Test 3_MAT 2030_11SS

# Test 3_MAT 2030_11SS - [8 pts(6 Find the equations of the...

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Name : MAT 2030, Calculus III, Test 3 No calculators needed or allowed! Questions : (50 + 5 points) (1) Verify that the conclusion of Clairaut’s Theorem holds, that is, u xy = u yx : u = x sin( x + 2 y ). [6 pts] (2) If z = 5 x 2 + y 2 and ( x, y ) changes from (1 , 2) to (1 . 05 , 2 . 1), compare the values of Δ z and dz . [6 pts] (3) If z = f ( x, y ), where x = r cos θ and y = r sin θ , (a). Find ∂z ∂r and ∂z ∂θ . [4 pts] (b). Show that ± ∂z ∂x ² 2 + ± ∂z ∂y ² 2 = ± ∂z ∂r ² 2 + 1 r 2 ± ∂z ∂θ ² 2 . [4 pts] . (4) If z = f ( x - y ), show that ∂z ∂x + ∂z ∂y = 0. [6 pts] (5) Find the maximum rate of change of f = sin( xy ) at (1 , 0) and the direction in which it occurs.
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Unformatted text preview: [8 pts] (6) Find the equations of the tangent plane and normal line at the point (-2 , 1 ,-3) to the ellipsoid x 2 4 + y 2 + z 2 9 = 3. [8 pts] (7) Find the local maximum and minimum values and saddle points of f ( x, y ) = x 4 + y 4-4 xy +1. [8 pts] Additional : Find three positive numbers whose sum is 27 and whose product is a maximum. [5 pts] (No partial credit) Show all your work in blue/green book! 1...
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