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

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

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First-Order Differential Equations

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1
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1 mark

If P open parentheses t close parentheses is the size of a population at time t, which of the following differential equations describes linear growth in the size of the population?

  • fraction numerator d P over denominator d t end fraction equals 50

  • fraction numerator d P over denominator d t end fraction equals 50 t

  • fraction numerator d P over denominator d t end fraction equals 50 P

  • fraction numerator d P over denominator d t end fraction equals 25 P squared

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2
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1 mark

If N open parentheses t close parentheses is the size of a population at time t, which of the following differential equations describes exponential growth in the size of the population?

  • fraction numerator d N over denominator d t end fraction equals 300

  • fraction numerator d N over denominator d t end fraction equals 300 t

  • fraction numerator d N over denominator d t end fraction equals 300 over N

  • fraction numerator d N over denominator d t end fraction equals 300 N

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3
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1 mark

The rate of change of the volume, V, of oil in a tank with respect to time, t, is directly proportional to the cube root of the volume. Which of the following is a differential equation that describes this relationship?

  • V open parentheses t close parentheses equals k cube root of V

  • V open parentheses t close parentheses equals k cube root of t

  • fraction numerator d V over denominator d t end fraction equals fraction numerator k over denominator cube root of V end fraction

  • fraction numerator d V over denominator d t end fraction equals k cube root of V

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4
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1 mark

Which of the following is the solution to the differential equation fraction numerator d y over denominator d x end fraction equals 3 cos x with the initial condition y open parentheses pi over 2 close parentheses equals negative 1?

  • y equals 3 sin x plus 2

  • y equals 3 sin x minus 4

  • y equals negative 3 sin x plus 2

  • y equals negative 3 sin x minus 4

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5
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1 mark

Which of the following is the solution to the differential equation fraction numerator d y over denominator d x end fraction equals 2 e to the power of 3 x end exponent with the initial condition y open parentheses ln 2 close parentheses equals 5?

  • y equals 2 over 3 e to the power of 3 x end exponent minus 1 third

  • y equals 2 over 3 e to the power of 3 x end exponent plus 1 third

  • y equals 31 over 3 minus 2 over 3 e to the power of 3 x end exponent

  • y equals 1 third minus 2 over 3 e to the power of 3 x end exponent

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1
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1 mark

A disease spreads among a population of N people at a rate which is proportional to the product of the number of people infected by the disease and the number of people not infected by the disease. If u denotes the number of people infected by the disease, which of the following differential equations could be used to model this situation with respect to time, t, where k is a positive constant.

  • fraction numerator d u over denominator d t end fraction equals k t open parentheses N minus t close parentheses

  • fraction numerator d u over denominator d t end fraction equals k t open parentheses t minus N close parentheses

  • fraction numerator d u over denominator d t end fraction equals k u open parentheses N minus u close parentheses

  • fraction numerator d u over denominator d t end fraction equals k u open parentheses u minus N close parentheses

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2
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1 mark

If y satisfies fraction numerator d y over denominator d t end fraction equals k y, where k is a non-zero constant, then y could be

  • 3 e to the power of k t end exponent

  • 3 e to the power of k t y end exponent

  • e to the power of k t end exponent plus 5

  • k t y plus 7

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3
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1 mark
Slope field for x and y each between -8 and 8. The tangent segments are horizontal at y=6 and y=-6.

The slope field shown above corresponds to which of the following differential equations?

  • fraction numerator d y over denominator d x end fraction equals fraction numerator y minus 6 over denominator 3 end fraction

  • fraction numerator d y over denominator d x end fraction equals fraction numerator y squared minus 36 over denominator 9 end fraction

  • fraction numerator d y over denominator d x end fraction equals fraction numerator x minus 6 over denominator 3 end fraction

  • fraction numerator d y over denominator d x end fraction equals fraction numerator x squared minus 36 over denominator 9 end fraction

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4
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1 mark

Which of the following is the solution to the differential equation fraction numerator d y over denominator d t end fraction equals 4 y with the initial condition y open parentheses 0 close parentheses equals 3?

  • y equals 2 y squared

  • y equals 4 t y plus 3

  • y equals 3 plus e to the power of 4 t end exponent

  • y equals 3 e to the power of 4 t end exponent

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5
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1 mark

If y satisfies fraction numerator d y over denominator d x end fraction equals fraction numerator 1 over denominator 2 y end fraction, then y could be

  • 1 half ln open vertical bar x close vertical bar plus 3

  • 1 half ln open vertical bar y close vertical bar plus 3

  • square root of x plus 3 end root

  • square root of x plus 3

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1
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1 mark

The growth of population y is described by the differential equation fraction numerator d y over denominator d t end fraction equals k y, where k is a constant and t is measured in hours. If the population doubles every 8 hours, then the value of k is

  • 0.087

  • 0.250

  • 0.271

  • 4.351

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2
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1 mark

Which of the following is the solution to the differential equation fraction numerator d y over denominator d x end fraction equals x cubed over y with the initial condition y open parentheses 2 close parentheses equals negative 3?

  • y equals 3 e to the power of x to the power of 4 over 4 minus 4 end exponent

  • y equals negative 3 e to the power of x to the power of 4 over 4 minus 4 end exponent

  • y equals square root of x to the power of 4 over 2 plus 1 end root

  • y equals negative square root of x to the power of 4 over 2 plus 1 end root

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3
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1 mark
Slope field for x and y each between -5 and 5. The tangent segments are horizontal at y=0 and x=2.

Shown above is a slope field for which of the following differential equations?

  • fraction numerator d y over denominator d x end fraction equals x y

  • fraction numerator d y over denominator d x end fraction equals x y minus 2 y

  • fraction numerator d y over denominator d x end fraction equals x y minus 2 x

  • fraction numerator d y over denominator d x end fraction equals open parentheses x minus 2 close parentheses cubed

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4
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1 mark

In the differential equation fraction numerator d y over denominator d x end fraction equals x to the power of k y squared, k is a positive integer. Which of the following is the solution to the differential equation with the initial condition y open parentheses 0 close parentheses equals 1?

  • y equals fraction numerator k plus 1 over denominator k plus 1 minus x to the power of k plus 1 end exponent end fraction

  • y equals fraction numerator k plus 1 over denominator k plus 1 plus x to the power of k plus 1 end exponent end fraction

  • y equals cube root of fraction numerator k plus 1 minus 3 x to the power of k plus 1 end exponent over denominator k plus 1 end fraction end root

  • y equals cube root of fraction numerator k plus 1 plus 3 x to the power of k plus 1 end exponent over denominator k plus 1 end fraction end root

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5
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1 mark

The number of radioactive atoms, N, in a sample is described by the differential equation fraction numerator d N over denominator d t end fraction equals negative k N, where k is a positive constant and t is measured in years. If the number of radioactive atoms is only one tenth of the original number after 1.255 years, then the value of k is

  • 0.545

  • 1.290

  • 1.835

  • 2.380

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