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10.1 Parametric functions

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

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10.1 Parametric functions

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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. 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.)

  12. Likewise, the equations for the surface area of a parametrized curve are similar to our previous “surface area” equations:

  13. This curve is: p

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