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In chapter 1, we talked about parametric equations.

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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.

The formula for finding the slope of a parametrized curve is:

This makes sense if we think about canceling dt.

The formula for finding the slope of a parametrized curve is:

We assume that the denominator is not zero.

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.

Example:

Example:

- Find the first derivative (dy/dx).

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

Quotient Rule

3. Divide by dx/dt.

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.)

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

This curve is:

p