Clicker Question 1 If x = e2t + 1 and y = 2t 2 + t , then what is y as a function of x ? A. y = (1/2)(ln2(x – 1) + ln(x – 1)) B. y = ln2(x – 1) + (1/2)ln(x – 1)) C. y = (1/2)(ln2(x) + ln(x) – 2) D. y = (1/2)(e2(x – 1) + e(x – 1)) E. y = e2(x – 1) + e(x – 1)

Clicker Question 2 What figure in the plane is given by the parametric equations x = 2 sin(t), y = 3 cos(t), 0  t   A. A circle B. The top half of a circle C. An ellipse D. The top half of an ellipse E. The right half of an ellipse

Calculus of Parametric Equations (11/1/13) Most of the ideas and techniques of calculus carry over quite easily into the parametric setting (derivatives, integrals, areas, volumes, arc length, surface area, etc.) The bottom line is that, by the ultra-important Chain Rule, (dy / dx )(dx / dt ) = dy / dt . Hence we have dy / dx = (dy / dt ) / (dx / dt ).

Example Suppose x = e2t + 1 and y = t 2 + t . What is dy / dx ? For what t is the tangent line to this curve horizontal? Vertical? At what point in the plane does this curve have a horizontal tangent? What is the equation of the tangent line to this curve when t = 0?

Clicker Question 3 If x = t cos(t) and y = t 3, what is the slope of the tangent line to the curve when t = /2 ? A. (-3/4) 2 B. (3/4) 2 C. (3/2)  D. (-3/2)  E. (-3/2)

Areas For example, the area enclosed by a curve and the x-axis is y dx (with appropriate endpoints on the integral). Example: Find the area enclosed by the curve x = ln(t + 1), y = t – t 2 and the x-axis. Note that since the variable of integration is t, the limits of integration must be in terms of t.

Arc Length Recall that using the Pythagorean Theorem, we had ds = (dx 2 + dy 2). This is easily computed in terms of t . Example: Find or estimate the length of the curve x = ln(t + 1), y = t – t 2 from t = 0 to t = 1. Similarly, surface area, volume, etc.

Assignment for Monday Read Section 10.2. In that section, please do Exercises 3, 7, 19, 31, 41, and 61. This last is challenging, but see if you can get it. The answer book is right.