Question: Set up an integral that represents the
Set up an integral that represents the length of the curve. Then use your calculator to find the length correct to four decimal places.
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x = t + e¯', y=t – e', 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. 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.
> (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.
> 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
> 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
> Find dy/dx and d2y/dx2. For which values of t is the curve concave upward? x = e', y=te
> Find dy/dx and d2y/dx2. For which values of t is the curve concave upward? х — + 1, у—r -1
> Find dy/dx and d2y/dx2. For which values of t is the curve concave upward? x = t? + 1, y= t? + t
> Find an equation of the tangent to the curve at the given point. Then graph the curve and the tangent. := sin mt, y= t² + t; (0, 2)
> Find an equation of the tangent to the curve at the given point. Then graph the curve and the tangent. x = t? – 1, y=t² + t + 1; (0, 3)
> Find an equation of the tangent to the curve at the given point by two methods: (a) without eliminating the parameter and (b) by first eliminating the parameter. x = 1 + Vi, y= e"; (2, e)
> What is a power series?
> Find an equation of the tangent to the curve at the given point by two methods: (a) without eliminating the parameter and (b) by first eliminating the parameter. x = 1 + In t, y = t² + 2; (1, 3)
> Find an equation of the tangent to the curve at the point corresponding to the given value of the parameter. х — е' sin mt, у — е"; 1— 0 ,21. t =
> Find an equation of the tangent to the curve at the point corresponding to the given value of the parameter. x =t cos t, y=t sin t; t= T
> Find an equation of the tangent to the curve at the point corresponding to the given value of the parameter. x= Vĩ, Vi, y = t? – 2t; t=4
> Find an equation of the tangent to the curve at the point corresponding to the given value of the parameter. x = t³ + 1, y= 1ª + t; t= -1
> Use a graphing device to graph the polar curve. Choose the parameter interval to make sure that you produce the entire curve. r = 2 + cos(90/4)
> Use a graphing device to graph the polar curve. Choose the parameter interval to make sure that you produce the entire curve. r= 1 + cos0 (Pac-Man curve)
> Use a graphing device to graph the polar curve. Choose the parameter interval to make sure that you produce the entire curve. r= | tan 0 |c* o| (valentine curve)
> Use a graphing device to graph the polar curve. Choose the parameter interval to make sure that you produce the entire curve. r= e sin o – 2 cos(40) (butterfly curve)
> Use a graphing device to graph the polar curve. Choose the parameter interval to make sure that you produce the entire curve. r= 1- 0.8 sin²0_ (hippopede) r =
> Use a graphing device to graph the polar curve. Choose the parameter interval to make sure that you produce the entire curve. r= 1 + 2 sin(0/2) (nephroid of Freeth)
> Find the points on the given curve where the tangent line is horizontal or vertical. r = e°
> Find the points on the given curve where the tangent line is horizontal or vertical. r=1+ cos 0
> Find the points on the given curve where the tangent line is horizontal or vertical. r=1- sin 0
> Find the points on the given curve where the tangent line is horizontal or vertical. -r= 3 cos 0
> Determine whether the series is absolutely convergent or conditionally convergent. (-1)* -1 In
> Match the polar equations with the graphs labeled I–VI. Give reasons for your choices. (Don’t use a graphing device.) (a) r= In 0, 1 < 0 < 6 (c) r= cos 30 (e) r= cos(0/2) (b) r= 0°, 0< 0 < 87 (d) r=2 + cos 30 (f)
> Sketch the curve (x2 + y2)3 = 4x2y2
> Test the series for convergence or divergence. E (v2 – 1)
> For each of the described curves, decide if the curve would be more easily given by a polar equation or a Cartesian equation. Then write an equation for the curve. (a) A circle with radius 5 and center (2, 3) (b) A circle centered at the origin with radi
> Find a polar equation for the curve represented by the given Cartesian equation. x? + y? = 2cx CX
> Find a polar equation for the curve represented by the given Cartesian equation. 4y2
> Use any test to determine whether the series is absolutely convergent, conditionally convergent, or divergent. (-1)* arctan n Σ n2
> Use a computer algebra system to find the Taylor polynomials Tn centered at a for n = 2, 3, 4, 5. Then graph these polynomials and f on the same screen. f(x) = V1 + x², a=0 %3D
> Find a polar equation for the curve represented by the given Cartesian equation. y = 1 + 3x
