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**Unformatted text preview: **1 Problem set 7 1. Compute the first-order partial derivatives of f ( x, y ) = xy 2 by explicitly evaluating the limits coming from the definition. 2. Let f ( x, y, z, w ) = (1 , z + w, 1 x 2 + y 2 ) and let a = (3 , 4 , , 0) . Compute D f ( a ) . 3. Let g ( s, t ) = (3 cot s t , 2 s ) and let a = (1 , 1) . Compute D g ( a ) . 4. Let F ( x, y, z ) = xz sin(2 y ) + ye z . (a) Find all the mixed partial derivatives of F and verify that Clairaut’s Theorem holds. (b) Is F zxy = F xzy ? Could you have known this without explicitly performing the calculation? (c) is F yzy = F yyz ? 5. For what real number a, b , and c does the function f ( x, y ) = ax 2 + bxy + cy 2 satisfy the partial differential equation f xx + f yy = 0? 6. Let S and P be the hemisphere and plane defined, respectively, by S = { ( x, y, z ) ∈ R 3 vextendsingle vextendsingle x 2 + y 2 + z 2 = 6 and z ≥ } P = { ( x, y, z ) ∈ R 3 vextendsingle vextendsingle x = 2 } ....

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