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AP®MathsCollege BoardCalculus BCExam QuestionsUnit 7: Differential EquationsFirst-Order Differential Equations

First-Order Differential Equations (College Board AP® Calculus BC): Exam Questions

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First-Order Differential Equations

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1
Sme Calculator1 mark

A slope field for the differential equation fraction numerator d A over denominator d t end fraction equals fraction numerator 50 minus A over denominator 3 end fraction is shown below.

A slope field for the differential equation dA/dt = (50-A)/3, between 0 and 20 on the horizontal (t) axis, and between 0 and 70 on the vertical (A) axis

Sketch the solution curve through the point open parentheses 0 comma space 10 close parentheses.

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2
Sme Calculator1 mark

A portion of the slope field for the differential equation fraction numerator d y over denominator d x end fraction equals sin open parentheses pi over 3 x close parentheses square root of y plus 5 end root is given below.

A slope field for the differential equation dy/dx =sin((pi/3)x)sqrt(y-5), between -10/3 and 10/3 on the x-axis, and between -2 and 4 on the y-axis

Sketch the solution curve through the point open parentheses 1 comma space minus 1 close parentheses.

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3
Sme Calculator2 marks

Consider the differential equation fraction numerator d y over denominator d x end fraction equals fraction numerator open parentheses y minus 1 close parentheses squared over denominator x minus 2 end fraction.

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

Graph displaying points with coordinates (1, 1), (1, 2), (1, 3), (3, 1), (3, 2) and (3, 3). Both axes are labelled from 0 to 4.

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4
Sme Calculator2 marks

Consider the differential equation fraction numerator d y over denominator d x end fraction equals open parentheses y minus 3 close parentheses squared cos open parentheses pi x close parentheses.

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

Graph displaying points with coordinates (-1, 1), (-1, 2), (-1, 3), (0, 1), (0, 2), (0, 3), (1, 1), (1, 2), and (1, 3). The x-axis is labelled from -1 to 1, and the y-axis is labelled from 0 to 3.

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5
Sme Calculator1 mark

Consider the differential equation fraction numerator d y over denominator d x end fraction equals fraction numerator x minus 3 over denominator y end fraction.

Describe all the points in the x y-plane, y not equal to 0, for which fraction numerator d y over denominator d x end fraction equals negative 1 half.

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6
Sme Calculator2 marks

Consider the differential equation fraction numerator d y over denominator d x end fraction equals fraction numerator 2 x y over denominator 5 end fraction. Let y equals f open parentheses x close parentheses be the particular solution to that differential equation with the initial condition f open parentheses 0 close parentheses equals 4. Use Euler's method with a step size of 0.1, starting at x equals 0, to approximate f open parentheses 0.2 close parentheses. Show the work that leads to your answer.

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7
Sme Calculator2 marks

A population is modeled by a function P that satisfies the logistic differential equation fraction numerator d P over denominator d t end fraction equals 6 P open parentheses 13 minus P close parentheses.

If P open parentheses 0 close parentheses equals 4, what is limit as t rightwards arrow infinity of P open parentheses t close parentheses?

If P open parentheses 0 close parentheses equals 15, what is limit as t rightwards arrow infinity of P open parentheses t close parentheses?

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1
Sme Calculator4 marks

A cylindrical tank contains water and is being used to fill up a garden pond. The tank is standing upright on its circular base. The rate of change of the height h of the water in the tank with respect to time t is modeled by fraction numerator d h over denominator d t end fraction equals negative 1 third square root of h cubed end root, where h is measured in feet and t is measured in seconds.

At time t equals 0 seconds, the height of water in the tank is 4 feet. Use separation of variables to find an expression for h in terms of t.

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2
Sme Calculator1 mark

Consider the differential equation fraction numerator d y over denominator d x end fraction equals open parentheses y minus 2 close parentheses cubed sin open parentheses pi x close parentheses.

There is a horizontal line with equation y equals c that satisfies this differential equation. Find the value of c.

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3
Sme Calculator5 marks

Consider the differential equation fraction numerator d y over denominator d x end fraction equals fraction numerator x minus 6 over denominator y squared end fraction.

Find the particular solution y equals f open parentheses x close parentheses to the differential equation with the initial condition f open parentheses 0 close parentheses equals 3.

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4
Sme Calculator1 mark

Consider the differential equation fraction numerator d y over denominator d x end fraction equals x cubed open parentheses y plus 3 close parentheses.

Describe all the points in the x y-plane for which the slopes of tangent lines drawn on a slope field for that differential equation will be negative.

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5
Sme Calculator5 marks

Consider the differential equation fraction numerator d y over denominator d x end fraction equals fraction numerator y squared over denominator 2 x minus 3 end fraction.

Find the particular solution y equals f open parentheses x close parentheses to the differential equation with the initial condition f open parentheses 2 close parentheses equals 2.

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6
Sme Calculator2 marks

bold italic x

0

pi over 4

pi over 2

fraction numerator 3 pi over denominator 4 end fraction

pi

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3

5

7

2

1

f is a twice-differentiable function for all values of x, with f open parentheses 0 close parentheses equals 0. The table above gives values of the derivative of f, f to the power of apostrophe, for selected values of x.

Use Euler's method, with two steps of equal size starting at x equals 0, to approximate f open parentheses pi close parentheses. Show the computations that lead to your answer.

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7a
Sme Calculator1 mark

Consider the logistic differential equation fraction numerator d y over denominator d t end fraction equals y open parentheses 4 minus y over 3 close parentheses for a population y. Let y equals f open parentheses t close parentheses be the particular solution to the differential equation for a given initial condition.

If f open parentheses 0 close parentheses equals 15, what is the range of f for t greater or equal than 0?

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7b
Sme Calculator1 mark

If, instead, f open parentheses 0 close parentheses equals 2, for what value of y is the population growing the fastest?

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1
Sme Calculator5 marks

Consider the differential equation fraction numerator d y over denominator d x end fraction equals fraction numerator 8 minus x cubed over denominator y end fraction.

Find the particular solution y equals f open parentheses x close parentheses to the differential equation such that the line y equals negative 4 is tangent to the graph of f.

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2
Sme Calculator5 marks

Scientists are studying a population of leopards in a nature reserve. The rate of change of the population is proportional to the difference between the current population and the maximum population that the reserve can support. If P open parentheses t close parentheses is the population at time t months after the start of the study, then

fraction numerator d P over denominator d t end fraction equals 1 over 20 open parentheses 90 minus P close parentheses

Use separation of variables to find the particular solution to the differential equation with initial condition P open parentheses 0 close parentheses equals 30. Give your answer as an expression for P in terms of t.

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3
Sme Calculator5 marks

Use separation of variables to find y equals f open parentheses x close parentheses, the particular solution to the differential equation fraction numerator d y over denominator d x end fraction equals cos open parentheses pi over 5 x close parentheses square root of y minus 2 end root with the initial condition f open parentheses 5 close parentheses equals 11.

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4
Sme Calculator2 marks

Consider the differential equation fraction numerator d y over denominator d x end fraction equals x minus 1 half y plus 1.

Find the values of the constants m and b, for which y equals m x plus b is a solution to the differential equation.

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5
Sme Calculator6 marks

Consider the differential equation fraction numerator d y over denominator d x end fraction equals open parentheses y plus 3 close parentheses cubed sin open parentheses pi x close parentheses.

Find the particular solution y equals f open parentheses x close parentheses to the differential equation with the initial condition f open parentheses 1 close parentheses equals 0

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6
Sme Calculator5 marks

Let y equals f open parentheses x close parentheses be the particular solution to the differential equation fraction numerator d y over denominator d x end fraction equals x y squared ln x with initial condition f open parentheses 1 close parentheses equals 2. Use Euler's method, with two equal steps starting at x equals 1, to approximate f open parentheses 3 over 2 close parentheses. Determine whether this approximation is an over- or underestimate, being sure to justify your answer.

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7a
Sme Calculator5 marks

A population is modeled by a function P that satisfies the logistic differential equation fraction numerator d P over denominator d t end fraction equals 4 P open parentheses 1 minus P over 16 close parentheses.

Use separation of variables to find P open parentheses t close parentheses if P open parentheses 0 close parentheses equals 5.

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7b
Sme Calculator2 marks

Determine the value of the time, t, at which the population, P, is changing the fastest.

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