91Ó°ÊÓ

CONCEPT CHECK Parametric Form of the Derivative What does the parametric form of the derivative represent?

Finding Slope and Concavity In Exercises \(9-18,\) 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. $$\begin{array}{ll}{\text { Parametric Equations }} & {\text { Parameter }} \\\ {\text { 9. } x=4 t,} {y=3 t-2} & {t=3}\end{array}$$

Finding Slope and Concavity In Exercises \(9-18,\) 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. $$\begin{array}{ll}{\text { Parametric Equations }} & {\text { Parameter }} \\\ {\text { } x=\sqrt{t}, \quad y=3 t-1} & {t=1}\end{array}$$

Finding the Area of a Polar Region In Exercises \(19-26,\) use a graphing utility to graph the polar equation. Find the area of the given region analytically. Inner loop of \(r=4-6 \sin \theta\)

Finding Equations of Tangent Lines In Exercises 27-30, find the equations of the tangent lines at the point where the curve crosses itself. $$x=t^{2}-t, \quad y=t^{3}-3 t-1$$

Rotated Parabola Write the equation for the parabola rotated \(\pi / 4\) radian counterclockwise from the parabola \(r=\frac{9}{1+\sin \theta}\)

Finding a Polar Equation In Exercises \(33-38\) , find a polar equation for the conic with its focus at the pole and the given eccentricity and directrix. (For convenience, the equation for the directrix is given in rectangular form.) $$\begin{array}{ll}{\text { Conic }} & {\text { Eccentricity }} \\ {\text { Parabola }} & {e=1}\end{array} \begin{array}{l}{\text { Directrix }} \\\ {x=-3}\end{array}$$

Polar-to-Rectangular Conversion In Exercises \(35-44\) , convert the polar equation to rectangular form and sketch its graph. $$r=4$$

Finding the Area of a Polar Region Between Two Curves In Exercises \(37-44\) , use a graphing utility to graph the polar equations. Find the area of the given region analytically. Common interior of \(r=2(1+\cos \theta)\) and \(r=2(1-\cos \theta)\)

Finding the Area of a Polar Region Between Two Curves In Exercises \(45-48\) , find the area of the region. $$Inside r=a(1+\cos \theta)\( and outside \)r=a \cos \theta$$