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Chapter 10, Problem 2(choose chapter or problem)
Consider the parametric equations \(x=4\ \cos^2 \theta\ \text{and}\ y=2\ \sin \theta\).
(a) Construct a table of values of \(\theta=-\frac{\pi}{2},\ -\frac{\pi}{4},\ 0,\ \frac{\pi}{4},\ \text{and}\ \frac{\pi}{2}\).
(b) Plot the points (x, y) generated in the table, and sketch a graph of the parametric equations. Indicate the orientation of the graph.
(c) Use a graphing utility to confirm your graph in part (b).
(d) Find the rectangular equation by eliminating the parameter, and sketch its graph. Compare the graph in part (b) with the graph of the rectangular equation.
(e) If values of \(\theta\) were selected from the interval \([\pi/2,\ 3\pi/2]\) for the table in part (a), would the graph in part (b) be different? Explain.
Questions & Answers
QUESTION:
Consider the parametric equations \(x=4\ \cos^2 \theta\ \text{and}\ y=2\ \sin \theta\).
(a) Construct a table of values of \(\theta=-\frac{\pi}{2},\ -\frac{\pi}{4},\ 0,\ \frac{\pi}{4},\ \text{and}\ \frac{\pi}{2}\).
(b) Plot the points (x, y) generated in the table, and sketch a graph of the parametric equations. Indicate the orientation of the graph.
(c) Use a graphing utility to confirm your graph in part (b).
(d) Find the rectangular equation by eliminating the parameter, and sketch its graph. Compare the graph in part (b) with the graph of the rectangular equation.
(e) If values of \(\theta\) were selected from the interval \([\pi/2,\ 3\pi/2]\) for the table in part (a), would the graph in part (b) be different? Explain.
ANSWER:Step 1 of 6
The equations are given as,
\(x = 4{\cos ^2}\theta ,y = 2\sin \theta ,0 \le x \le 4, - 2 \le y \le 2\)