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C Roettger, Fall 14 Math 265 Quiz 6B – 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) = x3 + xy . 2 Find an equation for the tangent plane to the graph of f at P0 = (−1, 2). Solution. First, the partial derivatives ∂f 3x2 + y = ∂x 2 ∂f x = . ∂y 2 At P0 , these have the values 5/2 and −1/2, respectively. Finally, we need the value f (P0 ) = −3/2. So the tangent plane to the graph of f at P0 is described by the equation 1 5 1 3 5 z = − + (x + 1) − (y − 2) = 2 + x − y. 2 2 2 2 2