Math 231E, 2013. Midterm 3.• This exam has 27 questions.• You must not communicate with other students during this tes t . No books, notes, calculators,or electronic devices allowed.• Please fill out all of the information bel ow. Make sure to fill out your Scantron form as directedin class; fill in name, UIN number, and NetID.1. Fill in your information:Full Name:UIN (Student Number):NetID:2. Fill out name, student number (UI N ) and NetID on Scantronsheet. Then fill in the followi ng answers on the Scantron form:89. D90. A91. A92. A93. A94. B95. A96. C11. (4 points) Evaluate limx!1sin(2x)x.(A) 0(B) 3(C) +1(D) does not exist(E) 22. (4 points) Let {an}1n=1denote the sequence an=1n3and S denote the series S =1Xn=11n3.Whichof the following statements is true?(A) the sequence diverges and the series converges.(B) The sequence and the series both diverge.(C) None of the other statements are true.(D) The sequence converges and the series diverges.(E) The sequence and the series both converge.23. (4 points) Give the correct form for the partial fractions expansion of the functionx3+2x2 2x +3x(x 1)2(x2+3x + 12)(A)Ax+Bx 1+C(x 1)2+Dx2+3x + 12(B)Ax+Bx 1+C(x 1)2+Dx + Ex2+3x + 12(C)Ax 1+Bx+Cx2+Dx + Ex2+3x + 12(D)Ax+Bx 1+C(x 1)2+Dx +3+Ex +4(E)Ax+B(x 1)2+Cx2+ x +1+D(x2+3x + 12)234. (4 points) Use polynomial division to express the following rational functionx3+7x 1x2+ x + 12as a polynomial plus a rational function where the degree of the numerator is less than t he degree ofthe denominator.(A) x 1+4x + 11x2+ x + 12(B) x +7+x + 11x2+ x + 12(C) x x2+5x +1x2+ x + 12(D)2x +3+7x 4(E)3x +3+4x 445. (4 points) Find the integral that is equivalent to the integralZx4(1 x2)3/2dx.(A)Zsin4(✓) d✓(B)Zsin4(✓)cos2(✓)d✓(C)Zsec5(✓)tan2(✓)d✓(D)Zsin4(✓)cos3(✓)d✓(E)Zsin5(✓)cos3(✓)d✓6. (4 points) Determine the type of the integral (prop er , impr oper type I, or improper typ e II), andpoint that makes the integral improper.Z10sin(x)2x2+ x 1(A) Proper(B) Type II — integrand undefined at x =12(C) Type I — Infinite domain(D) Type II — integrand undefined at x =1(E) Type II — integrand undefined at x =057. (4 points) Compute the following integralZsin2(x) cos(x)dx(A) cos3(x)3+ C(B)sin3(x)3+ C(C)12✓✓ +sin(2✓)2◆+ C(D)12✓✓ cos(2✓)2◆+ C(E)12✓✓ +cos(2✓)2◆+ C8. (4 points) Find the partial fractions expansion of the following functionf(x)=2x +1x2+3x +2.(A)1x +1+3x +2(B)1x +23x +1(C)1x +13x +2(D)1x +1+3x +2(E)3x +1+1x +269. (4 points) Evaluate the following improp er i ntegral for convergence. If i t converges find the value.Z10e(3/2)xdx(A) Diverges(B) Converges, value is 2/3(C) Converges, value is 1(D) Converges, value is e(3/2)1 1(E) Converges, value is 2/310. (4 points) ComputeL =limx!0cos✓⇡ sin(7x)7x◆.(A) L =1(B) L does not exist(C) L =0(D) L = 1(E) L = 1711. (4 points) Let an=1n ln nand S be the series S =P1n=2an. Which of the following stat eme ntsare true?(A) None of the other statements are true.(B) The series converges and the sequence diverges.(C) The sequence converges and the series diverges.(D) Both the series and sequence converge.(E) Both the series and sequence diverge.12. (3 points) A superball is dropped from a he i ght of 2 meters. After one bounce it reb oun ds to aheight of 1.5 meters. After a second bounce it rebounds to a height of 1.125 meters. Assuming thatthe successive bounce heights form a geometric sequence, calculate the total distance travelled by thesuperball. Remember to count both the distance travelled down and the dist an ce t ravelled up.(A) 21 meters.(B) 16 meters.(C) 8 meters.(D) 12 meters.(E) 14 meters.813. (4 points) Let R be the region contained between the curve y = x2+ 1, the x-axis, and t h e linesx = 0 and x = 1. Set up but do not evaluat e the integral for th e volume of the solid of revolutionobtained by rotating R about the y-axis.(A)Z102⇡(x3+ x) dx(B) None of the above(C)Z102⇡xp1+4⇡2xdx(D)Z10⇡(x4+2x2+ 1) dx(E)Z104⇡x2dx14. (4 points) Let an=n2n2+1and S be the series S =P1n=1an. Which of the following statementsare true.(A) None of the other statements are true.(B) The sequence converges and the series diverges.(C) The series converges and the sequence diverges.(D) Both the series and sequence converge.(E) Both the series and sequence diverge.915. (4 points) Compute the following integralZ321x(x + 1)(x 1)dx(A) ln✓3227◆(B)12ln✓2732◆(C) ln✓827◆(D)12ln✓278◆(E)12ln✓3227◆1016. (4 points) Evaluate the following indefinite integralZ1(1 + x2)3/2dx(A)xp1+x2+p1+x2+ C(B)xp1+x2+ C(C)1p1+x2+ arcsec(x) + C(D)1p1+x2+ C(E)xp1+x2+ arctan(px2+ 1) + C17. (4 points) Let y = f(x)bedefinedimplicitlybyy3 xy =8.and suppose that y(0) = 2. Find the first two terms of the Taylor series for y about the point 0.(A) y =2x6+ O(x2)(B) y =1+x +2x2+ O(x3)(C) y =2+x6+ O(x2)(D) y =223x + O(x2)(E) y =1+x6+ O(x2)1118. (4 points) Which of these five choices are the same asZ⇡ /20ecos2(x)sin(x) dx.(A)Z10eu2du(B) Z10eu2du(C)Z⇡ /20eudu(D)Z⇡ /20eu2du(E) eu2+ C19. (4 points) ComputeZ⇡ /20esin(x)cos(x) dx.(A) 0(B) e⇡ /2 e⇡ / 2(C) 1 e⇡(D) esin(x)+ C(E) e 11220. (3 points) Let us defineA =Z11x6x7+1dx, B =Z11x6x8+1dx.Which of the following are true?(A) A diverges, B di verges(B) A converges, B di verges(C) A converges, B converges(D) None of the other answers are true.(E) A diverges, B converges21. (3 points) Determi n e if the following i mp roper integral converges. If it converges, determine itsvalue.Z10x1+x4dx(A) Diverges(B) Converges to⇡2(C) Converges to 1(D) Converges to12(E) Converges to⇡41322. (4 points) Let R be the finite region bounded by the cur ve y =p25 4x2and the x-axis. Set upbut do not evaluate the integral for computing the surface area of the surface of rotation obtained byrotating R about the x-axis.(A)Z5/25/22⇡xp25 4x2dx(B)Z5/25/22⇡xs25 + 4x2p25 4x2dx(C)Z5/25/2⇡(25 4x2) dx(D)Z5/25/22⇡xp25 + 12x2dx(E)Z5/25/22⇡p25 + 12x2dx23. (3 points) Evaluate the following indefinite integralZx2(1 x2)3/2dx(A)p1 x2+ C(B)p1 x2+ arcsin(x)+C(C)xp1 x2 arcsin(x)+C(D)xp1 x2+ arcsin(x)+C(E)xp1 x2 arcsin(x)+C1424. (3 points) Compute the integralZ1x2(x2+ 4)dx(A)14✓1x12arctan(x/2)◆+ C(B)14✓1x+12arctan(x/2)◆+ C(C)14✓1x+ arctan(x/2)◆+ C(D) 14✓1x12arctan(x/2)◆+ C(E)
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