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Chapter 8: Problem 36
Determine the \(t\) intervals on which the curve is concave downward or concave upward. $$ x=2+t^{2}, \quad y=t^{2}+t^{3} $$
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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. $$ \text { Hypocycloid: } x=3 \cos ^{3} \theta, \quad y=3 \sin ^{3} \theta $$
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. \(r=\frac{4}{1+2 \cos \theta}\)
In Exercises \(27-38,\) find a polar equation for the conic with its focus at the pole. (For convenience, the equation for the directrix is given in rectangular form.) \(\frac{\text { Conic }}{\text { Parabola }} \quad \frac{\text { Eccentricity }}{e=1} \quad \frac{\text { Directrix }}{x=-1}\)
Sketch the curve represented by the parametric equations (indicate the orientation of the curve), and write the corresponding rectangular equation by eliminating the parameter. $$ x=2 t^{2}, \quad y=t^{4}+1 $$
Explain how the graph of each conic differs from the graph of \(r=\frac{4}{1+\sin \theta} .\) (a) \(r=\frac{4}{1-\cos \theta}\) (b) \(r=\frac{4}{1-\sin \theta}\) (c) \(r=\frac{4}{1+\cos \theta}\) (d) \(r=\frac{4}{1-\sin (\theta-\pi / 4)}\)
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