Math 1B, Final ExaminationN.Reshetikhin, May 18, 2004Student’s Name:TA’s name:Student’s i.d. number:1.10 pnts Evaluate the integralZx3ex2dx12.15 pnts Evaluate the integralZ1(t2− 1)(t − 1)dx23.15 pnts Indicate which of the following statements are true and which arefalse. Do not show your work.1.Z∞1sin2xx3dx converges by comparison test withZ∞11x3dx.2.Z∞1sin(1x)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 power series∞Xn=1n − 1(n + 2)(n + 3)(x − 12)n45.15 pnts State whether the following series is absolutely convergent, condi-tionally convergent, or divergent. Do not show your work.1.∞Xn=1cos(πn).2.∞Xn=1(−1)n27 + n327 + 4n4.3.∞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 andZ∞1000f(x)dx is convergentthen∞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 of the interval of convergence of thepower series∞Xn=1cnxn.2.∞Xn=1cnxnhas radius of convergence R, then∞Xn=1cnRnconverges condition-ally.3.∞Xn=1cnxnconverges for |x| < R, then R is the radius of convergence of thispower series.4.∞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.(x2+ 1)(2yy0+ 2y0) = x, y(0) = 0 .89.15 pnts Find the general solution to the differential equationdydx= 1 + x2+ y + x2y, .910.10 pnts Find the general solution to the differential equationy00− y = ex, y(0) = 0, y0(0) = 5/2 .1011.15 pnts In a box near each picture of directional fields indicate whichdifferential equation it is representing.Flow 00.511.522.53y0.5 1 1.5 2 2.5 3x11Flow 00.511.522.53y0.5 1 1.5 2 2.5 3x12Flow 00.511.522.53y0.5 1 1.5 2 2.5 3x13Flow 00.511.522.53y0.5 1 1.5 2 2.5 3x14Flow 00.511.522.53y0.5 1 1.5 2 2.5 3x1512.20 pnts Find the power series solution to the differential equation:y00− xy0− y = 0, y(0) = 1, y0(0) = 0
View Full Document
Unlocking...