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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First-Order Differential Equations

1
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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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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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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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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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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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6
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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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