MTH 234 Solutions to Exam 2 November 23 2015 Name Section Recitation Instructor READ THE FOLLOWING INSTRUCTIONS Do not open your exam until told to do so No calculators cell phones or any other electronic devices can be used on this exam Clear your desk of everything excepts pens pencils and erasers If you need scratch paper use the back of the previous page Without fully opening the exam check that you have pages 1 through 9 Fill in your name etc on this rst page Show all your work 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 There is no talking allowed during the exam You will be given exactly 90 minutes for this exam I have read and understand the above instructions SIGNATURE Page 1 of 9 MTH 234 Solutions to Exam 2 November 23 2015 Multiple Choice Circle the best answer No work needed No partial credit available 1 5 points Parameterize the part of the plane 6x 3y z 12 that lies in the rst octant A r s t B r s t C r s t 6s 3t 12 h s t 12 h s t 12 h s t 12 h s t i 6s s 6s t 3t i 3t i with s with s 0 2 and t 0 4 2s 0 2 and t 0 4 i with s 0 2 and t 0 4 with s 0 2 and t 0 4 2s D r s t E None of the above 2 5 points Which of the following vector eld plots could be F xy i y2 j B Extra Work Space Page 2 of 9 MTH 234 Solutions to Exam 2 November 23 2015 3 5 points Let F x2 xy 3z A curl F y and div F 3x 3 cid 10 Which of the following is true cid 11 B curl F y and div F 2x x 3 h i C curl F and div F 3x 3 D curl F and div F y 0 0 y i h 2x x 3 h i E None of the above Fill in the Blanks No work needed No partial credit available 4 5 points Write the spherical equation cos in rectangular coordinates z x2 y2 z2 5 10 points Convert the integral 1 1 y2 x 1 0 0 x2 y2 dz dx dy to an equivalent integral in cylindrical coordinates 2 r2 z2 1 0 0 r3 dz dr d where 1 2 2 2 r2 1 z2 r cos Extra Work Space Page 3 of 9 MTH 234 Solutions to Exam 2 November 23 2015 Standard Response Questions Show all work to receive credit Please BOX your nal answer 6 10 points Let f x y z x ex yz2 a Find f at P0 0 2 3 Solution Thus fx x 1 ex fx P0 fy z2 fy P0 9 fz 2yz fz P0 12 f P0 i 9 j 12 k b Find the derivative of f at P0 in the direction of the vector A 2 i j 2 k Solution Let u 2 i j 2 k Then A A 1 3 Duf f u i 9 j 12 k 2 i 1 3 31 j 2 k 3 7 14 points Let f x y x3 12xy 8y3 Find and classify each critical point of f as a local minimum a local maximum or a saddle point Solution i Find the critical points Notice that fx 3x2 12y and fy 24y2 12x So f has a critical points at P 0 0 and Q 2 1 ii Now let D x y fxx fyy f 2 xy 144 2xy 1 Notice that fxx Q 12 0 and that D P 144 0 and D Q 432 0 It follows that f has a local minimum at Q and a saddle point at P Page 4 of 9 MTH 234 Solutions to Exam 2 November 23 2015 8 12 points Sketch the region of integration for the integral below and evaluate the integral by reversing the order of integration Solution 1 1 0 y sin x2 dx dy sin x2 dy dx 1 x 0 0 1 0 1 2 x sin x2 dx cos x2 cid 0 cid 1 1 0 1 1 y x 1 9 12 points Evaluate the integral below 2 4 x2 0 4 x2 p 1 1 x2 y2 dy dx Solution We switch to polar coordinates r dr d 2 2 2 0 1 1 r2 2 1 1 r2 0 r dr 5 2 1 1 u du u 5 1 cid 17 cid 16 5 1 Page 5 of 9 MTH 234 Solutions to Exam 2 November 23 2015 10 12 points Find the volume of the part of the sphere 2 that lies between the cones 4 and 3 Solution 2 3 2 V 0 4 0 2 sin d d d 2 3 2 4 0 2 sin d d 3 16 3 4 sin d 16 3 3 cos cid 16 cos 4 cid 17 8 3 cid 16 2 1 cid 17 11 12 points Find the area of the surface z xy that lies inside the cylinder x2 y2 4 Solution Let r x y x i y j xy k Then So that and Thus rx i y k ry j x k rx ry y i x j k rx ry 2 1 x2 y2 SA x2 y2 rx 4 ry dA 1 x2 y2 dA x2 y2 2 4 p 2 1 r2 r dr 0 p 2 3 cid 16 53 2 1 cid 17 Page 6 of 9 MTH 234 Solutions to Exam 2 November 23 2015 12 12 points Find the work done by the force F 4y 2x along the straight line segment from 2 8 to 1 2 h i Let C be the indicated line segment Now let Solution Then it follows that r t 2 t i 8 10t j 0 t 1 dr i 10 j dt and F dr 8 20t dt 1 F dr 8 20t dt C 0 8t 10t2 cid 0 cid 1 1 0 18 13 Let F 2 2xyez i x2ez j x2yez k and answer the questions below a 6 points Find a function f so that f F f x y z x2yez 2x f 2 2xyez i x2ez j x2yez k b 6 points Let C be any path from 2 1 0 to 3 0 1 Evaluate the integral 2 2xyez dx x2ez dy x2yez dz I C Solution Let then Solution Observe that It follows that df 2 2xyez dx x2ez dy x2yez dz 3 …