Exam 2 November 20 2017 MTH 234 Name Section Recitation Instructor INSTRUCTIONS Fill in your name etc on this rst page Without fully opening the exam check that you have pages 1 through 12 Show all your work on the standard response questions Write your answers clearly Include enough steps for the grader to be able to follow your work Don t skip limits or equal signs etc Include words to clarify your reasoning Do rst all of the problems you know how to do immediately Do not spend too much time on any particular problem Return to di cult problems later If you have any questions please raise your hand and a proctor will come to you You will be given exactly 90 minutes for this exam Remove and utilize the formula sheet provided to you at the end of this exam ACADEMIC HONESTY Do not open the exam booklet until you are instructed to do so Do not seek or obtain any kind of help from anyone to answer questions on this exam If you have questions consult only the proctor s Books notes calculators phones or any other electronic devices are not allowed on the exam Students should store them in their backpacks No scratch paper is permitted If you need more room use the back of a page Anyone who violates these instructions will have committed an act of academic dishonesty Penalties for academic dishonesty can be very severe All cases of academic dishonesty will be reported immediately to the Dean of Undergraduate Studies and added to the student s academic record I have read and understand the above instructions and statements regarding academic honesty SIGNATURE Page 1 of 12 MTH 234 Exam 2 November 20 2017 Standard Response Questions Show all work to receive credit Please BOX your nal answer 1 5 points Let f x y x ey 2 y2 ln x Find f at P 1 2 2 9 points Let g x y 8x3 maximum or a saddle point 12xy y2 Find and classify each critical point of g as a local minimum a local Page 2 of 12 MTH 234 Exam 2 November 20 2017 3 Consider the integral below and answer the questions that follow 1 0 x sin y2 dy dx a 3 points Sketch the region of integration b 5 points Evaluate the integral above by reversing the order of integration 4 6 points Evaluate the integral below 3 9 x2 0 9 x2 p 1 5 x2 y2 dy dx Page 3 of 12 MTH 234 Exam 2 November 20 2017 5 6 points Set up but Do Not Evaluate the iterated integral for computing the volume of a region D if D is the right circular cylinder whose base is the disk r 2 sin in the xy plane and whose top lies on the surface z 8 x2 z z z 8 x2 r 2 sin x x y y 6 8 points Find the area of the surface of the cap cut from the paraboloid z 12 x2 y2 by the cone z x2 y2 p Page 4 of 12 MTH 234 Exam 2 November 20 2017 7 Let F 2xy i x2 cos z j y sin z k and answer the questions below a 7 points Find a function f so that f F b 4 points Let C be any path from 2 1 2 to 1 3 Evaluate the integral 2xy dx x2 cos z dy y sin z dz C c 3 points Calculate curl F Page 5 of 12 MTH 234 Exam 2 November 20 2017 8 7 points Let E be the portion of the ball x2 y2 z2 0 z rst octant x 0 y 0 1 that lies in the Rewrite the triple integral below as an iterated integral in spherical coordinates and evaluate 5y dV E 9 7 points Let F x y Use Green s Theorem to calculate the work done by F on a particle moving counter clockwise around the triangle bounded by x 0 y 0 x y 1 y2 x2 cid 10 cid 11 Page 6 of 12 MTH 234 Exam 2 November 20 2017 Multiple Choice Circle the best answer No work needed No partial credit available 10 4 points Let P P 1 2 2 and u j If f P 3 i 2 j 1 2 i 1 2 11 4 points Suppose that the line segment C is given by the parametrization r t t 1 i 2t j 0 t 3 Evaluate the integral below x2 y cid 11 dr C cid 10 then Du P A 11 2 3 B 2 2 C D 3 2 1 2 2 j i E None of the above A 57 B 39 C 21 D 3 E None of the above 12 4 points A sin 8 4 cos 8 2 cos 8 4 cos 4 2 B C D E None of the above 2 x2 4 x 0 0 0 sin 2z z 4 dy dz dx Page 7 of 12 MTH 234 Exam 2 November 20 2017 13 4 points Parameterize the part of the plane x z 3 that lies above the disk x 1 2 y2 1 A r s t s t 3 with s 0 2 cos t and t B r s t s t 3 with s C r s t s cos t s sin t 3 s cos t with s 0 2 and t s i s i D r s t s cos t s sin t 3 s cos t with s 1 1 1 1 0 2 and t 0 2 cos t and t 2 2 2 2 i i h h h h E None of the above 14 4 points Let F x y arctan cid 16 y y x cid 17 x2 y2 i x x2 y2 j If C is the unit circle parameterized by r t cos t i sin t j 0 t 2 then F dr C A 0 B C 2 D Unde ned E None of the above 15 4 points Let f be a di erentiable function of x and y with continuous second order partial derivatives Let f M i N j be a gradient eld and let C be the positively oriented ellipse as shown in the sketch F below Consider the following statements a F is a conservative vector eld M dx N dy 0 b c C curl F k 0 A All three statements are true B Only a and b are true C Only a and c are true D Only b and c are true E None of the above Page 8 of 12 MTH 234 Exam 2 November 20 2017 16 4 points Let F Which of the following is true x2y yz z2 cid 10 cid 11 A curl F 2xy z and div F y x2 B C curl F cid 10 curl F y y 0 x2 cid 11 2xy z 0 D curl F i E None of …