Math 231 C,D. Midterm 2. March 20, 2014Name: Section Code:Three points will be deducted if these instructions are not followed.1. Write your full name legibly above.2. Write your section code from the list in the boxes above.3. Code your name, netid, and test form D correctly on the scantron form.CDC – WF 10:00-10:50 -Instructor: Menezes, GlenCDD – WF 11:00-11:50 -Instructor: Menezes, GlenCDE – WF 12:00-12:50 –Instructor: Mastroeni, MatthewCDF – WF 1:00-1:50 -Instructor: Butler, StaceyCDG – WF 2:00-2:50 -Instructor: Butler, StaceyCDH – WF 3:00-3:50 -Instructor: Mastroeni, MatthewCDA – WF 8:00-8:50 -Instructor: Orlow, NathanCDJ – WF 11:00-11:50 -Instructor: Orlow, NathanDDG – WF 2:00-2:50 -Instructor: Jang, DonghoonDDH – WF 3:00-3:50 -Instructor: Golze, HiramDDA – WF 8:00-8:50 -Instructor: Ahmed, IftikharDDF – WF 1:00-1:50 -Instructor: Vellis, VyronDDD – WF 11:00-11:50 -Instructor: Heersink, ByronDDC – WF 10:00-10:50 -Instructor: Heersink, ByronDDE – WF 12:00-12:50 -Instructor: Golze, Hiram• Answers must be marked on scantron form.• You must not communicate with other students during this test.• No written materials of any kind allowed.• The only scratch paper allowed is the last page of the exam. It may be detached.• No phones, calculators, iPods or electronic devices of any kind are allowed for ANYreason, including checking the time (you may use a simple wristwatch).• Do not turn this page until instructed to.• There are several different versions of this exam.Violations of academic integrity (in other words, cheating) will be taken extremely seriously,and will be handled under the procedures of Article I, Part 4 of the student code.Mark answers on scantron form.Your test form is D. Code this on the scantron form now.(3 points each) Choose the correct answer for each given sequence {an}.1. an= n100e−n(A) Converges to 0(B) Converges to 1(C) Converges to e(D) Converges to e−1(E) Diverges2. an=n4+ 2n3+ 25n4+ 2.(A) Converges to 1(B) Converges to15(C) Converges to25(D) Converges to 0(E) Diverges3. an=sin(n)2√n + 1.(A) Converges to 1(B) Converges to1√2(C) Converges to12(D) Converges to 0(E) DivergesFor each series mark C if the series Converges or D if the series Diverges. Mark answerson Scantron form.4.∞Xn=1n4+ 15nn5+ n35.∞Xn=0n3+ 15n5n+ n36.∞Xn=1n3213 + n72.7.∞Xn=1arctan(n).8.∞Xn=1ln n3n2.9.∞Xn=2sin1n2.10. Complete the statement of the monotone convergence theorem: if a sequence {an} isthen it converges.(A) increasing and positive(B) positive and bounded(C) positive and continuous(D) increasing and bounded(E) continuous and bounded11. The velocity of a runner during the first6 seconds of running is shown. Use the mid-point rule with three intervals to estimate thedistance traveled by the runner.t (s) v (m/s)0 01 22 33 34 45 56 6(A) 10 m(B) 12 m(C) 20 m(D) 24 m(E) 30 m12. Find the sum of the series∞Xn=14n+132n−1.(A) 9/5(B) 56/5(C) 56/15(D) 144/15(E) DivergesFor the problems 13 ,14 and 15, consider the curve x =pa2− y2between the points0 ≤ y ≤ a.13. Which of the following is an integral which gives the length of the curve?(A)Za0a2a2− y2dy.(B)Za01pa2− y2dy.(C)Za0dy.(D)Za0apa2− y2dy.(E)Za0a dy.14. Which one is an integral which represents the surface area when the curve is rotatedabout the y-axis?(A) 2πZa0a2a2− y2dy.(B) 2πZa01pa2− y2dy.(C)Za02π dy.(D) 2πZa0apa2− y2dy.(E) 2πZa0a dy .15. What is the surface area?(A) 2πa(B) πa(C) 1(D) 2πa2(E) πa216. Test the series∞Xn=1(−3)n(2n + 1)!for convergence(A) Absolutely convergent(B) Conditionally convergent(C) Divergent17. Test the series∞Xn=1(−1)nn√n3+ 2for convergence.(A) Absolutely convergent(B) Conditionally convergent(C) Divergent18. If sn= 1 −23n, what is anfor n > 1?(A)23n(n−1)(B) 2 +23n(n−1)(C) 2 +2n(n−1)(D) 2 +29n(n−1)(E)29n(n−1)19. If sn= 1 −23n, what isXanfor n ≥ 1?(A)−23(B)−13(C)−53(D) 1(E) 020. What is the smallest value of n which guarantees that the partial sum Snapproximatesthe seriesP∞n=11n3to within1200? .(A) 11(B) 10(C) 19(D) 20(E) 2121. Find the sum of the series∞Xn=1(1√n−1√n + 2) if it is convergent.(A) Divergent(B)√2−1√2(C)−1√2(D)1√2(E) 122. Consider an infinite array of circles arranged in rows as shown. In row one there is acircle of radius 1. In row two there are two circles of radius 1/3. In row three there are 2circles of radius 1/9. In row four there are two circles of radius 1/27, and so on, forever.What is the total area enclosed by all of the circles?(A)π6(B)π4(C) 5π9(D) 5π3(E) 5π423. A vertical door has the shape of the parabola y = x2, −1 ≤ x ≤ 1. The top of the dooris 2 meters under water. What is the hydrostatic force on the door?(ρ kg/m3is the density of water and g m/s2is the acceleration due to gravity.)(A)85ρg(B) ρg83(C) ρg125(D) ρg165(E) ρg163Water levelScratch paper–may be
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