Mth 254 Exam Name: ID:Bent Petersen 254w2005-exam.tex March 15 2005, 1200 Time: 110 minutes.Instructions: =⇒If you do not read the in-structions, then how willyou know what to do?Read them now.Be sure to enter allrequired information onthe scantron and on thistest.Section Number: 001Form Number: 001• This test is mostly multiple-choice but may contain some workout problems. You must turn inboth the test and the scantron.• For the multiple-choice problems you must mark your answer on the provided scantron. Fillin the appropriate bubbles on the scantron very carefully.• For the workout problems you must show your work in reasonable detail on the test. Partialcredit is allocated only for clear and relevant work.• You may use one 8.5 × 11 inch note sheet prepared in advance. You may write on both sidesof your note sheet.• Note sheets may not be shared. If you do not bring a note sheet you will have to do withoutany help notes.• You may not use any books, notebooks, additional note sheets nor note cards.• You are expected to have a simple scientific calculator available for use on this test. Calcu-lators and other equipment may not be shared.• You may use a simple graphics calculator but not a laptop computer nor any device capableof extensive symbolic manipulation (other than your own brain).• There are 12 multiple-choice problems worth 8 points each and 0 work-out problems worth20 points each.Important Notes:• Note that log(x) means the natural logarithm of x, sometimes denoted by ln(x). The logarithm with base 10will be denoted by log10(x), the logarithm with base 2 will be denoted by log2(x), and so on.• If you are taking this test in the Mathematics Learning Center you will not need a scantron. Just be sure to writethe letters corresponding to your answers in the boxes provided below.Problem 1. Use the method of Lagrange multipliers to find the maximum of x3+ y3on the circle x2+ y2= 1.A.) 1 B.) 2C.)√2/2 D.)√2/4 E.) None of the above.←Write letter corresponding to your answer here and mark it on the scantron (Problem 1).Problem 2. Use the method of Lagrange multipliers (or a suitable parametrization) to find the maximum of xyon the ellipse 3x2+ 2y2= 5.A.) 1 B.)5√6C.)52√6D.)54√6E.) None of the above.←Write letter corresponding to your answer here and mark it on the scantron (Problem 2).Problem 3. Evaluate the double integralZZDx2dx dywhere D is the region below the parabola y = x2and above the graph of y = x4.A.) 4/35 B.) 1/9C.) 1/36 D.) 8/69 E.) None of the above.←Write letter corresponding to your answer here and mark it on the scantron (Problem 3).Problem 4. Let R be the square {(x, y) | 0 ≤ x ≤ 1, 0 ≤ y ≤ 1 }. Compute the double integralZZRx(xy + 1)3dx dy.A.) 1 B.) 3/8C.) 3/4 D.) 1/4 E.) None of the above.←Write letter corresponding to your answer here and mark it on the scantron (Problem 4).Problem 5. Evaluate the double integralZZTx2y dx dywhere T is the triangle 2x ≤ y ≤ 4, 0 ≤ x ≤ 2.A.) 17/2 B.) 8C.) 128/15 D.) 64/15 E.) None of the above.←Write letter corresponding to your answer here and mark it on the scantron (Problem 5).Problem 6. Compute the integralZZDx2+ y25/2dx dywhere D is the semidisk x2+ y2≤ 1, y ≥ 0.A.) π/6 B.) π/7C.) 2π/7 D.) π/3 E.) None of the above.←Write letter corresponding to your answer here and mark it on the scantron (Problem 6).Problem 7. Compute the integralZZZBx2+ y2+ z23/2dx dy dzwhere B is the ball x2+ y2+ z2≤ 4.A.) 128√2π/9 B.) 128π/3C.) 2π/3 D.) 40π E.) None of the above.←Write letter corresponding to your answer here and mark it on the scantron (Problem 7).Problem 8. Find the length of the helix~r(t) =Dcos(t), sin(t), 2√2tE, 0 ≤ t ≤ 2π.A.) 2π B.) 4πC.) 4√2π D.) 6π E.) None of the above.←Write letter corresponding to your answer here and mark it on the scantron (Problem 8).Problem 9. If~r(t) =Detcos(t), etsin(t),√2etEfind the unit tangent at t = π/3.A.)141, 1,√2B.)141 −√3, 1 +√3, 2√2C.)12√2−√3,√3,√2D.)1√6h1, 1, 2i E.) None of the above.←Write letter corresponding to your answer here and mark it on the scantron (Problem 9).Problem 10. If~r(t) =Detcos(t), etsin(t),√2etEcompute the curvature κ(t).A.)√2/4e−tB.)√2/4etC.)√2/4D.) 4e−tE.) None of the above.←Write letter corresponding to your answer here and mark it on the scantron (Problem 10).Problem 11. Find the maximum curvature of log(x) for x > 0.A.)√3/9 B.) 2√3/9C.) 8/(3√3) D.) 1/9 E.) None of the above.←Write letter corresponding to your answer here and mark it on the scantron (Problem 11).Problem 12. A point moves with constant speed 10 cm/sec along a curve with constant radius of curvature 10cm. The magnitude of it acceleration isA.) 0 cm/sec2B.) 10 cm/sec2C.) 100 cm/sec2D.) 1000 cm/sec2E.) None of the above.←Write letter corresponding to your answer here and mark it on the scantron (Problem 12).Use this page and the backs of all the pages for scratch