Calculus 8th Edition

Published by Cengage
ISBN 10: 1285740629
ISBN 13: 978-1-28574-062-1

Chapter 14 - Partial Derivatives - 14.1 Functions of Several Variables - 14.1 Exercises - Page 940: 18


The domain is $$\mathcal{D}=\{(x,y)|x<2,x^2+y^2\neq1\},$$ and it is presented as a blue region on the graph below with red lines excluded from it.

Work Step by Step

We have two constraints: 1. The argument of the logarithm has to be positive i.e. $$2-x>0\Rightarrow x<2.$$ 2. The denominator has to be different than zero i.e. $$1-x^2-y^2\neq0\Rightarrow x^2+y^2\neq1$$ The first constraint says that we count in only the region of the $xy$ plane that is left from the vertical line $x=2$, excluding the point of this line. The second constraint says that we have to exclude the points from the circle $x^2+y^2=1$ i.e. the circle with the center at the origin and the radius of $1$. So the domain is $$\mathcal{D}=\{(x,y)|x<2,x^2+y^2\neq1\},$$ and it is shown on the graph below (the red circle and the red line are excluded from the domain)
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