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Calculus IIChapter 6: 3-Dimensional SpaceSection 6-5

Functions of Several Variables — Question 5

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Question 5

Let f(u,v)=u−vf(u,v)=\sqrt{u-v} and g(x,y)=(x2+y,x−y2)g(x,y)=(x^2+y,\,x-y^2). Find (f∘g)(x,y)(f\circ g)(x,y) and its domain. Determine the boundary and whether the point (1,1)(1,1) is interior, exterior, or on the boundary.

Original worksheet page 1: question and worked solution for 6-5-005
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Question 5 – Solution

Strategy Substitute both components of gg into ff and require a nonnegative radicand.

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Composition We obtain (f∘g)(x,y)=x2+y−x+y2.(f\circ g)(x,y)=\sqrt{x^2+y-x+y^2}. Thus the domain is x2−x+y2+y≥0x^2-x+y^2+y\ge 0, or (x−1/2)2+(y+1/2)2≥1/2.\boxed{(x-1/2)^2+(y+1/2)^2\ge 1/2}. It is the exterior of a disk, including its circular boundary.

Point test At (1,1)(1,1) the original radicand is 2>02>0, so the point is interior to the domain.

Original worksheet page 2: question and worked solution for 6-5-005

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