Math 241 Midterm exam 2 Spring 2014Name:Section (circle one):AD1 AD2 ADA ADB ADCADD ADE ADF ADG ADHADI ADJ ADK ADL ADMBDA BDB BDC BDD BDEBDF BDG BDH BDI BDJBDK BDL BDM BDN BDOBDP BDQREAD ALL INSTRUCTIONS CAREFULLY. Write legibly, and use the boxes for your final answers whereprovided.Be sure to use correct notation; in particular, distinguish vectors from scalars by arrow notation, use explicitclearly visible dots for dot products, etc.An answer alone, without justification, will not earn full credit (with the exception of the multiple choiceScantron problems 1, 4, 5, and 7). If you make a mistake, cross out all of your incorrect work. We willtake points off for incorrect work that is not crossed out, even if the correct answer is given elsewhere.Problem Point Value Test Score1 5 Scantron2 63 54 4 Scantron5 6 Scantron6 47 3 Scantron8 7Total 40Math 241 Midterm exam 2 Spring 20141. (5 points) Fill in the blanks in the integrals below that compute the length L of the curve C given bythe top half of the ellipse x2+ 4y2= 4 (that is, the part of the ellipse that lies on or above the x-axisand connects the two points (2, 0) and (−2, 0)). Use the parametrization r(t) = ⟨2 cos t, sin t⟩ for theellipse. You do NOT have to evaluate the integral. Hint: Sketch the ellipse and the curve C.L =∫CLine..1ds =∫Second number in line..2First number in line..2Line..3dtPencil your answers into the corresponding lines in your Scantron bubble sheet.Line..1 :•..A = x2+ 4y2•..B = cos2t + 4 sin2t•..C = 1•..D = F(r(t)) • r′(t)•..E = 4Line..2 :•..A = 0 to π•..B = 0 to 2π•..C = 0 to 1•..D = −π2toπ2•..E = −π2to πLine..3 :•..A = 2•..B = −2 sin t + cos t•..C =√−2 sin t + cos t•..D =√1 + 3 sin2t•..E = 1 + 3 sin2tMath 241 Midterm exam 2 Spring 20142. (6 points) Find the curvature κ of the curve given by the vector functionr(t) =⟨1, t, t2⟩at the point(1,12,14).A few words of advice (which in principle apply to all exam problems): if you find yourself entangledin tedious calculations, you are most likely on the wrong track or are missing an easier solution, andit may be best to try a different approach. If necessary, move on to the next problem, and come backto this problem later if you have time left at the end.κ = at the point(1,12,14)Math 241 Midterm exam 2 Spring 20143. (5 points) Evaluate the line integral of the vector fieldF(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 aretwo such arcs; C is the one that lies on and to the right of the y-axis.) Show your work.∫CF • dr =Math 241 Midterm exam 2 Spring 20144. (2 points each = 4 points) Below are the plots of two vector fields F and the graphs of curves Csuperimposed on each plot. For both figures, determine whether the line integral of the vector fieldalong the curve is:..A = positive or..B = negative or..C = zeroand pencil your answers into the corresponding lines in your Scantron bubble sheet. (Options D andE are not valid answers in this problem.).The figure on the LEFT corresponds to line..4 , and the figure 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 fields F in the above figures withtheir 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 field on the LEFT is:..A F(x, y) = yı −ȷ or..B F(x, y) = xı + yȷ or..C F(x, y) = xyı + (x − y)ȷIn line..7 of your Scantron bubble sheet, mark whether the vector field on the RIGHT is:..A F(x, y) = 0.5yı or..B F(x, y) =ı or..C F(x, y) = 0.5xȷAlso match the two curves C in the same figures with their vector equations, and pencil your answersinto 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 ≤ π or..B r(t) = ⟨t, 4 cos t + 4 sin t⟩, 0 ≤ t ≤ π/2 or..C r(t) = ⟨4 cos t, 4 sin t⟩, 0 ≤ t ≤ π/2In 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 ≤ 5Math 241 Midterm exam 2 Spring 20146. (2 points each = 4 points) For each of the two given vector fields F, find a potential function. That is,find a function f such that F = ∇f. If such a potential function does not exist, enter DNE and explainwhy.(a) F(x, y) =⟨3x2y2, 2x3y⟩f =f DNE because:(b) F(x, y) =⟨ex+y, ey⟩f =f DNE because:7. (3 points) Evaluate the line integral of the vector field F(x, y) = ⟨3x2y2, 2x3y⟩ along the curve Cshown 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).)Pencil your answer into line..10 of your Scantron bubble sheet among the choices..A −..E below...·.·.2.·.1.·.2.·.−1.·.−2.·.−2.·.−1.·.−3.·.3.·.3.·.−3.1..A = 0..B = 31..C = −31..D = 46..E = −46Math 241 Midterm exam 2 Spring 20148. (7 points) Evaluate the double integral∫∫D3y dA,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.∫∫D3y dA
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