Clicker Question 1 At what point (x, y) does the curve x = sin(t), y = t 3 – 24 ln(t) have a horizontal tangent ? A. (1, 2) B. (0, 2) C. (0, 8(1 – 3 ln(2)) D. (1, 8(1 – 3 ln(2)) E. (1, 1/8 – 24 ln(1/2))

Polar Coordinates (11/5/12) Normally we label points in the plane in “Cartesian” or “rectangular” coordinates (x, y). (How far over, how far up or down?) Another alternative is polar coordinates (r, ) where r is the distance from the origin and  is the angle (measured in radians) from the positive x-axis to the point, moving counter-clockwise.

Non-Uniqueness of Coordinates Note that unlike rectangular coordinates, polar coordinates for a given point are not unique. Example: Give 5 different ways of describing the point (x, y) = (2, 2) in polar coordinates. We use here that by definition, (-r, ) is the same as (r,  + ).

Converting Polar  Rectangular Note at x = r cos() and y = r sin(). On the other hand, r = (x2 + y2) and  = arctan(y / x). Example: What are the rectangular coordinates of the point (r, ) = (-3, /2) ? Example: What are the polar coordinates of the point (x, y) = (27, 3) ?

Clicker Question 2 If the polar coordinates of a point in the plane are (4, /3), what are the rectangular coordinates? A. (2, 23) B. (23, 2) C. (4, 43) D. (1, 3) E. (43, 4)

Polar Curves To plot a curve in polar coordinates, we normally write r as a function of . What do these curves look like? r = 5  = /4 r = 2 cos() r = 1 + sin() r = cos(2 )

Tangents to Polar Curves Thinking of r as a function of  , we get Example: What is the slope of r = 1 + sin() when  = /3 ?

Assignment for Wednesday Read Section 10.3. In that section please do Exercises 1, 3, 5, 7, 13, 17, 21, 23, 29, 33, 55, 61. (Don’t panic! Most of these are pretty quick.)