Chapter 10

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 \(t.\) $$x=\cos t, y=8 \sin t ; t=\pi / 2$$

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

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 \(t.\) $$x=\cos t, y=8 \sin t ; t=\pi / 2$$

Let a polar curve be described by \(r=f(\theta)\) and let \(\ell\) be the line tangent to the curve at the point \(P(x, y)=P(r, \theta)\) (see figure). a. Explain why \(\tan \alpha=\frac{d y}{d x}\). b. Explain why \(\tan \theta=y / x\). c. Let \(\varphi\) be the angle between \(\ell\) and the line through \(O\) and \(P\). Prove that \(\tan \varphi=f(\theta) / f^{\prime}(\theta)\). d. Prove that the values of \(\theta\) for which \(\ell\) is parallel to the \(x\) -axis satisfy \(\tan \theta=-f(\theta) / f^{\prime}(\theta)\). e. Prove that the values of \(\theta\) for which \(\ell\) is parallel to the \(y\) -axis satisfy \(\tan \theta=f^{\prime}(\theta) / f(\theta)\).

Graph the following equations. Then use arrows and labeled points to indicate how the curve is generated as \(\theta\) increases from 0 to \(2 \pi\). $$r=\frac{3}{1-\cos \theta}$$

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 \(t.\) $$x=t+1 / t, y=t-1 / t ; t=1$$

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

Convert the following equations to polar coordinates. \((x-1)^{2}+y^{2}=1\)

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 \(t.\) $$x=\sqrt{t}, y=2 t ; t=4$$

Use a graphing utility to graph the parabolas \(y^{2}=4 p x,\) for \(p=-5,-2,-1,1,2,\) and 5 on the same set of axes. Explain how the shapes of the curves vary as \(p\) changes.

Determine whether the following statements are true and give an explanation or counterexample. a. The equations \(x=-\cos t, y=-\sin t,\) for \(0 \leq t \leq 2 \pi,\) generate a circle in the clockwise direction. b. An object following the parametric curve \(x=2 \cos 2 \pi t\) \(y=2 \sin 2 \pi t\) circles the origin once every 1 time unit. c. The parametric equations \(x=t, y=t^{2},\) for \(t \geq 0,\) describe the complete parabola \(y=x^{2}\) d. The parametric equations \(x=\cos t, y=\sin t,\) for \(-\pi / 2 \leq t \leq \pi / 2,\) describe a semicircle. e. There are two points on the curve \(x=-4 \cos t, y=\sin t,\) for \(0 \leq t \leq 2 \pi,\) at which there is a vertical tangent line.

Use a graphing utility to graph the hyperbolas \(r=\frac{e}{1+e \cos \theta},\) for \(e=1.1,1.3,1.5,1.7\) and 2 on the same set of axes. Explain how the shapes of the curves vary as \(e\) changes.

Find an equation of the line tangent to the curve at the point corresponding to the given value of \(t.\) $$x=\sin t, y=\cos t ; t=\pi / 4$$

Explain why or why not Determine whether the following statements are true and give an explanation or counterexample. a. The hyperbola \(x^{2} / 4-y^{2} / 9=1\) has no \(y\) -intercepts. b. On every ellipse, there are exactly two points at which the curve has slope \(s,\) where \(s\) is any real number. c. Given the directrices and foci of a standard hyperbola, it is possible to find its vertices, eccentricity, and asymptotes. d. The point on a parabola closest to the focus is the vertex.

Sketch the following sets of points \((r, \theta)\). \(\theta=\frac{2 \pi}{3}\)

Find an equation of the line tangent to the curve at the point corresponding to the given value of \(t.\) $$x=t^{2}-1, y=t^{3}+t ; t=2$$

Find an equation of the line tangent to the curve at the point corresponding to the given value of \(t.\) $$x=e^{t}, y=\ln (t+1) ; t=0$$

Sketch the following sets of points \((r, \theta)\). \(2 \leq r \leq 8\)

Find an equation of the line tangent to the following curves at the given point. $$x^{2}=-6 y ;(-6,-6)$$

Find an equation of the line tangent to the curve at the point corresponding to the given value of \(t.\) $$x=\cos t+t \sin t, y=\sin t-t \cos t ; t=\pi / 4$$

Sketch the following sets of points \((r, \theta)\). \(\frac{\pi}{2} \leq \theta \leq \frac{3 \pi}{4}\)

Find an equation of the line tangent to the following curves at the given point. $$r=\frac{1}{1+\sin \theta} ;\left(\frac{2}{3}, \frac{\pi}{6}\right)$$

Sketch the following sets of points \((r, \theta)\). \(1

Find an equation of the line tangent to the following curves at the given point. $$y^{2}-\frac{x^{2}}{64}=1 ;\left(6,-\frac{5}{4}\right)$$

Find parametric equations for the following curves. Include an interval for the parameter values. Answers are not unique. The line that passes through the points (1,1) and \((3,5),\) oriented in the direction of increasing \(x\)

Sketch the following sets of points \((r, \theta)\). \(|\theta| \leq \frac{\pi}{3}\)

Find a polar equation for each conic section. Assume one focus is at the origin.

Find parametric equations for the following curves. Include an interval for the parameter values. Answers are not unique. The lower half of the circle centered at \((-2,2)\) with radius 6 oriented in the counterclockwise direction

Sketch the following sets of points \((r, \theta)\). \(0

Find a polar equation for each conic section. Assume one focus is at the origin.

Find parametric equations for the following curves. Include an interval for the parameter values. Answers are not unique. The upper half of the parabola \(x=y^{2}\), originating at \((0,0)\)

Circles in general $$r^{2}-2 r(a \cos \theta+b \sin \theta)=R^{2}-a^{2}-b^{2}$$ describes a circle of radius \(R\) centered at \((a, b)\).

Suppose two circles, whose centers are at least \(2 a\) units apart (see figure), are centered at \(F_{1}\) and \(F_{2},\) respectively. The radius of one circle is \(2 a+r\) and the radius of the other circle is \(r,\) where \(r \geq 0 .\) Show that as \(r\) increases, the intersection point \(P\) of the two circles describes one branch of a hyperbola with foci at \(F_{1}\) and \(F_{2}\)

An ellipse (discussed in detail in Section 10.4 ) is generated by the parametric equations \(x=a \cos t, y=b \sin t.\) If \(0 < a < b,\) then the long axis (or major axis) lies on the \(y\) -axis and the short axis (or minor axis) lies on the \(x\) -axis. If \(0 < b < a,\) the axes are reversed. The lengths of the axes in the \(x\) - and \(y\) -directions are \(2 a\) and \(2 b,\) respectively. Sketch the graph of the following ellipses. Specify an interval in t over which the entire curve is generated. $$x=4 \cos t, y=9 \sin t$$

Show that the polar equation $$r^{2}-2 r r_{0} \cos \left(\theta-\theta_{0}\right)=R^{2}-r_{0}^{2}$$ describes a circle of radius \(R\) whose center has polar coordinates \(\left(r_{0}, \theta_{0}\right)\).

Let \(R\) be the region bounded by the upper half of the ellipse \(x^{2} / 2+y^{2}=1\) and the parabola \(y=x^{2} / \sqrt{2}\) a. Find the area of \(R\). b. Which is greater, the volume of the solid generated when \(R\) is revolved about the \(x\) -axis or the volume of the solid generated when \(R\) is revolved about the \(y\) -axis?

Show that an equation of the line tangent to the ellipse \(x^{2} / a^{2}+y^{2} / b^{2}=1\) at the point \(\left(x_{0}, y_{0}\right)\) is $$ \frac{x x_{0}}{a^{2}}+\frac{y y_{0}}{b^{2}}=1 $$

Find parametric equations (not unique) of the following ellipses (see Exercises \(75-76\) ). Graph the ellipse and find a description in terms of \(x\) and \(y.\) An ellipse centered at the origin with major axis of length 6 on the \(x\) -axis and minor axis of length 3 on the \(y\) -axis, generated counterclockwise

Find an equation of the line tangent to the hyperbola \(x^{2} / a^{2}-y^{2} / b^{2}=1\) at the point \(\left(x_{0}, y_{0}\right)\)

Find parametric equations (not unique) of the following ellipses (see Exercises \(75-76\) ). Graph the ellipse and find a description in terms of \(x\) and \(y.\) An ellipse centered at the origin with major and minor axes of lengths 12 and \(2,\) on the \(x\) - and \(y\) -axes, respectively, generated clockwise

Suppose that the ellipse \(x^{2} / a^{2}+y^{2} / b^{2}=1\) is revolved about the \(x\) -axis. What is the volume of the solid enclosed by the ellipsoid that is generated? Is the volume different if the same ellipse is revolved about the \(y\) -axis?

Find parametric equations (not unique) of the following ellipses (see Exercises \(75-76\) ). Graph the ellipse and find a description in terms of \(x\) and \(y.\) An ellipse centered at (-2,-3) with major and minor axes of lengths 30 and \(20,\) parallel to the \(x\) - and \(y\) -axes, respectively, generated counterclockwise (Hint: Shift the parametric equations.)

Find parametric equations (not unique) of the following ellipses (see Exercises \(75-76\) ). Graph the ellipse and find a description in terms of \(x\) and \(y.\) An ellipse centered at (0,-4) with major and minor axes of lengths 10 and \(3,\) parallel to the \(x\) - and \(y\) -axes, respectively, generated clockwise (Hint: Shift the parametric equations.)

Consider the region \(R\) bounded by the right branch of the hyperbola \(x^{2} / a^{2}-y^{2} / b^{2}=1\) and the vertical line through the right focus. a. What is the volume of the solid that is generated when \(R\) is revolved about the \(x\) -axis? b. What is the volume of the solid that is generated when \(R\) is revolved about the \(y\) -axis?

Consider the following pairs of lines. Determine whether the lines are parallel or intersecting. If the lines intersect, then determine the point of intersection. a. \(x=1+s, y=2 s\) and \(x=1+2 t, y=3 t\) b. \(x=2+5 s, y=1+s\) and \(x=4+10 t, y=3+2 t\) c. \(x=1+3 s, y=4+2 s\) and \(x=4-3 t, y=6+4 t\)

The region bounded by the parabola \(y=a x^{2}\) and the horizontal line \(y=h\) is revolved about the \(y\) -axis to generate a solid bounded by a surface called a paraboloid (where \(a>0\) and \(h>0\) ). Show that the volume of the solid is \(\frac{3}{2}\) the volume of the cone with the same base and vertex.

Which of the following parametric equations describe the same curve? a. \(x=2 t^{2}, y=4+t ;-4 \leq t \leq 4\) b. \(x=2 t^{4}, y=4+t^{2} ;-2 \leq t \leq 2\) c. \(x=2 t^{2 / 3}, y=4+t^{1 / 3} ;-64 \leq t \leq 64\)

Eliminate the parameter to express the following parametric equations as a single equation in \(x\) and \(y.\) $$x=2 \sin 8 t, y=2 \cos 8 t$$

Consider the polar curve \(r=2 \sec \theta\). a. Graph the curve on the intervals \((\pi / 2,3 \pi / 2),(3 \pi / 2,5 \pi / 2)\) and \((5 \pi / 2,7 \pi / 2) .\) In each case, state the direction in which the curve is generated as \(\theta\) increases. b. Show that on any interval \((n \pi / 2,(n+2) \pi / 2),\) where \(n\) is an odd integer, the graph is the vertical line \(x=2\).

Completed in 1937, San Francisco's Golden Gate Bridge is \(2.7 \mathrm{km}\) long and weighs about 890,000 tons. The length of the span between the two central towers is \(1280 \mathrm{m}\) the towers themselves extend \(152 \mathrm{m}\) above the roadway. The cables that support the deck of the bridge between the two towers hang in a parabola (see figure). Assuming the origin is midway between the towers on the deck of the bridge, find an equation that describes the cables. How long is a guy wire that hangs vertically from the cables to the roadway \(500 \mathrm{m}\) from the center of the bridge?