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Question: Determine whether the series converges or

Determine whether the series converges or diverges.
Determine whether the series converges or diverges.





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> Use multiplication or division of power series to find the first three nonzero terms in the Maclaurin series for each function. y - e* In(1 + x)

> Find the Taylor polynomial T3(x) for the function f centered at the number a. Graph f and T3 on the same screen. f) — In x, а — 1

> Find an equation of the ellipse with foci (±4, 0) and vertices (±5, 0).

> Find the foci and vertices and sketch the graph. 25x? + 4y? + 50x – 16y = 59

> Evaluate the integral. dx X - 1

> Find the values of x for which the series converges. Find the sum of the series for those values of x. E (-5)"x"

> Find the foci and vertices and sketch the graph. 4x? - у? — 16 %3D

> Find the foci and vertices and sketch the graph. x2 y? 1 8 9

> Determine whether the sequence converges or diverges. If it converges, find the limit. (-3)" an n!

> The curves defined by the parametric equations are called strophoids (from a Greek word meaning “to turn or twist”). Investigate how these curves vary as c varies. (? — с) y = 1? + 1 1? - c 1? + 1

> Use the method of cylindrical shells to find the volume generated by rotating the region bounded by the curves about the given axis. у —е", у— 0, х — —1, х — 0; about x— 1 %3D

> Find the area of the surface obtained by rotating the given curve about the x-axis. x = 2 + 3t, y = cosh 31, 0<i<1

> Find the area of the surface obtained by rotating the given curve about the x-axis. x = 4 T, y=+, x = 4 /t, y=- 3 212"

> Find the length of the curve. r= sin'(0/3), 0< 0<

> Sketch the parametric curve and eliminate the parameter to find the Cartesian equation of the curve. x = 2 cos 0, y =1 + sin0 = 1 + sin@

> Use multiplication or division of power series to find the first&Acirc;&nbsp;three nonzero terms in the Maclaurin series for each function. y SI sin x

> Express the number as a ratio of integers. 5.71358

> Evaluate the integral.

> Determine whether the series is convergent or divergent. If it is convergent, find its sum. 1 Σ 1 + (?)"

> Calculate, to four decimal places, the first ten terms of the sequence and use them to plot the graph of the sequence by hand. Does the sequence appear to have a limit? If so, calculate it. If not, explain why. 10" a, = 1 + 9"

> Find the length of the curve. r= 1/0, 7< 0 < 27

> Find the length of the curve. x = 2 + 3t, y = cosh 31, 0< t<1

> Find the length of the curve. x = 312, y = 21°, 0<t<2

> Find the area of the region that lies inside both of the circles /

> Use the method of cylindrical shells to find the volume generated by rotating the region bounded by the curves about the given axis. у —е", у—е", х — 1;B about the y-аxis

> At what points does the curve have vertical or horizontal tangents? Use this information to help sketch the curve. X= 2a cos t - a cos 21 y = 2a sin t – a sin 2t

> Find the area enclosed by the loop of the curve in Exercise 27. Data from Exercise 27: Use a graph to estimate the coordinates of the lowest point on the curve x = t3 - 3t, y = t2 + t + 1. Then use calculus to find the exact coordinates.

> Determine whether the geometric series is convergent or divergent. If it is convergent, find its sum. 3 – 4 + – 4 - +

> Determine whether the sequence converges or diverges. If it converges, find the limit. a„ = In(n + 1) – In n

> Use a graph to estimate the coordinates of the lowest point on the curve x = t3 - 3t, y = t2 + t + 1. Then use calculus to find the exact coordinates.

> Find the volume obtained by rotating the region bounded by the curves about the given axis. у — sin x, у — 0, п/2 <x< п; about the x-axis

> Find dy/dx and d2y/dx2. x = 1 + t2, y = t – t3

> Use multiplication or division of power series to find the first&Acirc;&nbsp;three nonzero terms in the Maclaurin series for each function. y - sec x

> Find dy/dx and d2y/dx2. x = t + sint. y = t – cos t

> Find the slope of the tangent line to the given curve at the point corresponding to the specified value of the parameter. r = e e "; 0 = T

> Populations of birds and insects are modeled by the equations (a) Which of the variables, x or y, represents the bird population and which represents the insect population? Explain. (b) Find the equilibrium solutions and explain their significance. (c) F

> Find the slope of the tangent line to the given curve at the point corresponding to the specified value of the parameter. x = In 1, y = 1 + t²; t=1

> Use the method of cylindrical shells to find the volume generated by rotating the region bounded by the curves about the given axis. y = cos(Tx/2), y= 0, 0<x< 1; about the y-axis

> Determine whether the sequence converges or diverges. If it converges, find the limit. {{东春 !!!!!!! ..} 3> 4> 6

> Express the number as a ratio of integers. 1.234567

> A tank contains 100 L of pure water. Brine that contains 0.1 kg of salt per liter enters the tank at a rate of 10 L/min. The solution is kept thoroughly mixed and drains from the tank at the same rate. How much salt is in the tank after 6 minutes?

> Determine whether the sequence converges or diverges. If it converges, find the limit. (2n – 1)! (2n + 1)!

> The von Bertalanffy growth model is used to predict the length L(t) of a fish over a period of time. If L is the largest length for a species, then the hypothesis is that the rate of growth in length is proportional to L` 2 L, the length yet to be achiev

> Use multiplication or division of power series to find the first&Acirc;&nbsp;three nonzero terms in the Maclaurin series for each function. y = e * cos x

> Determine whether the sequence converges or diverges. If it converges, find the limit. {0, 1, 0, 0, 1, 0, 0, 0, 1, . }

> Express the number as a ratio of integers. 10.135 = 10.135353535 ...

> Use a graph to find approximate x coordinates of the points of intersection of the given curves. Then find (approximately) the area of the region bounded by the curves. y = x у — х Iп(x + 1), у3 Зх — .2

> Evaluate the integral. ( In Va dx

> Solve the initial-value problem. dr + 2tr = r, r(0) = 5 dt

> Solve the differential equation. x?y' – y = 2x'e-1/½

> Solve the differential equation. 2ye "'y' = 2x + 3 /x %3D

> Solve the differential equation. dx 1 - t + x - tx dt

> Determine whether the sequence converges or diverges. If it converges, find the limit. а, — п — уп +1 ул + 3

> Use the Table of Integrals on the Reference Pages to evaluate the integral. cot x - dx 1 + 2 sin x

> Use the Table of Integrals on the Reference Pages to evaluate the integral. cos x /4 + sin²x dx

> Express the number as a ratio of integers. 2.516 = 2.516516516 ...

> Use the Table of Integrals on the Reference Pages to evaluate the integral. | csc't dt

> Use the series in Example 13(b) to evaluate We found this limit in Example 4.4.4 using l&acirc;&#128;&#153; Hospital&acirc;&#128;&#153;s Rule three times. Which method do you prefer? tan x - x lim

> Graph the function f(x) = cos2x sin3x and use the graph to guess the value of the integral / . Then evaluate the integral to confirm your guess.

> Use a graph to find approximate x&Acirc;&shy; coordinates of the points of intersection of the given curves. Then find (approximately) the area of the region bounded by the curves. y = arcsin n(4x), у — 2 — х?

> Evaluate the indefinite integral. Illustrate and check that your answer is reasonable by graphing both the function and its antiderivative (take C = 0). dx /x² + 1

> Determine whether the sequence converges or diverges. If it converges, find the limit. an arctan(In n)

> Evaluate the indefinite integral. Illustrate and check that your answer is reasonable by graphing both the function and its antiderivative (take C = 0). In(x² + 2x + 2) dx

> Evaluate the integral or show that it is divergent. tan 'x - dx x?

> Solve the differential equation. sin x y' = xe y cos x

> Test the series for convergence or divergence. E n’e E n²e n-

> Evaluate the integral or show that it is divergent. dx 4x2 + 4x + 5

> Evaluate the integral or show that it is divergent. dx x? 2x

> Evaluate the integral or show that it is divergent. - : dx

> Evaluate the integral or show that it is divergent. ах 2 - Зх · dx

> Use series to evaluate the limit. x3 – 3x + 3 tan 'x lim

> Evaluate the integral or show that it is divergent. In x

> Determine whether the series is convergent or divergent. If it is convergent, find its sum. 2" + 4" Σ e"

> Find a polar equation for the curve represented by the given Cartesian equation. x? – y? = 4

> Find the area of the region bounded by the given curves. y - x'e *, y- xe * хе

> Evaluate the integral or show that it is divergent. y dy - 2 /y – 2

> Evaluate the integral or show that it is divergent. dx ½ x In x

> A sequence of terms is defined by aj = 1 а, — (5 — п)а, I Calculate Σa.

> Evaluate the integral or show that it is divergent. In x dx a.

> Evaluate the integral or show that it is divergent. 1 · dx. (2x + 1)'

> Evaluate the integral. Ls Vtan o Ju/4 sin 20 *w OP

> (a) Use Euler&acirc;&#128;&#153;s method with step size 0.2 to estimate y(0.4), where y(x) is the solution of the initial-value problem (b) Repeat part (a) with step size 0.1. (c) Find the exact solution of the differential equation and compare the value

> Evaluate the integral. 2x xe 1/2 dx Jo (1 + 2x)²

> Determine whether the sequence converges or diverges. If it converges, find the limit. an Vn

> Evaluate the integral. ·dx

> Evaluate the integral. (cos x + sin x)? cos 2x dx

> Evaluate the integral. arctan(1/x) dx

> Use series to evaluate the limit. Vī +x - 1 - }x lim x?

> Find the area of the region bounded by the given curves. y = у — х' In x, у— 4 lnx y у — 4 Inx

> Let x = 0.99999 .... (a) Do you think that x < 1 or x = 1? (b) Sum a geometric series to find the value of x. (c) How many decimal representations does the number 1 have. (d) Which numbers have more than one decimal representation?

> Determine whether the geometric series is convergent or divergent. If it is convergent, find its (-3)"-| 4"

> Evaluate the integral. dx

> Evaluate the integral. 1 dx Vx + x3/2

> Evaluate the integral. | (arcsin x)° dx

> Determine whether the sequence converges or diverges. If it converges, find the limit. an 1 +

> Evaluate the integral. (4 – x²)/2

> Evaluate the integral. *w/4 X sin x dx cos'x CoS X

> Evaluate the integral. e*Ve* – 1 CIn 10 dx e* + 8

> Evaluate the integral. dx e*/1 – e-2x

> (a) A direction field for the differential equation y&acirc;&#128;&#153; = x2 &acirc;&#128;&#147; y2 is shown. Sketch the solution of the initial-value problem Use your graph to estimate the value of y(0.3). (b) Use Euler&acirc;&#128;&#153;s method wit

1.99

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