59 Samplefinal-bypublisher

59 Samplefinal-bypublisher - Page 123 Practice Final...

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Page 123 Practice Final Examination This is a practice examination, consisting of 10 questions that requires approximately 3 hours to complete. Full solutions are at the end of the answer section following this exam, so you can check your solutions and self-mark the exam. It is always important to make sure you know how to solve the problem completely if you only got it partly correct. Of course there might be correct solutions that are slightly different from those given. Consult your instructor if you have special concerns. Note that there are also practice quizzes and exams in the student guide. Good luck! 1. (a) Let a, b > 0 and let a point of mass m move along the curve in R 3 defined by c ( t ) = ( a cos t, b sin t, t 2 ). Describe the curve and find the tangent line at t = π . (b) Find the force acting on the particle at t = π . (c) Define the function E ( z ) by E ( z ) = π/ 2 0 1 z 2 sin 2 t dt. Find a formula for the circumference of the ellipse x 2 a 2 + y 2 b 2 = 1 , where 0 < a < b , in terms of E ( z ).
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124 Practice Final Examination (d) Let f : R 3 R be defined by f ( x, y, z ) = z + exp [ x 2 y 2 ] . In what directions starting from (1, 1, 1) is f decreasing at 50% of its maximum rate of change?
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