Question: Use the Table of Integrals on the
Use the Table of Integrals on the Reference Pages to evaluate the integral.
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> 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 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 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 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
> Determine whether the series converges or diverges. 1/m Σ 1 п
> 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 series in Example 13(b) to evaluate We found this limit in Example 4.4.4 using l’ Hospital’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 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’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’ = x2 – 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’s method wit
> Find a polar equation for the curve represented by the given Cartesian equation. y =x
> Evaluate the integral. dx 4x + e
> Four bugs are placed at the four corners of a square with side length a. 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
> Evaluate the integral. Vx + 1 dx Vx - 1
> Use Exercise 52 to find Data from Exercise 52: Use integration by parts to prove the reduction formula. fx*e* dx. x"e*dx = x"e* – n |x" 'e*dx
> Determine whether the sequence converges or diverges. If it converges, find the limit. an sin(1/n)
> Use series to evaluate the limit. sin x – x + a lim .3
> Evaluate the integral. 2 dx X1
