Math 132 - Final Exam - Spring 2008 1This exam contains 20 multiple choice questions. Each question is worth 5points.1. Find the arc length ofy =x312+1xover the interval [1, 3]. (Hint: Use the identity 1 + (y0)2= (x2/4 + x−2)2.)A) 8/13B) 13/12C) 7/9D) 5/12E) 2/5F) 13/6*G) 17/6H) 15/12I) 5/6J) 17/12Math 132 - Final Exam - Spring 2008 22. Which of the following integrals correctly represents the surface area ofrevolution of y = exabout the x-axis over the interval [0, 1] ?A)R10u√1 + u2duB)R10√1 + u2duC) πRe1u√1 + u2duD) πRe1√1 + u2duE) 2πR10u√1 + u2duF) 2πR10√1 + u2duG) 2πRe1u√1 + u2du*H) 2πRe1√1 + u2duI)Re1u√1 + u2duJ)Re1√1 + u2duMath 132 - Final Exam - Spring 2008 33. A thin triangular plate is submerged vertically in water so that one side islevel with the water’s surface. The plate is a right-triangle with two sidesof length 1. Let w b e the weight density of water. Calculate the force ofthe water on the surface of the plate.*A) w/6B) w/5C) w/4D) w/3E) w/2F) wG) 2w/3H) 2w/5I) 2w/7J) 3w/5Math 132 - Final Exam - Spring 2008 44. Calculate the Taylor polynomial T2(x) at x = 1 for f(x) = ln x.A) (x − 1) − (x − 1)2/6B) x + x2/3C) x − x2/2D) x + x2/6E) x + x2/2F) (x − 1) − 2(x − 1)2G) (x − 1) − (x − 1)2*H) (x − 1) − (x − 1)2/2I) x + 2x2J) (x − 1) + (x − 1)2/2Math 132 - Final Exam - Spring 2008 55. Use the error bound formula to find an upper bound on the error|f(1.5) − T3(1.5)|in approximating f(1.5) by its Taylor polynomial ce ntered at x = 1. As-sume that |f(4)(u)| ≤ 2 for all u between 1 and 1.5.A) 1/269B) 1/136*C) 1/192D) 1/428E) 1/320F) 1/284G) 1/216H) 1/36I) 1/196J) 1/96Math 132 - Final Exam - Spring 2008 66. Solve the initial value problem:yy0= xe−y2, y(0) = 1.A) y =pln(x2)B) y = cosh(x2) + 2x + 1C) y = ln(√x2+ e)D) y = ln(√x2+ e)*E) y =pln(x2+ e)F) y = ex2+ 2x + CG) y =12ln(√x2+ e)H) y =12ln(x2+ e)I) y = e−x2+ CJ) y =12ln(x2)Math 132 - Final Exam - Spring 2008 77. A cup of coffee, cooling off in a room at temperature 30◦C, has coolingconstant k = 0.08 min−1. If the coffee is served at a temp e rature of 90◦C,how long should you wait until its temperature drops to 60◦C?A) 15.2 minB) 20.3 minC) 3.9 minD) 10.2 minE) 16.6 min*F) 8.7 minG) 4.8 minH) 5.1 minI) 2.7 minJ) 12.5 minMath 132 - Final Exam - Spring 2008 88. Find the solution of the initial value problem:y0= 3y1 −y4, y(0) = 2.*A) y = 4/(1 + e−3t)B) y = 4/(1 − e−3t)C) y = 2/(1 + e−3t)D) y = 2/(1 − e−3t)E) y = 4/(1 + e−4t)F) y = 2/(1 + 2e−2t)G) y = 3/(1 + 4e−2t)H) y = 2/(1 − 4e−3t)I) y = 2/(1 + 4e−3t)J) y = 4/(1 + e−2t)Math 132 - Final Exam - Spring 2008 99. Find the solution of the initial value problem:y0+ 3y = e−3x, y(0) = −1.A) (x + 1)e−3xB) (1 − x)e−3xC) (2x − 1)e−3xD) xe−3x+ e3x*E) (x − 1)e−3xF) e−3x+ e3xG) e−3x− e3xH) e−3x+ xe3xI) (x − 1)e3xJ) (1 − x)e3xMath 132 - Final Exam - Spring 2008 1010. A 200-gal tank contains 100 gal of water with a salt concentration of 0.1lb/gal. Water with a salt concentration of 0.4 lb/gal flows into the tankat a rate of 20 gal/min. The fluid is mixed instantaneously, and water inpumped out at the same rate it flows into the tank. What is the limitingsalt concentration for large t?A) 31.1 lb/galB) 1.3 lb/galC) 0.9 lb/galD) 7.8 lb/galE) 45.7 lb/galF) 0.6 lb/galG) 0.5 lb/gal*H) 0.4 lb/galI) 0.3 lb/galJ) 32.2 lb/galMath 132 - Final Exam - Spring 2008 1111. Find the term a4of the sequence defined recursively by the equations:a0= 0, a1= 1, an= an−1+ an−2for n ≥ 2.A) 12B) 11C) 10D) 9E) 8F) 7G) 6H) 5I) 4*J) 3Math 132 - Final Exam - Spring 2008 1212. Let the n-th term of a sequence be defined byan=nn + 1.Find the smallest number M such that |an− 1| ≤ 0.0001 for all n ≥ M.A) 2000B) 200C) 20D) 1000000E) 100000F) 10000G) 99999*H) 9999I) 999J) 99Math 132 - Final Exam - Spring 2008 1313. Find the limitlimn→∞1 +2nn.A) e*B) e2C) e−1D) e−2E) 0F) 1G) 2H) πnI) nJ) e2/nMath 132 - Final Exam - Spring 2008 1414. Consider the two series(a)∞Xn=21ln n−1ln(n + 1), (b)∞X1√n −√n + 1.Do these series converge? If they do, what is their value?A) (a) converges to 1/ ln 2; (b) converges to 1*B) (a) converges to 1/ ln 2; (b) divergesC) (a) converges to 0; (b) divergesD) (a) diverges; (b) converges to 1E) (a) diverges; (b) divergesF) (a) converges to 1/ ln 2; (b) converges to 0G) (a) converges to 1/2; (b) divergesH) (a) converges to 1/2; (b) converges to√2I) (a) diverges; (b) converges to√2J) (a) diverges; (b) converges to√5Math 132 - Final Exam - Spring 2008 1515. Find the value of the series∞Xn=3e3−2n.*A) e−3/(1 − e−2)B) e/(e2− 1)C) e−2/(1 − e−3)D) e/(e3− 2)E) e−1/(1 − e−3)F) e3/(e2− 1)G) 1/(1 − e−2)H) 1/(e2− 1)I) 3/(1 − e−2)J) The series divergesMath 132 - Final Exam - Spring 2008 1616. Determine whether the s eries(a)∞Xn=1n−1/3, (b)∞Xn=3n2(n3+ 9)5/2, (c)∞Xn=12n +√n, (d)∞Xn=1sin2kk2converge or diverge. (c = converges and d = diverges.)A) c, d, c, dB) c, c, d, dC) d, d, c, cD) d, c, d, dE) d, c, c, d*F) d, c, d, cG) c, d, d, cH) c, d, d, dI) d, d, d, cJ) d, d, d, dMath 132 - Final Exam - Spring 2008 1717. Using the error formula for alte rnating series, what error do you make byapproximating the sumS =∞Xn=1(−1)nn(n + 2)(n + 3)by S3(i.e., the sum of the first three terms)?A) 1/10B) 1/90C) 1/100*D) 1/168E) 1/24F) 1/356G) 1/1000H) 1/10000I) 1/500J) 1/350Math 132 - Final Exam - Spring 2008 1818. For the following two series, determine the range of values of x for w hicheach series converges.(a)∞Xn=13nxn; (b)∞Xn=1xnn.A) (a) |x| < 3; (b) |x| < 4B) (a) |x| < 3; (b) |x| < 3C) (a) |x| < 3; (b) |x| < 2D) (a) |x| < 3; (b) |x| < 1E) (a) |x| < 1/3; (b) converges for all x*F) (a) |x| < 1/3; (b) |x| < 1G) (a) |x| < 1/2; (b) |x| < 2H) (a) |x| < 1; (b) |x| < 3I) (a) converges for all x; (b) |x| < 1J) (a) and (b) convergeMath 132 - Final Exam - Spring 2008 1919. Find the radius of convergence of the power series∞Xn=1xnnen.A) 1B) ∞C) 2*D) eE) e2F) 2eG) 3eH) e/2I) 4J) 0Math 132 - Final Exam - Spring 2008 2020. The function f(x) = 1/(1 + x9) can be expanded as a p ower series of theform∞Xn=0(−1)nx9n.For what values of x is the expansion valid?A) for all xB) |x| < 9C) |x| < 3D) |x| < 2E) |x| > 1F) |x| > 2G) |x| > 9*H) |x| < 1I) |x| = 1J) the series diverges for all
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