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Math 2110Q Sec 100, Spring 2023 Exam 2 Version A Loc Hoang TOTAL POINTS 37 / 48 QUESTION 1 Multiple Choice 16 pts 1.1 Local Extrema 4 / 4 ✓ - 0 pts Correct! This is a local minimum since $$D>0$$ and $$f_{xx}>0$$. 1.2 Bead Integral Setup 4 / 4 ✓ - 0 pts Correct! The fourth option gives the volume of the bead. 1.3 Half a Sphere 4 / 4 ✓ - 0 pts Correct! The second and third options are correct. 1.4 Change of Variables 4 / 4 ✓ - 0 pts Correct! The Jacobian is 1/5, so we get the second one. QUESTION 2 2 Matching 4 / 8 ✓ - 2 pts First one should be A. ✓ - 2 pts Third one should be B. QUESTION 3 3 Tangent Plane 8 / 8 ✓ + 8 pts [Full Credit] Correct! The tangent plane is $$z=-4-3(x-1)-4(y-1)$$. + 1 pts [Partial credit] Correct $$f_x$$. Should get $$f_x=2xy-5y$$. + 1 pts [Partial credit] Correct $$f_y$$. Should get $$f_y=x^2-5x$$. + 1 pts [Partial credit] Correct or consistent $$f_x(1,1)$$. Should get -3 or an answer consistent with a prior mistake. + 1 pts [Partial credit] Correct or consistent $$f_y(1,1)$$. Should get -4 or an answer consistent with a prior mistake. + 1 pts [Partial credit] Correct $$f(1,1)$$. Should get $$-4$$. + 3 pts [Partial credit] Correctly applied the formula $$z - z_0 = f_x(x_0,y_0)(x-x_0) + f_y(x_0,y_0)(y - y_0)$$. + 2 pts [Partial credit] Correct formula except for one small issue or mistake in application + 1 pts [Partial credit] Partially correct formula + 0 pts none of the above/no work shown + 0 pts Incorrect Formula QUESTION 4 4 Triple Integral Setup 6 / 8 Error in Spherical. ✓ - 1 pts Wrong upper $$\phi$$ bound. Should be $$\pi$$. ✓ - 1 pts Wrong upper $$\theta$$ bound. Should be $$\pi$$ QUESTION 5

5 Double Integral Switch Order 1 / 0 + 0 pts No work/not very far into solving the question ✓ + 1 pts Not fully completed but good start and work on solving the question + 2 pts Correct, with perhaps a small mistake. QUESTION 6 Tent Problem 8 pts 6.1 Equations of Planes: SIDE 1 0 / 1 ✓ - 1 pts Incorrect or blank. You should have $$z=y$$. 6.2 Equation of Planes: SIDE 2 0 / 1 ✓ - 1 pts Blank or incorrect. You should have $$z=2- y$$. 6.3 Swapping Order 2 / 6 ✓ - 1 pts Incorrect lower $$y$$ bound. It should be $$z$$. ✓ - 1 pts Incorrect upper $$y$$ bound. It should be $$2-z$$. ✓ - 1 pts Incorrect lower $$z$$ bound. It should be 0. ✓ - 1 pts Incorrect upper $$z$$ bound. It should be 1. QUESTION 7 7 Bonus (cone) 0 / 0 + 2 pts Correct! By changing this to spherical we can get $$\phi=\pi/6$$. + 1 pts Efforts to simplify the equation or correct answer without justification or minor mistake! Should end up with $$\phi=\pi/6$$ ✓ + 0 pts No bonus points awarded. Page 2