1. 9.1 Parametric Curves 9.2 Calculus with Parametric Curves

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)

3. Circle: If we let t = the angle, then: We could identify the parametric equations as a circle.

4. Ellipse: This is the equation of an ellipse.

5. 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: Tangents

6. Example: Tangents

7. Example (cont.): 1.Find the first derivative ( dy/dx ).

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

9. 3. Divide by dx/dt.

10. Areas under parametric curves If a curve is given by parametric equations x=f(t), y=g(t) and is traversed once as t increases from α to β, then the area under the curve is Examples on the board

11. Lengths of parametric curves If a curve C is described by the parametric equations x=f(t), y=g(t), α ≤ t ≤ β, where f’ and g’ are continuous on [α, β] and C is traversed exactly once as t increases from α to β, then the length of the curve is Examples on the board