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

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

49 mins30 questions

Select a download format for Parametric Equations

Scroll for more

Select an answer set to view for
Parametric Equations

Easy
Medium
Hard
1
Sme Calculator2 marks

A particle moves along a curve in the x y-plane with position open parentheses x open parentheses t close parentheses comma space y open parentheses t close parentheses close parentheses where time t is measured in seconds, and x open parentheses t close parentheses and y open parentheses t close parentheses are measured in meters. It is known that x apostrophe open parentheses t close parentheses equals t cubed minus 2 t squared and y apostrophe open parentheses t close parentheses equals 2 t plus square root of 4 plus t to the power of 5 end root.

Find the speed of the particle at time t equals 5 seconds. Show the setup for your calculations.

How did you do?

Did this page help you?

2
Sme Calculator1 mark

A particle moves along a curve in the x y-plane with position open parentheses x open parentheses t close parentheses comma space y open parentheses t close parentheses close parentheses at time t greater than 0. The particle is moving such that fraction numerator d x over denominator d t end fraction equals square root of 1 plus e to the power of t end root and fraction numerator d y over denominator d t end fraction equals ln open parentheses t cubed plus 1 close parentheses.

Find the slope of the line tangent to the path of the particle at time t equals 2.

How did you do?

Did this page help you?

3
Sme Calculator2 marks

A particle is traveling along a curve in the x y-plane with its position given by open parentheses x open parentheses t close parentheses comma space y open parentheses t close parentheses close parentheses where time t is measured in seconds, and x open parentheses t close parentheses and y open parentheses t close parentheses are measured in feet. It is known that x apostrophe open parentheses t close parentheses equals square root of 5 t cubed plus 2 t end root and y apostrophe open parentheses t close parentheses equals 2 t plus t to the power of 1.5 end exponent.

Find the total distance traveled by the particle over the time interval 0 less or equal than t less or equal than 3. Show the setup for your calculations.

How did you do?

Did this page help you?

4
Sme Calculator2 marks

A particle follows a curve so that its position at time t greater or equal than 0 seconds is open parentheses x open parentheses t close parentheses comma space y open parentheses t close parentheses close parentheses measured in centimeters, where x open parentheses t close parentheses equals t cubed minus t squared plus 8 and y open parentheses t close parentheses is not explicitly given but it is known that fraction numerator d y over denominator d t end fraction equals t squared space ln open parentheses t plus 1 close parentheses.

Find the speed of the particle at time t equals 1 second. Show the setup for your calculations.

How did you do?

Did this page help you?

5
Sme Calculator2 marks

A particle moves along a curve in the x y-plane with position open parentheses x open parentheses t close parentheses comma space y open parentheses t close parentheses close parentheses at time t greater than 0. The particle is moving in such a way thatfraction numerator d x over denominator d t end fraction equals sin open parentheses 2 t squared close parentheses and fraction numerator d y over denominator d t end fraction equals 4 t plus square root of t.

Find the total distance the particle travels along the curve from time t equals 1 to time t equals 4. Show the setup for your calculations.

How did you do?

Did this page help you?

1
Sme Calculator2 marks

For 0 less or equal than t less or equal than 10 a particle moves 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 ln open parentheses 4 plus square root of t close parentheses. At time t equals 0 the position of the particle is open parentheses 3 comma space 6 close parentheses.

Find the x-coordinate of the position of the particle at time t equals 2. Show the work that leads to your answer.

How did you do?

Did this page help you?

2
Sme Calculator2 marks

A particle is traveling 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 t e to the power of t plus 1 and y open parentheses t close parentheses equals 5 t to the power of 4 minus 2 t squared.

Find the time at which the speed of the particle is 4. Show the work that leads to your answer.

How did you do?

Did this page help you?

3
Sme Calculator2 marks

At time t greater or equal than 0, a particle is moving along a curve in the x y-plane with its position given by open parentheses x open parentheses t close parentheses comma space y open parentheses t close parentheses close parentheses where x open parentheses t close parentheses equals 6 t plus e to the power of t. The equation for y open parentheses t close parentheses is not explicitly given, but it is known that y apostrophe open parentheses t close parentheses equals 30 plus 5 ln open parentheses 2 t plus 1 close parentheses.

There is a point on the curve at which the line tangent to the curve has a slope of 1. Find the time at which the particle is at this point.

How did you do?

Did this page help you?

4
Sme Calculator2 marks

A particle is moving along a curve in the x y-plane with position open parentheses x open parentheses t close parentheses comma space y open parentheses t close parentheses close parentheses at time t greater or equal than 0, where fraction numerator d y over denominator d t end fraction equals e to the power of cos space t end exponent. At time t equals 0 the position of the particle is open parentheses 5 comma space 7 close parentheses.

Find the y-coordinate of the position of the particle at time t equals pi over 2. Show the work that leads to your answer.

How did you do?

Did this page help you?

5
Sme Calculator2 marks

A particle travels 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. At time t equals 2, the particle has a position of open parentheses negative 3 comma space minus 1 close parentheses. It is known that fraction numerator d x over denominator d t end fraction equals sin open parentheses fraction numerator 2 t over denominator 1 plus t end fraction close parentheses and fraction numerator d y over denominator d t end fraction equals ln open parentheses 3 t plus e to the power of t close parentheses.

Write an equation for the line tangent to the curve at the point where the particle is at time t equals 2.

How did you do?

Did this page help you?

1
Sme Calculator2 marks

For 0 less or equal than t less or equal than 4, a particle is moving along a curve in the x y-plane with its position given by open parentheses x open parentheses t close parentheses comma space y open parentheses t close parentheses close parentheses. The equation for y open parentheses t close parentheses is not explicitly given, but it is known that y apostrophe open parentheses t close parentheses equals negative t squared plus 2 ln open parentheses 1 plus 6 t close parentheses and that the particle remains in the first quadrant during this time.

Find all the times t in the interval 0 less or equal than t less or equal than 4 when the particle is moving towards the x-axis. Give a reason for your answer.

How did you do?

Did this page help you?

2
Sme Calculator3 marks

A particle moves along a curve in the x y-plane with position open parentheses x open parentheses t close parentheses comma space y open parentheses t close parentheses close parentheses where time t greater or equal than 0 is measured in seconds, and x open parentheses t close parentheses and y open parentheses t close parentheses are measured in meters. It is known that x apostrophe open parentheses t close parentheses equals square root of t plus sin open parentheses fraction numerator t over denominator 1 plus t end fraction close parentheses. At time t equals 5 seconds, the particle is at the point open parentheses negative 2 comma space minus 3 close parentheses.

Find the x-coordinate of the position of the particle at the time t equals 0. Show the setup for your calculation.

How did you do?

Did this page help you?

3
Sme Calculator3 marks

At time t greater or equal than 0, the position of a particle moving along a curve in the x y-plane is given by the parametric functions 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 square root of t squared plus 3 cos space t end root. The graph of y open parentheses t close parentheses is shown below, formed using two straight line segments.

Line graph with axes labelled y and t. Plot starts at (0,10), rises uniformly to (5,30), then is horizontal up to (10,30), with grid lines and tick marks.

Find the total distance traveled by the particle over the time interval 0 less or equal than t less or equal than 10.

How did you do?

Did this page help you?

4
Sme Calculator3 marks

A curve is defined by the parametric equations x open parentheses t close parentheses equals 2 e to the power of t minus t to the power of 5 and y open parentheses t close parentheses equals 4 plus 3 e to the power of t.

Find an expression for fraction numerator d squared y over denominator d x squared end fraction in terms of t. Write your answer in the form fraction numerator a t cubed e to the power of t open parentheses b minus t close parentheses over denominator open parentheses g open parentheses t close parentheses close parentheses cubed end fraction where a and b are positive integers and g open parentheses t close parentheses is a function to be found.

How did you do?

Did this page help you?

5
Sme Calculator4 marks

A curve is defined parametrically by x equals e to the power of t space cos space t and y equals e to the power of t space sin space t.

Find the length of the curve from t equals 0 to t equals pi. Give your answer in the form p open parentheses e to the power of q minus 1 close parentheses where p and q are real numbers to be found.

How did you do?

Did this page help you?