15.1Ex20 - S E C T I O N 15.1 Functions of Two or More...

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SECTION 15.1 Functions of Two or More Variables (ET Section 14.1) 607 Raising to the power of two and transfering sides gives 9 x 2 y 2 z 2 = ω 2 x 2 + y 2 + z 2 = 9 2 The left-hand side is nonnegative, hence also 9 2 0or 2 9. Therefore, 3 3. By (1), 0, hence we must satisfy 0 3. We obtain the following range: { R : 0 3 } . 20. Match the functions (a)–(f) with their graphs (A)–(F) in Figure 21. (a) f ( x , y ) =| x |+| y | (b) f ( x , y ) = cos ( x y ) (c) f ( x , y ) = 1 1 + 9 x 2 + y 2 (d) f ( x , y ) = cos ( x 2 ) e 0 . 1 ( x 2 + y 2 ) (e) f ( x , y ) = 1 1 + 9 x 2 + 9 y 2 (f) f ( x , y ) = cos ( x 2 + y 2 ) e 0 . 1 ( x 2 + y 2 ) (B) (C) (D) x y x y z (A) (E) (F) z x y z x y z x y z x y z FIGURE 21 SOLUTION (a) | x y | . The level curves are | x y |= c , y = c −| x | ,o r y =− c +| x | . The graph (D) corresponds to the function with these level curves. (b) cos ( x y ) . The vertical trace in the plane x = c is the curve z = cos ( c y ) in the plane x = c . These traces correspond to the graph (C).
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608 CHAPTER 15 DIFFERENTIATION IN SEVERAL VARIABLES (ET CHAPTER 14) (c) 1 1 + 9 x 2 + y 2 (e) 1 1 + 9 x 2 + 9 y 2 . The level curves of the two functions are:
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15.1Ex20 - S E C T I O N 15.1 Functions of Two or More...

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