Calculus 10th Edition

Published by Brooks Cole
ISBN 10: 1-28505-709-0
ISBN 13: 978-1-28505-709-5

Chapter 13 - Functions of Several Variables - 13.6 Exercises - Page 924: 15

Answer

$gradZ(2,3)=4i-j$

Work Step by Step

To find the gradient of $f(x,y)$ denoted $gradf(x,y)$ we use the formula: $gradf(x,y)=f_{x}(x,y)i+f_{y}(x,y)j$ Note that $f_{x}(x,y)$ and $f_{y}(x,y)$ are partial derivates of the function with respect to $x$ and $y$, respectfully. $f(x,y)=Z(x,y)=ln(x^2-y)$ $gradZ(x,y)=\frac{2x}{x^2-y}i+\frac{-1}{x^2-y}j$ Substituting in the point $(2,3)$ we have $gradZ(2,3)=\frac{2(2)}{(2)^2-(3)}i+\frac{-1}{(2)^2-(3)}j$ $gradZ(2,3)=4i-j$
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