Chapter 10

Find equations of the tangents to the curve \(x=3 t^{2}+1\) \(y=2 t^{3}+1\) that pass through the point \((4,3)\)

\(29-34\) Find the area of the region that lies inside both curves. $$ r=1+\cos \theta, \quad r=1-\cos \theta $$

\(29-48\) Sketch the curve with the given polar equation. $$r^{2}-3 r+2=0$$

Graph the curves \(y=x^{5}\) and \(x=y(y-1)^{2}\) and find their points of intersection correct to one decimal place.

Find an equation for the conic that satisfies the given conditions. Parabola, vertex \((0,0), \quad\) focus \((0,-2)\)

Use the parametric equations of an ellipse, \(x=a \cos \theta\) \(y=b \sin \theta, 0 \leqslant \theta \leqslant 2 \pi,\) to find the area that it encloses.

\(29-34\) Find the area of the region that lies inside both curves. $$ r=\sin 2 \theta, \quad r=\cos 2 \theta $$

\(29-48\) Sketch the curve with the given polar equation. $$r=\sin \theta$$

(a) Show that the parametric equations $$x=x_{1}+\left(x_{2}-x_{1}\right) t \quad y=y_{1}+\left(y_{2}-y_{1}\right) t$$ where \(0 \leqslant t \leqslant 1,\) describe the line segment that joins the points \(P_{1}\left(x_{1}, y_{1}\right)\) and \(P_{2}\left(x_{2}, y_{2}\right).\) (b) Find parametric equations to represent the line segment from \((-2,7)\) to \((3,-1)\).

Find an equation for the conic that satisfies the given conditions. Parabola, vertex \((1,0), \quad\) directrix \(x=-5\)

\(29-48\) Sketch the curve with the given polar equation. $$r=-3 \cos \theta$$

Find an equation for the conic that satisfies the given conditions.\(y^{2}=-12(x+1)\) Parabola, focus \((-4,0), \quad\) directrix \(x=2\)

\(29-48\) Sketch the curve with the given polar equation. $$r=2(1-\sin \theta), \theta \geqslant 0$$

Find parametric equations for the path of a particle that moves along the circle \(x^{2}+(y-1)^{2}=4\) in the manner described. (a) Once around clockwise, starting at \((2,1)\) (b) Three times around counterclockwise, starting at \((2,1)\) (c) Halfway around counterclockwise, starting at \((0,3)\)

Find an equation for the conic that satisfies the given conditions. Parabola, focus \((3,6), \quad\) vertex \((3,2)\)

Find the area of the region enclosed by the astroid \(x=a \cos ^{3} \theta, y=a \sin ^{3} \theta .\) (Astroids are explored in the Laboratory Project on page \(629 .\) )

\(29-34\) Find the area of the region that lies inside both curves. $$ r=a \sin \theta, \quad r=b \cos \theta, \quad a>0, b>0 $$

\(29-48\) Sketch the curve with the given polar equation. $$r=1-3 \cos \theta$$

(a) Find parametric equations for the ellipse \(x^{2} / a^{2}+y^{2} / b^{2}=1 .\) [Hint: Modify the equations of the circle in Example \(2 . ]\) (b) Use these parametric equations to graph the ellipse when \(a=3\) and \(b=1,2,4,\) and \(8 .\) (c) How does the shape of the ellipse change as \(b\) varies?

Find an equation for the conic that satisfies the given conditions. Parabola, vertex \((2,3), \quad\) vertical axis, passing through \((1,5)\)

\(29-48\) Sketch the curve with the given polar equation. $$r=\theta, \quad \theta \geqslant 0$$

Find an equation for the conic that satisfies the given conditions. Parabola, horizontal axis, passing through \((-1,0),(1,-1),\) and \((3,1)\)

\(29-48\) Sketch the curve with the given polar equation. $$r=\ln \theta, \quad \theta \geqslant 1$$

Find an equation for the conic that satisfies the given conditions. Ellipse, foci \((\pm 2,0), \quad\) vertices \((\pm 5,0)\)

Set up an integral that represents the length of the curve. Then use your calculator to find the length correct to four decimal places. $$x=t-t^{2}, \quad y=\frac{4}{3} t^{3 / 2}, \quad 1 \leqslant t \leqslant 2 $$

\(37-42\) Find all points of intersection of the given curves. $$ r=1+\sin \theta, \quad r=3 \sin \theta $$

\(29-48\) Sketch the curve with the given polar equation. $$r=4 \sin 3 \theta$$

Compare the curves represented by the parametric equations. How do they differ? (a) \(x=t^{3}, \quad y=t^{2}\) (b) \(\quad x=t^{6}, \quad y=t^{4}\) (c) \(x=e^{-3 t}, \quad y=e^{-2 t}\)

Find an equation for the conic that satisfies the given conditions. Ellipse, foci \((0, \pm 5), \quad\) vertices \((0, \pm 13)\)

Set up an integral that represents the length of the curve. Then use your calculator to find the length correct to four decimal places. $$x=1+e^{t}, \quad y=t^{2}, \quad-3 \leq t \leqslant 3$$

\(37-42\) Find all points of intersection of the given curves. $$ r=1-\cos \theta, \quad r=1+\sin \theta $$

\(29-48\) Sketch the curve with the given polar equation. $$r=\cos 5 \theta$$

Compare the curves represented by the parametric equations. How do they differ? (a) \(x=t, \quad y=t^{-2}\) (b) \(x=\cos t, \quad y=\sec ^{2} t\) (c) \(x=e^{t}, \quad y=e^{-2 t}\)

Find an equation for the conic that satisfies the given conditions. Ellipse, foci\((0,2),(0,6), \quad\) vertices \((0,0),(0,8)\)

\(37-42\) Find all points of intersection of the given curves. $$ r=2 \sin 2 \theta, \quad r=1 $$

\(29-48\) Sketch the curve with the given polar equation. $$r=2 \cos 4 \theta$$

Find an equation for the conic that satisfies the given conditions. Ellipse, foci \((0,-1),(8,-1), \quad\) vertex \((9,-1)\)

Set up an integral that represents the length of the curve. Then use your calculator to find the length correct to four decimal places. $$x=\ln t, \quad y=\sqrt{t+1}, \quad 1 \leq t \leq 5$$

\(37-42\) Find all points of intersection of the given curves. $$ r=\cos 3 \theta, \quad r=\sin 3 \theta $$

\(29-48\) Sketch the curve with the given polar equation. $$r=3 \cos 6 \theta$$

Let \(P\) be a point at a distance \(d\) from the center of a circle of radius \(r .\) The curve traced out by \(P\) as the circle rolls along a straight line is called a trochoid. (Think of the motion of a point on a spoke of a bicycle wheel.) The cycloid is the special case of a trochoid with \(d=r .\) Using the same parameter \(\theta\) as for the cycloid and, assuming the line is the \(x\) -axis and \(\theta=0\) when \(P\) is at one of its lowest points, show that parametric equations of the trochoid are $$x=r \theta-d \sin \theta \quad y=r-d \cos \theta$$ Sketch the trochoid for the cases \(d < r\) and \(d > r.\)

Find an equation for the conic that satisfies the given conditions. Ellipse, center \((-1,4), \quad\) vertex \((-1,0),\) focus \((-1,6)\)

Find the exact length of the curve. $$x=1+3 t^{2}, \quad y=4+2 t^{3}, \quad 0 \leqslant t \leqslant 1$$

\(37-42\) Find all points of intersection of the given curves. $$ r=\sin \theta, \quad r=\sin 2 \theta $$

\(29-48\) Sketch the curve with the given polar equation. $$r=1-2 \sin \theta$$

Find an equation for the conic that satisfies the given conditions. Ellipse, \(\quad\) foci \((\pm 4,0), \quad\) passing through \((-4,1.8)\)

\(37-42\) Find all points of intersection of the given curves. $$ r^{2}=\sin 2 \theta, \quad r^{2}=\cos 2 \theta $$

\(29-48\) Sketch the curve with the given polar equation. $$r=2+\sin \theta$$

Find an equation for the conic that satisfies the given conditions. Hyperbola, vertices \((\pm 3,0), \quad\) foci \((\pm 5,0)\)

Find the exact length of the curve. $$x=\frac{t}{1+t^{\prime}}, \quad y=\ln (1+t), \quad 0 \leqslant t \leqslant 2$$