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s10wk05 - Math 23 B Dodson Week 5 functions of several...

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Math 23 B. Dodson Week 5: functions of several variables 14.1 graphs, level curves/surfaces, contour maps 14.2 limits 14.3 Partial Derivatives Problem 14.1.9b: Find the domain of f ( x, y, z ) = e z - x 2 - y 2 . Solution: By definition, the domain of a function defined by a formula is the collection of points for which the formula makes sense (rather than by specifiying a particular subset).
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2 We analyze the formula in pieces (as a composite of functions). For the exponential function, e w is defined for all w , so this gives no restriction. Next, the squareroot is only defined on non-negative input, so we must restrict ( x, y, z ) so that z - x 2 - y 2 is non-negative. We check the boundary points, ones for which 0 = z - x 2 - y 2 , or z = x 2 + y 2 ; which we recognize as an elliptic paraboloid. This boundary surface divides 3-space into two regions, and we see z - x 2 - y 2 positive on the region above the paraboloid, so domain( f ) = { ( x, y, z ) so z x 2 + y 2 } . We also find the range; the squareroot non-negative gives values of e w for which w is non-negative,
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