Math 1B, Final ExaminationN.Reshetikhin, May 13, 2005P roblem 1 2 3 4 5 6 7 8 9 10 11 12 T otalP oints 10 15 15 15 15 15 15 15 15 10 15 20 175GradeStudent’s Name:GSI’s name:Student’s i.d. number:1.(10 pnts) Evaluate the integralZe√xdx12.(15 pnts) Evaluate the integralZ1(t2− 1)(t − 1)dt23.(15 pnts) Indicate which of the following statements are true and whichare false. Do not show your work.1.Z∞1sin2xx3dx converges by comparison test withZ∞11x3dx.2.Z∞1sin(x−2)xdx diverges by comparison test withZ∞11xdx.3.Z21dx(x − 1)2dx is a convergent improper integral.4.Z∞−∞1x2dx is a divergent improper integral.5.Z∞0ln(x)(x + 1)3/2dx is a convergent improper integral.34.(15 pnts) Find the radius and the interval of convergence of the powerseries∞Xn=1n − 1(n + 2)(2n + 5)x2n45.(15 pnts) State whether the following series is absolutely convergent, con-ditionally convergent, or divergent. Do not show your work.1.∞Xn=1cos(πn2).2.∞Xn=1(−1)n13 + n13 + 4n2.3.∞Xn=11√n + 1−1√n.4.∞Xn=13nn!(−1)n.5.∞Xn=2cos(πn)1n2ln(n)56.(15 pnts) For each statement indicate whether it is true or false. Do notshow your work.1. If∞Xn=1c2nconverges, then∞Xn=1(−1)nc2nalso converges.2. If f(x) < 0 is monotonically increasing, continuos, andZ∞1000f(x)dx isconvergent then∞Xn=1f(n) converges.3. If the sequence {an} converges and the sequence {bn} diverges then{an+ bn} diverges.4. If the sequence {an} converges and and the sequence {bn} diverges then{anbn} diverges.5. IfXn≥0an5nconverges andXn≥0an(−6)ndiverges, thenXn≥0an8ndiverges.67.(15 pnts) For each statement indicate whether it is true or false. Do notshow your work.1.∞Xn=1ncnxnconverges absolutely inside (excluding boundary points) of theinterval of convergence of the power series∞Xn=1cnxn.2. If∞Xn=1cnxnhas radius of convergence R, then∞Xn=1cnRnconverges condi-tionally.3. If∞Xn=1cnxnconverges for |x| < R, then R is the radius of convergence ofthis power series.4. If∞Xn=1cnxndiverges for |x| > a > 0, then a > R where R is the radius ofconvergence of this power series.5. The radius of convergence of∞Xn=1xnn!+∞Xn=1xnn100is 1.78.(15 pnts) Solve the initial-value problem.y0= ex+2y.89.(15 pnts) Find the general solution to the differential equationdydx= 2xy + ln(x)ex2, .910.(10 pnts) Find the general solution to the differential equationy00+ y = cos x, y(0) = 0, y0(0) = 5/2 .1011.(15 pnts) Match pictures to differential equations.1.dydx= y22.dydx= y + x 3.dydx= y1/24.dydx= y−25.dydx= xyFlow 00.511.522.53y0.5 1 1.5 2 2.5 3xFigure 1: Equation number ....Flow 00.511.522.53y0.5 1 1.5 2 2.5 3xFigure 2: Equation number ....11Flow 00.511.522.53y0.5 1 1.5 2 2.5 3xFigure 3: Equation number ....Flow –2–1012y–2 –1 1 2xFigure 4: Equation number ....Flow 00.511.522.53y0.5 1 1.5 2 2.5 3xFigure 5: Equation number ....1212.(20 pnts) Find the power series solution to the differential equation:y00− xy0− y = 0, y(0) = 1, y0(0) = 0
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