#26#4 VTUIOR O89OA2JA T In Exercises 11-14, find a value for the parameter a that makes10.2 Delecting Convergence, Estimating Limitsy lar1+e"Show that f (x) > 1/2 for all x > 0.IHINT: e' > 1 for all x > 0.]Ise the inequality in part (a) to show thatthe value of the improper integral less than 10$37we'll1.varke dx12. e dadiverges.13.dx5 Let f(x) =r + sin.x.Show that x – 1< S(x) for all x > 0.14. -In Exercises I5-18, find a proper integral whose value approt-dsimates that of the given (convergent) improper integral withinx (In x)Usc the inequality in part (a) to show thatdx10-5S(x)diverges.15.dx6 Let f(x)=x² + sin x.a Show that r – 1s S(x) for all x > 0dxarctan xdx17.dx(1+x)18.O) Does T converge? Justify your answer.f(x)19. Suppose that f(x)20 for all x21 and that7. Let (x) = x² + /ĩ.(2) Show that x2 < [(x) for all x > 0.converges. Explain why there is a number a such that| S(x) dx < 10 ".dx(h) Use the inequality in part (a) to show thatf(x)20. Suppose that g(x) 20 for all x21 and thatconverges.diverges. Explain why there is a number b'such thatLet (a)=r-.g(x)dx 2 10".6) Show that x2/2 < f(x) for all x > 2.dx(b) Does , T)converge? Justify your answer.1%3 S(x)dx comverges and that21. Suppose thatdxconverge? Justify your answer.)d<0.0001.Show that|7 – " sc«) da|s0.0S0.001.9. DoesメーVズdxconverge? Justify your answer.22. Show that 0.8 <sin (e") dx < 1.10. Does[HINT: Use the substitution u = e.]orFURTHER EXERCISEStaldxdxL e“ dx converges. [HINT: See Example 529.VI(1+x)23. Show thatbut be careful: e- >e- for some x 2 0.dxedx converge? Justify your answer.31. Does24. Consider the integral I =(In x) "cos Xdx diverges.in(a) Show that In xs VI for all x> 2.32. Show that the integral(b) Does I converge? Justify your answer.cosxdxdx converges.31, TInx)33. (a) Show thatK. Suppose that p>1 is a constant and that / =(b) Use integration by parts and part (a) to show thatFor which values of p, if any, does I converge? Justify your(In x)"= 0.]sin.xdx converges.answer. [HINT: limdxShow that 0<3.34. Let I = |ja) Show that 0 <x/2< sin.x if 0 <x<l.[HINT: 1/2 < cos.x if 0 <x < 1.](b) Use the inequality in part (a) to show that26.dxVitediVsin x35. Evaluate limconverges.36. Evaluate lim e le" dt.mproper integral converges or diverges.dx27.dx28.

Name: #26#4 VTUIOR O89OA2JA T In Exercises 11-14, find a value for the param
Uploaded: 2024-03-20T07:06:41.000Z
Channel: Bartleby
Description: #26#4 VTUIOR O89OA2JA T In Exercises 11-14, find a value for the parameter a that makes10.2 Delecting Convergence, Estimating Limitsy lar1+e"Show that f (x) > 1/2 for all x > 0.IHINT: e' > 1 for all x > 0.]Ise the inequality in part (a) to show thatt