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All Textbook Solutions for Calculus: Early Transcendentals (3rd Edition)

Use the result of Example 2 to evaluate 11x4 dx. Example 2 The Family f(x) = 1/xp Consider the family of functions f(x) = 1/xp, where p is a real number. For what values of p does 1f(x) dx converge?Explain why the one-sided limit c 0+ (instead of a two-sided limit) must be used in the previous calculation.4QC What are the two general ways in which an improper integral may occur?Evaluate 2dxx3 after writing the expression as a limit.Rewrite 2dxx1/5 as a limit and then show that the integral diverges.Evaluate 01dxx1/5 after writing the integral as a limit.Write limaa0f(x)dx+limb0bf(x)dxas an improper integral.Improper integrals Evaluate the following integrals or state that they diverge. 7. 3dxx2 Improper integrals Evaluate the following integrals or state that they diverge. 8. 2dxx Infinite intervals of integration Evaluate the following integrals or state that they diverge. 9. 2dxx Infinite intervals of integration Evaluate the following integrals or state that they diverge. 11. 0e2xdx Infinite intervals of integration Evaluate the following integrals or state that they diverge. 13. 0eaxdx,a0 Improper integrals Evaluate the following integrals or state that they diverge. 12. dxx3 Infinite intervals of integration Evaluate the following integrals or state that they diverge. 18. 0cosxdx Improper integrals Evaluate the following integrals or state that they diverge. 14. 1dxx3 Improper integrals Evaluate the following integrals or state that they diverge. 15. dxx2+100 Improper integrals Evaluate the following integrals or state that they diverge. 16. dxx2+a2,a0 Improper integrals Evaluate the following integrals or state that they diverge. 17. 7dx(x+1)1/3 Infinite intervals of integration Evaluate the following integrals or state that they diverge. 27. 2dx(x+2)2 Infinite intervals of integration Evaluate the following integrals or state that they diverge. 25. 13x2+1x3+xdx Improper integrals Evaluate the following integrals or state that they diverge. 20. 12xdx Infinite intervals of integration Evaluate the following integrals or state that they diverge. 19. 2cos(/x)x2dx Infinite intervals of integration Evaluate the following integrals or state that they diverge. 26. 11z2sinzdz Infinite intervals of integration Evaluate the following integrals or state that they diverge. 21. 0eue2u+1du Infinite intervals of integration Evaluate the following integrals or state that they diverge. 22. aexdx,areal Improper integrals Evaluate the following integrals or state that they diverge. 25. e3x1+e6xdx Improper integrals Evaluate the following integrals or state that they diverge. 26. e3x(x2+1)2dx Improper integrals Evaluate the following integrals or state that they diverge. 27. xex2dx Infinite intervals of integration Evaluate the following integrals or state that they diverge. 28. 1tan1ss2+1ds Improper integrals Evaluate the following integrals or state that they diverge. 29. (tan1t)2t2+1dt Infinite intervals of integration Evaluate the following integrals or state that they diverge. 7. 0exdx Infinite intervals of integration Evaluate the following integrals or state that they diverge. 23. 1duv(v+1)Improper integrals Evaluate the following integrals or state that they diverge. 32.1dxx2(x1)Improper integrals Evaluate the following integrals or state that they diverge. 33. 2dyylny Improper integrals Evaluate the following integrals or state that they diverge. 34. 4/1x2sec2(1x)dx Infinite intervals of integration Evaluate the following integrals or state that they diverge. 10. 0dx2x3 Infinite intervals of integration Evaluate the following integrals or state that they diverge. 15. e2dxxlnpx,p1 Improper integrals Evaluate the following integrals or state that they diverge. 37. 08dxx3 Improper integrals Evaluate the following integrals or state that they diverge. 38. 12dxx1 Improper integrals Evaluate the following integrals or state that they diverge. 39. 0/2tand Improper integrals Evaluate the following integrals or state that they diverge. 40. 31dx(2x+6)2/3 Integrals with unbounded integrands Evaluate the following integrals or state that they diverge. 39. 0/2secxtanxdx 42E Improper integrals Evaluate the following integrals or state that they diverge. 43. 01exxdx Improper integrals Evaluate the following integrals or state that they diverge. 44. 0ln3ey(ey1)2/3dy Improper integrals Evaluate the following integrals or state that they diverge. 45. 01x3x41dx Improper integrals Evaluate the following integrals or state that they diverge. 46. 1dxx13 Integrals with unbounded integrands Evaluate the following integrals or state that they diverge. 45. 010dx10x4 Integrals with unbounded integrands Evaluate the following integrals or state that they diverge. 46. 111dx(x3)2/3 Improper integrals Evaluate the following integrals or state that they diverge. 49. 02dx(x1)2 Integrals with unbounded integrands Evaluate the following integrals or state that they diverge. 50. 09dx(x1)1/3 Integrals with unbounded integrands Evaluate the following integrals or state that they diverge. 49. 22dp4p2 Improper integrals Evaluate the following integrals or state that they diverge. 52. 0xexdx Improper integrals Evaluate the following integrals or state that they diverge. 53. 01lnxdx Improper integrals Evaluate the following integrals or state that they diverge. 54. 0lnxx2dx Improper integrals Evaluate the following integrals or state that they diverge. 55. 0ln2exe2x1dx 56E Improper integrals Evaluate the following integrals or state that they diverge. 57. e|x|dx Improper integrals Evaluate the following integrals or state that they diverge. 58. dxx2+2x+5 Perpetual annuity Imagine that today you deposit B in a savings account that earns interest at a rate of p% per year compounded continuously (Section 6.9). The goal is to draw an income of I per year from the account forever. The amount of money that must be deposited is B=I0ertdt, where r = p/100. Suppose you find an account that earns 12% interest annually and you wish to have an income from the account of 5000 per year. How much must you deposit today?Draining a pool Water is drained from a swimming pool at a rate given by R(t) = 100 e0.05t gal/hr. If the drain is left open indefinitely, how much water drains from the pool?Bioavailability When a drug is given intravenously, the concentration of the drug in the blood is Ci(t) = 250e0.08t, for t 0. When the same drug is given orally, the concentration of the drug in the blood is Co(t) = 200(e0.08t e1.8t), for t 0. Compute the bioavailability of the drug.Electronic chips Suppose the probability that a particular computer chip fails after a hours of operation is 0.00005 ae0.00005tdt. a. Find the probability that the computer chip fails after 15,000 hr of operation. b. Of the chips that are still operating after 15,000 hr, what fraction of these will operate for at least another 15,000 hr? c. Evaluate 0.00005 0eaxcosbxdx=aa2+b2 and interpret its meaning.Average lifetime The average time until a computer chip fails (see Exercise 62) is 0.00005 0te0.00005tdt. Find this value. Electronic chips Suppose the probability that a particular computer chip fails after a hours of operation is 0.00005 ae0.00005tdt a. Find the probability that the computer chip fails after 15,000 hr of operation. b. Of the chips that are still operating after 15,000 hr, what fraction of these will operate for at least another 15,000 hr? c. Evaluate 0.00005 ae0.00005tdtand interpret its meaning.Maximum distance An object moves on a line with velocity v(t) = 10/(t + 1)2 mi/hr, for t 0. What is the maximum distance the object can travel?Volumes on infinite intervals Find the volume of the described solid of revolution or state that it does not exist. 29. The region bounded by f(x) = x2 and the x-axis on the interval [1, ) is revolved about the x-axis.Volumes on infinite intervals Find the volume of the described solid of revolution or state that it does not exist. 30. The region bounded by f(x) = (x2 + 1)1/2 and the x-axis on the interval [2, ) is revolved about the x-axis.Volumes on infinite intervals Find the volume of the described solid of revolution or state that it does not exist. 31. The region bounded by f(x)=x+1x3 and the x-axis on the interval [1, ) is revolved about the x-axis.Volumes on infinite intervals Find the volume of the described solid of revolution or state that it does not exist. 32. The region bounded by f(x) = (x + 1)3 and the x-axis on the interval [0, ) is revolved about the y-axis.Volumes on infinite intervals Find the volume of the described solid of revolution or state that it does not exist. 33. The region bounded by f(x)=1xlnx and the x-axis on the interval [2, ) is revolved about the x-axis.Volumes on infinite intervals Find the volume of the described solid of revolution or state that it does not exist. 34. The region bounded by f(x)=xx2+13 and the x-axis on the interval [0, ) is revolved about the x-axis.Volumes with infinite integrands Find the volume of the described solid of revolution or stare that it does not exist. 51. The region bounded by f(x) = (x 1)1/4 and the x-axis on the interval (1, 2] is revolved about the x-axis.Volumes with infinite integrands Find the volume of the described solid of revolution or stare that it does not exist. 54. The region bounded by f(x) = (x + 1)3/2 and the x-axis on the interval (1, 1] is revolved about the line y = 1.Volumes with infinite integrands Find the volume of the described solid of revolution or stare that it does not exist. 55. The region bounded by f(x) = tan x and the x-axis on the interval [0, /2) is revolved about the x-axis.Volumes with infinite integrands Find the volume of the described solid of revolution or stare that it does not exist. 56. The region bounded by f(x) = ln x and the x-axis on the interval (0, 1] is revolved about the x-axis.Volumes with infinite integrands Find the volume of the described solid of revolution or stare that it does not exist. 53. The region bounded by f(x) = (4 x)1/3 and the x-axis on the interval [0, 4) is revolved about the y-axis.Volumes with infinite integrands Find the volume of the described solid of revolution or stare that it does not exist. 52. The region bounded by f(x) = (x3 1)1/4 and the x-axis on the interval (1, 2] is revolved about the y-axis.Comparison Test Determine whether the following integrals converge or diverge. 77. 1dxx3+1 Comparison Test Determine whether the following integrals converge or diverge. 78. 0dxex+x+1 Comparison Test Determine whether the following integrals converge or diverge. 79. 3dxlnx(Hint: ln x x.)Comparison Test Determine whether the following integrals converge or diverge. 80. 2x3x4x1dx Comparison Test Determine whether the following integrals converge or diverge. 81. 1sin2xx2dx Comparison Test Determine whether the following integrals converge or diverge. 82. 11ex(1+x2)dx Comparison Test Determine whether the following integrals converge or diverge. 83. 12+cosxxdx Comparison Test Determine whether the following integrals converge or diverge. 84. 12+cosxx2dx Comparison Test Determine whether the following integrals converge or diverge. 85. 01dxx1/3+x Comparison Test Determine whether the following integrals converge or diverge. 86. 01sinx+1x5dx Explain why or why not Determine whether the following statements are true and give an explanation or counterexample. a. If f is continuous and 0 f(x) g(x) on the interval [0, ), and 0g(x)dx=M, then 0f(x)dx exists. b. If limxf(x)=1, then 0f(x)dx exists. c. If 01xpdx exists, then 01xqdx exists, where q p. d. If 1xpdx exists, then 1xqdx exists, where q p. e. 1dxx3p+2 exists, for p13.Incorrect calculation a. What is wrong with this calculation? 11dxx=ln|x||11=ln1ln1=0 b. Evaluate 11dxx or show that the integral does not exist.Area between curves Let R be the region bounded by the graphs of y = eax and y = ebx, for x 0, where a b 0. Find the area of R.Area between curves Let R be the region bounded by the graphs of y = xp and y = xq, for x 1, where q p 1. Find the area of R.Regions bounded by exponentials Let a 0 and let R be the region bounded by the graph of y = eax and the x-axis on the interval [b, ). a. Find A(a, b), the area of R as a function of a and b. b. Find the relationship b = g(a) such that A(a, b) = 2. c. What is the minimum value of b (call it b) such that when b b, A(a, b) = 2 for some value of a 0?Improper integrals with infinite intervals and unbounded integrands For a real number a. suppose limxa+f(x)= or limxa+f(x)=. In these cases, the integral af(x)dx is improper for two reasons: appears in the upper limit and f is unbounded at x = a. It can be shown that af(x)dx=acf(x)dx+cf(x)dx, for any c a. Use this result to evaluate the following improper integrals. 92. 0xx(lnx+1)dx Improper integrals with infinite intervals and unbounded integrands For a real number a. suppose limxa+f(x)= or limxa+f(x)=. In these cases, the integral af(x)dx is improper for two reasons: appears in the upper limit and f is unbounded at x = a. It can be shown that af(x)dx=acf(x)dx+cf(x)dx, for any c a. Use this resjlt to evaluate the following improper integrals. 93. 1dxxx1 94E 95E 96E 97E 98E 99E The Eiffel Tower property Let R be the region between the curves y = ecx and y = ecx on the interval [a, ), where a 0 and c 0. The center of mass of R is located at (x,0), where x=axecxdxaecxdx. (The profile of the Eiffel Tower is modeled by the two exponential curves; see the Guided Project The exponential Eiffel Tower.) a. For a = 0 and c = 2, sketch the curves that define R and find the center of mass of R. Indicate the location of the center of mass. b. With a = 0 and c = 2, find equations of the lines tangent to the curves at the points corresponding to x = 0. c. Show that the tangent lines intersect at the center of mass. d. Show that this same property holds for any a 0 and any c 0; that is, the tangent lines to the curves y = ecx at x = a intersect at the center of mass of R.Many methods needed Show that 0xlnx(1+x)2dx = in the following steps. a. Integrate by parts with u = x ln x. b. Change variables by letting y = 1/x. c. Show that 01lnxx(1+x)dx=1lnxx(1+x)dx(and that both integrals converge). Conclude that 0lnxx(1+x)dx=0. d. Evaluate the remaining integral using the change of variables z = x.Laplace transforms A powerful tool in solving problems in engineering and physics is the Laplace transform. Given a function f(t), the Laplace transform is a new function F(s) defined by F(s)=0estf(t)dt, where we assume that s is a positive real number. For example, to find the Laplace transform of f(t) = et, the following improper integral is evaluated: F(s)=0estetdt=0e(s+1)tdt=1s+1. Verify the following Laplace transforms, where a is a real number. 90. f(t)=1F(s)=1s Laplace transforms A powerful tool in solving problems in engineering and physics is the Laplace transform. Given a function f(t), the Laplace transform is a new function F(s) defined by F(s)=0estf(t)dt, where we assume that s is a positive real number. For example, to find the Laplace transform of f(t) = et, the following improper integral is evaluated: F(s)=0estetdt=0e(s+1)tdt=1s+1. Verify the following Laplace transforms, where a is a real number. 91. f(t)=eatF(s)=1sa Laplace transforms A powerful tool in solving problems in engineering and physics is the Laplace transform. Given a function f(t), the Laplace transform is a new function F(s) defined by F(s)=0estf(t)dt, where we assume that s is a positive real number. For example, to find the Laplace transform of f(t) = et, the following improper integral is evaluated: F(s)=0estetdt=0e(s+1)tdt=1s+1. Verify the following Laplace transforms, where a is a real number. 92. f(t)=tF(s)=1s2 Laplace transforms A powerful tool in solving problems in engineering and physics is the Laplace transform. Given a function f(t), the Laplace transform is a new function F(s) defined by F(s)=0estf(t)dt, where we assume that s is a positive real number. For example, to find the Laplace transform of f(t) = et, the following improper integral is evaluated: F(s)=0estetdt=0e(s+1)tdt=1s+1. Verify the following Laplace transforms, where a is a real number. 93. f(t)=sinatF(s)=as2+a2 Laplace transforms A powerful tool in solving problems in engineering and physics is the Laplace transform. Given a function f(t), the Laplace transform is a new function F(s) defined by F(s)=0estf(t)dt, where we assume that s is a positive real number. For example, to find the Laplace transform of f(t) = et, the following improper integral is evaluated: F(s)=0estetdt=0e(s+1)tdt=1s+1. Verify the following Laplace transforms, where a is a real number. 94. f(t)=cosatF(s)=ss2+a2 Improper integrals Evaluate the following improper integrals (Putnam Exam, 1939). a. 13dx(x1)(3x) b. 1dxex+1+e3x Draining a tank Water is drained from a 3000-gal tank at a rate that starts at 100 gal/hr and decreases continuously by 5%/hr. If the drain is left open indefinitely, how much water drains from the tank? Can a full tank be emptied at this rate?Escape velocity and black holes The work required to launch an object from the surface of Earth to outer space is given by W=RF(x)dx, where R = 6370 km is the approximate radius of Earth, F(x) = GMm/x2 is the gravitational force between Earth and the object, G is the gravitational constant, M is the mass of Earth, m is the mass of the object, and GM = 4 1014 m3/s2. a. Find the work required to launch an object in terms of m. b. What escape velocity ve is required to give the object a kinetic energy 12mve2 equal to W? c. The French scientist Laplace anticipated the existence of black holes in the 18th century with the following argument: If a body has an escape velocity that equals or exceeds the speed of light, c = 300,000 km/s, then light cannot escape the body and it cannot be seen. Show that such a body has a radius R 2GM/c2. For Earth to be a black hole, what would its radius need to be?Adding a proton to a nucleus The nucleus of an atom is positively charged because it consists of positively charged protons and uncharged neutrons. To bring a free proton toward a nucleus, a repulsive force F(r) = kqQ/r2 must be overcome, where q = 1.6 1019 C (coulombs) is the charge on the proton, k = 9 109 N-m2/C2, Q is the charge on the nucleus, and r is the distance between the center of the nucleus and the proton. Find the work required to bring a free proton (assumed to be a point mass) from a large distance (r ) to the edge of a nucleus that has a charge Q = 50q and a radius of 6 1011 m.Gamma function The gamma function is defined by (p)=0xp1exdx, for p not equal to zero or a negative integer. a. Use the reduction formula 0xpexdx=p0xp1exdx,forp=1,2,3, to show that (p + 1) = p! (p factorial). b. Use the substitution x = u2 and the fact that 0eu2du=2 to show that (12)=.112E Explain why or why not Determine whether the following statements are true and give an explanation or counterexample. a. The integral x2e2xdx can be evaluated using integration by parts. b. To evaluate the integral dxx2100 analytically, it is best to use partial fractions. c. One computer algebra system produces 2sinxcosxdx=sin2x. Another computer algebra system produces 2sinxcosxdx=cos2x. One computer algebra system is wrong (apart from a missing constant of integration). d. 2sinxcosxdx=12cos2x+C. e. The best approach to evaluating x3+13x2dx is to use the change of variables u = x3 + 1.Basic integration techniques Use the methods introduced in Section 7.1 to evaluate the following integrals. 2. cos(x2+3)dx Basic integration techniques Use the methods introduced in Section 7.1 to evaluate the following integrals. 3. 3xx+4dx Integration by parts Use integration by parts to evaluate the following integrals. 8. 1ln23tetdt Integration by parts Use integration by parts to evaluate the following integrals. 9. x2x+2dx Basic integration techniques Use the methods introduced in Section 7.1 to evaluate the following integrals. 4. 2sin2cos22d Basic integration techniques Use the methods introduced in Section 7.1 to evaluate the following integrals. 5. 213x2+4x+13dx Trigonometric integrals Evaluate the following trigonometric integrals. 12. 2cotx3dx Trigonometric integrals Evaluate the following trigonometric integrals. 13. 0/4cos52xsin22xdx Basic integration techniques Use the methods introduced in Section 7.1 to evaluate the following integrals. 6. x3+3x2+1x3+1dx Basic integration techniques Use the methods introduced in Section 7.1 to evaluate the following integrals. 7. t12tdt(Hint: Let u=t1.)Partial fractions Use partial fractions to evaluate the following integrals. 22. 8x+52x2+3x+1dx Integration techniques Use the methods introduced in Sections 8.1 through 8.5 to evaluate the following integrals. 13. 0e3xsin6xdx Integration techniques Use the methods introduced in Sections 8.1 through 8.5 to evaluate the following integrals. 14. 23(6w10)e3wdw Integration techniques Use the methods introduced in Sections 8.1 through 8.5 to evaluate the following integrals. 15. 123x5+48x3+3x2+16x3+16xdx Integration techniques Use the methods introduced in Sections 8.1 through 8.5 to evaluate the following integrals. 16. x6+2x4+x2+1(x2+1)2dx Partial fractions Use partial fractions to evaluate the following integrals. 23. 2x2+7x+4x3+2x2+2xdx Integration techniques Use the methods introduced in Sections 8.1 through 8.5 to evaluate the following integrals. 18. 02x+13x2+6dx Integration techniques Use the methods introduced in Sections 8.1 through 8.5 to evaluate the following integrals. 19. x2+4x+7(x+3)3dx Integration techniques Use the methods introduced in Sections 8.1 through 8.5 to evaluate the following integrals. 20. 1x2xdx Trigonometric substitutions Evaluate the following integrals using a trigonometric substitution. 19. 22x21xdx Integration techniques Use the methods introduced in Sections 8.1 through 8.5 to evaluate the following integrals. 22. tan35d Trigonometric integrals Evaluate the following trigonometric integrals. 15. sin4tcos6tdt Integration techniques Use the methods introduced in Sections 8.1 through 8.5 to evaluate the following integrals. 24. dx18xx2 Integration techniques Use the methods introduced in Sections 8.1 through 8.5 to evaluate the following integrals. 25. 3/21dx4x2+12x+10 Integration techniques Use the methods introduced in Sections 8.1 through 8.5 to evaluate the following integrals. 26. tan10xdx Integration techniques Use the methods introduced in Sections 8.1 through 8.5 to evaluate the following integrals. 27. dw(w+1)2w2+2w8 Integration techniques Use the methods introduced in Sections 8.1 through 8.5 to evaluate the following integrals. 28. sin3xcos5xdx Integration techniques Use the methods introduced in Sections 8.1 through 8.5 to evaluate the following integrals. 29. cos4xsin6xdx Integration techniques Use the methods introduced in Sections 8.1 through 8.5 to evaluate the following integrals. 30. xtan17xdx Integration techniques Use the methods introduced in Sections 8.1 through 8.5 to evaluate the following integrals. 31. xsinh2xdx Integration techniques Use the methods introduced in Sections 8.1 through 8.5 to evaluate the following integrals. 32. csc26xcot6xdx Integration techniques Use the methods introduced in Sections 8.1 through 8.5 to evaluate the following integrals. 33. tan33sec33d Integration techniques Use the methods introduced in Sections 8.1 through 8.5 to evaluate the following integrals. 34. w3369w2dw Integration techniques Use the methods introduced in Sections 8.1 through 8.5 to evaluate the following integrals. 35. x34x2+16dx Partial fractions Use partial fractions to evaluate the following integrals. 24. 1/21/2u2+1u21du Partial fractions Use partial fractions to evaluate the following integrals. 25. 3x3+4x2+6x(x+1)2(x2+4)dx Integration techniques Use the methods introduced in Sections 8.1 through 8.5 to evaluate the following integrals. 38. /4/2xcsc2xdx Integration techniques Use the methods introduced in Sections 8.1 through 8.5 to evaluate the following integrals. 39. dt2+et Miscellaneous Integrals Evaluate the following integrals analytically. 38. x24x+4dx Miscellaneous Integrals Evaluate the following integrals analytically. 39. d1+cos Miscellaneous Integrals Evaluate the following integrals analytically. 40. x2cosxdx Miscellaneous Integrals Evaluate the following integrals analytically. 41. exsinxdx Miscellaneous Integrals Evaluate the following integrals analytically. 42. 1ex2lnxdx 2-74. Integration techniques Use the methods introduced in Sections 8.1 through 8.5 to evaluate the following integrals. 45. Integration techniques Use the methods introduced in Sections 8.1 through 8.5 to evaluate the following integrals. 46. x3+4x2+12x+4(x2+4x+10)2dx Integration techniques Use the methods introduced in Sections 8.1 through 8.5 to evaluate the following integrals. 47. cos24d Integration techniques Use the methods introduced in Sections 8.1 through 8.5 to evaluate the following integrals. 48. sin3xcos63xdx Integration techniques Use the methods introduced in Sections 8.1 through 8.5 to evaluate the following integrals. 49. sec492ztan2zdz Integration techniques Use the methods introduced in Sections 8.1 through 8.5 to evaluate the following integrals. 50. 0/6cos43xdx Integration techniques Use the methods introduced in Sections 8.1 through 8.5 to evaluate the following integrals. 51. 0/4sin54d Integration techniques Use the methods introduced in Sections 8.1 through 8.5 to evaluate the following integrals. 52. tan42udu Integration techniques Use the methods introduced in Sections 8.1 through 8.5 to evaluate the following integrals. 53. dxx+x3 (Hint: Let u = x6.)Integration techniques Use the methods introduced in Sections 8.1 through 8.5 to evaluate the following integrals. 54. dx9x225, x53 Integration techniques Use the methods introduced in Sections 8.1 through 8.5 to evaluate the following integrals. 55. dyy2182y2 Integration techniques Use the methods introduced in Sections 8.1 through 8.5 to evaluate the following integrals. 56. 03/2x2(1x2)3/2dx Integration techniques Use the methods introduced in Sections 8.1 through 8.5 to evaluate the following integrals. 57. 03/249+4x2dx Integration techniques Use the methods introduced in Sections 8.1 through 8.5 to evaluate the following integrals. 58. (1u2)5/2u8du Integration techniques Use the methods introduced in Sections 8.1 through 8.5 to evaluate the following integrals. 59. sech2xsinhxdx Integration techniques Use the methods introduced in Sections 8.1 through 8.5 to evaluate the following integrals. 60. x2coshxdx Integration techniques Use the methods introduced in Sections 8.1 through 8.5 to evaluate the following integrals. 61. 0ln(3+2)coshx4sinh2xdx Miscellaneous Integrals Evaluate the following integrals analytically. 58. sinh1xdx Miscellaneous Integrals Evaluate the following integrals analytically. 59. dxx22x15 Miscellaneous Integrals Evaluate the following integrals analytically. 60. dxx32x2 Miscellaneous Integrals Evaluate the following integrals analytically. 61. 01dy(y+1)(y2+1)Miscellaneous Integrals Evaluate the following integrals analytically. 62. 06x1+x6dx Miscellaneous Integrals Evaluate the following integrals analytically. 63.Preliminary work Make a change of variables or use an algebra step before evaluating the following integrals. 64. 11dxx2+2x+5 Preliminary work Make a change of variables or use an algebra step before evaluating the following integrals. 65. dxx2x2 Preliminary work Make a change of variables or use an algebra step before evaluating the following integrals. 66. 2x24xx24dx Preliminary work Make a change of variables or use an algebra step before evaluating the following integrals. 67. 2x24xx24dx Preliminary work Make a change of variables or use an algebra step before evaluating the following integrals. 68. 1/121/4dxx(1+4x)Preliminary work Make a change of variables or use an algebra step before evaluating the following integrals. 69. e2t(1+e4t)3/2dt Integration techniques Use the methods introduced in Sections 8.1 through 8.5 to evaluate the following integrals. 74. dx1+x Evaluate the integral in part (a) and then use this result to evaluate the integral in part (b). a. exsecextanexdx b. e2xsecextanexdx Table of integrals Use a table of integrals to evaluate the following integrals. 26. x(2x+3)5dx Table of integrals Use a table of integrals to evaluate the following integrals. 27. dxx4x6 Table of integrals Use a table of integrals to evaluate the following integrals. 28. 0/2d1+sin2 Table of integrals Use a table of integrals to evaluate the following integrals. 29. sec5xdx Table of integrals Use a table of integrals to evaluate the following integrals. 80. 01exe2x16e2xdx Table of integrals Use a table of integrals to evaluate the following integrals. 81. 0/2cosxln3(1+sinx)dx Improper integrals Evaluate the following integrals. 34. 1dx(x1)4 Improper integrals Evaluate the following integrals. 35. 0xexdx Improper integrals Evaluate the following integrals. 36. 0sec2xdx Improper integrals Evaluate the following integrals. 37. 03dx9x2 Improper integrals Evaluate the following integrals or show that the integral diverges. 86. x31+x8dx Improper integrals Evaluate the following integrals or show that the integral diverges. 87. (Hint: First write the integral as the sum of two improper integrals.) Improper integrals Evaluate the following integrals or show that the integral diverges. 88. 2x2+2x+2dx Comparison Test Determine whether the following integrals converge or diverge. 89.1dxx5+x4+x3+1 Comparison Test Determine whether the following integrals converge or diverge. 90. 01dxx+sinx Comparison Test Determine whether the following integrals converge or diverge. 91. 3x3x71dx Integral with a parameter For what values of p does the integral 2dxxlnpx converge and what is its value (in terms of p)?Approximations Use a computer algebra system to approximate the value of the following integrals. 93. 11e2x2dx Approximations Use a computer algebra system to approximate the value of the following integrals. 94. 1ex3ln3xdx 95-98. Numerical integration Estimate the following integrals using the Midpoint Rule M(n), the Trapezoidal Rule T(n), and Simpson’s Rule S(n) for the given values of n. 95. n = 4 (see figure) Numerical integration Estimate the following integrals using the Midpoint Rule M(n), the Trapezoidal Rule T(n), and Simpsons Rule S(n) for the given values of n. 96. 13dxx3+x+1; n 4 Numerical integration Estimate the following integrals using the Midpoint Rule M(n), the Trapezoidal Rule T(n), and Simpsons Rule S(n) for the given values of n. 97. 01tanx2dx; n = 40 Numerical integration Estimate the following integrals using the Midpoint Rule M(n), the Trapezoidal Rule T(n), and Simpsons Rule S(n) for the given values of n. 98. 18esinxdx; n = 60 Improper integrals by numerical methods Use the Trapezoid Rule (Section 8.8) to approximate 0Rex2 with R = 2, 4, and 8. For each value of R, take n = 4, 8, 16, and 32, and compare approximations I=0ex2dx.Comparing areas Show that the area of the region bounded by the graph of y = aeax and the x-axis on the interval [0, ) is the same for all values of a 0.Comparing volumes Let R be the region bounded by the graph of y = sin x and the x-axis on the interval [0, ]. Which is greater, the volume of the solid generated when R is revolved about the x-axis or the y-axis?Volumes The region R is bounded by the curve y = ln x and the x-axis on the interval [1, e]. Find the volume of the solid that is generated when R is revolved in the following ways. 72. About the y-axis Volumes The region R is bounded by the curve y = ln x and the x-axis on the interval [1, e]. Find the volume of the solid that is generated when R is revolved in the following ways. 71. About the x-axis Volumes The region R is bounded by the curve y = ln x and the x-axis on the interval [1, e]. Find the volume of the solid that is generated when R is revolved in the following ways. 74. About the line y = 1 Volumes The region R is bounded by the curve y = ln x and the x-axis on the interval [1, e]. Find the volume of the solid that is generated when R is revolved in the following ways. 73. About the line x = 1 Arc length Find the length of the curve y=x23x2+32sin1x3 from x = 0 to x = 1.Zero log integral It is evident from the graph of y = ln x that for every real number a with 0 a 1, there is a unique real number b = g(a) with b 1, such that ablnxdx=0 (the net area bounded by the graph of y = ln x on [a, b] is 0). a. Approximate b=g(12). b. Approximate b=g(13). c. Find the equation satisfied by all pairs of numbers (a, b) such that b = g(a). d. Is g an increasing or decreasing function of a? Explain.Arc length Find the length of the curve y = ln x on the interval [1, e2].Average velocity Find the average velocity of a projectile whose velocity over the interval 0 t is given by v(t) = 10 sin 3t.Comparing distances Starting at the same time and place (t = 0 and s = 0), the velocity of car A (in mi/hr) is given by u(t) = 40/(t + 1) and the velocity of car B (in mi/hr) is given by v(t) = 40et/2. a. After t = 2 hr, which car has traveled farther? b. After t = 3 hr, which car has traveled farther? c. If allowed to travel indefinitely (t ), which car will travel a finite distance?Traffic flow When data from a traffic study are fitted to a curve, the flow rate of cars past a point on a highway is approximated by R(t) = 800tet/2 cars/hr. How many cars pass the measuring site during the time interval 0 t 4?Comparing integrals Graph the functions f(x) = 1/x2, g(x) = (cos x)/x2, and h(x) = (cos2 x)/x2. Without evaluating integrals and knowing that 1f(x)dx has a finite value, determine whether 1g(x)dx and 1h(x)dx have finite values.A family of logarithm integrals Let I(p)=1elnxxpdx, where p is a real number. a. Find an expression for I(p), for all real values of p. b. Evaluate limpI(p) and limpI(p). c. For what value of p is I(p) = 1?114RE Best approximation Let I=01x2xlnxdx. Use any method you choose to find a good approximation to I. You may use the facts that limx0+x2xlnx=0 and limx1x2xlnx=1.Numerical integration Use a calculator to determine the integer n that satisfies 01/2ln(1+2x)xdx=2n.Numerical integration Use a calculator to determine the integer n that satisfies 01sin1xxdx=ln2n.Two worthy integrals a. Let I(a)=0dx(1+xa)(1+x2), where a is a real number. Evaluate I(a) and show that its value is independent of a. (Hint: Split the integral into two integrals over [0, 1] and [1, ); then use a change of variables to convert the second integral into an integral over [0, 1].) b. Let f be any positive continuous function on [0, /2]. Evaluate 0/2f(cosx)f(cosx)+f(sinx)dx. (Hint: Use the identity cos (/2 x) = sin x.) (Source: Mathematics Magazine 81, 2, Apr 2008)Comparing volumes Let R be the region bounded by y = ln x, the x-axis, and the line x = a, where a 1. a. Find the volume V1(a) of the solid generated when R is revolved about the x-axis (as a function of a). b. Find the volume V2(a) of the solid generated when R is revolved about the y-axis (as a function of a). c. Graph and V1 and V2. For what values of a 1 is V1(a) V2(a)?Equal volumes a. Let R be the region bounded by the graph of f(x) = xp and the x-axis, for x 1. Let V1 and V2 be the volumes of the solids generated when R is revolved about the x-axis and the y-axis, respectively, if they exist. For what values of p (if any) is V1 = V2? b. Repeat part (a) on the interval (0, 1].Equal volumes Let R1 be the region bounded by the graph of y = eax and the x-axis on the interval [0, b] where a 0 and b 0. Let R2 be the region bounded by the graph of y = eax and the x-axis on the interval [b, ). Let V1 and V2 be the volumes of the solids generated when R1 and R2 are revolved about the x-axis. Find and graph the relationship between a and b for which V1 = V2.Comparing areas The region R1 is bounded by the graph of y = tan x and the x-axis on the interval [0, /3]. The region R2 is bounded by the graph of y = sec x and the x-axis on the interval [0, /6]. Which region has the greater area?Region between curves Find the area of the region bounded by the graphs of y = tan x and y = sec x on the interval [0, /4].Mercator map projection The Mercator map projection was proposed by the Flemish geographer Gerard us Mercator (15121594). The stretching factor of the Mercator map as a function of the latitude is given by the function G()=0secxdx. Graph G, for 0 /2. (See the Guided Project Mercator projections for a derivation of this integral.)Wallis products Complete the following steps to prove a well-known formula discovered by the 17th-century English mathematician John Wallis. Use a reduction formula to show that , for any integer m ≥ 2. Show that , for any integer n ≥ 1. The product on the right is called a Wallis product. Show that , for any integer n ≥ 1. The product on the right is another example of a Wallis product. Use the inequality 0 ≤ sin x ≤ 1 on [0, π] to show that , for any integer m ≥ 0. Use part (d) to show that for any integer n ≥ 2. Show that , for any integer n ≥ 2. Use part (f) to conclude that . What are the orders of the equations in Example 2? Are they linear or nonlinear? Example 2 General Solutions Find the general solution of the following differential equations. a. y(t) = 5 cos t + 6 sin 3t b. y(t) = 10t3 144t7 + 12t What is the solution of the initial value problem in Example 3 with the initial condition y(0) = 16? Example 3 An Initial Value Problem Solve the initial value problem y(t) = 10 et/2, y(0) = 4.Solve the initial value problem in Example 4a with an initial condition of y(1) = 4. What is the domain of the solution? Example 4 Determining The Domain Solve each initial value problem and determine the domain of the solution. a. y(t)=1+2t3, y(1) = 4 Suppose the initial conditions in Example 5a are v0 = 14.7 in/s and s0 = 49 m. Write the position function s(t), and state its domain. At what time will the stone reach its maximum height? What is the maximum height at that time? Example 5 Motion In A Gravitational Field A stone is launched vertically upward with a velocity of v0 m/s from a point s0 meters above the ground, where v0 0 and s0 0. Assume the stone is launched at time t = 0 and that s(t) is the position of the store at time t 0; the positive s-axis points upward with the origin at the ground. By Newtons Second Law of Motion, assuming no air resistance, the position of the stone is governed by the differential equation s(t) = g, where g = 9.8 m/s2 is the acceleration due to gravity (in the downward direction) a. Find the position s(t) of the stone for all times at which the stone is above the ground.In Example 7, if the height function were given by h(t) = (4.2 0.14t)2, at what time would the tank be empty? What does your answer say about the domain of this solution? Example 7 Flow From a Tank Imagine a large cylindrical tank with cross-sectional area A. The bottom of the tank has a circular drain with cross-sectional area a. Assume the tank is initially filed with water to a height (in meters) of h(0) = H (Figure 9.7). According to Torricellis law, the height of the water t seconds after the drain is opened is described by the differential equation h(t)=kh, where t 0, k=aA2g, Figure 9.7 and g = 9.8 m/s2 is the acceleration due to gravity. a. According to the differential equation, is h an increasing or decreasing function of t, for t 0? b. Verify by substitution that the solution of the initial value problem is h(t)=(Hkt2)2. c. Graph the solution for H = 1.44 m, A = 1 m2, and a = 0.05 m2, d. After how many seconds is the tank in part (c) empty?Consider the differential equation y(t) + 9y(t) = 10. a. How many arbitrary constants appear in the general solution of the differential equation? b. Is the differential equation linear or nonlinear?If the general solution of a differential equation is y(t) = Ce3t + 10, what is the solution that satisfies the initial condition y(0) = 5?Does the function y(t) = 2t satisfy the differential equation y(t) + y(t) = 2?Does the function y(t) = 6e3t satisfy the initial value problem y(t) 3y(t) = 0, y(0) = 6?The solution to the initial value problem y(t) = 2 sec t tan t, y() = 2 is y(t) = 2 sec t. What is the domain of this solution? (Hint: Sketch a graph of y).Explain why the graph of the solution to the initial value problem y(t)=t21t, y( 1) = ln 2 cannot cross the line t = 1.Verifying general solutions Verify that the given function y is a solution of the differential equation that follows it. Assume that C is an arbitrary constant. 7.y(t) = Ce5t; y(t) + 5y(t) = 0 Verifying general solutions Verify that the given function y is a solution of the differential equation that follows it. Assume that C is an arbitrary constant. 8.y(t) = Ct3; ty(t) + 3y(t) = 0 Verifying general solutions Verify that the given function y is a solution of the differential equation that follows it. Assume that C is an arbitrary constant. 9.y(t) = C1 sin 4t + C2 cos 4t; y(t) + 16y(t) = 0 Verifying general solutions Verify that the given function y is a solution of the differential equation that follows it. Assume that C is an arbitrary constant. 10.y(x) = C1ex + C2 ex; y(x) y(x) = 0 Verifying general solutions Verify that the given function is a solution of the differential equation that follows it. 44.u(t)=Ce1/(4t4);u(t)+1t5u(t)=0 Verifying general solutions Verify that the given function is a solution of the differential equation that follows it. 45.u(t)=C1et+C2tet;u(t)2u(t)+u(t)=0 Verifying general solutions Verify that the given function is a solution of the differential equation that follows it. 46.g(x)=C1e2x+C2xe2x+2;g(x)+4g(x)+4g(x)=8 Verifying general solutions Verify that the given function is a solution of the differential equation that follows it. 47.u(t)=C1t2+C2t3;t2u(t)4tu(t)+6u(t)=0 Verifying general solutions Verify that the given function is a solution of the differential equation that follows it. 48.u(t)=C1t5+C2t4t3;t2u(t)20u(t)=14t3 Verifying general solutions Verify that the given function is a solution of the differential equation that follows it. 49.z(t)=C1et+C2e2t+C3e3tet;z(t)+2z(t)5z(t)6z(t)=8et Verifying solutions of initial value problems Verify that the given function y is a solution of the initial value problem that follows it. 11.y(t) = 16e2t 10; y(t) 2y(t) = 20, y(0) = 6 Verifying solutions of initial value problems Verify that the given function y is a solution of the initial value problem that follows it. 12.y(t) = 8t6 3; ty(t) 6y(t) = 18, y(1) = 5 Verifying solutions of initial value problems Verify that the given function y is a solution of the initial value problem that follows it. 13.y(t) = 3 cos 3t; y(t) + 9y(t) = 0, y(0) = 3, y(0) = 0 Verifying solutions of initial value problems Verify that the given function y is a solution of the initial value problem that follows it. 14.y(x)=14(e2xe2x);y(x)4y(x)=0,y(0)=0,y(0)=1 Finding general solutions Find the general solution of each differential equation. Use C, C1, C2, to denote arbitrary constants. 15.y(t) = 3 + e2t Finding general solutions Find the general solution of each differential equation. Use C, C1, C2, to denote arbitrary constants. 16.y(t)=12t520t4+26t2 Finding general solutions Find the general solution of each differential equation. Use C, C1, C2, to denote arbitrary constants. 17.y(x) = 4 tan 2x 3 cos x Finding general solutions Find the general solution of each differential equation. Use C, C1, C2, to denote arbitrary constants. 18.p(x)=16x95+14x6 Finding general solutions Find the general solution of each differential equation. Use C, C1, C2, to denote arbitrary constants. 19.y(t) = 60t4 4 + 12t3 Finding general solutions Find the general solution of each differential equation. Use C, C1, C2, to denote arbitrary constants. 20.y(t) = 15e3t + sin 4t Finding general solutions Find the general solution of each differential equation. Use C, C1, C2, to denote arbitrary constants. 21.u(x)=55x9+36x721x5+10x3 Finding general solutions Find the general solution of each differential equation. Use C, C1, C2, ... to denote arbitrary constants. v(x)=5x2 General solutions Find the general solution of the following differential equations. 37.u(x)=2(x1)x2+4 General solutions Find the general solution of the following differential equations. 36.y(t) = t ln t + 1 General solutions Find the general solution of the following differential equations. 39.y(x)=x(1x2)3/2 General solutions Find the general solution of the following differential equations. 38.v(t)=4t24 Solving initial value problems Solve the following initial value problems. 23.y(t)=1+et,y(0)=4 Solving initial value problems Solve the following initial value problems. 24.y(t) = sin t + cos 2t, y(0) = 4 Solving initial value problems Solve the following initial value problems. 25.y(x) = 3x2 3x4, y(1) = 0 Solving initial value problems Solve the following initial value problems. 26.y(x) = 4 sec2 2x, y(0) = 8 Solving initial value problems Solve the following initial value problems. 27.y(t) = 12t 20t3, y(0) = 1, y(0) = 0 Solving initial value problems Solve the following initial value problems. 28.u(x) = 4e2x 8e2x, u(0) = 1, u(0) = 3 Solving initial value problems Find the solution of the following initial value problems. 43.y(t) = tet, y(0) = 0, y(0) = 1 Solving initial value problems Find the solution of the following initial value problems. 40.y(t) = tet, y(0) = 1 Solving initial value problems Find the solution of the following initial value problems. 41.u(x)=1x2+164,u(0)=2 Solving initial value problems Find the solution of the following initial value problems. 42.p(x)=2x2+x,p(1)=0 Motion in a gravitational field An object is fired vertically upward with an initial velocity v(0) = v0 from an initial position s(0) = s0. a.For the following values of v0 and s0, find the position and velocity functions for all times at which the object is above the ground. b.Find the time at which the highest point of the trajectory is reached and the height of the object at that time. 29.v0=29.4m/s,s0=30m 44E Harvesting problems Consider the harvesting problem in Example 6. If r = 0.05 and p0 = 1500, for what values of H is the amount of the resource increasing? For what value of H is the amount of the resource constant? If H = 100, when does the resource vanish? Example 6 A Harvesting Model A simple model of o harvested resource (for example, timber or fish) assumes a competitor between the harvesting and the natural growth of the resource. This process may be described by the differential equation where p(t) is the amount (or population) of the resource at time t 0, r 0 is the natural growth rate of the resource, and H 0 is the constant harvesting rate. An initial condition p(0) = p0 is also specified to create an initial value problem. Notice that the rate of change p(t) has a positive contribution from the natural growth rate one a negative contribution from the harvesting term. a. For given constants p0, r, and H. verify that the function p(t)=(p0Hr)ert+Hr is a solution of the initial value problem b. Let p0 = 1000 and r = 0.1. Graph the solutions for the harvesting rates H = 50, 90, 130, and 170. Describe and interpret the four curves. c. What value of H gives a constant value of p, for all t 0?Harvesting problems Consider the harvesting problem in Example 6. If r = 0.05 and H = 500, for what values of p0 is the amount of the resource decreasing? For what value of p0 is the amount of the resource constant? If p0 = 9000, when does the resource vanish? Example 6 A Harvesting Model A simple model of o harvested resource (for example, timber or fish) assumes a competitor between the harvesting and the natural growth of the resource. This process may be described by the differential equation where p(t) is the amount (or population) of the resource at time t 0, r 0 is the natural growth rate of the resource, and H 0 is the constant harvesting rate. An initial condition p(0) = p0 is also specified to create an initial value problem. Notice that the rate of change p(t) has a positive contribution from the natural growth rate one a negative contribution from the harvesting term. a. For given constants p0, r, and H. verify that the function p(t)=(p0Hr)ert+Hr is a solution of the initial value problem b. Let p0 = 1000 and r = 0.1. Graph the solutions for the harvesting rates H = 50, 90, 130, and 170. Describe and interpret the four curves. c. What value of H gives a constant value of p, for all t 0?Draining tanks Consider the tank problem in Example 7. For the following parameter values, find the water height function. Then determine the approximate time at which the tank is first empty and graph the solution. 47. H = 1.96 m, A = 1.5 m2, a = 0.3 m2 Example 7 Flow From a Tank Imagine a large cylindrical tank with cross-sectional area A. The bottom of the tank has a circular drain with cross-sectional area a. Assume the tank is initially filed with water to a height (in meters) of h(0) = H (Figure 9.7). According to Torricellis law, the height of the water t seconds after the drain is opened is described by the differential equation h(t)=kh, where t 0, k=aA2g, Figure 9.7 and g = 9.8 m/s2 is the acceleration due to gravity. a. According to the differential equation, is h an increasing or decreasing function of t, for t 0? b. Verify by substitution that the solution of the initial value problem is h(t)=(Hkt2)2. c. Graph the solution for H = 1.44 m, A = 1 m2, and a = 0.05 m2, d. After how many seconds is the tank in part (c) empty?48E Explain why or why not Determine whether the following statements arc true and give an explanation or counterexample. a.The general solution of the differential equation y(t) = 1 is y(t) = t. b.The differential equation y(t) y(t)y(t) = 0 is second order and linear. c.To find the solution of an initial value problem, you usually begin by finding a general solution of the differential equation.A second-order equation Consider the differential equation y(t) k2y(t) = 0, where k 0 is a real number. a.Verify by substitution that when k = 1, a solution of the equation is y(t) = C1et + C2et. You may assume that this function is the general solution. b.Verify by substitution that when k = 2, the general solution of the equation is y(t) = C1e2t + C2e2t. c.Give the general solution of the equation for arbitrary k 0 and verify your conjecture. d.For a positive real number k, verify that the general solution of the equation may also be expressed in the form y(t) = C1 cosh kt + C2 sinh kt, where cosh and sinh are the hyperbolic cosine and hyperbolic sine, respectively.Another second-order equation Consider the differential equation y(t) + k2y(t) = 0, where k is a positive real number. a.Verify by substitution that when k = 1, a solution of the equation is y(t) = C1 sin t + C2 cos t. You may assume that this function is the general solution. b.Verify by substitution that when k = 2, the general solution of the equation is y(t) = C1 sin 2t + C2 cos 2t. c.Give the general solution of the equation for arbitrary k 0 and verify your conjecture.Drug infusion The delivery of a drug (such as an antibiotic) through an intravenous line may be modeled by the differential equation m(t) + km(t) = I, where m(t) is the mass of the drug in the blood at time t 0, k is a constant that describes the rate at which the drug is absorbed, and I is the infusion rate. a.Show by substitution that if the initial mass of drug in the blood is zero (m(0) = 0), then the solution of the initial value problem is m(t)=Ik(1ekt). b.Graph the solution for I = 10 mg/hr and k = 0.05 hr1. c.Evaluate limtm(t), the steady-state drug level, and verify the result using the graph in part (b).Logistic population growth Widely used models for population growth involve the logistic equation P(t)=rP(1PK), where P(t) is the population, for t 0, and r 0 and K 0 are given constants. a.Verify by substitution that the general solution of the equation is P(t)=K1+Cert, where C is an arbitrary constant. b.Find that value of C that corresponds to the initial condition P(0) = 50. c.Graph the solution for P(0) = 50, r = 0.1, and K = 300. d.Find limtP(t) and check that the result is consistent with the graph in part (c).Free fall One possible model that describes the free fall of an object in a gravitational field subject to air resistance uses the equation v(t) = g bv, where v(t) is the velocity of the object for t 0, g = 9.8 m/s2 is the acceleration due to gravity, and b 0 is a constant that involves the mass of the object and the air resistance. a.Verify by substitution that a solution of the equation, subject to the initial condition v(0) = 0, isv(t)=gb(1ebt). b.Graph the solution with b = 0.1 s1. c.Using the graph in part (c), estimate the terminal velocitylimtv(t).Chemical rate equations The reaction of certain chemical compounds can be modeled using a differential equation of the form y(t) = kyn(t), where y(t) is the concentration of the compound for t 0, k 0 is a constant that determines the speed of the reaction, and n is a positive integer called the order of the reaction. Assume that the initial concentration of the compound is y(0) = y0 0. a.Consider a first-order reaction (n = 1) and show that the solution of the initial value problem is y(t) = y0ekt. b.Consider a second-order reaction (n = 2) and show that the solution of the initial value problem is y(t)=y0y0kt+1. c.Let y0 = 1 and k = 0.1. Graph the first-order and second-order solutions found in parts (a) and (b). Compare the two reactions.Tumor growth The growth of cancer tumors may be modeled by the Gompertz growth equation. Let M(t) be the mass of a tumor, for t 0. The relevant initial value problem is dMdt=rM(t)ln(M(t)K),M(0)=M0, where r and K are positive constants and 0 M0 K. a.Show by substitution that the solution of the initial value problem is M(t)=K(M0K)exp(rt). b.Graph the solution for M0 = 100 and r = 0.05. c.Using the graph in part (b), estimate limtM(t), the limiting size of the tumor.

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