University:
California State University, Northridge
Course:
MATH 150BL | Calculus II Laboratory
Academic year:

2024
Views:

37

Pages:

3
Author:

texans105bvn

Assignment 10 -- Polar Coordinates So far, we have been representing graphs as collections of points on the rectangular coordinate system. The corresponding equations for these graphs have been in either rectangular or parametric form. Here we work with the coordinate system called the polar coordinate system. To form the polar coordinate system in the plane, we fix a point origin, and construct from point , called the an initial ray called the polar axis. Then each in the plane can be assigned polar coordinates directed distance from as follows. to directed angle, counterclockwise from polar axis to segment With rectangular coordinates, each point has a unique representation. This is not true with polar coordinates. For instance, the coordinates represent the same point. Also, because and is a directed distance, the coordinates and represent the same point. To establish the relationship between polar and rectangular coordinates, let the polar axis coincide with the positive lies on a circle of radius for axis and the pole with the origin. Because , it follows that . Moreover, , the definition of the trigonometric functions implies that , , and . That is, the polar coordinates related to the rectangular coordinates , , of a point are of the point as follows. , and . Example: Polar-to-Rectangular Conversion For the point , and . Thus the rectangular coordinates are . Example: Rectangular-to-Polar Conversion For the point , . Thus, (since the original point is in the second quadrant). . Thus, one set of polar coordinates is: are not unique). (recall that polar coordinates Problems 1. Convert the following rectangular coordinates to polar coordinates. a. b. 2. Convert the following polar coordinates to rectangular coordinates. a. b. The following will be useful for the next problem: If of , then the slope of the tangent line to the graph of is a differentiable function at the point is provided that at . Problems 3. Graph the following polar equation and find all points of horizontal tangency (that is, find the values of such that and 4. Graph the following polar equation. Find an interval for is traced only once. ). over which the graph 5. Graph the following polar equation so that you get the curve depicted below. The curve is given by to produce the curve? . Over what interval must vary 6. Heart to Bell Graph the polar equation the integers to for . What values of of the curve? What values of produce the "heart" portion produce the "bell"? The following will be useful for the next problem: If on the interval is continuous and nonnegative , then the area of the region bounded by the graph of between the radial lines and is given by . Problems 7. Graph the following polar equation and find the area of the indicated region. a. Inner loop of . b. Between the loops of c. Inside for and outside

Assignment 10 - Polar Coordinates

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