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

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S E C T I O N 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 ω 2 The left-hand side is nonnegative, hence also 9 ω 2 0 or ω 2 9. Therefore, 3 ω 3. By (1), ω 0, hence we must satisfy 0 ω 3. We obtain the following range: { ω R : 0 ω 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 | , or 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 C H A P T E R 15 DIFFERENTIATION IN SEVERAL VARIABLES (ET CHAPTER 14) (c)
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