1. Section 9.2: Parametric Equations – Slope, Arc Length, and Surface Area Slope and Tangent Lines: Theorem. 9.4 – If a smooth curve C is given by the equations x = f(t) and y = g(t), then the slope of C at (x, y) is

2. Example 1 Curve C is given by a)Find the slope at the point (2, 3). b)Find the equation of the tangent line and normal line at the point (2, 3).

3. Example 2 The prolate cycloid is given by The curve crosses itself at the point (0, 2). Find the equation of the two tangent lines at this point.

4. Finding Higher Order Derivatives If a smooth curve C is given by the equations x = f(t) and y = g(t) and is a differentiable function of t, then theorem 9.4 can be applied to as follows to find. Note:

5. Example 3 Let C be the curve with parametrization a)Sketch the graph of C and indicate orientation. b)Find. c)Find a function k(x) that has the same graph as C, and use k’(x) and k’’(x) to check the answers to part b. d)Discuss the concavity of C.

6. Section 5.5 Reminder Arc Length of f(x) from x=a to x=b: Arc Length of f(y) from y=c to y=d: Ex. Set up integral for arc length of f(x) = 2x 3 from (0,0) to (3,54).

7. Section 5.5 Extension The surface area of a solid of revolution can be found using the formula for the surface area of a frustrum of a cone. If f is smooth and f(x) ≥ 0 on [a, b], then the area S of the surface generated by revolving the graph of f about the x-axis is and about the y-axis is

8. Example 4 Find the surface area formed by rotating About the x-axis from (0, 0) to (4, 2).