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Chapter 8: Problem 16
Use the angle feature of a graphing utility to find one set of polar coordinates for the point given in rectangular coordinates. $$ (3 \sqrt{2}, 3 \sqrt{2}) $$
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In Exercises \(7-16,\) find the eccentricity and the distance from the pole to the directrix of the conic. Then sketch and identify the graph. Use a graphing utility to confirm your results. \(r=\frac{-6}{3+7 \sin \theta}\)
Use a graphing utility to graph the curve represented by the parametric equations. Indicate the direction of the curve. Identify any points at which the curve is not smooth. $$ \text { Witch of Agnesi: } x=2 \cot \theta, \quad y=2 \sin ^{2} \theta $$
A curve called the folium of Descartes can be represented by the parametric equations \(x=\frac{3 t}{1+t^{3}} \quad\) and \(y=\frac{3 t^{2}}{1+t^{3}}\) (a) Convert the parametric equations to polar form. (b) Sketch the graph of the polar equation from part (a). (c) Use a graphing utility to approximate the area enclosed by the loop of the curve.
In Exercises \(17-20,\) use a graphing utility to graph the polar equation. Identify the graph. \(r=\frac{3}{-4+2 \sin \theta}\)
Sketch the curve represented by the parametric equations (indicate the orientation of the curve), and write the corresponding rectangular equation by eliminating the parameter. $$ x=e^{t}, \quad y=e^{3 t}+1 $$
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