Math 241 Midterm exam 2 Spring 2014 Name Section circle one AD1 ADD ADI AD2 ADE ADJ ADA ADF ADK ADB ADG ADL ADC ADH ADM BDA BDF BDK BDP BDB BDG BDL BDQ BDC BDH BDM BDD BDI BDN BDE BDJ BDO READ ALL INSTRUCTIONS CAREFULLY Write legibly and use the boxes for your nal answers where provided Be sure to use correct notation in particular distinguish vectors from scalars by arrow notation use explicit clearly visible dots for dot products etc An answer alone without justi cation will not earn full credit with the exception of the multiple choice Scantron problems 1 4 5 and 7 If you make a mistake cross out all of your incorrect work We will take points off for incorrect work that is not crossed out even if the correct answer is given elsewhere Problem Point Value Test Score 1 5 Scantron 2 6 3 5 4 4 Scantron 5 6 Scantron 6 4 7 3 8 7 Total 40 Scantron Math 241 Midterm exam 2 Spring 2014 1 5 points Fill in the blanks in the integrals below that compute the length L of the curve C given by the top half of the ellipse x2 4y2 4 that is the part of the ellipse that lies on or above the x axis and connects the two points 2 0 and 2 0 Use the parametrization r t 2 cos t sin t for the ellipse You do NOT have to evaluate the integral Hint Sketch the ellipse and the curve C Line 1 ds L C Second number in line 2 Line 3 dt First number in line 2 Pencil your answers into the corresponding lines in your Scantron bubble sheet Line 1 A x2 4y2 B cos2 t 4 sin2 t C 1 D F r t r t E 4 A 0 to B 0 to 2 C 0 to 1 Line 2 2 2 2 D E A 2 B C D 2 sin t cos t 2 sin t cos t 1 3 sin2 t E 1 3 sin2 t to to Line 3 Math 241 Midterm exam 2 Spring 2014 2 6 points Find the curvature of the curve given by the vector function r t 1 t t2 at the point 1 1 1 2 4 A few words of advice which in principle apply to all exam problems if you nd yourself entangled in tedious calculations you are most likely on the wrong track or are missing an easier solution and it may be best to try a different approach If necessary move on to the next problem and come back to this problem later if you have time left at the end at the point 1 1 1 2 4 Math 241 Midterm exam 2 Spring 2014 3 5 points Evaluate the line integral of the vector eld F x y y x along the curve C given by the arc of the circle of radius 2 from 0 2 to 0 2 Of course there are two such arcs C is the one that lies on and to the right of the y axis Show your work F dr C Math 241 Midterm exam 2 Spring 2014 4 2 points each 4 points Below are the plots of two vector elds F and the graphs of curves C superimposed on each plot For both gures determine whether the line integral of the vector eld along the curve is A positive or B negative C zero or and pencil your answers into the corresponding lines in your Scantron bubble sheet Options D and E are not valid answers in this problem The gure on the LEFT corresponds to line 4 and the gure on the RIGHT corresponds to line 5 in your Scantron bubble sheet 5 2 2 1 1 points 6 points Match the plots of the vector elds F in the above gures with their equations and pencil your answers into the corresponding lines in your Scantron bubble sheet Options D and E are not valid answers in this problem In line 6 of your Scantron bubble sheet mark whether the vector eld on the LEFT is A F x y y or B F x y x y or C F x y xy x y In line 7 of your Scantron bubble sheet mark whether the vector eld on the RIGHT is A F x y 0 5y or B F x y or C F x y 0 5x Also match the two curves C in the same gures with their vector equations and pencil your answers into the corresponding lines in your Scantron bubble sheet In line 8 of your Scantron form mark whether the curve on the LEFT is given by the vector equation A r t 4 cos t 4 sin t 0 t or B r t t 4 cos t 4 sin t 0 t 2 C r t 4 cos t 4 sin t 0 t 2 or In line 9 of your Scantron form mark whether the curve on the RIGHT is given by the vector equation A r t t t2 0 t 5 or B r t t t 0 t 5 or C r t t t 0 t 5 Math 241 Midterm exam 2 Spring 2014 6 2 points each 4 points For each of the two given vector elds F nd a potential function That is nd a function f such that F f If such a potential function does not exist enter DNE and explain why a F x y 3x2 y2 2x3 y f DNE because f b F x y ex y ey f DNE because f 7 3 points Evaluate the line integral of the vector eld F x y 3x2 y2 2x3 y along the curve C shown below C is the union of the four line segments from 1 1 to 2 1 from 2 1 to 3 2 from 3 2 to 3 3 and then from 3 3 to 2 2 of your Scantron bubble sheet among the choices A E below Pencil your answer into line 10 3 2 A 0 1 3 2 1 1 2 3 B 31 1 2 3 C 31 D 46 E 46 Math 241 Midterm exam 2 8 7 points Evaluate the double integral Spring 2014 3y dA D where D is the region bounded by the parabola y2 x and the line x 2y 0 First sketch the region D Then set up an iterated integral and evaluate it Show your work 3y dA D
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