Answer to Question #135760 in Calculus for Promise Omiponle

Question #135760
1. For the function f(x,y) = (25 − x^2 − y^2)^(−1/2), please sketch the level curves of the function, and then list the domain and the range of the function, write down a curve that represents the boundary, and state whether the domain is an open or closed region.
1
Expert's answer
2020-09-29T18:35:44-0400

"f(x,y) = \\dfrac{1}{\\sqrt{25-x^2-y^2}} = \\dfrac{1}{\\sqrt{5^2-(x^2+y^2)}} ."


Let us determine the level curve "f(x,y) = \\dfrac15," "\\dfrac15 = \\dfrac{1}{\\sqrt{5^2-(x^2+y^2)}} \\Rightarrow x^2+y^2 = 0, \\; x=y=0."


Let us determine the level curve "f(x,y) = \\dfrac13,\\;\\; \\dfrac19 = \\dfrac{1}{{5^2-(x^2+y^2)}} \\Rightarrow x^2+y^2 = 4^2."





The square root may be calculated only for non-negative numbers, so "5^2-(x^2+y^2) \\ge 0." Moreover, the square root is placed in the denominator, so it should not be equal to 0. Therefore, "x^2+y^2 < 5^2," so all (x,y) are situated in the circle with boundary "x^2+y^2=5^2" and the domain is open.


Next, the largest value of f(x,y) can be obtained if the denominator is smallest, namely if "x^2+y^2 = 0," so "f(x,y) = f(0,0) = \\dfrac{1}{\\sqrt{5^2-0}}= \\dfrac{1}{5}."

But if "x^2+y^2" is very close to "5^2" , the denominator is close to 0, so f(x,y) tends to infinity. Therefore, the range of function is "\\left[ \\dfrac15, +\\infty \\right)."



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