1. 10.2 – 10.3 Parametric Equations

2. There are times when we need to describe motion (or a curve) that is not a function. We can do this by writing equations for the x and y coordinates in terms of a third variable (usually t or ). These are called parametric equations. “ t ” is the parameter. (It is also the independent variable) Parametric Equations

3. Example Sketch the curve described by the parametric equations. When t = -1, we have x =4 and y = -5. The point (4,-5) is called the initial point. When t = 4, we have x = -1 and y = 5. The point (-1,5) is called the terminal point.

4. Example Sketch the curve defined by parametric equations Eliminate the parameter to write in rectangular equation.

5. Examples Eliminate the parameter to find a Cartesian equation of the curve. Then sketch the curve defined by parametric equations.

6. Parametric Form of the Derivative Example: Find the slope of the tangent line to the curve given by at the point (5, -3). If a smooth curve is given by the equations x = f ( t ) and y = g ( t ), then the slope of the tangent line to the curve at ( x, y ) is: This makes sense if we think about canceling dt.

7. To find the second derivative of a curve, we find the derivative of the first derivative: 1.Find the first derivative ( dy/dx ). 2. Find the derivative of dy/dx with respect to t. 3. Divide by dx/dt. The Second Derivative

8. a. Is the parametric curve concave up or down at the origin? Example b. Find the equation of the tangent line at the origin.

9. 1.Find dy/dx. 2. Find the derivative of dy/dx with respect to t. Quotient Rule 3. Divide by dx/dt. Example

10. If a smooth curve does not intersect itself on an interval [a, b] (except possibly at the endpoints), then the arc length of the curve over the interval is: This formula can be derived from Arc Length in Parametric Form

11. If a smooth curve C does not cross itself on an interval [a, b], then the area of the surface formed by revolving C about the coordinated axes is given by Surface Area in Parametric Form

12. Examples 1.Write the equation in parametric form. 2.Find the circumference of the circle. 3.Find the area of the surface formed by revolving the arc of the circle from (5,-1) to (2,2) about the y-axis.