Chapter 10

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 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. $$x^{2}=-10 y,(2 \sqrt{5},-2)$$

In Problems 17-22, find the Cartesian equations of the graphs of the given polar equations. \(r \sin \theta-1=0\)

In Problems \(21-30\), find \(d y / d x\) and \(d^{2} y / d x^{2}\) without eliminating the parameter. $$ x=3 \tau^{2}, y=4 \tau^{3} ; \tau \neq 0 $$

Sketch the graph of the given equation. $$ (y-1)^{2}=16 $$

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

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. $$x^{2}=2 y,(4,8)$$

In Problems 17-22, find the Cartesian equations of the graphs of the given polar equations. \(r^{2}-6 r \cos \theta-4 r \sin \theta+9=0\)

In Problems \(21-30\), find \(d y / d x\) and \(d^{2} y / d x^{2}\) without eliminating the parameter. $$ x=6 s^{2}, y=-2 s^{3} ; s \neq 0 $$

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 \(r=2+2 \sin \theta\) and outside the cardioid \(r=2+2 \cos \theta\), and find its area.

In Problems \(1-32\), sketch the graph of the given polar equation and verify its symmetry (see Examples 1-3). \(r=3 \sin 3 \theta\) (three-leaved rose)

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. $$y^{2}=-9 x,(-1,-3)$$

In Problems 23-36, name the curve with the given polar equation. If it is a conic, give its eccentricity. Sketch the graph. \(r=6\)

In Problems \(21-30\), find \(d y / d x\) and \(d^{2} y / d x^{2}\) without eliminating the parameter. $$ x=2 \theta^{2}, y=\sqrt{5} \theta^{3} ; \theta \neq 0 $$

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) \(r=4-3 \cos \theta\)

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. $$y^{2}=-15 x,(-3,-3 \sqrt{5})$$

In Problems \(21-30\), 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 \(\frac{3}{2}\)

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.

In Problems \(1-32\), sketch the graph of the given polar equation and verify its symmetry (see Examples 1-3). \(r=4 \cos 2 \theta\) (four-leaved rose)

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. $$x^{2}=4 y,(4,4)$$

In Problems 23-36, name the curve with the given polar equation. If it is a conic, give its eccentricity. Sketch the graph. \(r=\frac{3}{\sin \theta}\)

In Problems \(21-30\), find \(d y / d x\) and \(d^{2} y / d x^{2}\) without eliminating the parameter. $$ x=1-\cos t, y=1+\sin t ; t \neq n \pi $$

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

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.

In Problems \(1-32\), sketch the graph of the given polar equation and verify its symmetry (see Examples 1-3). \(r=7 \cos 5 \theta\) (five-leaved rose)

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. $$x^{2}=-6 y,(3 \sqrt{2},-3)$$

In Problems 23-36, name the curve with the given polar equation. If it is a conic, give its eccentricity. Sketch the graph. \(r=\frac{-4}{\cos \theta}\)

In Problems \(21-30\), find \(d y / d x\) and \(d^{2} y / d x^{2}\) without eliminating the parameter. $$ x=3-2 \cos t, y=-1+5 \sin t ; t \neq n \pi $$

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 polar curve from \(\theta=\alpha\) to \(\theta=\beta\). $$ L=\int_{\alpha}^{\beta} \sqrt{[f(\theta)]^{2}+\left[f^{\prime}(\theta)\right]^{2}} d \theta $$

In Problems \(1-32\), sketch the graph of the given polar equation and verify its symmetry (see Examples 1-3). \(r=3 \sin 5 \theta\) (five-leaved rose)

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. $$y^{2}=20 x,(2,-2 \sqrt{10})$$

In Problems 23-36, name the curve with the given polar equation. If it is a conic, give its eccentricity. Sketch the graph. \(r=4 \sin \theta\)

In Problems \(21-30\), find \(d y / d x\) and \(d^{2} y / d x^{2}\) without eliminating the parameter. $$ x=3 \tan t-1, y=5 \sec t+2 ; t \neq \frac{(2 n+1) \pi}{2} $$

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

The slope of the tangent line to the parabola \(y^{2}=5 x\) at a certain point on the parabola is \(\sqrt{5} / 4\). Find the coordinates of that point. Make a sketch.

In Problems 23-36, name the curve with the given polar equation. If it is a conic, give its eccentricity. Sketch the graph. \(r=-4 \cos \theta\)

In Problems \(21-30\), find \(d y / d x\) and \(d^{2} y / d x^{2}\) without eliminating the parameter. $$ x=\cot t-2, y=-2 \csc t+5 ; 0

Hyperbola with foci \((\pm 4,0)\) and directrices \(x=\pm 1\)

Sketch the graph of the given equation. $$ x^{2}-4 x+8 y=0 $$

Find the length of the logarithmic spiral \(r=e^{\theta / 2}\) from \(\theta=0 \operatorname{to} \theta=2 \pi\).

The slope of the tangent line to the parabola \(x^{2}=-14 y\) at a certain point on the parabola is \(-2 \sqrt{7} / 7\). Find the coordinates of that point.

In Problems 23-36, name the curve with the given polar equation. If it is a conic, give its eccentricity. Sketch the graph. \(r=\frac{4}{1+\cos \theta}\)

In Problems \(21-30\), find \(d y / d x\) and \(d^{2} y / d x^{2}\) without eliminating the parameter. $$ x=\frac{1}{1+t^{2}}, y=\frac{1}{t(1-t)} ; 0

Hyperbola whose asymptotes are \(x \pm 2 y=0\) and that goes through the point \((4,3)\)