91Ó°ÊÓ

Polar-to-Rectangular Conversion In Exercises \(1-10,\) plot the point in polar coordinates and find the corresponding rectangular coordinates for the point. $$ \left(-4,-\frac{3 \pi}{4}\right) $$

Graphical Reasoning In Exercises \(1-4,\) use a graphing utility to graph the polar equation when (a) \(e=1,\) (b) \(e=0.5,\) and \((\mathrm{c}) e=1.5 .\) Identify the conic. $$ r=\frac{2 e}{1-e \sin \theta} $$

Graphical Reasoning In Exercises \(1-4,\) use a graphing utility to graph the polar equation when (a) \(e=1,\) (b) \(e=0.5,\) and \((\mathrm{c}) e=1.5 .\) Identify the conic. $$ r=\frac{2 e}{1+e \sin \theta} $$

In Exercises 1–18, sketch the curve represented by the parametric equations (indicate the orientation of the curve), and write the corresponding rectangular equation by eliminating the parameter. $$ x=2 t^{2}, \quad y=t^{4}+1 $$

Finding a Derivative In Exercises \(1-4,\) find \(d y / d x\) $$ x=2 e^{\theta}, y=e^{-\theta / 2} $$

Polar-to-Rectangular Conversion In Exercises \(1-10,\) plot the point in polar coordinates and find the corresponding rectangular coordinates for the point. $$ \left(0,-\frac{7 \pi}{6}\right) $$

In Exercises 1–18, sketch the curve represented by the parametric equations (indicate the orientation of the curve), and write the corresponding rectangular equation by eliminating the parameter. $$ x=t^{3}, \quad y=\frac{t^{2}}{2} $$

Finding Slope and Concavity In Exercises \(5-14\) , find \(d y / d x\) and \(d^{2} y / d x^{2},\) and find the slope and concavity (if possible) at the given value of the parameter. $$ \text{Parametric Equations} \quad \text{Parameter} $$ $$ x=4 t, y=3 t-2 \quad t=3 $$

Polar-to-Rectangular Conversion In Exercises \(1-10,\) plot the point in polar coordinates and find the corresponding rectangular coordinates for the point. $$ \left(7, \frac{5 \pi}{4}\right) $$

Writing Consider the polar equation \(r=\frac{4}{1+e \sin \theta}\). (a) Use a graphing utility to graph the equation for \(e=0.1\) \(e=0.25, e=0.5, e=0.75,\) and \(e=0.9 .\) Identify the conic and discuss the change in its shape as \(e \rightarrow 1^{-}\) and \(e \rightarrow 0^{+}\) (b) Use a graphing utility to graph the equation for \(e=1\) Identify the conic. (c) Use a graphing utility to graph the equation for \(e=1.1\) \(e=1.5,\) and \(e=2 .\) Identify the conic and discuss the change in its shape as \(e \rightarrow 1^{+}\) and \(e \rightarrow \infty\)