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All Textbook Solutions for Calculus (MindTap Course List)
Evaluating an Improper Integral In Exercises 1732, determine whether the improper integral diverges or converges. Evaluate the integral if it converges. 1lnxxdx27E28E29E30E31E32E33E34E35EEvaluating an Improper Integral In Exercises 3348, determine whether the improper integral diverges or converges. Evaluate the integral if it converges, and check your results with the results obtained by using the integration capabilities of a graphing utility. 0838xdxEvaluating an Improper Integral In Exercises 3348, determine whether the improper integral diverges or converges. Evaluate the integral if it converges, and check your results with the results obtained by using the integration capabilities of a graphing utility. 01xlnxdxEvaluating an Improper Integral In Exercises 3348, determine whether the improper integral diverges or converges. Evaluate the integral if it converges, and check your results with the results obtained by using the integration capabilities of a graphing utility. 0elnx2dxEvaluating an Improper Integral In Exercises 3348, determine whether the improper integral diverges or converges. Evaluate the integral if it converges, and check your results with the results obtained by using the integration capabilities of a graphing utility. 0/2tand40E41E42EEvaluating an Improper Integral In Exercises 3348, determine whether the improper integral diverges or converges. Evaluate the integral if it converges, and check your results with the results obtained by using the integration capabilities of a graphing utility. 351x29dxEvaluating an Improper Integral In Exercises 3348, determine whether the improper integral diverges or converges. Evaluate the integral if it converges, and check your results with the results obtained by using the integration capabilities of a graphing utility. 05125x2dxEvaluating an Improper Integral In Exercises 3348, determine whether the improper integral diverges or converges. Evaluate the integral if it converges, and check your results with the results obtained by using the integration capabilities of a graphing utility. 31xx29dx46EEvaluating an Improper Integral In Exercises 3348, determine whether the improper integral diverges or converges. Evaluate the integral if it converges, and check your results with the results obtained by using the integration capabilities of a graphing utility. 04x(x+6)dx48EFinding Values In Exercises 49 and 50, determine all values of p for which the improper integral converges. 11xpdx50E51E52E53EConvergence or Divergence In Exercises 5360, use the results of Exercises 4952 to determine whether the improper integral converges or diverges. 011x9dx55EConvergence or Divergence In Exercises 5360, use the results of Exercises 4952 to determine whether the improper integral converges or diverges. 0x4exdx57E58EConvergence or Divergence In Exercises 5360, use the results of Exercises 4952 to determine whether the improper integral converges or diverges. 11sinxx2dx60E61E62E63E64EArea In Exercises 63-66, find the area of the unbounded shaded region. Witch of Agnesi:Area In Exercises 63-66, find the area of the unbounded shaded region. Witch of Agensi: y=8x2+4Area and Volume In Exercises 67 and 68, consider the region satisfying the inequalities (a) Find the area of the region. (b) Find the volume of the solid generated by revolving the region about the x -axis. (c) Find the volume of the solid generated by revolving the region about the y -axis. yex,y0,x068E69E70EPropulsion In Exercises 71 and 72, use the weight of the rocket to answer each question. (Use 4000 miles as the radius of Earth and do not consider the effect of air resistance.) How much work is required to propel the rocket an unlimited distance away from Earths surface? How far has the rocket traveled when half of the total work has occurred? 5-mctric-ton rocket Probability A nonnegative function f is called a probability density function if f(t)dt=1. The probability that x lies between a and b is given by P(axb)=abf(t)dt.72E73E74ENormal Probability The mean height of American men between 20 and 29 years old is 69 inches, and the standard deviation is 3 inches. A 20- to 29-year-old man is chosen at random from the population. The probability that he is 6 feet tall or taller is P(72x)=72132e(x69)2/18dx. (Source: National Center for Health Statistics) (a) Use a graphing utility to graph the integrand. Use the graphing utility to convince yourself that the area between the x-axis and the integrand is 1. (b) Use a graphing utility to approximate P(72x). (c) Approximate 0.5P(69x72) using a graphing utility. Use the graph in part (a) to explain why this result is the same as the answer in part (b).76E77E78E79E80E81E82E83E84E85E86E87E88E89E90E91E92E93E94E95E96E97E98E99E100E101E102E103E104Eu -Substitution In Exercises 105 and 106, rewrite the improper integral as a proper integral using the given u -substitution. Then use the Trapezoidal Rule with n = 5 to approximate the integral. 01sinxxdx,u=x106E107EWriting the Terms of a Sequence In Exercises 1-4, write the first five terms of the sequence with the given n th term. an=6n22RE3RE4RE5RE6RE7RE8RE9RE10RE11RE12RE13RE14RE15REDetermining Convergence or Divergence In Exercises 11-18, determine the convergence or divergence of the sequence with the given nth term. If the sequence converges, find its limit. an=nlnn17RE18RE19RE20RE21RE22RE23RE24RE25RE26RE27REFinding Partial Sums In Exercises 27 and 28, find the sequence of partial sums S1,S2,S3,S4andS5, 7+117+1491343+29RE30RE31RE32RE33RE34RE35RE36RE37RE38RE39RE40RE41RE42RE43RE44RE45RE46RE47RE48RE49RE50RE51RE52RE53RE54RE55RE56RE57RE58RE59RE60RE61RE62RE63RE64RE65RE66RE67RE68RE69RE70RE71RE72RE73RE74RE75RE76RE77RE78RE79RE80RE81RE82RE83RE84RE85RE86RE87RE88RE89REFinding Intervals of Convergence In Exercises 89 and 90, find the intervals of convergence of (a) f(x). (b) f(x) (c) f(x) and (d) f(x)dx . (Be sure to include a check for convergence at the endpoints of the intervals.) f(x)=n=1(1)n+1(x4)nn91RE92REFinding a Geometric Power Series In Exercises 93 and 94, find a geometric power series for the function, centered at 0. g(x)=23x94RE95RE96RE97RE98RE99RE100RE101RE102REFinding a Taylor Series In Exercises 103-110, use the definition of Taylor series to find the Taylor series, centered at c, for the function. f(x)=sinx,c=34Finding a Taylor Series In Exercises 103-110, use the definition of Taylor series to find the Taylor series, centered at c, for the function. f(x)=cosx,c=4105RE106RE107RE108REFinding a Taylor SeriesIn Exercises 103-110, use the definition of Taylor series to find the Taylor series, centered at c, for the function. g(x)=1+x5,c=0110RE111RE112RE113RE114RE115RE116RE117RE118RE1PS2PS3PSFinding a Limit Let T be an equilateral triangle with sides of length 1. Let an be the number of circles that can be packed tightly in n rows inside the triangle. For example, a1 = 1, a2 = 3, and a3 = 6, as shown in the figure. Let An be the combined area of the an circles. Find limnAn.5PS6PS7PS8PS9PS10PS11PS12PS13PS14PS15PS16PSCONCEPT CHECK Recursively Defined Sequence What does it mean for a sequence to be defined recursively?2E3E4EWriting the Terms of a Sequence In Exercises 510, write the first five terms of the sequence with the given n th term. an=2n6EWriting the Terms of a Sequence In Exercises 510, write the first five terms of the sequence with the given n th term. an=sinn28E9E10EWriting the Terms of a Sequence In Exercises 11 and 12, write the first five terms of the recursively defined sequence. a1=3,ak1=2(a11)12E13EMatching In Exercises 13-16, match the sequence with the given nth term with its graph. [The graphs are labeled (a), (b), (c). and (d).] an=10nn+115E16E17E18E19ESimplifying Factorials In Exercises 1720, simplify the ratio of factorials. (4n+1)!(4n+3)!21E22E23EFinding the Limit of a Sequence In Exercises 2124, find the limit of the sequence with the given n th term. an=cos2n25E26E27E28EDetermining Convergence or Divergence In Exercises 2944, determine the convergence or divergence of the sequence with the given n th term. If the sequence converges, find its limit. an=5n+2Determining Convergence or Divergence In Exercises 2944, determine the convergence or divergence of the sequence with the given n th term. If the sequence converges, find its limit. an=n1n!31E32E33EDetermining Convergence or Divergence In Exercises 2944, determine the convergence or divergence of the sequence with the given n th term. If the sequence converges, find its limit. an=n3n3+135E36E37E38E39E40E41EDetermining Convergence or Divergence In Exercises 2944. determine the convergence or divergence of the sequence with the given n th term. If the sequence converges, find its limit an=3n43EDetermining Convergence or Divergence In Exercises 29-44, determine the convergence or divergence of the sequence with the given nth term. If the sequence converges, find its limit. an=cos2n3nFinding the n th Term of a Sequence In Exercises 4552, write an expression for the nth term of the sequence and then determine whether the sequence you have chosen converges or diverges. (There is more than one correct answer.) 2, 8, 14, 20, ...46EFinding the n th Term of a Sequence In Exercises 4552, write an expression for the nth term of the sequence and then determine whether the sequence you have chosen converges or diverges. (There is more than one correct answer.) 2, 1, 6, 13, 22, ...48EFinding the nth Term of a Sequence In Exercises 45-52, write an expression for the nth term of the sequence and then determine whether the sequence you have chosen converges or diverges. (There is more than one correct answer.) 23,34,45,56,Finding the n th Term of a Sequence In Exercises 4552, write an expression for the nth term of the sequence and then determine whether the sequence you have chosen converges or diverges. (There is more than one correct answer.) 2, 24, 720, 40, 320, 3,628, 800, ...Finding the nth Term of a Sequence In Exercises 4552, write an expression for the n th term of the sequence and then determine whether the sequence you have chosen converges or diverges. (There is more than one correct answer.) 2,1+121+171+141+15,...Finding the nth Term of a Sequence In Exercises 4552, write an expression for the n th term of the sequence and then determine whether the sequence you have chosen converges or diverges. (There is more than one correct answer.) 123,234,345,456,...53E54E55EMonotonic and Bounded Sequences In Exercises 53-60, determine whether the sequence with the given nth term is monotonic and whether it is bounded. Use a graphing utility to confirm your results. an=(23)n57E58E59E60E61E62E63E64E65E66E67EGovernment Expenditures A government program that currently costs taxpayers$4.5 billion per year is cut back by 6% per year. (a) Write an expression for the amount budgeted for this program after n years. (b) Compute the budgets for the first 4 years. (c) Determine the convergence or divergence of the sequence of reduced budgets. If the sequence converges, find its limit.EXPLORING CONCEPTS Writing a Sequence Give an example of a sequence satisfying the condition. (a) A monotonically increasing sequence that converges to 10 (b) A sequence that converges to 3470E71EHOW DO YOU SEE IT? The graph, of two sequences are shown in the figures. Which graph represents the sequence with alternating signs? Explain.73E74E75E76E77E78E79E80EFibonacci Sequence In a study of the progeny of rabbits, Fibonacci (ca. 1170ca. 1240) encountered the sequence now bearing his name. The sequence is defined recursively as an+2=an+an+1 where a1. = 1 and a2 = 1. (a) Write the first 12 terms of the sequence. (b) Write the first 10 terms of the sequence defined by bn=an+1an,n1. (c) Using the definition in part (b). show that bn=1+1bn1. (d) The golden ratio can be defined by limnbn=. Show that =1+(1/) and solve this equation for .82E83EUsing a Sequence Consider the sequence an, where a1=k,an1=k+an, and k0 (a) Show that an is increasing and bounded. (b) Prove that limnan exists. (c) Find limnan