NAME: MA 23200Sample Final ExamPUID:INSTRUCTIONS• There are 25 problems on 14 pages.• Record all your answers on the answer sheet provided. The answer sheet is the onlything that will be graded.• No books or notes are allowed.• You may use a one-line scientific calculator. No other electronic device is allowed.Be sure to turn off your cellphone.• Show all your work on the exam. If you need more space, use the backs of the pages.• The last page is a formula sheet. You may detach this page from the exam for easyreference, but you must hand it with your exam booklet.1MA 23200 - Sample Final Exam1. EvaluateZe3x7 − e3xdx.A.13(7−e3x)2+ CB.−13e3x7x−13e3x+ CC.13e3x7x−13e3x+ CD. −13ln7 − e3x+ CE.13ln7 − e3x+ C2. EvaluateZ41ln x√xdx.A. 2 ln 4 + 1B. 2 ln 4 − 1C.12(ln 4)2D. 4 ln 4 + 4E. 4 ln 4 − 42MA 23200 - Sample Final Exam3. Use the Trapezoid Rule with n = 4 trapezoids to estimateZ31xx + 1dx.Round your answer to three decimal places.A. 0.808B. 1.490C. 1.303D. 2.606E. 2.9814. Find the volume of the solid generated by rotating about the x-axis the regionbounded by the curves y = x +1x, x = 1 and x = 4 and the x-axis.A. 21.75πB. 26.25πC. 27.75πD. (7.5 − ln 4)πE. (7.5 + ln 4)π3MA 23200 - Sample Final Exam5. A volume has cross-sections which are rectangles with length x + 3 and width√x for0 ≤ x ≤ 4. Find the volume of this solid.A. 24B.1445C. 48D. 98E.32036. EvaluateZ∞0xe−12xdx.A. 0B.14C. 2D. 4E. This integral diverges.4MA 23200 - Sample Final Exam7.f(x, y) =yx2+ y2.Compute fx(3, −4).A.24625B.12625C. 0D. −12625E. −246258.f(x, y) = ex cos y.Compute fyy(x, y).A. (x2cos2y −x sin y)ex cos yB. (x sin y cos y − sin y)ex cos yC. x2cos y sin yex cos yD. −x sin yex cos yE. (x2sin2y −x cos y)ex cos y5MA 23200 - Sample Final Exam9. The function f(x, y) has partial derivativesfx(x, y) = 3y − x − 7, fy(x, y) = 3y2− 6y + 3x + 3and critical points(−1, 2), (−16, −3) .(There are no other critical points.) Which statement best describes these criticalpoints?A. Both points are saddle points.B. (−1, 2) is a saddle point and (−16, −3) is a relative maximum.C. (−1, 2) is a saddle point and (−16, −3) is a relative minimum.D. (−1, 2) is a relative minimum and (−16, −3) is a relative maximum.E. (−1, 2) is a relative maximum and (−16, −3) is a relative minimum.10. Sally sells seashells down by the seashore. She has found that if, in the morning,she spends x hours looking for seashells, and y hours polishing and decorating theseashells, she will sellS = 4xy + 4y − 4x2− 2y2+ 196seashells that day. How many hours should Sally spend looking for seashells, andhow many polishing, in order to sell the highest number possible of shells that day?A. 1 hour looking and 3 hours polishing.B. 3 hours looking and 1 hour polishing.C. 1 hour looking and 1 hour polishing.D. 1 hour looking and 2 hours polishing.E. 2 hours looking and 1 hour polishing.6MA 23200 - Sample Final Exam11. EvaluateZ10Zx20(y + 2x2)5dydx.A.272B.66578C.66554D.24326E.745612. Supposef00(x) =1√x+ xwithf(1) = 1 and f0(1) = 0.Find f(4).A.353B.403C.1256D. 25E.6527MA 23200 - Sample Final Exam13. Supposey0− 4y = e6xy(0) = 3.Compute y(1).A.12e6+52e4B. e6+ 2e4C. e6+ 12D.111e6+3211e−4E.269−59e614. Find the general solution to the differential equationxy0+ 6y = x2.A. y =x33+x39 ln x+Cln xB. y =x33−x39 ln x+Cln xC. y =13x4+Cx6D. y =x28+Cx6E. y =x39+Cx68MA 23200 - Sample Final Exam15. The autonomous differential equation,y0= y4− 4y3− 3y2+ 18yhas exactly three equilibrium solutions:y = −2, y = 0 and y = 3.Which statement best describes the stability of these equilibrium solutions?A. y = −2 is asymptotically stable, y = 0 is unstable and y = 3 is semi-stable.B. y = −2 is asymptotically stable, y = 0 is semi-stable and y = 3 is unstable.C. y = −2 is semi-stable, y = 0 is unstable and y = 3 is asymptotically stable.D. y = −2 is unstable, y = 0 is semi-stable and y = 3 is asymptotically stable.E. y = −2 is unstable, y = 0 is asymptotically stable and y = 3 is semi-stable.16. Find the general solution to the differential equationy0= x2y2.A. Ce−13x3B. ±pCe13x3C.3C−x3D. −3x3+ CE. lnC −x339MA 23200 - Sample Final Exam17. Supposey0= x + y2y(0) = 1.Use Euler’s method with ∆x = 0.5 to approximate y(1).A. 1.5B. 1.75C. 2.5D. 2.875E. 3.7812518. A certain vine grows in such a way so that its length satisfies the differential equationL0= 3√Lwhere L is the length of the vine, in feet, t years from now. If a vine is initiallymeasured at 4 feet long, how long will the vine be in 10 years?A. 225 ftB. 229 ftC. 289 ftD. 904 ftE. 1024 ft10MA 23200 - Sample Final Exam19. Which of the following matrices is singular?A.1 23 4B.3 −4−5 7C.3 4−6 −8D.5 −1−1 −4E.105 01 320. Find det(A), whereA =7 6 23 4 1−2 5 −1.A. 59B. 23C. 9D. −11E. −1911MA 23200 - Sample Final Exam21. Find the inverse of the matrixA =4 −2−13 7.A.72121321B.7211322C.721122D.7 213 4E.14 126 222. Find the eigenvalues for the matrixA =3 22 6.A. 3 and 6B. −2 and 11C. −11 and 2D. −3 and 2E. 2 and 712MA 23200 - Sample Final Exam23. The matrixA =−1 −1 1−4 0 2−14 −6 8has r = 4 as one of its eigenvalues. Which of the following is an eigenvectorassociated to this matrix and eigenvalue?A.−114B.021C.102D.138E.01113MA 23200 - Sample Final Exam24. The matrix A has eigenvalues r1= 2 and r2= 3, with eigenvectors v1=10andv2=11. Compute A545.A.−15B.1315C.211243D.371243E.1183121525. Consider the difference equationxn+1= xn+ 6xn−1.If x0= 0 and x1= 20, find x15.A. 57526700B. 57395628C. 57264556D. 19066340E. 1438167514MA 23200 - Sample Final ExamFormula SheetThe Trapezoid Rule The Trapezoid Rule for estimating the integralRbaf(x)dx with ntrapezoids is given byTn=12∆x [f(x0) + 2f(x1) + ··· + 2f(xn−1) + f(xn)]where ∆x =b−an, x0= a, x1= a + ∆x, x2= a + 2∆x, . . . , xn= a + n∆x = b.D-Test To find the relative maximum and minimum values of f:1. Find fx, fy, fxx, fyyand fxy.2. Solve fx(x, y) = 0 and fy(x, y) = 0.3. Evaluate D = fxxfyy− [fxy]2at each point (a, b) found in Step 2.(a) If D(a, b) > 0 and fxx(a, b) < 0 then f has a relative maximum at (a, b).(b) If D(a, b) > 0 and fxx(a, b) > 0 then f has a relative