MATH 152 FALL 2011 COMMON EXAM III VERSION AMATH 152 FALL 2011 COMMON EXAM III VERSION AMATH 152 FALL 2011 COMMON EXAM III VERSION APRINTLAST NAME FIRST NAMEINSTRUCTOR: SECTION NUMBER:UIN: SEAT NUMBER:Directions1. The use of all electronic devices is prohibited.2. In Part 1 (Problems 1-12), mark the correct choice on your Scantron using a No. 2 pencil. Recordyour choices on your exam. Scantrons will not be returned.3. In Part 2 (Problems 13-17), present your solutions in the space provided. Show all your workneatly and concisely and clearly indicate your final answer. You will be graded not merelyon the final answer, but also on the quality and correctness of the work leading up to it.4. Be sure to write your name, section and version letter of the exam on the Scantron form.5. Any scratch paper used must be handed in with the exam.6. Good Luck!THE AGGIE CODE OF HONOR“An Aggie does not lie, cheat, or steal, or tolerate those who do.”Signature:http://foxtrot.comQuestion 1-12 13 14 15 16 17 TOTALPoints AwardedPoints Possible 48 10 14 7 11 10 1001MATH 152 FALL 2011 COMMON EXAM III VERSION AMATH 152 FALL 2011 COMMON EXAM III VERSION AMATH 152 FALL 2011 COMMON EXAM III VERSION A1. Find a unit vector in the direction of the vector h√3, −3, 2i.(a) 4h√3, −3, 2i(b)116h√3, −3, 2i(c) 16h√3, −3, 2i(d)116h−√3, 3, −2i(e)14h√3, −3, 2i2. What is the distance between points (−2, −1, 4) and (1, −3, −2)?(a) 3(b)√29(c) 7(d)√61(e) 493. What is the Maclaurin series for ex22?(a)∞Xn=0xn+22(n!)(b)∞Xn=0xn2(n!)(c)∞Xn=0xn2n(n!)(d)∞Xn=0x2n2n(n!)(e)∞Xn=0x2n2n(2n!)2MATH 152 FALL 2011 COMMON EXAM III VERSION AMATH 152 FALL 2011 COMMON EXAM III VERSION AMATH 152 FALL 2011 COMMON EXAM III VERSION A4. Which series diverges?(a)∞Xn=2n2− 2n − 1n3+ 4n(b)∞Xn=01n2+ 2n + 4(c)∞Xn=1ne−n2(d)∞Xn=11n!(e)∞Xn=11n√n5. Which series converges absolutely?(a)∞Xn=1(−1)n√n(b)∞Xn=1(−1)nenen+1(c)∞Xn=1√n + 2 −√n(d)∞Xn=1(−1)nn(e)∞Xn=1(−2)nn!6. What is the power series representation of f(x) =19 − 4x2at x = 0?(a)∞Xn=0(−1)n4nx2n9n+1(b)∞Xn=04nx2n9n+1(c)∞Xn=09nx2n4n+1(d)∞Xn=0(−1)n9nx2n4n+1(e)∞Xn=14nx2n93MATH 152 FALL 2011 COMMON EXAM III VERSION AMATH 152 FALL 2011 COMMON EXAM III VERSION AMATH 152 FALL 2011 COMMON EXAM III VERSION A7. Given that the power series∞Xn=0cnxnconverges when x = 3 and diverges when x = 6, which of thestatements is certain to be true?(a)∞Xn=0cn(−6)nis divergent(b)∞Xn=0cn(−4)nis convergent(c)∞Xn=0cn(−3)nis convergent(d)∞Xn=0cn(−2)nis convergent(e) None of these statements is certain to be true.8. What is the cosine of the angle between the vectors h−1, 2, −3i and h3, −2, −1i?(a)−5√7(b)5√7(c)−57(d)−27(e)2√79. Let a, b, c and d be nonzero vectors where kak is the length of a. Givena · b = kakkbkand0 < |c · d| < kckkdkWhich of these statements is true?(a) a and b are parallel; c and d are neither orthogonal nor parallel(b) a and b are orthogonal; c and d are neither orthogonal nor parallel(c) a and b are neither orthogonal nor parallel; c and d are orthogonal.(d) a and b are neither orthogonal nor parallel; c and d are parallel.(e) None of the above statements is true.4MATH 152 FALL 2011 COMMON EXAM III VERSION AMATH 152 FALL 2011 COMMON EXAM III VERSION AMATH 152 FALL 2011 COMMON EXAM III VERSION A10. Find the vector projection of the vector h3, 2, 1i onto the vector h1, 2, 3i.(a)57(b)10√14(c)114h30, 20, 10i(d)114h10, 20, 30i(e)1√14h30, 20, 10i11. Find the radius of convergence of the power series∞Xn=13n(x − 5)nn!.(a) 0(b)13(c) 3(d)163(e) ∞12. The equation of the sphere passing through the point (1, 2, 3) with center (2, −1, 3) is(a) (x − 1)2+ (y − 2)2+ (z − 3)2= 10(b) (x − 2)2+ (y + 1)2+ (z − 3)2= 10(c) (x − 1)2+ (y − 2)2+ (z − 3)2=√10(d) (x − 2)2+ (y + 1)2+ (z − 3)2=√10(e) (x − 1)2+ (y − 2)2+ (z − 3)2= 1005MATH 152 FALL 2011 COMMON EXAM III VERSION AMATH 152 FALL 2011 COMMON EXAM III VERSION AMATH 152 FALL 2011 COMMON EXAM III VERSION APART II WORK OUTDirections: Present your solutions in the space provided. Show all your work neatly andconcisely and box your final answer. You will be graded not merely on the final answer, butalso on the quality and correctness of the work leading up to it.13. (10 points) Find the radius and center of the spherex2+ y2+ z2− 4x + 2y − 6z = 116MATH 152 FALL 2011 COMMON EXAM III VERSION AMATH 152 FALL 2011 COMMON EXAM III VERSION AMATH 152 FALL 2011 COMMON EXAM III VERSION A14. (8 points) Find the first four terms of the Taylor series for f(x) = x32centered at a = 4.(6 points) Use Taylor’s Inequality to give a bound for the error for when using T1(x) (the first degreeTaylor polynomial) centered at a = 4 to approximate f (x) = x32on [3, 5].Taylor’s Inequality: |Rn(x)| ≤M(n + 1)!|x − a|n+1on the interval [3, 5] where M = max |f(n+1)(x)|on [3, 5].7MATH 152 FALL 2011 COMMON EXAM III VERSION AMATH 152 FALL 2011 COMMON EXAM III VERSION AMATH 152 FALL 2011 COMMON EXAM III VERSION A15. (7 points) Compute the Maclaurin series forcos(x) − 1x2.8MATH 152 FALL 2011 COMMON EXAM III VERSION AMATH 152 FALL 2011 COMMON EXAM III VERSION AMATH 152 FALL 2011 COMMON EXAM III VERSION A16. (6 points) The series∞Xn=1(−1)nn(3n)(2n2− n)converges conditionally. Determine whether this seriesalso converges absolutely. Clearly explain your reasoning.(5 points) What is a bound on the error if we sum the first 3 terms of the series?9MATH 152 FALL 2011 COMMON EXAM III VERSION AMATH 152 FALL 2011 COMMON EXAM III VERSION AMATH 152 FALL 2011 COMMON EXAM III VERSION A17. (10 points) Find the interval of convergence for the power series∞Xn=0(x − 5)n3n(n2+
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