 for all values of t.

A slope field for the differential equation \displaystyle \frac{\text{d} M}{\text{d} t} = \frac{1}{4} (40 - M) is shown. Sketch the solution curve through the point (0, 5).

Slope field for M(t) versus time t, showing solution curves bending towards the horizontal equilibrium line midway between M=5 and M=70 marks.

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1b
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2 marks

Use the line tangent to the graph of M at t = 0 to approximate M(2), the temperature of the milk at time t = 2 minutes.

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1c
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2 marks

Write an expression for \displaystyle \frac{\text{d}^2 M}{\text{d} t^2} in terms of M. Use \displaystyle \frac{\text{d}^2 M}{\text{d} t^2} to determine whether the approximation from part (b) is an underestimate or an overestimate for the actual value of M(2). Give a reason for your answer.

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1d
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4 marks

Use separation of variables to find an expression for M(t), the particular solution to the differential equation \displaystyle \frac{\text{d} M}{\text{d} t} = \frac{1}{4} (40 - M) with initial condition M(0) = 5.

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2a
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1 mark

Consider the differential equation \frac{\text{d} y}{\text{d} x} = \frac{1}{2} \text{sin} \; \left(\frac{\pi}{2} x\right) \sqrt{y + 7}. Let y = f(x) be the particular solution to the differential equation with the initial condition f(1) = 2. The function f is defined for all real numbers.

A portion of the slope field for the differential equation is given below. Sketch the solution curve through the point (1, 2).

Slope field on x–y axes from −3 to 3 and −2 to 4, with short line segments tilting; a single point is marked near (1.5, 2).

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2b
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2 marks

Write an equation for the line tangent to the solution curve in part (a) at the point (1, 2). Use the equation to approximate f(0.8).

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2c
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1 mark

It is known that f''(x) > 0 for -1 \leq x \leq 1. Is the approximation found in part (b) an overestimate or an underestimate for f(0.8)? Give a reason for your answer.

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2d
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5 marks

Use separation of variables to find y = f(x), the particular solution to the differential equation \frac{\text{d} y}{\text{d} x} = \frac{1}{2} \text{sin}\left(\frac{\pi}{2} x\right) \sqrt{y + 7} with the initial condition f(1) = 2.

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3a
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1 mark

The depth of seawater at a location can be modeled by the function H that satisfies the differential equation

\frac{\text{d}H}{\text{d}t} = \frac{1}{2}(H - 1)\cos\frac{t}{2}

where H(t) is measured in feet and t is measured in hours after noon (t = 0). It is known that H(0) = 4.

A portion of the slope field for the differential equation is provided below.

Slope field on t–y axes from t=0–5 and y≈0–11, with short line segments curving upward; a point is marked at (t=0, y=4).

Sketch the solution curve y = H(t) through the point (0, 4) on the slope field above.

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3b
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3 marks

For 0 < t < 5, it can be shown that H(t) > 1. Find the value of t, for 0 < t < 5, at which H has a critical point. Determine whether the critical point corresponds to a relative minimum, a relative maximum, or neither. Justify your answer.

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3c
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5 marks

Use separation of variables to find y = H(t), the particular solution to the differential equation

\frac{\text{d}H}{\text{d}t} = \frac{1}{2}(H - 1)\text{cos}\frac{t}{2}

with initial condition H(0) = 4.

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4a
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1 mark

A medication is administered to a patient. The amount, in milligrams, of the medication in the patient at time t hours is modeled by a function y = A(t) that satisfies the differential equation \dfrac{\text{d}y}{\text{d}t} = \dfrac{12 - y}{3}. At time t = 0 hours, there are 0 milligrams of the medication in the patient.

A portion of the slope field for the differential equation \dfrac{\text{d}y}{\text{d}t} = \dfrac{12-y}{3} is given below. Sketch the solution curve through the point (0, 0).

Slope field on a t–y graph, with short line segments slanting upward; above a dashed horizontal line slopes are shallower than those below it

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4b
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1 mark

Using correct units, interpret the statement \displaystyle\lim_{t \to \infty} A(t) = 12 in the context of this problem.

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4c
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4 marks

Use separation of variables to find y = A(t), the particular solution to the differential equation \dfrac{\text{d}y}{\text{d}t} = \dfrac{12-y}{3} with initial condition A(0) = 0.

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4d
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3 marks

A different procedure is used to administer the medication to a second patient. The amount, in milligrams, of the medication in the second patient at time t hours is modeled by a function y = B(t) that satisfies the differential equation \dfrac{\text{d}y}{\text{d}t} = 3 - \dfrac{y}{t+2}. At time t = 1 hour, there are 2.5 milligrams of the medication in the second patient. Is the rate of change of the amount of medication in the second patient increasing or decreasing at time t = 1? Give a reason for your answer.

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5a
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2 marks

A cylindrical barrel with a diameter of 2 feet contains collected rainwater, as shown in the figure below. The water drains out through a valve (not shown) at the bottom of the barrel. The rate of change of the height h of the water in the barrel with respect to time t is modelled by

\frac{dh}{dt} = -\frac{1}{10}\sqrt{h}

where h is measured in feet and t is measured in seconds. The volume V of a cylinder with radius r and height h is V = \pi r^{2} h.

Diagram of a vertical cylinder, 2 ft in diameter, partially filled with liquid of height h ft, with arrows marking 2 ft across the top and h up the side

Find the rate of change of the volume of water in the barrel with respect to time when the height of the water is 4 feet. Indicate units of measure.

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5b
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3 marks

When the height of the water is 3 feet, is the rate of change of the height of the water with respect to time increasing or decreasing? Explain your reasoning.

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5c
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4 marks

At time t = 0 seconds, the height of the water is 5 feet. Use separation of variables to find an expression for h in terms of t.

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6a
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2 marks

Consider the differential equation

\frac{dy}{dx} = \frac{y^{2}}{x - 1}

On the axes provided, sketch a slope field for the given differential equation at the six points indicated.

Scatter plot on x–y axes showing points at (0,0), (0,1), (0,2), (2,0), (2,1) and (2,2); axes labelled x and y with ticks at 1, 2 and 3

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6b
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2 marks

Let y = f(x) be the particular solution to the given differential equation with the initial condition f(2) = 3. Write an equation for the line tangent to the graph of y = f(x) at x = 2. Use your equation to approximate f(2.1).

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6c
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5 marks

Find the particular solution y = f(x) to the given differential equation with the initial condition f(2) = 3.

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7a
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3 marks

At time t = 0, a boiled potato is taken from a pot on a stove and left to cool in a kitchen. The internal temperature of the potato is 91\,^{\circ}\text{C} at time t = 0, and the internal temperature of the potato is greater than 27\,^{\circ}\text{C} for all times t > 0. The internal temperature of the potato at time t minutes is modeled by the function H satisfying the differential equation \dfrac{dH}{dt} = -\dfrac{1}{4}(H - 27), where H(t) is measured in degrees Celsius and H(0) = 91.

Write an equation for the line tangent to the graph of H at t = 0. Use this equation to approximate the internal temperature of the potato at time t = 3.

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7b
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1 mark

Use \dfrac{d^2H}{dt^2} to determine whether your answer in part (a) is an underestimate or an overestimate of the internal temperature of the potato at time t = 3.

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7c
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5 marks

For t < 10, an alternate model for the internal temperature of the potato at time t minutes is the function G satisfying the differential equation \dfrac{dG}{dt} = -(G - 27)^{2/3}, where G(t) is measured in degrees Celsius and G(0) = 91. Find an expression for G(t). Based on this model, what is the internal temperature of the potato at time t = 3?

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8a
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2 marks

Consider the differential equation \dfrac{dy}{dx} = \dfrac{1}{3}x(y-2)^2.

A slope field for the given differential equation is shown below. Sketch the solution curve that passes through the point (0, 2), and sketch the solution curve that passes through the point (1, 0).

Slope field on x–y axes with short dashed line segments curving outward from the y-axis, showing changing gradient above and below the origin.

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8b
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2 marks

Let y = f(x) be the particular solution to the given differential equation with initial condition f(1) = 0. Write an equation for the line tangent to the graph of y = f(x) at x = 1. Use your equation to approximate f(0.7).

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8c
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5 marks

Find the particular solution y = f(x) to the given differential equation with initial condition f(1) = 0.

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