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Exploring Parametric Functions: A Comprehensive Guide

Dive into understanding parametric equations and their applications in describing non-function motion. Learn how to find derivatives and slopes of parametrized curves, exploring the Quotient Rule and second derivatives. Discover how to calculate arc length of parameterized curves using Pythagorean Theorem. Get ready to master these concepts with examples and practice problems.

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Exploring Parametric Functions: A Comprehensive Guide

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  1. Photo by Vickie Kelly, 2008 Greg Kelly, Hanford High School, Richland, Washington 10.1 Parametric functions Mark Twain’s Boyhood Home Hannibal, Missouri

  2. Photo by Vickie Kelly, 2008 Greg Kelly, Hanford High School, Richland, Washington Mark Twain’s Home Hartford, Connecticut

  3. In chapter 1, we talked about parametric equations. Parametric equations can be used to describe motion that is not a function. If f and g have derivatives at t, then the parametrized curve also has a derivative at t.

  4. The formula for finding the slope of a parametrized curve is: This makes sense if we think about canceling dt.

  5. The formula for finding the slope of a parametrized curve is: We assume that the denominator is not zero.

  6. To find the second derivative of a parametrized curve, we find the derivative of the first derivative: • Find the first derivative (dy/dx). 2. Find the derivative of dy/dx with respect to t. 3. Divide by dx/dt.

  7. Example:

  8. Example: • Find the first derivative (dy/dx).

  9. 2. Find the derivative of dy/dx with respect to t. Quotient Rule

  10. 3. Divide by dx/dt.

  11. Example 2. Find

  12. Topic 2 • Arc length of parameterized curve

  13. The equation for the length of a parametrized curve is similar to our previous “length of curve” equation: (Notice the use of the Pythagorean Theorem.)

  14. Example: 1 arc length • Find the arc length of

  15. Solution.

  16. Classwork/Homework: • Page 535 (7-16,23-33)

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