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C Roettger, Fall 14 Math 265 Quiz 6D – Solutions Graph of f (x, y) in blue, tangent plane red. Intersection curves of the graph with planes x = x0 and y = y0 green and yellow, respectively. Problem 1 Consider the function f (x, y) = x2 + xy . 3 Find an equation for the tangent plane to the graph of f at P0 = (2, −1). Solution. First, the partial derivatives ∂f 2x + y = ∂x 3 ∂f x = . ∂y 3 At P0 , these have the values 1 and 2/3, respectively. Finally, we need the value f (P0 ) = 2/3. So the tangent plane to the graph of f at P0 is described by the equation z= 2 2 2 2 + (x − 2) + (y + 1) = − + x + y. 3 3 3 3