1MATH 132 FINAL 12/11/07 3-hour closed-bookNO CALCULATORS; SHOW ALL WORK; Put ANSWERS in BOXESINSTRUCTOR: Drupieski Khongsap Malek McCrimmon Snider QuertermousName : SCORE /300 =Missed p1 p2 p3 p4 p5 p6 p7 p8 Total missedProblem 1: Find the following definite and indefinite integrals:(a)Z10ln(1 + x2) dx =(b)Zsin2(x) cos3(x) dx = +C(c)Zdx(a2− x2)3/2= +C2Problem 2: Write the form of the partial fraction decomposition of the following rationalfunctions (DO NOT SOLVE FOR THE CONSTANTS):(a)2x + 3x(x2+ 2x + 1)=(b)2x + 3x(x2+ 2x + 2)=Problem 3: (a) If f is a continuous function on (−∞, a], then the integralRa−∞f(x)dx isdefined to be convergent if:(b) Determine whether the improper integralZ∞1esin(x)x2+ 3dx converges; explain yourreasoning, but do not evaluate the integral if it converges.converges diverges(c) Evaluate the improper integralZ4112x − 3dx (write DIV if it does not converge). Explainyour reasoning.Z4112x − 3dx =3Problem 4: Answer the following questions for the parameterized curve x = et, y = e−tfor 0 ≤ t ≤ 1; explain your reasoning.(a) Set up (but do not evaluate) an integral for the length of the curve.L =(b) Find the equation of the tangent line to the curve at time t =12.y = x +(c) Find the area between the curve and the x-axis from t = 0 to t = 1;Area =(d) Find a Cartesian equation y = f(x) for this curve.y =4Problem 5: (a) Solve the differential equation y0+y =11 + exwith initial condition y(0) = 0.y(x) =(b) Find the most general solution ofdydx= 5yx4.y(x) =Problem 6: (a) Find the area of the wedge-shaped region inside the polar curve r = cos(θ)between θ = 0 and θ =π4.Area =(b) Give the Cartesian equation of the polar curve r =1cos(θ) + sin(θ ).5(c) For the circle r = 3 cos(θ) and the cardioid r = 1 +cos(θ), s et up (but do not evaluate)the integral for the area inside the circle but outside the cardioid (the unshaded crescentregion sketched below):Area =Problem 7: Write the iterated integralR10R3x0f(x, y)dydx as an iterated integral with theorder of integration interchanged.Problem 8: In the following 3 parts, state the definitions of convergence, absolute conver-gence, and conditional convergence of an infinite se riesP∞n=1an. The seriesP∞1anisdefined to be(a) convergent if:(b) absolutely convergent if:(c) conditionally convergent if6Problem 9: Determine whether the following infinite series c onverge absolutely, conditionally,or diverge (check one). Mention which tests you use.(a)∞Xn=1ln2n2+ 1n2+ 3convergesabsolutelyconvergesconditionallydiverges(b)∞Xn=12 + cos(n)√nconvergesabsolutelyconvergesconditionallydivergesProblem 10: Find all values of q such that the following infinite series converge:(a)∞Xn=1nqconverges all q with(b)∞Xn=1(−1)nnqconverges f or all q with(c)∞Xn=1qnconverges f or all q with(d)Find the interval of convergence of∞Xn=0n!(x + 5)n4n(n + 1)!. Interval7Problem 11. (a) For a general differentiable function f, the Taylor series centered at a isdefined to beP∞n=0cn(x − a)nwhere the coefficients arecn=(b) Find the Taylor series for the function f(x) = excentered at ln(5).(c) Find the fourth-degree Taylor polynomial T4(x) centered at 0 for the function f (x) =Zx0e−t2dt:T4(x) =P4n=0cnxnfor coefficients c0= c1= c2= c3= c4=Problem 12: Identify the following power series with the indicated functions (put the letterof the correct function next to the series):(i)∞Xn=0(−1)nxnn!(ii)∞Xn=1(−1)nxnn(iii)∞Xn=0(−1)nx2n(2n + 1)!(iv)∞Xn=0(−1)nx2n+12n + 1A: arctan(x)B: arctan(−x)C: −ln(1 + x)D:1exE: −exF:sin(x)xG: sin(x)H:cos(x)2n+18Problem 13: Find the sum of the following series:(a)∞Xn=0(−1)nπ2n+262n+1(2n)!=(b) −ln(2) +(ln(2))22!−(ln(2))33!+ ... =True-False QuestionsEach is worth 4 points; circle the correct answer T or F (you do not need to justify youranswer).1. T F If limb→∞Rb−bf(x)dx exists, thenR∞−∞f(x)dx converges.2. T F The graph of the parametric curve x = 2t3/5, y = t3/5is a straight line .3. T F If an≥ 0 for all n and∞Xn=1anconverges, then∞Xn=1(−1)nanmust converge.4. T F Every power seriesP∞n=0cn(x − a)nconverges for at least one value of x.5. T F If the series∞X1cnxnconverges for all −3 < x < −1, then∞X1cn2nmustconverge as well.PLEDGE IN