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Math 227-104 (CRN 24680) Carter Study Guide Spring 2015 General Instructions. 1. Find equations of the tangent plane and normal line to the w = 0 level surface of f (x, y, z), at a specific point. 2. For a function f (x, y), (a) Find ∇f (b) Find the directional derivative of f at a point P0 , in a specific direction v — observe that you’ll need to normalize v. (c) Find the maximum rate of change of f at the point P0 , and state the direction in which it occurs. 3. (a) Find and classify the critical points of some function f (x, y). (b) Use Lagrange multipliers to find the maximum and minimum values of some function f (x, y), subject to some constraint g(x, y) = 0. 4. Evaluate the integral Z ~b M dx + N dy + P dz. ~a 5. Use Green’s Theorem to evaluate I M dx + N dy C where C is the boundary of some standard object. 6. A question about the area form. 7. Study Example 5 page 907. 1