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AP®MathsCollege BoardCalculus BCExam QuestionsUnit 9: Parametric Equations, Vector-Valued Functions & Polar CoordinatesParametric Equations

Parametric Equations (College Board AP® Calculus BC): Exam Questions

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Parametric Equations

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Parametric Equations

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1
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A curve is given by the parametric equations

table row x equals cell 2 t minus 4 end cell row y equals cell t squared minus 1 end cell end table

The coordinates of the point at which the curve crosses the y-axis are

  • open parentheses 0 comma space minus 1 close parentheses

  • open parentheses 0 comma space 1 close parentheses

  • open parentheses 0 comma space 2 close parentheses

  • open parentheses 0 comma space 3 close parentheses

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2
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A curve is defined by the parametric equations

table row cell x open parentheses t close parentheses end cell equals cell t cubed plus e to the power of t end cell row cell y open parentheses t close parentheses end cell equals cell 5 plus t squared end cell end table

Which of following is an expression for fraction numerator d y over denominator d x end fraction?

  • fraction numerator t cubed plus e to the power of t over denominator 5 plus t squared end fraction

  • fraction numerator 5 plus t squared over denominator t cubed plus e to the power of t end fraction

  • fraction numerator 3 t squared plus e to the power of t over denominator 2 t end fraction

  • fraction numerator 2 t over denominator 3 t squared plus e to the power of t end fraction

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3
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The parametric equations

table row x equals cell fraction numerator t minus 1 over denominator 2 end fraction end cell row y equals cell negative 3 t end cell end table

represent which one out of the following straight-line equations?

  • y equals negative 3 x

  • y equals negative 6 x minus 3

  • y equals 2 x plus 1

  • y equals fraction numerator x minus 1 over denominator 2 end fraction

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4
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A particle moves along a curve so that its position at time t is open parentheses x open parentheses t close parentheses comma space y open parentheses t close parentheses close parentheses, where

fraction numerator d x over denominator d t end fraction equals 5 t squared minus 4 fraction numerator d y over denominator d t end fraction equals t to the power of 4 minus 3 t squared

What is the acceleration in the y-direction at time t equals 3?

  • 30

  • 54

  • 90

  • 102

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5
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A particle travels along a curve in the x y-plane so that its position at time t is open parentheses x open parentheses t close parentheses comma space y open parentheses t close parentheses close parentheses, where

fraction numerator d x over denominator d t end fraction equals 2 t squared minus 3 t fraction numerator d y over denominator d t end fraction equals e to the power of t minus t cubed

What is the speed of the particle at time t equals 5?

  • 42.109

  • 46.499

  • 58.413

  • 75.356

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1
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A curve in the x y-plane is defined parametrically as

table row x equals cell t squared plus cos space open parentheses t close parentheses end cell row y equals cell 1 plus sin open parentheses t close parentheses end cell end table

What is the slope of the line tangent to the curve at t equals pi?

  • negative fraction numerator 1 over denominator 2 pi end fraction

  • fraction numerator 1 over denominator 2 pi end fraction

  • negative 2 pi

  • 2 pi

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2
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A curve is defined parametrically by

x equals 5 space cos space t y equals 4 space sin space t

The equation of the curve in terms of x and y only is

  • x squared plus y squared equals 41

  • x squared over 25 plus y squared over 16 equals 1

  • 25 x squared plus 16 y squared equals 1

  • y squared over x squared equals 16 over 25

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3
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A curve is defined parametrically by

table row x equals cell t cubed minus 3 t end cell row y equals cell t cubed minus 12 t end cell end table

where 0 less or equal than t less or equal than 10.

Find the coordinates of the point on the curve at which the tangent is horizontal.

  • open parentheses negative 2 comma space minus 11 close parentheses

  • open parentheses 2 comma space 11 close parentheses

  • open parentheses negative 2 comma space 16 close parentheses

  • open parentheses 2 comma space minus 16 close parentheses

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4
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A particle moves along a curve so that its position at time t is open parentheses x open parentheses t close parentheses comma space y open parentheses t close parentheses close parentheses, where

table row cell fraction numerator d x over denominator d t end fraction end cell equals cell 24 t cubed minus 6 t squared plus 1 end cell row cell fraction numerator d y over denominator d t end fraction end cell equals cell 12 t squared plus 2 t minus 1 end cell end table

The particle is at the point open parentheses 3 comma space 9 close parentheses at time t equals 1.

An expression for x open parentheses t close parentheses in terms of t is

  • 6 t to the power of 4 minus 2 t cubed plus t minus 2

  • 6 t to the power of 4 minus 2 t cubed plus t

  • 6 t to the power of 4 minus 2 t cubed plus t plus 3

  • 6 t to the power of 4 minus 2 t cubed plus t plus 4

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5
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The motion of a particle at position open parentheses x open parentheses t close parentheses comma space y open parentheses t close parentheses close parentheses at time t where t greater or equal than 0 is defined by

table row cell fraction numerator d x over denominator d t end fraction end cell equals cell 1 minus 5 space sin space t end cell row cell fraction numerator d y over denominator d t end fraction end cell equals cell 4 plus cos space t end cell end table

The total distance travelled by the particle at time t equals pi is

  • integral subscript 0 superscript pi square root of open parentheses t plus 5 space cos space t close parentheses squared plus open parentheses 4 t plus sin space t close parentheses squared end root space d t

  • integral subscript 0 superscript pi square root of open parentheses t minus 5 space cos space t close parentheses squared plus open parentheses 4 t minus sin space t close parentheses squared end root space d t

  • integral subscript 0 superscript pi square root of open parentheses 1 minus 5 space sin space t close parentheses squared plus open parentheses 4 plus cos space t close parentheses squared end root space d t

  • integral subscript 0 superscript pi square root of open parentheses negative 5 space cos space t close parentheses squared plus open parentheses negative sin space t close parentheses squared end root space d t

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1
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A curve is defined by the parametric equations

x equals 1 fourth t to the power of 4 y equals t cubed minus t

An expression for fraction numerator d squared y over denominator d x squared end fraction is

  • 2 over t

  • negative 3 t plus 3 over t

  • negative 3 over t squared plus 3 over t to the power of 4

  • negative 3 over t to the power of 5 plus 3 over t to the power of 7

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2
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What is the equation of the vertical tangent to the curve

table row x equals cell t plus sin open parentheses 2 t close parentheses end cell row y equals cell negative cos open parentheses 2 t close parentheses end cell end table

where 0 less or equal than t less or equal than pi over 2?

  • x equals pi over 3 minus fraction numerator square root of 3 over denominator 2 end fraction

  • x equals pi over 3 plus fraction numerator square root of 3 over denominator 2 end fraction

  • x equals fraction numerator 2 pi over denominator 3 end fraction minus fraction numerator square root of 3 over denominator 2 end fraction

  • x equals fraction numerator 2 pi over denominator 3 end fraction plus fraction numerator square root of 3 over denominator 2 end fraction

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3
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A curve is defined parametrically by

table row x equals cell 1 half t squared minus ln space t end cell row y equals cell 2 t minus 1 end cell end table

where t greater or equal than 1.

Which of the following integrals gives the correct arc length from t equals 1 to t equals 8?

  • integral subscript 1 superscript 8 square root of open parentheses t plus 1 over t close parentheses squared end root space d t

  • integral subscript 1 superscript 8 square root of t squared plus 1 over t squared space end root space d t

  • integral subscript 1 superscript 8 square root of t squared plus 1 over t squared space plus 4 space end root space d t

  • integral subscript 1 superscript 8 square root of t minus 1 over t plus 2 space end root space d t

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4
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A curve defined parametrically is shown below, with parameter t where 0 less or equal than t less or equal than 2.

Graph of a curve on an xy-plane in the third quadrant with a y-intercept at the origin and a non-zero negative y-intercept.

Which of the following parametric equations could represent the curve above?

  • table row x equals cell negative t cubed plus 4 t end cell row y equals cell negative 2 t end cell end table

  • table row x equals cell negative t cubed plus 4 t end cell row y equals cell 2 t end cell end table

  • table row x equals cell t cubed minus 4 t end cell row y equals cell negative 2 t end cell end table

  • table row x equals cell t cubed minus 4 t end cell row y equals cell 2 t end cell end table

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5
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A particle is moving along a curve in the x y-plane with its position at time t given by open parentheses x open parentheses t close parentheses comma space y open parentheses t close parentheses close parentheses, where t greater or equal than 0 and

table row cell fraction numerator d x over denominator d t end fraction end cell equals cell e to the power of square root of x end exponent end cell row cell fraction numerator d y over denominator d t end fraction end cell equals cell square root of 1 plus t cubed end root end cell end table

The initial position of the particle is open parentheses 12 comma space 10 close parentheses.

The position of the y-coordinate of the particle at time t equals 5 is

  • 23.596

  • 25.13

  • 33.596

  • 37.13

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