Chapter 10

Graph the following equations. Use a graphing utility to check your work and produce a final graph. \(r=\sin ^{2}(\theta / 2)\)

In Exercises \(37-40,\) you found the intersection points of pairs of curves. Find the area of the entire region that lies within both of the following pairs of curves. $$r=1+\sin \theta \text { and } r=1+\cos \theta$$

Find a parametric description of the line segment from the point \(P\) to the point \(Q\). Solutions are not unique. $$P(-1,-3), Q(6,-16)$$

Sketch a graph of the following hyperbolas. Specify the coordinates of the vertices and foci, and find the equations of the asymptotes. Use a graphing utility to check your work. $$10 x^{2}-7 y^{2}=140$$

Graph the following equations. Use a graphing utility to check your work and produce a final graph. \(r^{2}=4 \sin \theta\)

Find a parametric description of the line segment from the point \(P\) to the point \(Q\). Solutions are not unique. $$P(8,2), Q(-2,-3)$$

Find an equation of the following hyperbolas, assuming the center is at the origin. Sketch a graph labeling the vertices, foci, and asymptotes. Use a graphing utility to check your work. A hyperbola with vertices (±4,0) and foci (±6,0)

Determine whether the following statements are true and give an explanation or counterexample. a. The area of the region bounded by the polar graph of \(r=f(\theta)\) on the interval \([\alpha, \beta]\) is \(\int_{\alpha}^{\beta} f(\theta) d \theta\). b. The slope of the line tangent to the polar curve \(r=f(\theta)\) at a point \((r, \theta)\) is \(f^{\prime}(\theta)\).

Give a set of parametric equations that describes the following curves. Graph the curve and indicate the positive orientation. If not given, specify the interval over which the parameter varies. The segment of the parabola \(y=2 x^{2}-4,\) where \(-1 \leq x \leq 5\)

Find an equation of the following hyperbolas, assuming the center is at the origin. Sketch a graph labeling the vertices, foci, and asymptotes. Use a graphing utility to check your work. A hyperbola with vertices (±1,0) that passes through \(\left(\frac{5}{3}, 8\right)\)

Graph the following equations. Use a graphing utility to check your work and produce a final graph. \(r^{2}=16 \sin 2 \theta\)

Find an equation of the following hyperbolas, assuming the center is at the origin. Sketch a graph labeling the vertices, foci, and asymptotes. Use a graphing utility to check your work. A hyperbola with vertices (±2,0) and asymptotes \(y=\pm 3 x / 2\)

Graph the following equations. Use a graphing utility to check your work and produce a final graph. \(r=\sin 3 \theta\)

Find the areas of the following regions. The region common to the circles \(r=2 \sin \theta\) and \(r=1\)

Give a set of parametric equations that describes the following curves. Graph the curve and indicate the positive orientation. If not given, specify the interval over which the parameter varies. The piecewise linear path from \(P(-2,3)\) to \(Q(2,-3)\) to \(R(3,5)\) using parameter values \(0 \leq t \leq 2\)

Graph the following equations. Use a graphing utility to check your work and produce a final graph. \(r=2 \sin 5 \theta\)

Find the areas of the following regions. The region inside the inner loop of the limaçon \(r=2+4 \cos \theta\)

Give a set of parametric equations that describes the following curves. Graph the curve and indicate the positive orientation. If not given, specify the interval over which the parameter varies. The path consisting of the line segment from \((-4,4)\) to \((0,8)\) followed by the segment of the parabola \(y=8-2 x^{2}\) from \((0,8)\) to \((2,0),\) using parameter values \(0 \leq t \leq 3\)

Find the areas of the following regions. The region inside the outer loop but outside the inner loop of the limaçon \(r=3-6 \sin \theta\)

Use a graphing utility to graph the following curves. Be sure to choose an interval for the parameter that generates all features of interest. $$\text { Spiral } x=t \cos t, y=t \sin t ; t \geq 0$$

A Cartesian and a polar graph of \(r=f(\theta)\) are given in the figures. Mark the points on the polar graph that correspond to the points shown on the Cartesian graph. \(r=\sin (1+3 \cos \theta)\)

Find an equation of the following curves, assuming the center is at the origin. Sketch a graph labeling the vertices, foci, asymptotes (if they exist), and directrices. Use a graphing utility to check your work. An ellipse with vertices (±9,0) and eccentricity \(\frac{1}{3}\)

Use a graphing utility to graph the following curves. Be sure to choose an interval for the parameter that generates all features of interest. $$\text { Folium of Descartes } x=\frac{3 t}{1+t^{3}}, y=\frac{3 t^{2}}{1+t^{3}}$$

Use a graphing utility to determine the first three points with \(\theta \geq 0\) at which the spiral \(r=2 \theta\) has a horizontal tangent line. Find the first three points with \(\theta \geq 0\) at which the spiral \(r=2 \theta\) has a vertical tangent line.

Use a graphing utility to graph the following curves. Be sure to choose an interval for the parameter that generates all features of interest. $$\text { Involute of a circle } x=\cos t+t \sin t, y=\sin t-t \cos t$$

Assume \(m\) is a positive integer. a. Even number of leaves: What is the relationship between the total area enclosed by the \(4 m\) -leaf rose \(r=\cos (2 m \theta)\) and \(m ?\) b. Odd number of leaves: What is the relationship between the total area enclosed by the \((2 m+1)\) -leaf rose \(r=\cos ((2 m+1) \theta)\) and \(m ?\)

Find an equation of the following curves, assuming the center is at the origin. Sketch a graph labeling the vertices, foci, asymptotes (if they exist), and directrices. Use a graphing utility to check your work. A hyperbola with vertices (±1,0) and eccentricity 3

Use a graphing utility to graph the following equations. In each case, give the smallest interval \([0, P]\) that generates the entire curve. \(r=\sin \frac{\theta}{4}\)

Use a graphing utility to graph the following curves. Be sure to choose an interval for the parameter that generates all features of interest. Evolute of an ellipse \(x=\frac{a^{2}-b^{2}}{a} \cos ^{3} t, y=\frac{a^{2}-b^{2}}{b} \sin ^{3} t\) \(a=4\) and \(b=3.\)

Find an equation of the following curves, assuming the center is at the origin. Sketch a graph labeling the vertices, foci, asymptotes (if they exist), and directrices. Use a graphing utility to check your work. A hyperbola with vertices (0,±4) and eccentricity 2

Use a graphing utility to graph the following curves. Be sure to choose an interval for the parameter that generates all features of interest. $$\text { Cissoid of Diocles } x=2 \sin 2 t, y=\frac{2 \sin ^{3} t}{\cos t}$$

Find the area of the regions bounded by the following curves. The complete three-leaf rose \(r=2 \cos 3 \theta\)

Find the area of the regions bounded by the following curves. \text { The lemniscate } r^{2}=6 \sin 2 \theta

Use a graphing utility to graph the following equations. In each case, give the smallest interval \([0, P]\) that generates the entire curve. \(r=2 \sin \frac{2 \theta}{3}\)

Consider the family of curves $$\begin{aligned} &x=\left(2+\frac{1}{2} \sin a t\right) \cos \left(t+\frac{\sin b t}{c}\right).\\\ &y=\left(2+\frac{1}{2} \sin a t\right) \sin \left(t+\frac{\sin b t}{c}\right) \end{aligned}.$$ Plot the curve for the given values of \(a, b,\) and \(c\) with \(0 \leq t \leq 2 \pi\) (Source: Mathematica in Action, Stan Wagon, Springer, 2010 ; created by Norton Starr, Amherst College). $$a=6, b=12, c=3$$

Find the area of the regions bounded by the following curves. \text { The limaçon } r=2-4 \sin \theta

Graph the following conic sections, labeling the vertices, foci, directrices, and asymptotes (if they exist). Use a graphing utility to check your work. $$r=\frac{1}{2-\cos \theta}$$

Use a graphing utility to graph the following equations. In each case, give the smallest interval \([0, P]\) that generates the entire curve. \(r=\cos \frac{3 \theta}{5}\)

Consider the family of curves $$\begin{aligned} &x=\left(2+\frac{1}{2} \sin a t\right) \cos \left(t+\frac{\sin b t}{c}\right).\\\ &y=\left(2+\frac{1}{2} \sin a t\right) \sin \left(t+\frac{\sin b t}{c}\right) \end{aligned}.$$ Plot the curve for the given values of \(a, b,\) and \(c\) with \(0 \leq t \leq 2 \pi\) (Source: Mathematica in Action, Stan Wagon, Springer, 2010 ; created by Norton Starr, Amherst College). $$a=18, b=18, c=7$$

Find the area of the regions bounded by the following curves. \text { The limaçon } r=4-2 \cos \theta

Graph the following conic sections, labeling the vertices, foci, directrices, and asymptotes (if they exist). Use a graphing utility to check your work. $$r=\frac{6}{3+2 \sin \theta}$$

Use a graphing utility to graph the following equations. In each case, give the smallest interval \([0, P]\) that generates the entire curve. \(r=\sin \frac{3 \theta}{7}\)

Consider the family of curves $$\begin{aligned} &x=\left(2+\frac{1}{2} \sin a t\right) \cos \left(t+\frac{\sin b t}{c}\right).\\\ &y=\left(2+\frac{1}{2} \sin a t\right) \sin \left(t+\frac{\sin b t}{c}\right) \end{aligned}.$$ Plot the curve for the given values of \(a, b,\) and \(c\) with \(0 \leq t \leq 2 \pi\) (Source: Mathematica in Action, Stan Wagon, Springer, 2010 ; created by Norton Starr, Amherst College). $$a=7, b=4, c=1$$

A blood vessel with a circular cross section of constant radius \(R\) carries blood that flows parallel to the axis of the vessel with a velocity of \(v(r)=V\left(1-r^{2} / R^{2}\right),\) where \(V\) is a constant and \(r\) is the distance from the axis of the vessel. a. Where is the velocity a maximum? A minimum? b. Find the average velocity of the blood over a cross section of the vessel. c. Suppose the velocity in the vessel is given by \(v(r)=V\left(1-r^{2} / R^{2}\right)^{1 / p},\) where \(p \geq 1 .\) Graph the velocity profiles for \(p=1,2,\) and 6 on the interval \(0 \leq r \leq R .\) Find the average velocity in the vessel as a function of \(p .\) How does the average velocity behave as \(p \rightarrow \infty ?\)

Graph the following conic sections, labeling the vertices, foci, directrices, and asymptotes (if they exist). Use a graphing utility to check your work. $$r=\frac{1}{2-2 \sin \theta}$$

Consider the following parametric curves. a. Determine \(d y / d x\) in terms of \(t\) and evaluate it at the given value of \(t.\) b. Make a sketch of the curve showing the tangent line at the point corresponding to the given value of \(t.\) $$x=2+4 t, y=4-8 t ; t=2$$

Use a graphing utility to graph the following equations. In each case, give the smallest interval \([0, P]\) that generates the entire curve. \(r=1-2 \sin 5 \theta\)

Consider the following sequence of problems related to grazing goats tied to a rope. (See the Guided Project Grazing goat problems.) A circular concrete slab of unit radius is surrounded by grass. A goat is tied to the edge of the slab with a rope of length \(0 \leq a \leq 2\) (see figure). What is the area of the grassy region that the goat can graze? Note that the rope can extend over the concrete slab. Check your answer with the special cases \(a=0\) and \(a=2\).

Graph the following equations. Then use arrows and labeled points to indicate how the curve is generated as \(\theta\) increases from 0 to \(2 \pi\). $$r=\frac{1}{1+\sin \theta}$$

Determine whether the following statements are true and give an explanation or counterexample. a. The point with Cartesian coordinates (-2,2) has polar coordinates \((2 \sqrt{2}, 3 \pi / 4),(2 \sqrt{2}, 11 \pi / 4),(2 \sqrt{2},-5 \pi / 4),\) and \((-2 \sqrt{2},-\pi / 4)\) b. The graphs of \(r \cos \theta=4\) and \(r \sin \theta=-2\) intersect exactly once. c. The graphs of \(r=2\) and \(\theta=\pi / 4\) intersect exactly once. d. The point \((3, \pi / 2)\) lies on the graph of \(r=3 \cos 2 \theta\). e. The graphs of \(r=2 \sec \theta\) and \(r=3 \csc \theta\) are lines.