Practice Problem Set: Parametric Equations

How to Use This Problem Set

Parametric-equation questions often look unfamiliar because and are given separately as functions of a third variable, usually . The key is to work systematically with the parameter rather than trying to eliminate it immediately.

For AP Calculus BC, make sure you can move comfortably among four ideas: finding , finding the second derivative, writing tangent-line equations, and interpreting motion from and .

Essential Formulas

If and , then the first derivative is obtained by dividing the derivatives with respect to the parameter:

The second derivative requires one more differentiation with respect to the parameter:

Notice that the second derivative is not simply . You must divide by again.

Worked Example

Example: Tangent Line and Concavity

Suppose and . Find the slope of the curve at , and determine whether the curve is concave up or concave down there.

Step 1: Differentiate both equations with respect to : and .

Step 2: Find the first derivative: .

Step 3: Evaluate at : .

Step 4: Find the second derivative: .

Step 5: Evaluate the second derivative at . Since the result is positive, the curve is concave up.

Practice Problems

Try each problem before opening the solution. Several questions deliberately combine parametric differentiation with the kinds of reasoning that appear on AP free-response questions.

Question 1 — First Derivative

A curve is defined by and . Find in terms of .

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Question 2 — Slope at a Point

For and , find the slope of the curve at .

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Question 3 — Horizontal Tangent

The curve is given by and . Find the value of at which the curve has a horizontal tangent, assuming .

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Question 4 — Vertical Tangent

For and , determine the value of at which the curve has a vertical tangent.

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Question 5 — Second Derivative

A curve is defined by and . Find .

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Question 6 — Concavity

For and , determine whether the curve is concave up or concave down at .

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Question 7 — Tangent Line

The curve is defined by and . Find the equation of the tangent line at .

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Question 8 — Motion

A particle moves according to and . Find the particle's velocity vector at and its speed at that time.

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Question 9 — Position and Direction

A particle has position and . At , is the particle moving to the right or left? Is it moving upward or downward?

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Question 10 — AP-Style Combined Problem

A curve is defined by and for . Find the coordinates of the point corresponding to , find the slope of the tangent line there, and determine whether the curve is concave up or down at that point.

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Common Mistakes to Avoid

Dividing in the wrong order

The parametric derivative is , not . Think "change in divided by change in ."

Forgetting that the second derivative is a derivative with respect to x

After finding , differentiate it with respect to , then multiply by . This is the step that makes problems different from ordinary second derivatives.

Using a parameter value as though it were an x-coordinate

If the problem says , that does not mean the point has . You must substitute into both parametric equations to find the actual point.

Confusing slope with speed

For a moving particle, describes the slope of the path, while speed is the magnitude of the velocity vector: .

AP Exam Checklist

Can you calculate quickly and accurately?
Can you identify when a tangent is horizontal or vertical?
Can you find without confusing differentiation with respect to and differentiation with respect to ?
Can you evaluate the position of a particle from ?
Can you interpret the signs of and as direction of motion?
Can you distinguish the slope of a path from the particle's speed?

Test yourself: AP Calculus practice exams

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AB Multiple Choice AB Free Response BC Multiple Choice BC Free Response

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