?Determine whether the statement is true or false. If it is true, explain why

(a) What is a parametric curve? (b) How do you sketch a parametric curve?

(a) How do you find the slope of a tangent to a parametric curve? (b) How do you find the area under a parametric curve?

Write an expression for each of the following: (a) The length of a parametric curve (b) The area of the surface obtained by rotating a parametric curve about the x-axis (c) The speed of a particle traveling along a parametric curve

(a) Use a diagram to explain the meaning of the polar coordinates of a point. (b) Write equations that express the Cartesian coordinates of a point in terms of the polar coordinates. (c) What equations would you use to find the polar coordinates of a point if you knew the Cartesian coordinates?

(a) How do you find the area of a region bounded by a polar curve? (b) How do you find the length of a polar curve? (c) How do you find the slope of a tangent line to a polar curve?

(a) Give a geometric definition of a parabola. (b) Write an equation of a parabola with focus and directrix What if the focus is and the directrix is

a) Give a definition of an ellipse in terms of foci. (b) Write an equation for the ellipse with foci and vertices

(a) Give a definition of a hyperbola in terms of foci. (b) Write an equation for the hyperbola with foci \((\pm c, 0)\) and vertices \((\pm a, 0)\). (c) Write equations for the asymptotes of the hyperbola in part (b).

(a) What is the eccentricity of a conic section? (b) What can you say about the eccentricity if the conic section is an ellipse? A hyperbola? A parabola? (c) Write a polar equation for a conic section with eccentricity e and directrix x = d. What if the directrix is \(x=-d\)? \(y=d\)? \(y=-d\)?

Determine whether the statement is true or false. If it is true, explain why. If it is false, explain why or give an example that disproves the statement. If the parametric curve satisfies , then it has a horizontal tangent when .

Determine whether the statement is true or false. If it is true, explain why. If it is false, explain why or give an example that disproves the statement. If and are twice differentiable, then

Determine whether the statement is true or false. If it is true, explain why. If it is false, explain why or give an example that disproves the statement. The length of the curve , is

Determine whether the statement is true or false. If it is true, explain why. If it is false, explain why or give an example that disproves the statement. If the position of a particle at time is given by the parametric equations , then the speed of the particle at time is the value of when .

Determine whether the statement is true or false. If it is true, explain why. If it is false, explain why or give an example that disproves the statement. If a point is represented by (x, y) in Cartesian coordinates (where \(x \neq 0\) ) and \((r, \theta)\) in polar coordinates, then \(\theta=\tan ^{-1}(y / x)\)

Determine whether the statement is true or false. If it is true, explain why. If it is false, explain why or give an example that disproves the statement. The polar curves \(r=1-\sin 2 \theta \quad r=\sin 2 \theta-1\) have the same graph. Equation Transcription: Text Transcription: r=1 - sin 2theta r=sin 2theta -1

Determine whether the statement is true or false. If it is true, explain why. If it is false, explain why or give an example that disproves the statement. The equations \(r=2, x^{2}+y^{2}=4\), and \(x=2 \sin 3 t, \quad y=2 \cos 3 t(0 \leq t \leq 2 \pi)\) all have the same graph. Equation Transcription: Text Transcription: r=2, x^2 + y^2 =4 x= 2 sin 3t, y= 2 cos 3t(0 < or = t < or = 2pi)

Determine whether the statement is true or false. If it is true, explain why. If it is false, explain why or give an example that disproves the statement. The parametric equations have the same graph as .

Determine whether the statement is true or false. If it is true, explain why. If it is false, explain why or give an example that disproves the statement. The graph of is a parabola.

Determine whether the statement is true or false. If it is true, explain why. If it is false, explain why or give an example that disproves the statement. A tangent line to a parabola intersects the parabola only once.

Determine whether the statement is true or false. If it is true, explain why. If it is false, explain why or give an example that disproves the statement. A hyperbola never intersects its directrix.

Sketch the parametric curve and eliminate the parameter to find a Cartesian equation of the curve. \(x=t^2+4 t, \quad y=2-t, \quad-4 \leqslant t \leqslant 1\)

(a) Find the highest and lowest points on the curve \(x^4+y^4=x^2+y^2\). (b) Sketch the curve. (Notice that it is symmetric with respect to both axes and both of the lines \(y=\pm x\), so it suffices to consider \(y \geqslant x \geqslant 0\) initially.) (c) Use polar coordinates and a computer algebra system to find the area enclosed by the curve.

Sketch the parametric curve and eliminate the parameter to find a Cartesian equation of the curve. \(x=\ln t, \quad y=t^{2}\) Equation Transcription: Text Transcription: x=ln t, y=t^2

Sketch the parametric curve and eliminate the parameter to find a Cartesian equation of the curve. \(x=2 \cos \theta, \ \ y=1+\sin \theta\)

Sketch the parametric curve and eliminate the parameter to find a Cartesian equation of the curve. \(x=\cos \theta, \quad y=\sec \theta, \quad 0 \leq \theta<\pi / 2\) Equation Transcription: Text Transcription: x= cos theta, y=sec theta, 0 < or = theta < pi/2

Describe the motion of a particle with position \((x,y)\), where \(x=2+4 \cos \pi t\) and \(y=-3+4 \sin \pi t\), as \(t\) increases from 0 to 4 . Equation Transcription: Text Transcription: (x,y) x=2 +4 cos pi t y= -3 + 4 sin pi t t

Write three different sets of parametric equations for the curve \(y=\sqrt{x}\). Equation Transcription: Text Transcription: y=sqrt x

Use the graphs of \(x=f(t)\) and \(y=g(t)\) to sketch the parametric curve \(x=f(t), y=g(t)\). Indicate with arrows the direction in which the curve is traced as $t$ increases. Equation Transcription: Text Transcription: x=f(t) y=g(t) x=f(t), y=g(t) t

(a) Plot the point with polar coordinates \((4,2 \pi / 3)\). Then find its Cartesian coordinates. (b) The Cartesian coordinates of a point are \((-3,3)\). Find two sets of polar coordinates for the point. Equation Transcription: Text Transcription: (4,2pi/3) (-3,3)

Sketch the region consisting of points whose polar coordinates satisfy \(1 \leqslant r<2\) and \(\pi / 6 \leqslant \theta \leqslant 5 \pi / 6\). Equation Transcription: ? ? ? ? Text Transcription: 1 leqslant r leqslant 2 pi/6 leqslant theta leqslant 5pi/6

Sketch the polar curve. \(r=1+\sin \theta\) Equation Transcription: Text Transcription: r=1+sin theta

Sketch the polar curve. \(r=\sin 4 \theta\)

Sketch the polar curve. \(r=3+\cos 3 \theta\) Equation Transcription: Text Transcription: r=3+cos 3theta

Sketch the polar curve.

Sketch the polar curve. \(r=1+\cos 2 \theta\) Equation Transcription: Text Transcription: r=1+cos 2theta

Sketch the polar curve. \(r=2 \cos (\theta / 2)\) Equation Transcription: Text Transcription: r=2 cos (theta/2)

Sketch the polar curve. \(r=\frac{3}{1+2 \sin \theta}\) Equation Transcription: Text Transcription: r=3/1+2 sin theta

Sketch the polar curve. \(r=\frac{3}{2-2 \cos \theta}\)

Find a polar equation for the curve represented by the given Cartesian equation. x+y=2

Find a polar equation for the curve represented by the given Cartesian equation. \(x^{2}+y^{2}=2\)

The curve with polar equation \(r=\sin \theta / \theta\) is called a cochleoid. Use a graph of \(r\) as a function of \(\theta\) in Cartesian coordinates to sketch the cochleoid by hand. Then graph it with a calculator or computer to check your sketch. Equation Transcription: Text Transcription: r= sin theta /theta r theta

The figure shows a graph of \(r\) as a function of \(\theta\) in Cartesian coordinates. Use it to sketch the corresponding polar curve. Equation Transcription: Text Transcription: r theta

Find the slope of the tangent line to the given curve at the point corresponding to the specified value of the parameter. \(x=\ln t, \quad y=1+t^{2} ; \quad t=1\) Equation Transcription: Text Transcription: x=ln t, y=1 +t^2; t=1

Find the slope of the tangent line to the given curve at the point corresponding to the specified value of the parameter. \(x=t^{3}+6 t+1, \quad y=2 t-t^{2} ; \quad t=-1\) Equation Transcription: Text Transcription: x=t^3+6t+1, y=2t-t^2; t=-1

Find the slope of the tangent line to the given curve at the point corresponding to the specified value of the parameter. \(r=e^{-\theta}; \ \ \theta=\pi\)

Find the slope of the tangent line to the given curve at the point corresponding to the specified value of the parameter. \(r=3+\cos 3 \theta ; \quad \theta=\pi / 2\) Equation Transcription: Text Transcription: r=3 + cos 3theta; theta =pi/2

Find \(d y / d x\) and \(d^{2} y / d x^{2}\). \(x=t+\sin t, \quad y=t-\cos t\) Equation Transcription: Text Transcription: dx/dy d^2y/dx^2 x=t + sint, y=t -cost

Find \(d y / d x\) and \(d^{2} y / d x^{2}\) \(x=1+t^{2}, \quad y=t-t^{3}\) Equation Transcription: Text Transcription: dy/dx d^2 y/dx^2 x=1+t^2, y=t-t^3

Use a graph to estimate the coordinates of the lowest point on the curve \(x=t^{3}-3 t, y=t^{2}+t+1\). Then use calculus to find the exact coordinates. Equation Transcription: Text Transcription: x=t^3-3t,y=t^2+t+1

Find the area enclosed by the loop of the curve in Exercise

At what points does the curve x=2a cos t - a cos 2t y=2a sin 2t have vertical or horizontal tangents? Use this information to help sketch the curve.

Find the area enclosed by the curve in Exercise 31.

Find the area enclosed by the curve \(r^{2}=9 \cos 5 \theta\). Equation Transcription: Text Transcription: r^2 =9 cos 5 theta

Find the area enclosed by the inner loop of the curve \(r=1-3 \sin \theta\). Equation Transcription: Text Transcription: r=1-3 sin theta

Find the points of intersection of the curves \(r=2\) and \(r=4 \cos \theta\). Equation Transcription: Text Transcription: r=2 r=4 cos theta

Find the points of intersection of the curves \(r=\cot \theta\) and \(r=2 \cos \theta\). Equation Transcription: Text Transcription: r = cot theta R = 2 cos theta

Find the area of the region that lies inside both of the circles \(r=2 \sin \theta\) and \(r=\sin \theta+\cos \theta\). Equation Transcription: Text Transcription: r=2 sin theta r=sin theta + cos theta

Find the area of the region that lies inside the curve \(r=2+\cos 2 \theta\) but outside the curve \(r=2+\sin \theta\).

Find the length of the curve. \(x=3 t^{2}, \ \ y=2 t^{3}, \ \ 0 \leqslant t \leqslant 2\)

Find the length of the curve. \(x=2+3 t, \quad y=\cosh 3 t, \quad 0 \leqslant t \leqslant 1\) Equation Transcription: ? ? Text Transcription: x=2+3t, y=cosh 3t, 0 leqslant t leqslant 1

Find the length of the curve. \(r=1 / \theta, \quad \pi \leqslant \theta \leqslant 2 \pi\) Equation Transcription: ? ? Text Transcription: r=1/theta ,pi leqslant theta leqslant 2pi

Find the length of the curve. \(r=\sin ^{3}(\theta / 3), \quad 0 \leqslant \theta \leqslant \pi\) Equation Transcription: ? ? Text Transcription: r=sin^3(theta/3), 0 leqslant theta leqslant pi

The position (in meters) of a particle at time \(t\) seconds is given by the parametric equations \(x=\frac{1}{2}\left(t^{2}+3\right) \quad y=5-\frac{1}{3} t^{3}\) (a) Find the speed of the particle at the point \((6,-4)\). (b) What is the average speed of the particle for \(0 \leqslant t \leqslant 8\)? Equation Transcription: Text Transcription: t x=1/2(t^02+3) y=5-1/3 t^3 (6,-4) 0<t<8

(a) Find the exact length of the portion of the curve shown in blue. (b) Find the area of the shaded region. \(r=2 \cos ^{2}(\theta / 2)\) Equation Transcription: Text Transcription: r=2 cos^2(theta/2)

Find the area of the surface obtained by rotating the given curve about the \(x\)-axis. \(x=4 \sqrt{t}\), \(y=\frac{t^{3}}{3}+\frac{1}{2 t^{2}}\), \(1 \leqslant t \leqslant 4\)

Find the area of the surface obtained by rotating the given curve about the \(x\)-axis. \(x=2+3 t, \quad y=\cosh 3 t, \quad 0 \leqslant t \leqslant 1\) Equation Transcription: ? ? Text Transcription: x-axis x=2+3t, y=cosh 3t, 0 leqslant t leqslant 1

The curves defined by the parametric equations \(x=\frac{t^{2}-c}{t^{2}+1} \quad y=\frac{t\left(t^{2}-c\right)}{t^{2}+1}\) are called strophoids (from a Greek word meaning "to turn or twist"). Investigate how these curves vary as \(c\) varies. Equation Transcription: Text Transcription: x=t^2-c/t^2+1 y=t(t^2-c)/t^2+1 c

A family of curves has polar equations \(r^{a}=|\sin 2 \theta|\) where \(a\) is a positive number. Investigate how the curves change as \(a\) changes. Equation Transcription: Text Transcription: r^a =|sin2theta| a a

Find the foci and vertices and sketch the graph. \(\frac{x^{2}}{9}+\frac{y^{2}}{8}=1\) Equation Transcription: Text Transcription: x^2/9+y^2/8=1

Find the foci and vertices and sketch the graph. \(4 x^{2}-y^{2}=16\) Equation Transcription: Text Transcription: 4x^2-y^2=16

Find the foci and vertices and sketch the graph. \(6 y^{2}+x-36 y+55=0\) Equation Transcription: Text Transcription: 6y^2+x-36y+55=0

Find the foci and vertices and sketch the graph. \(25 x^{2}+4 y^{2}+50 x-16 y=59\) Equation Transcription: Text Transcription: 25x^2+4y^2+50x-16y=59

Find an equation of the ellipse with foci \((\pm 4,0)\) and vertices \((\pm 5,0)\) Equation Transcription: Text Transcription: (pm 4,0) (pm 5,0)

Find an equation of the parabola with focus \((2,1)\) and directrix \(x=-4\). Equation Transcription: Text Transcription: (2,1) X = -4

Find an equation of the hyperbola with foci \((0, \pm 4)\) and asymptotes \(y=\pm 3 x\). Equation Transcription: Text Transcription: (0,pm4) y=pm3x

Find an equation of the ellipse with foci \((3, \pm 2)\) and major axis with length 8 .

Find an equation for the ellipse that shares a vertex and a focus with the parabola \(x^{2}+y=100\) and that has its other focus at the origin. Equation Transcription: Text Transcription: x^2+y=100

Show that if m is any real number, then there are exactly two lines of slope m that are tangent to the ellipse \(x^{2} / a^{2}+y^{2} / b^{2}=1\) and their equations are \(y=m x \pm \sqrt{a^{2} m^{2}+b^{2}}\)

Find a polar equation for the ellipse with focus at the origin, eccentricity \(\frac{1}{3}\) and directrix with equation \(r=4 \sec \theta\)

Graph the ellipse \(r=2 /(4-3 \cos \theta)\) and its directrix. Also graph the ellipse obtained by rotation about the origin through an angle \(2 \pi / 3\). Equation Transcription: Text Transcription: r=2/(4-3 cos theta) 2pi/3

Show that the angles between the polar axis and the asymptotes of the hyperbola \(r=e d /(1-e \cos \theta), \ e>1\), are given by \(\cos ^{-1}(\pm 1 / e)\).

A curve called the folium of Descartes is defined by the parametric equations \(x=\frac{3 t}{1+t^{3}}\) \(\quad\) \(y=\frac{3 t^{2}}{1+t^{3}}\) (a) Show that if \((a, b)\) lies on the curve, then so does \((a, b)\); that is, the curve is symmetric with respect to the line \((y, x)\). Where does the curve intersect this line? (b) Find the points on the curve where the tangent lines are horizontal or vertical. (c) Show that the line \(y=-x-1\) is a slant asymptote. (d) Sketch the curve. (e) Show that a Cartesian equation of this curve is \(x^{3}+y^{3}=3 x y\). (f) Show that the polar equation can be written in the form \(r=\frac{3 \sec \theta \tan \theta}{1+\tan ^{3} \theta}\) (g) Find the area enclosed by the loop of this curve. (h) Show that the area of the loop is the same as the area that lies between the asymptote and the infinite branches of the curve. (Use a computer algebra system to evaluate the integral.) Equation Transcription: Text Transcription: x=3t/1+t^3 y=3t^2/1+t^3 (a,b) (b,a) (y,x) y=-x-1 x^3 +y^3 = 3xy R = 3sec theta tan theta/1+tan^3 theta

Sketch the parametric curve and eliminate the parameter to find a Cartesian equation of the curve. \(x=t^{2}+4 t, \quad y=2-t, \quad-4 \leqslant t \leqslant 1\) Equation Transcription: ? ? Text Transcription: x=t^2+4t, y=2-t, -4 leqslant t leqslant 1

(a) Find the highest and lowest points on the curve . (b) Sketch the curve. (Notice that it is symmetric with respect to both axes and both of the lines , so it suffices to consider initially.) (c) Use polar coordinates and a computer algebra system to find the area enclosed by the curve.

What is the smallest viewing rectangle that contains every member of the family of polar curves \(r=1+c \sin \theta\), where \(0 \leq c \leq 1\)? Illustrate your answer by graphing several members of the family in this viewing rectangle. Equation Transcription: Text Transcription: r=1+c sin theta 0 < or = c < or = 1

Four bugs are placed at the four corners of a square with side length . The bugs crawl counterclockwise at the same speed and each bug crawls directly toward the next bug at all times. They approach the center of the square along spiral paths. (a) Find a polar equation of a bug's path assuming the pole is at the center of the square. (Use the fact that the line joining one bug to the next is tangent to the bug's path.) (b) Find the distance traveled by a bug by the time it meets the other bugs at the center.

Show that any tangent line to a hyperbola touches the hyperbola halfway between the points of intersection of the tangent and the asymptotes.

A circle \(C\) of radius \(2r\) has its center at the origin. A circle of radius \(r\) rolls without slipping in the counterclockwise direction around \(C\) A point \(P\) is located on a fixed radius of the rolling circle at a distance \(b\) from its center, \(0<b<r\). [See parts (i) and (ii) of the figure below.] Let \(L\) be the line from the center of \(C\) to the center of the rolling circle and let \(\theta\) be the angle that \(L\) makes with the positive \(x\) -axis. (a) Using as a parameter, show that parametric equations of the path traced out by \(P\) are $$x=b \cos 3 \theta+3 r \cos \theta y=b \sin 3 \theta+3 r \sin \theta$$ Note: If \(b=0\), the path is a circle of radius \(3r\); if \(b=r\), the path is an epicycloid. The path traced out by \(P\) for \(0<b<r\) is called an epitrochoid. (b) Graph the curve for various values of \(b\) between 0 and \(r\). (c) Show that an equilateral triangle can be inscribed in the epitrochoid and that its centroid is on the circle of radius \(b\) centered at the origin. Note: This is the principle of the Wankel rotary engine. When the equilateral triangle rotates with its vertices on the epitrochoid, its centroid sweeps out a circle whose center is at the center of the curve. (d) In most rotary engines the sides of the equilateral triangles are replaced by arcs of circles centered at the opposite vertices as in part (iii) of the figure. (Then the diameter of the rotor is constant.) Show that the rotor will fit in the epitrochoid if \(b \leqslant \frac{3}{2}(2-\sqrt{3}) r\). Equation Transcription: Text Transcription: C 2r r C P b 0<b<r L C Theta L x P x= b cos 3 theta + 3rcos theta y= b sin 3theta +3r sin theta b=0 3r b=r P 0<b<r b r b b<or =3/2(2-square root of 3) r