91Ó°ÊÓ

Sketch the curve by using the parametric equations to plot points. Indicate with an arrow the direction in which the curve is traced as \(t\) increases. \(x=2 \cos t, \quad y=t-\cos t, \quad 0 \leq t \leqslant 2 \pi\)

Write a polar equation of a conic with the focus at the origin and the given data. Parabola, directrix \(x=4\)

Find an equation of the tangent to the curve at the point corresponding to the given value of the parameter. $$x=t^{4}+1, \quad y=t^{3}+t ; \quad t=-1$$

1-4 Find the area of the region that is bounded by the given curve and lies in the specified sector. $$r=\sin \theta, \quad \pi / 3 \leqslant \theta \leqslant 2 \pi / 3$$

\(3-4\) Plot the point whose polar coordinates are given. Then find the Cartesian coordinates of the point. $$ \begin{array}{lll}{\text { (a) }(1, \pi)} & {\text { (b) }(2,-2 \pi / 3)} & {\text { (c) }(-2,3 \pi / 4)}\end{array} $$

Find the vertex, focus, and directrix of the parabola and sketch its graph. $$4 x^{2}=-y$$

Sketch the curve by using the parametric equations to plot points. Indicate with an arrow the direction in which the curve is traced as \(t\) increases. \(x=5 \sin t, \quad y=t^{2}, \quad-\pi \leqslant t \leqslant \pi\)

Write a polar equation of a conic with the focus at the origin and the given data. Ellipse, eccentricity \(\frac{3}{4}, \quad\) directrix \(x=-5\)

Find an equation of the tangent to the curve at the point corresponding to the given value of the parameter. $$x=t-t^{-1}, \quad y=1+t^{2} ; \quad t=1$$

1-4 Find the area of the region that is bounded by the given curve and lies in the specified sector. $$r=\sqrt{\sin \theta}, \quad 0 \leqslant \theta \leqslant \pi$$