MTH 234Michigan State UniversityDepartment of MathematicsName: PID: Section No:Problem Total Score1 162 163 174 175 166 177 178 179 1710 1611 1712 17Total 200MTH 234 Practice Final Exam Fall 2010 1Michigan State UniversityDepartment of MathematicsName: PID: Section No:Signature:Total Score:1. Check that you have pages 1 through 16 and that none are blank.2. Fill in the information at the top of the page.3. You will need a pen or pencil and this booklet for the exam. Please clear everything else fromyour desk.4. The use of calculators, cell phones, or any other electronic device as an aid to writing thisexam is strictly prohibited.5. The grading of this exam is based on your method. Show all of your work. (There areproblems however that will be graded right or wrong.) If you need additional space, use thebacks of the exam pages.6. If you present different answers, the worst answer will be graded.7. Box your answers.MTH 234 Practice Final Exam Fall 2010 21. (16 points)(a) Find a unit vector in the opposite direction of v = h1, 2, 3i.(b) Find the scalar projection of w = h1, −1, 2i onto v.(c) Find the vector projection of w onto v.MTH 234 Practice Final Exam Fall 2010 32. (16 points) Find the equation of the plane that contains the lines r1(t) = h1, 2, 3it andr2(t) = h1, 1, 0i + h1, 2, 3it.MTH 234 Practice Final Exam Fall 2010 43. (17 points) A particle moves along the curve r(t) = hsin(2t2), t3, cos(2t2)i, for t ≥ 0.(a) Find the velocity v(t) and acceleration a(t) functions of the particle.(b) Find the arc length function for the curve r(t) measured from the point where t = 0, in thedirection of increasing t.MTH 234 Practice Final Exam Fall 2010 54. (17 points)(a) Find and sketch the domain of the function f (x, t) = ln(t − x2).(b) Determine whether the function f above is solution of the wave equation ftt− fxx= 0.MTH 234 Practice Final Exam Fall 2010 65. (16 points)(a) Find the tangent plane approximation of f(x, y) = sin(2x + 5y) at the point (−5, 2).(b) Use the linear approximation computed above to approximate the value of f(−4.8, 2.1).MTH 234 Practice Final Exam Fall 2010 76. (17 points) Find the absolute maximum and absolute minimum of the function f(x, y) =x2+ 3y2− 2xy in the triangle formed by the lines y = 0, x = 1 and y = x.MTH 234 Practice Final Exam Fall 2010 87. (17 points)(a) Sketch the region of integration, D, whose area is given by the double integralZ ZDdA =Z20Z3√x/232xdy dx.(b) Compute the double integral given in (a).(c) Change the order of integration in the integral given in (a).MTH 234 Practice Final Exam Fall 2010 98. (17 points) Transform to polar coordinates and then evaluate the integralI =Z1−1Z0−√1−y2ln(x2+ y2+ 1) dx dy.MTH 234 Practice Final Exam Fall 2010 109. (17 points) Find the component z of the centroid for a wire lying along the the curve givenby r(t) = ht cos(t), t sin(t), (2√2/3)t3/2i, for t ∈ [0, 1].MTH 234 Practice Final Exam Fall 2010 1110. (16 points) Use the Green Theorem area formula to find the area of the region enclosed bythe curve r(t) = hcos2(t), sin2(t)i for t ∈ [0, π/2]. (16.4.23).MTH 234 Practice Final Exam Fall 2010 1211. (17 points) Find the flux of ∇ × F outward through the s urface S, where F = h−y, x, x2iand S = {x2+ y2= a2, z ∈ [0, h]} ∪{x2+ y26 a2, z = h}.MTH 234 Practice Final Exam Fall 2010 1312. (17 points) Find the outward flux of the field F = hx2, −2xy, 3xzi across the boundary ofthe region D = {x2+ y2+ z26 4, x > 0, y > 0, z >