Math 132, Fall 2003Final ExamName: ID#No calculators with a CAS are allowed. Be sure your calculator is set for ”radians”,not ”degrees”, if you do any calculus computations with trig functions. For Parts I,II, and III, please mark your answer on the answer card.Part I, Multiple Choice, 5 points/problem:1. Solve the initial value problemdydx= 4x3y, y(0) = 6.A) y = 6ex4B) y = e7x4C) y = ex4D) y = 4ex7E) y = 7x4F) y = x4G) y = x7H) y = 4x7I) y = 4xe7xJ) No Solution12. Find the radius of convergence and the interval of convergence of the series∞Xn=1nbn(x − 1)n, b > 0.A) R = 0; I = {1}B) R = 1; I = (−1, 1)C) R = 1; I = [−1, 1]D) R = 1; I = (−1, 1]E) R = 1; I = [−1, 1)F) R = b; I = [1 − b, 1 + b)G) R = b; I = (1 − b, 1 + b]H) R = b; I = (1 − b, 1 + b)I) R = b; I = [1 − b, 1 + b]J) R = ∞; I = (−∞, ∞)23. Find the volume of the following solid: The base is the region in the xy− plane bounded by thecurvesx + 3y = 3, x − 3y = 3, and x = 0.Cross sections perpendicular to the x-axis are isosceles right triangles with one side on the base.(Answers are in terms of units3.)A) 1 B)13C) 4 D)32E)43F) 3 G)23H) 6 I) 2 J)3434. A bacteria population grows with constant relative growth rate (i.e. its growth rate is propor-tional to its size). At noon, there are 1000 bacteria. By 3:00 PM, there are 10,000 bacteria. Atapproximately what time will the population be 100,000?A) 5:00 PM B) 5:30 PM C) 6:00 PM D) 6:30 PM E) 7:00 PMF) 7:30 PM G) 8:00 PM H) 8:30 PM I) 9:00 PM J) 9:30 PM5. Determine the values of p for which the integralZ∞2(ln x)pxdx converges.A) p ≤ −1 B) p ≥ −1 C) p < 0 D) p > 0 E) p < −1F) p > −1 G) p ≤ 0 H) p ≥ 0 I) −1 < p < 0 J) p < 146. Which of the following describes the area bounded by the curves y =10xand y = 7 − x.A) 7 −x22−10x252B) −1 +10x252C) 7x −x22− 10 ln x52D)10x2− 7 +x2252E) π−13(7 − x)3+10x52F) π(7 − x)2−10x252G) π10x2− (7 − x)252H) 10 ln x − 7x +x2252I)10x2+ 152J) 7x −x225257. Using an infinite series, we can approximateZ10x cos√xdx. How many non-zero terms mustwe add so the approximation has error less than 0.00001?A) 1 term B) 2 terms C) 3 terms D) 4 terms E) 5 termsF) 6 terms G) 7 terms H) 8 terms I) 9 terms J) 10 terms8. A rocket full of fuel weighs 10,000 lbs at launch. After launch, the rocket gains altitude andloses weight as the fuel burns. Assume that the rocket loses 1 lb of fuel for every 15 feet of altitudegained. What is the work done in raising the rocket 30,000 feet? Answers are given in millions offoot-pounds. (Note: You need not account for the change in gravity that occurs with the changein altitude.)A) 10 B) 30 C) 95 D) 105 E) 145F) 255 G) 270 H) 300 I) 330 J) 45069. Find the power series representation for the function f (x) =1(4 + x)2.A)∞Xn=0(−1)nxn+1(n + 1)4n+2=x42−x22 (43)+x33 (44)−x44 (45)+ ···B)∞Xn=0(−1)nxn4n+2=142−x43+x244−x345+ ···C)∞Xn=1nxn4n=x4+2x242+3x343+4x444+ ···D)∞Xn=1(−1)n+1xn4n+1=x42−x243+x344−x445+ ···E)∞Xn=0(−1)nxn+14n= x −x24+x342−x443+ ···F)∞Xn=0(n + 1)xnn!4n+2=142+243x +32!44x2+43!45x3+ ···G)∞Xn=1nxn4n=x4+2x242+3x343+4x444+ ···H)∞Xn=1xn−14n+1=142+x43+x244+x345+ ···I)∞Xn=1(−1)n+114n+1nxn−1=142−243x +344x2−445x3+ ···J)∞Xn=0(−1)nxn4n= 1 −x4+x242−x343+ ···710. Determine which of the following series are absolutely convergent (AC), conditionally convergent(CC), or divergent (D):(i)∞Xn=1sinnπ3n√n(ii)∞Xn=1(−1)nn!en(iii)∞Xn=2(−1)nn2n3− 1A) i) CC ii) AC iii) DB) i) CC ii) CC iii) ACC) i) AC ii) D iii) DD) i) D ii) CC iii) ACE) i) CC ii) AC iii) ACF) i) AC ii) D iii) CCG) i) AC ii) AC iii) CCH) i) D ii) D iii) DI) i) AC ii) AC iii) DJ) i) CC ii) D iii) CC811. Determine the following:(i) Find the sum of the series1 − e +e22!−e33!+e44!− ···(ii) The function f (x) = ln (5 + 2x) has a Taylor series expansion centered at the pointx = 1, soln (5 + 2x) =∞Xn=0cn(x − 1)n.Find c2.A) i)1eeii) c2= −249B) i) eeii) c2= −249C) i) cos (ee) ii) c2= −225D) i) sin (ee) ii) c2= −225E) i)11 + eii) c2=20343F) i)11 − eii) c2= ln 7G) i) sin1eeii) c2=25H) i) cos1eeii) c2=25I) i) eeii) c2=27J) i)1eeii) c2=27912. Find the arc length of the curve defined by the parametric equationsx(t) =Zt1cos wwdw , y(t) =Zt1sin wwdw , 1 ≤ t ≤π2.A) ∞ B) 1 C) 1 −2πD) cos 1 + sin 1 E)4π2− cos 1 − sin 1F)π2− 1 G) ln3π2H) 1 −16π3I) 1 −4π2J) lnπ213. If g(x) =Z3x2p1 + t3dt, then what is the value of g0(1)?A) -6√1 + 36B) −√28 C) −32√2D) −2√2 E) −√2F)23(2)3/2G)12√2H)32√2I)√2 J)√1 + 361014. Suppose we approximate the function f(x) = ex/4using the Taylor polynomial of degree 2centered at a = 0. How accurate is this approximation when 0 ≤ x ≤ 2. (Pick the “best” answerbelow.)A) |R2(x)| ≤ .20609016 B) |R2(x)| ≤ .03434836 C) |R2(x)| ≤ .18976224D) |R2(x)| ≤ .00347222 E) |R2(x)| ≤ 1.29319601 F) |R2(x)| ≤ .02333333G) |R2(x)| ≤ .01717418 H) |R2(x)| ≤ .076531128 I) |R2(x)| ≤ .57358037J) |R2(x)| ≤ 1.648721271115. Calculate the mean of a random variable with the probability density functionf(x) =(4π(x2+1), if 0 ≤ x ≤ 10, otherwise.A)12πB)1π ln 2C)2ln 2D)49E)ln 2πF)2πG) ln 2 H)4ln 2I)4π2J)2 ln 2π12Part II: Three-Choice (3 points each)16. If r > 1 and a 6= 0, then∞Xn=1arn.A) Converges B) Diverges C) Can’t Tell17. If∞Xn=1cn(−2)nis convergent, then∞Xn=1cn2n.A) Converges B) Diverges C) Can’t Tell18. If∞Xn=1cn(x − 10)nconverges for all x, then∞Xn=1cnn(x − 10)n−1for all x.A) Converges B) Diverges C) Can’t Tell19. Let Sn=nXj=1ajbe the nthpartial sum. If limn→∞Sn= 2, then∞Xn=1an.A) Converges B) Diverges C) Can’t Tell20. If∞Xn=1anconverges, then∞Xn=1(−1)nan.A) Converges B) Diverges C) Can’t Tell13Part III: True/False (2 points each)21.ddxZe21px5+ 1 dx = e10− 1.A) True B) False22. If the value of n used in Simpson’s Rule is doubled, then the error in the estimate is cut byabout a factor of 4.A) True B) False23.Zbaf(x)g(x)dx =Rbaf(x) dxRbag(x) dxA) True B) False24. limn→∞nXi=12πnsin2πin= 0A) True B) False25. IfRbaf(x) dx ≥ 0 andRbag(x) dx ≥ 0, thenRba(f(x) + g(x)) dx ≥ 0A) True B)
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