(a) Show that the curvature at each point of a straight line is / = 0. (b) Show that the curvature at each point of a circle of radius r is / = 1/r.
> The curves with equations x = asin nt, y = bcos t are called Lissajous figures. Investigate how these curves vary when a, b, and n vary. (Take n to be a positive integer.)
> Graph several members of the family of curves x = sin t + sin nt, y = cos t + cos nt, where n is a positive integer. What features do the curves have in common? What happens as n increases?
> Graph several members of the family of curves with parametric equations x = t + a cos t, y = t + a sin t, where a > 0. How does the shape change as a increases? For what values of a does the curve have a loop?
> The swallowtail catastrophe curves are defined by the parametric equations x = 2ct - 4t3, y = 2ct2 + 3t4. Graph several of these curves. What features do the curves have in common? How do they change when c increases?
> Investigate the family of curves defined by the parametric equations x = t2, y = t3 - ct. How does the shape change as c increases? Illustrate by graphing several members of the family.
> Suppose that the position of one particle at time t is given by and the position of a second particle is given by (a) Graph the paths of both particles. How many points of intersection are there? (b) Are any of these points of intersection collision poin
> Find the radius of convergence and interval of convergence of the series. (-1)"-1 Σ n5" n-1
> (a) Find parametric equations for the set of all points P as shown in the figure such that |OP|−|AB|. (This curve is called the cissoid of Diocles after the Greek scholar Diocles, who introduced the cissoid as a graphical method for con
> A curve, called a witch of Maria Agnesi, consists of all possible positions of the point P in the figure. Show that parametric equations for this curve can be written as Sketch the curve. х — 2а сot 0 y = 2a sin?e C yA y=2a -- a+
> Compare the curves represented by the parametric equations. How do they differ? (a) x= t, y= t² (с) х — е', у—е " (b) х — сos t, у — sec't
> Compare the curves represented by the parametric equations. How do they differ? (a) x = t°, y= t² (c) x= e , y = e " (b) х — 19, у — 31 21
> Use a graphing calculator or computer to reproduce the picture. yA 3 8 4. 2.
> Use a graphing calculator or computer to reproduce the picture. YA 2- 2
> Find the radius of convergence and interval of convergence of the series. (-1)"4" n-
> (a) Find parametric equations for the ellipse x2/a2 + y2/b2 = 1. (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 parametric equations for the path of a particle that moves along the circle x2 + (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 count
> Use a graphing device and the result of Exercise 31(a) to draw the triangle with vertices A(1, 1), B(4, 2), and C(1, 5). Data from Exercise 31: (a) Show that the parametric equations where 0 (b) Find parametric equations to represent the line segment f
> (a) Show that the parametric equations where 0 (b) Find parametric equations to represent the line segment from (22, 7) to (3, 21). x= x, + (x2 – xi)t y = yı + (y2 – yı)t
> Graph the curves y = x3 - 4x and x = y3 - 4y and find their points of intersection correct to one decimal place.
> Match the parametric equations with the graphs labeled I–VI. Give reasons for your choices. (Do not use a graphing device.) (a) x = 1ª – t + 1, y=t² (b) х — ? - 21, у — (c) x = sin 2t, y= sin(t + sin 2i) (d) x = cos 5t, y= sin 2t
> Use the graphs of x = f(t) and y = t(t) to sketch the parametric curve x = f(t), y = t(t). Indicate with arrows the direction in which the curve is traced as t increases. XA 1
> Use the graphs of x = f(t) and y = t(t) to sketch the parametric curve x = f(t), y = t(t). Indicate with arrows the direction in which the curve is traced as t increases. XA 1. -1 1 1
> Use the graphs of x = f(t) and y = t(t) to sketch the parametric curve x = f(t), y = t(t). Indicate with arrows the direction in which the curve is traced as t increases. XA
> Find the radius of convergence and interval of convergence of the series. E 2"n?x" n=1
> Match the graphs of the parametric equations x = f(t) and y = t(t) in (a)–(d) with the parametric curves labeled I–IV. Give reasons for your choices. (a) I 2 1 2 x (b) II XA YA 2. y 2- 2 x (c) III y 1 2 x (d) I
> Describe the motion of a particle with position sx, yd as t varies in the given interval. x = sin 1, y = cos?t, -27 < t< 2m
> Describe the motion of a particle with position sx, yd as t varies in the given interval. x = 5 sin t, y = 2 cos t, -T<I< 5T
> Describe the motion of a particle with position sx, yd as t varies in the given interval. x = 2 + sin t, y=1 + 3 cos t, T/2<1<27
> Describe the motion of a particle with position sx, yd as t varies in the given interval. x = 5 + 2 cos T t, y=3 + 2 sin mt, 1<t<2
> (a) Eliminate the parameter to find a Cartesian equation of the curve. (b) Sketch the curve and indicate with an arrow the direction in which the curve is traced as the parameter increases. х — tan'o, у — sec 0, —п/2 <0 < п/2
> (a) Eliminate the parameter to find a Cartesian equation of the curve. (b) Sketch the curve and indicate with an arrow the direction in which the curve is traced as the parameter increases. x = sinh t, y= cosh t
> (a) Eliminate the parameter to find a Cartesian equation of the curve. (b) Sketch the curve and indicate with an arrow the direction in which the curve is traced as the parameter increases. x = /i + 1, y= vi - 1
> (a) Eliminate the parameter to find a Cartesian equation of the curve. (b) Sketch the curve and indicate with an arrow the direction in which the curve is traced as the parameter increases. x = t, y= In t
> (a) Eliminate the parameter to find a Cartesian equation of the curve. (b) Sketch the curve and indicate with an arrow the direction in which the curve is traced as the parameter increases. 21 x = e', y
> Find the radius of convergence and interval of convergence of the series. Σ n*4"
> (a) Eliminate the parameter to find a Cartesian equation of the curve. (b) Sketch the curve and indicate with an arrow the direction in which the curve is traced as the parameter increases. х — sin t, у - csc i, 0 <1< п/2
> (a) Eliminate the parameter to find a Cartesian equation of the curve. (b) Sketch the curve and indicate with an arrow the direction in which the curve is traced as the parameter increases. x =} cos 0, y = 2 sin 0, 0< 0 <
> (a) Eliminate the parameter to find a Cartesian equation of the curve. (b) Sketch the curve and indicate with an arrow the direction in which the curve is traced as the parameter increases. х — sin 0, y — сos s0, -<0 < T
> (a) Sketch the curve by using the parametric equations to plot points. Indicate with an arrow the direction in which the curve is traced as t increases. (b) Eliminate the parameter to find a Cartesian equation of the curve. x = t', y = t .2 3. х
> (a) Sketch the curve by using the parametric equations to plot points. Indicate with an arrow the direction in which the curve is traced as t increases. (b) Eliminate the parameter to find a Cartesian equation of the curve. x = /i, y=1 – t Vi,
> (a) Sketch the curve by using the parametric equations to plot points. Indicate with an arrow the direction in which the curve is traced as t increases. (b) Eliminate the parameter to find a Cartesian equation of the curve. x = sin 1, y =1 - cos t, 0
> (a) Sketch the curve by using the parametric equations to plot points. Indicate with an arrow the direction in which the curve is traced as t increases. (b) Eliminate the parameter to find a Cartesian equation of the curve. х — 12 — 3, у—1+2, —3<1s 3
> (a) Sketch the curve by using the parametric equations to plot points. Indicate with an arrow the direction in which the curve is traced as t increases. (b) Eliminate the parameter to find a Cartesian equation of the curve. x = 3t + 2, y = 2t + 3
> (a) Sketch the curve by using the parametric equations to plot points. Indicate with an arrow the direction in which the curve is traced as t increases. (b) Eliminate the parameter to find a Cartesian equation of the curve. х — 21 — 1, у— +1 x =
> Sketch the curve by using the parametric equations to plot points. Indicate with an arrow the direction in which the curve is traced as t increases. x = e + t, y= e' – t, -2 <t< 2
> Test the series for convergence or divergence. E (-1)*-1 4" n-
> Sketch the curve by using the parametric equations to plot points. Indicate with an arrow the direction in which the curve is traced as t increases. x = t³ + t, _y= t² + 2, -2<t< 2
> Sketch the curve by using the parametric equations to plot points. Indicate with an arrow the direction in which the curve is traced as t increases. x= 1 - t, y= 2t – t°, -1 <t<2
> A cow is tied to a silo with radius r by a rope just long enough to reach the opposite side of the silo. Find the grazing area available for the cow.
> Use Formula 1 to derive Formula 6 from Formula 8.2.5 for the case in which the curve can be represented in the form /
> Find the radius of convergence and interval of convergence of the series. o n! -0
> Find the surface area generated by rotating the given curve about the y-axis. x = e' – t, y= 4e?, 0<t<1 = 4e2, 0<t<1
> Find the surface area generated by rotating the given curve about the y-axis. x = 3t, y = 2t², 0<t<5
> Find the exact area of the surface obtained by rotating the given curve about the x-axis. х —а сos'0, у —asin'0, 0<0< п/2
> Find the exact area of the surface obtained by rotating the given curve about the x-axis. x = 212 + 1/1, y= 8/t, 1<t< 3
> Find the exact area of the surface obtained by rotating the given curve about the x-axis. x = t, y = t, 0<t<1
> Set up an integral that represents the area of the surface obtained by rotating the given curve about the x-axis. Then use your calculator to find the surface area correct to four decimal places. x = t² – t, y=t+ t*, 0<t<1
> Set up an integral that represents the area of the surface obtained by rotating the given curve about the x-axis. Then use your calculator to find the surface area correct to four decimal places. x=t+ e', y= e ', 0<t< 1
> Set up an integral that represents the area of the surface obtained by rotating the given curve about the x-axis. Then use your calculator to find the surface area correct to four decimal places. x = sin t, y = sin 2t, 0<t< T/2
> Find the radius of convergence and interval of convergence of the series. (-1)"x" Σ n2
> Set up an integral that represents the area of the surface obtained by rotating the given curve about the x-axis. Then use your calculator to find the surface area correct to four decimal places. x = t sin t, y =t cos 1, 0<1<T/2
> A curve called Cornu’s spiral is defined by the parametric equations where C and S are the Fresnel functions that were introduced in Chapter 5. (a) Graph this curve. What happens as / and as / (b) Find the length of Cornuâ€&#
> (a) Graph the epitrochoid with equations What parameter interval gives the complete curve? (b) Use your CAS to find the approximate length of this curve. x= 11 cos t – 4 cos(11t/2) y = 11 sin t – 4 sin(11t/2)
> Find the distance traveled by a particle with position (x, y) as t varies in the given time interval. Compare with the length of the curve. x = cos't, y = cos t, 0 I< 4T
> Find the distance traveled by a particle with position (x, y) as t varies in the given time interval. Compare with the length of the curve. x = sin?t, y = cos?t, 0<t<3T
> Use Simpson’s Rule with n = 6 to estimate the length of the curve x = t – e', y = t + e', -6 <t< 6.
> Find the length of the loop of the curve x = 3t - t3 , y = 3t2.
> Find the radius of convergence and interval of convergence of the series. x" Σ 2n – 1
> (a) Approximate f by a Taylor polynomial with degree n at the number a. (b) Use Taylor’s Inequality to estimate the accuracy of the approximation / when x lies in the given interval. (c) Check your result in part (b) by graphing / S
> Graph the curve x = sin t + sin +1.5t, y = cos t and find its length correct to four decimal places.
> Graph the curve and find its exact length. x = cos t + In(tan t), y= sin t, T/4 <t< 3/4
> Graph the curve and find its exact length. x = e' cos t, y = e' sin t, 0 <t<
> Find the exact length of the curve. х — 3 3 cos t - cos Зі, cos 31, y = 3 sin t – sin 3t, 0 <t<+
> Find the exact length of the curve. x = t sin t, y =t cos 1, 0 <1<1
> Find the exact length of the curve. x = e' – t, y = 4e"?, 0<t<2
> Find the exact length of the curve. x = 1 + 3t², y= 4 + 2t³, 0 <<t<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 = t + vt, y=t- Vi, 0<t<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 = t – 2 sin t, y=1-2 cos t, 0<t<4m
> 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, y= t“, 1<t<4
> Find the radius of convergence and interval of convergence of the series. (-1)"x" Σ 'n
> 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 + e¯', y=t – e', 0<t< 2
> Let R be the region enclosed by the loop of the curve in Example 1. (a) Find the area of R. (b) If 5 is rotated about the x-axis, find the volume of the resulting solid. (c) Find the centroid of R.
> Find the area enclosed by the x-axis and the curve x =t3 + 1, y = 2t – t2.
> Find equations of the tangents to the curve x = 3t2 + 1, y = 2t3 + 1 that pass through the point (4, 3).
> At what point(s) on the curve x = 3t2 + 1, y = t3 - 1 does the tangent line have slope 1 2 ?
> Find the radius of convergence and interval of convergence of the series. E(-1)"nx" 'x' -1
> Graph the curve x = 22 cos t, y = sin t + sin 2t to discover where it crosses itself. Then find equations of both tangents at that point.
> Show that the curve x = cos t, y = sin t cos t has two tangents at (0, 0) and find their equations. Sketch the curve.
> Graph the curve in a viewing rectangle that displays all the important aspects of the curve. x= 14 + 41° – 81², y=21² – t
> Graph the curve in a viewing rectangle that displays all the important aspects of the curve. x = t* – 2t3 – 2t², y=t³ – t
> Use a graph to estimate the coordinates of the lowest point and the leftmost point on the curve x = t4 - 2t, y = t + t4. Then find the exact coordinates.
> Use a graph to estimate the coordinates of the rightmost point on the curve x = t - t6, y = et. Then use calculus to find the exact coordinates.
> Find the points on the curve where the tangent is horizontal or vertical. If you have a graphing device, graph the curve to check your work. *= e sino sinº, y = e cs0 cos0
> Find the points on the curve where the tangent is horizontal or vertical. If you have a graphing device, graph the curve to check your work. x = cos 0, y= cos 30
> Find the points on the curve where the tangent is horizontal or vertical. If you have a graphing device, graph the curve to check your work. x = t³ – 31, y =t} - 3t²
> (a) What is the radius of convergence of a power series? How do you find it? (b) What is the interval of convergence of a power series? How do you find it?
> Find the points on the curve where the tangent is horizontal or vertical. If you have a graphing device, graph the curve to check your work. х — — 31, у— ? - 3 x =
> Find dy/dx and d2y/dx2. For which values of t is the curve concave upward? x = cos t, y = sin 2t, 0<t<
> Find dy/dx and d2y/dx2. For which values of t is the curve concave upward? x = t - In t, y = t + In t
> Find dy/dx and d2y/dx2. For which values of t is the curve concave upward? х — 1? + 1, у — е' — 1
