Chapter 10

Sketch the graph of the given equation. \(4(x+3)=(y+2)^{2}\)

Sketch the region that is inside the circle \(r=3 \sin \theta\) and outside the cardioid \(r=1+\sin \theta\), and find its area.

Find the Cartesian equations of the graphs of the given polar equations. $$ r=3 $$

Find the equation of the parabola whose vertex is the origin and whose axis is the \(y\) -axis if the parabola passes through the point \((-3,5)\). Make a sketch.

Sketch the graph of the given equation. \((x+2)^{2}=8(y-1)\)

Sketch the region that is outside the circle \(r=2\) and inside the lemniscate \(r^{2}=8 \cos 2 \theta\), and find its area.

Find the Cartesian equations of the graphs of the given polar equations. $$ r \cos \theta+3=0 $$

a parametric representation of a curve is given. $$ x=\cos \theta, y=-2 \sin ^{2} 2 \theta ;-\infty<\theta<\infty $$

Find the equations of the tangent and the normal lines to the given parabola at the given point. Sketch the parabola, the tangent line, and the normal line. $$ y^{2}=16 x,(1,-4) $$

Sketch the graph of the given equation. \((x+2)^{2}=4\)

Sketch the limaçon \(r=3-6 \sin \theta\), and find the area of the region that is inside its large loop, but outside its small loop.

Find the Cartesian equations of the graphs of the given polar equations. $$ r-5 \cos \theta=0 $$

a parametric representation of a curve is given. $$ x=\sin \theta, y=2 \cos ^{2} 2 \theta ;-\infty<\theta<\infty $$

Find the equations of the tangent and the normal lines to the given parabola at the given point. Sketch the parabola, the tangent line, and the normal line. $$ x^{2}=-10 y,(2 \sqrt{5},-2) $$

Sketch the region in the first quadrant that is inside the cardioid \(\quad r=3+3 \cos \theta\) and outside the cardioid \(r=3+3 \sin \theta\), and find its area.

Sketch the graph of the given polar equation and verify its symmetry (see Examples \(1-3)\). \(r=5 \cos 3 \theta\) (three-leaved rose)

Find the Cartesian equations of the graphs of the given polar equations. $$ r \sin \theta-1=0 $$

Find the equations of the tangent and the normal lines to the given parabola at the given point. Sketch the parabola, the tangent line, and the normal line. $$ x^{2}=2 y,(4,8) $$

Sketch the graph of the given equation. \(\frac{(x+3)^{2}}{4}+\frac{(y-2)^{2}}{8}=0\)

Sketch the region in the second quadrant that is inside the cardioid \(\quad r=2+2 \sin \theta\) and outside the cardioid \(r=2+2 \cos \theta\), and find its area.

Find the Cartesian equations of the graphs of the given polar equations. $$ r^{2}-6 r \cos \theta-4 r \sin \theta+9=0 $$

find \(d y / d x\) and \(d^{2} y / d x^{2}\) without eliminating the parameter. $$ x=6 s^{2}, y=-2 s^{3} ; s \neq 0 $$

Find the equations of the tangent and the normal lines to the given parabola at the given point. Sketch the parabola, the tangent line, and the normal line. $$ y^{2}=-9 x,(-1,-3) $$

Sketch the graph of the given equation. \(x^{2}+4 y^{2}-2 x+16 y+1=0\)

Find the slope of the tangent line to each of the following curves at \(\theta=\pi / 3\). (a) \(r=2 \cos \theta\) (b) \(r=1+\sin \theta\) (c) \(r=\sin 2 \theta\) (d) \(r=4-3 \cos \theta\)

Name the curve with the given polar equation. If it is a conic, give its eccentricity. Sketch the graph. $$ r=6 $$

find \(d y / d x\) and \(d^{2} y / d x^{2}\) without eliminating the parameter. $$ x=2 \theta^{2}, y=\sqrt{5} \theta^{3} ; \theta \neq 0 $$

Find the equation of the given central conic. Hyperbola with a vertex at \((0,-4)\) and a focus at \((0,-5)\)

Find the equations of the tangent and the normal lines to the given parabola at the given point. Sketch the parabola, the tangent line, and the normal line. $$ y^{2}=-15 x,(-3,-3 \sqrt{5}) $$

Sketch the graph of the given equation. \(25 x^{2}+9 y^{2}+150 x-18 y+9=0\)

Find all points on the cardioid \(r=a(1+\cos \theta)\) where the tangent line is (a) horizontal, and (b) vertical.

find \(d y / d x\) and \(d^{2} y / d x^{2}\) without eliminating the parameter. $$ x=\sqrt{3} \theta^{2}, y=-\sqrt{3} \theta^{3} ; \theta \neq 0 $$

Find the equation of the given central conic. Hyperbola with a vertex at \((0,-3)\) and eccentricity \(\frac{3}{2}\)

Find the equations of the tangent and the normal lines to the given parabola at the given point. Sketch the parabola, the tangent line, and the normal line. $$ x^{2}=4 y,(4,4) $$

Sketch the graph of the given equation. \(9 x^{2}-16 y^{2}+54 x+64 y-127=0\)

Find all points on the limaçon \(r=1-2 \sin \theta\) where the tangent line is horizontal.

Sketch the graph of the given polar equation and verify its symmetry (see Examples \(1-3)\). \(r=7 \cos 5 \theta\) (five-leaved rose)

Name the curve with the given polar equation. If it is a conic, give its eccentricity. Sketch the graph. $$ r=\frac{3}{\sin \theta} $$

Find the equation of the given central conic. Hyperbola with asymptotes \(2 x \pm 4 y=0\) and a vertex at \((8,0)\)

Find the equations of the tangent and the normal lines to the given parabola at the given point. Sketch the parabola, the tangent line, and the normal line. $$ x^{2}=-6 y,(3 \sqrt{2},-3) $$

Sketch the graph of the given equation. \(x^{2}-4 y^{2}-14 x-32 y-11=0\)

Let \(r=f(\theta)\), where \(f\) is continuous on the closed interval \([\alpha, \beta] .\) Derive the following formula for the length \(L\) of the corresponding 26. polar curve from \(\theta=\alpha\) to \(\theta=\beta\). $$ L=\int_{\alpha}^{\beta} \sqrt{[f(\theta)]^{2}+\left[f^{\prime}(\theta)\right]^{2}} d \theta $$

Name the curve with the given polar equation. If it is a conic, give its eccentricity. Sketch the graph. $$ r=\frac{-4}{\cos \theta} $$

find \(d y / d x\) and \(d^{2} y / d x^{2}\) without eliminating the parameter. $$ x=3-2 \cos t, y=-1+5 \sin t ; t \neq n \pi $$

Find the equation of the given central conic. Vertical hyperbola with eccentricity \(\sqrt{6} / 2\) that passes through \((2,4)\)

Find the equations of the tangent and the normal lines to the given parabola at the given point. Sketch the parabola, the tangent line, and the normal line. $$ y^{2}=20 x,(2,-2 \sqrt{10}) $$

Sketch the graph of the given equation. \(4 x^{2}+16 x-16 y+32=0\)

Name the curve with the given polar equation. If it is a conic, give its eccentricity. Sketch the graph. $$ r=4 \sin \theta $$

find \(d y / d x\) and \(d^{2} y / d x^{2}\) without eliminating the parameter. $$ x=3 \tan t-1, y=5 \sec t+2 ; t \neq \frac{(2 n+1) \pi}{2} $$

Find the equation of the given central conic. Ellipse with foci \((\pm 2,0)\) and directrices \(x=\pm 8\)