Chapter 10

\(7-12\) Sketch the region in the plane consisting of points whose polar coordinates satisfy the given conditions. $$ 2

(a) 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. (b) Eliminate the parameter to find a Cartesian equation of the curve. \(x=t^{2}, \quad y=t^{3}\)

(a) Find the eccentricity, (b) identify the conic, (c) give an equation of the directrix, and (d) sketch the conic. $$I^{*}=\frac{12}{4-\sin \theta}$$

Find the vertices and foci of the ellipse and sketch its graph. $$\frac{x^{2}}{9}+\frac{y^{2}}{5}=1$$

\(9-14\) Sketch the curve and find the area that it encloses. $$ r^{2}=4 \cos 2 \theta $$

\(7-12\) Sketch the region in the plane consisting of points whose polar coordinates satisfy the given conditions. $$2

Find \(d y / d x\) and \(d^{2} y / d x^{2} .\) For which values of \(t\) is the curve concave upward? $$x=4+t^{2}, \quad y=t^{2}+t^{3}$$

(a) Eliminate the parameter to find a Cartesian equation of the curve. (b) Sketch the curve and indicate with an arrow the direction in which the curve is traced as the parameter increases. \(x=\sin \theta, \quad y=\cos \theta, \quad 0 \leqslant \theta \leqslant \pi\)

(a) Find the eccentricity, (b) identify the conic, (c) give an equation of the directrix, and (d) sketch the conic. $$r=\frac{3}{2+2 \cos \theta}$$

Find the vertices and foci of the ellipse and sketch its graph. $$\frac{x^{2}}{64}+\frac{y^{2}}{100}=1$$

\(9-14\) Sketch the curve and find the area that it encloses. $$ r=2-\sin \theta $$

\(7-12\) Sketch the region in the plane consisting of points whose polar coordinates satisfy the given conditions. $$r \geqslant 1, \quad \pi \leqslant \theta \leqslant 2 \pi$$

Find \(d y / d x\) and \(d^{2} y / d x^{2} .\) For which values of \(t\) is the curve concave upward? $$x=t^{3}-12 t, \quad y=t^{2}-1$$

(a) Eliminate the parameter to find a Cartesian equation of the curve. (b) Sketch the curve and indicate with an arrow the direction in which the curve is traced as the parameter increases. \(x=4 \cos \theta, \quad y=5 \sin \theta, \quad-\pi / 2 \leqslant \theta \leqslant \pi / 2\)

(a) Find the eccentricity, (b) identify the conic, (c) give an equation of the directrix, and (d) sketch the conic. $$r=\frac{9}{6+2 \cos \theta}$$

Find the vertices and foci of the ellipse and sketch its graph. $$4 x^{2}+y^{2}=16$$

\(9-14\) Sketch the curve and find the area that it encloses. $$ r=2 \cos 3 \theta $$

Find the distance between the points with polar coordinates \((2, \pi / 3)\) and \((4,2 \pi / 3) .\)

Find \(d y / d x\) and \(d^{2} y / d x^{2} .\) For which values of \(t\) is the curve concave upward? $$x=t-e^{t}, \quad y=t+e^{-t}$$

(a) Eliminate the parameter to find a Cartesian equation of the curve. (b) Sketch the curve and indicate with an arrow the direction in which the curve is traced as the parameter increases. \(x=\sin t, \quad y=\csc t, \quad 0< t<\pi / 2\)

(a) Find the eccentricity, (b) identify the conic, (c) give an equation of the directrix, and (d) sketch the conic. $$r=\frac{8}{4+5 \sin \theta}$$

Find the vertices and foci of the ellipse and sketch its graph. $$4 x^{2}+25 y^{2}=25$$

\(9-14\) Sketch the curve and find the area that it encloses. $$ r=2+\cos 2 \theta $$

Find a formula for the distance between the points with polar coordinates \(\left(r_{1}, \theta_{1}\right)\) and \(\left(r_{2}, \theta_{2}\right)\)

Find \(d y / d x\) and \(d^{2} y / d x^{2} .\) For which values of \(t\) is the curve concave upward? $$x=t+\ln t, \quad y=t-\ln t$$

(a) Eliminate the parameter to find a Cartesian equation of the curve. (b) Sketch the curve and indicate with an arrow the direction in which the curve is traced as the parameter increases. \(x=e^{t}-1, \quad y=e^{2 t}\)

(a) Find the eccentricity, (b) identify the conic, (c) give an equation of the directrix, and (d) sketch the conic. $$r=\frac{3}{4-8 \cos \theta}$$

Find the vertices and foci of the ellipse and sketch its graph. $$9 x^{2}-18 x+4 y^{2}=27$$

15-16 Graph the curve and find the area that it encloses. $$ r=1+2 \sin 6 \theta $$

Find \(d y / d x\) and \(d^{2} y / d x^{2} .\) For which values of \(t\) is the curve concave upward? \(x=2 \sin t$$y=3 \cos t$$0<\)t\(<2 \pi\)

\(15-20\) Identify the curve by finding a Cartesian equation for the curve. $$r=2$$

(a) Eliminate the parameter to find a Cartesian equation of the curve. (b) Sketch the curve and indicate with an arrow the direction in which the curve is traced as the parameter increases. \(x=e^{2 t}, \quad y=t+1\)

(a) Find the eccentricity, (b) identify the conic, (c) give an equation of the directrix, and (d) sketch the conic. $$r=\frac{10}{5-6 \sin \theta}$$

Find the vertices and foci of the ellipse and sketch its graph. $$x^{2}+3 y^{2}+2 x-12 y+10=0$$

Find \(d y / d x\) and \(d^{2} y / d x^{2} .\) For which values of \(t\) is the curve concave upward? \(x=\cos 2 t\), \(y=\cos t\), \(0<\)t\(<\pi\)

\(15-20\) Identify the curve by finding a Cartesian equation for the curve. $$ r \cos \theta=1 $$

(a) Eliminate the parameter to find a Cartesian equation of the curve. (b) Sketch the curve and indicate with an arrow the direction in which the curve is traced as the parameter increases. \(x=\ln t, \quad y=\sqrt{t}, \quad t \geqslant 1\)

(a) Find the eccentricity and directrix of the conic \(r=1 /(1-2 \sin \theta)\) and graph the conic and its directrix. (b) If this conic is rotated counterclockwise about the origin through an angle 3\(\pi / 4\) . write the resulting equation and graph its curve.

Find the points on the curve where the tangent is horizontal or vertical. If you have a graphing device, graph the curve to check your work. $$x=10-t^{2}, \quad y=t^{3}-12 t$$

\(15-20\) Identify the curve by finding a Cartesian equation for the curve. $$r=3 \sin \theta$$

(a) Eliminate the parameter to find a Cartesian equation of the curve. (b) Sketch the curve and indicate with an arrow the direction in which the curve is traced as the parameter increases. \(x=\sinh t, \quad y=\cosh t\)

Graph the conic \(r=4 /(5+6 \cos \theta)\) and its directrix. Also graph the conic obtained by rotating this curve about the origin through an angle \(\pi / 3\) .

Find the points on the curve where the tangent is horizontal or vertical. If you have a graphing device, graph the curve to check your work. $$x=2 t^{3}+3 t^{2}-12 t, \quad y=2 t^{3}+3 t^{2}+1$$

\(17-21\) Find the area of the region enclosed by one loop of the curve. $$ r=4 \sin 3 \theta $$

\(15-20\) Identify the curve by finding a Cartesian equation for the curve. $$r=2 \sin \theta+2 \cos \theta$$

(a) Eliminate the parameter to find a Cartesian equation of the curve. (b) Sketch the curve and indicate with an arrow the direction in which the curve is traced as the parameter increases. \(x=2 \cosh t, \quad y=5 \sinh t\)

Graph the conics \(r=e /(1-e \cos \theta)\) with \(e=0.4,0.6\) . \(0.8,\) and 1.0 on a common screen. How does the value of \(e\) affect the shape of the curve?

Find the vertices, foci, and asymptotes of the hyperbola and sketch its graph. $$\frac{x^{2}}{144}-\frac{y^{2}}{25}=1$$

Find the points on the curve where the tangent is horizontal or vertical. If you have a graphing device, graph the curve to check your work. $$x=2 \cos \theta, \quad y=\sin 2 \theta$$

\(17-21\) Find the area of the region enclosed by one loop of the curve. $$ r=3 \cos 5 \theta $$