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# Parametric Curves for AP Calculus

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By McGraw-Hill Professional
Updated on Oct 24, 2011

Practice problems for these concepts can be found at:  Graphs of Functions and Derivatives Practice Problems for AP Calculus

Parametric curves are relations (x (t), y (t)) for which both x and y are defined as functions of a third variable, t, that is, x = f (t) and y =g (t).

### Example 1

A particle is moving in the coordinate plane in such a way that x (t)=2t – 5 and y (t) for 0 ≤ t ≤ 5. Sketch the path of the particle and indicate the direction of motion.

Step 1:  Create a table of values.

Step 2:  Plot the points and sketch the path of a particle as a smooth curve. Place arrows to indicate the direction of motion.

### Example 2

A parametric curve is defined by x = 2+et and y = e3t. Find the Cartesian equation of the curve.

Step 1:  Solve x =2+et for t. x – 2= et so t = ln(x – 2).

Step 2:  Substitute t = ln(x – 2) into y = e3t. y = e3 ln(x –2) = (x – 2)3.

Step 3:  Note that t = ln(x – 2) is defined only when x > 2. The equation of the curve is y = (x – 2)3 with domain (2,∞).

Practice problems for these concepts can be found at:  Graphs of Functions and Derivatives Practice Problems for AP Calculus

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