Calculus (3rd Edition)

Published by W. H. Freeman
ISBN 10: 1464125260
ISBN 13: 978-1-46412-526-3

Chapter 15 - Differentiation in Several Variables - 15.4 Differentiability and Tangent Planes - Exercises - Page 789: 1


$$z=-34-20 x+16 y$$

Work Step by Step

Given $$f(x, y)=2 x^{2}-4 x y^{2} \quad \text { at }(-1,2)$$ Since \begin{aligned} f_{x}(x, y) &=4 x-4 y^{2} \\ f_{x}(-1,2) &=-20 \end{aligned} and \begin{aligned} f_{y}(x, y) &=-8xy \\ f_{y}(-1,2) &=16 \end{aligned} Then the tangent plane is given by \begin{align*} z &=f(-1,2)+f_{x}(-1,2)(x+1)+f_{y}(-1,2)(y-2) \\ &=18-20(x+1)+16(y-2) \\ &=18-20 x-20+16 y-32 \\ &=-34-20 x+16 y \end{align*}
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