Math 231/EL1 FinalUIUC, May 7, 2013Q. Pt Score Q. Pt Score Q. Pt Score Q. Pt Score1 16 6 22 11 8 16 92 16 7 22 12 10 17 43 16 8 5 13 20 18 54 5 9 3 14 10 TA 25 7 10 8 15 12 ExCr 10Tot 60 Tot 60 Tot 60 Tot 301. (8 points each)Evaluatetheintegral.(a)Zx sin(3x) dx(b)Zsec4(5x) dx2. (8 points each)Evaluatetheintegral.(a)Z3cos5↵psin ↵d↵(b)Zdx(25 + x2)323. (8 points each)Evaluatetheintegral.(a)Zx2x2+4(b)Zx + ax2 xdx4. (5 points)Determinewhethertheintegralisconvergentordivergent. Ifitisconvergent, evaluate it.Z21dxx115. (7 points)Determinewhethertheintegralisconvergentordivergent. Ifitisconvergent, evaluate it.Z10xe3xdx6. (11 points each)Determine if the series is absolutely convergent, conditionally convergentor diverge nt.Besuretoshowyourreasoning.Nowork,nocredit.(a)1Xn=51pn3+30n(b)P1n=2(1)nn+3n7. (11 points each)Determine if the series is absolutely convergent, conditionally convergentor diverge nt.Besuretoshowyourreasoning.Nowork,nocredit.(a)1Xn=1n27n(b)1Xn=21ln(nn)8. (5 points)Showthatforanynumberr 6=1andpositiveintegerk,1+r + r2+ ···+ rk=1 rk+11 r9. (3 points)Drawonthediagramandgiveabriefexplanationwhy6Xn=21n(n +1)Z61dxx(x +1)2f(x)=1x(x +1)10. (8 points, 2/3/3)This problem concerns the curvey =2sinx +sin2x, 0 x ⇡(a) Give an integral for the length of the curve. You do not n eed to evaluatethe integral.(b) Give an integral for the area of the surface obtained by rotating th e curveabout the x-axis. You do not need to evaluate the integra l .(c) Give an integral fo r the area of the surface obtained by rotating the curveabout the y-axis. You do not need to evaluate the integral.11. (8 points)Shortanswer.(a) Suppose that c(x)=P1n=0cnxnconverges for x = 4butdivergesforx =6.i.P1n=0(1)ncn(absolutely converges/ conditionally converges/diverges).ii.P1n=0(7)ncn(absolutely converges/ conditionally converges/diverges).3(b) Calculate the binomial coefficient✓34◆=(c) Recall that⇡4=tan1(1) =P1n=0(1)n2n+1. By the Alternating Series Esti-mation, how accurate is 1 13+1517+19=263315to the actual value of⇡4?12. (10 points) Find a series solution to the integralZ11sin(x2) dx13. (10 points each) Find the radius and interval of convergence for the powerseries. Be sure to indicate which p oi nts converge absolu t ely and which convergeconditionally.(a)1Xn=0(x 2)n1 · 3 · 5 ···(2n +1)(b)1Xn=0(3x 2)nn14. (10 points)GivetheTaylorpolynomialwithdegree2centeredat1forf(x)=3px.ThenuseTaylor’sInequalitytoestimatetheaccuracyofthisapproxima-tion for3p2.15. (12 points)Determineapowerseriescenteredat0forf(x)=sin1(x2)and use it to determine the 104-th derivative of sin1x at 0. You may find thefollowing useful✓12n◆=(1)n(1)(3)(5) ···(2n 1)2nn!416. (9 points, 3 points each)Recallthatthefollowparametricequationsmodelaparticle traveling counter-clockwise about an ellipse at one rev/2⇡ unit timestarting at the point (a, 0).x = a cos ty= b sin t(a) What is the slope of the tangent line when t =⇡4?(b) Give an integral for the area of the ellipse using this set of parametricequations. You do not need to solve the integral.(c) Give an integral for the circumference of the el l i p se using this set of para-metric equations. You do not need to solve the integral.17. (4 points)Matchthegraphstotheircorrespondingpolarequations.r =sin✓r=cos✓r=1+2sin✓r=1 2cos✓18. (5 points) Find the area of the shaded region of r = e✓.5Extra Credit (10 points)Recall that hyperbolic cosine is defined bycosh(x)=ex+ ex2If we let the imaginary number i =p1, use power series to esta bl i sh theidentitycos(ix)=cosh(x
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