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10.1

Parametric Functions. 10.1. 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.

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10.1

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  1. Parametric Functions 10.1

  2. 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.

  3. Since the derivatives of each parametric equation are: The formula for finding the slope of a parametrized curve is: This makes sense if we think about canceling dt.

  4. Example 2 (page 514): The parametric curve below is given by the equations Find the values of t for which the line tangent to this curve is a) vertical b) horizontal When is this equal to 0?

  5. Example 2 (page 514): The parametric curve below is given by the equations Find the values of t for which the line tangent to this curve is a) vertical b) horizontal When is this equal to 0?

  6. Remember since this is still a parametric function that will be given in terms of t In the next slides, we will be using y in place of for simplicity. The formula for finding the slope of a parametrized curve is:

  7. 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.

  8. Example 2 (page 514):

  9. Example 2 (page 514): • Find the first derivative (dy/dx).

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

  11. 3. Divide by dx/dt.

  12. The length of a segment of a parametric curve can be approximated using the Pythagorean theorem: L dy dx

  13. Multiply by dt/dt After some algebra As dt gets smaller, the approximation gets better. As dtgoes to zero, we can determine the exact length of the curve.

  14. If we wanted to add up many segments over an interval of t, we can add an infinite amount of infinitely small segments to get… An Integral: (Notice the use of the Pythagorean Theorem.) This equation for the length of a parametrized curve is similar to our previous “length of curve” equation.

  15. Example 2 (page 514): The parametric curve below is given by the equations Find the length of this curve over the given interval 5.69

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

  17. By the way, this is an actual curve whose equations are: p

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