Ch-15-sol - DIFFERENTIATION 12 IN SEVERAL VARIABLES 12.1 Functions of Two or More Variables Preliminary Questions 1 What is the difference between a

Ch-15-sol - DIFFERENTIATION 12 IN SEVERAL VARIABLES 12.1...

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12 DIFFERENTIATION IN SEVERAL VARIABLES 12.1 Functions of Two or More Variables Preliminary Questions 1.What is the difference between a horizontal trace and a level curve? How are they related? 2.Describe the trace off .x;y/Dx2sin.x3y/in thexz-plane. 3.Is it possible for two different level curves of a function to intersect? Explain. 4.Describe the contour map off .x; y/Dxwith contour interval 1. 5. How will the contour maps of f .x;y/ D x and g.x;y/ D 2x with contour interval 1 look different? SOLUTION The level curves of f .x;y/ D x are the vertical lines x D c , and the level curves of g.x; y/ D 2x are the vertical lines 2x D c or x D c 2 . Therefore, the contour map of f .x; y/ D x with contour interval 1 consists of vertical lines with distance one unit between adjacent lines, whereas in the contour map of g.x; y/ D 2x (with contour interval 1 ) the distance between two adjacent vertical lines is 1 2 . Exercises In Exercises 1–4, evaluate the function at the specified points. 1. f .x;y/ D x C yx 3 , .2; 2/ , . 1; 4/ SOLUTION We substitute the values for x and y in f .x;y/ and compute the values of f at the given points. This gives f .2; 2/ D 2 C 2 2 3 D 18 f . 1; 4/ D 1 C 4 . 1/ 3 D 5 2. g.x;y/ D y x 2 C y 2 , .1; 3/ , .3; 2/ SOLUTION We substitute .x; y/ D .1; 3/ and .x;y/ D .3; 2/ in the function to obtain g.1; 3/ D 3 1 2 C 3 2 D 3 10 I g.3; 2/ D 2 3 2 C . 2/ 2 D 2 13 3. h.x; y; z/ D xyz 2 , .3; 8; 2/ , .3; 2; 6/ SOLUTION Substituting .x; y; z/ D .3; 8; 2/ and .x; y; z/ D .3; 2; 6/ in the function, we obtain h.3; 8; 2/ D 3 8 2 2 D 3 8 1 4 D 6 h.3; 2; 6/ D 3 . 2/ . 6/ 2 D 6 1 36 D 1 6
1506 C H A P T E R 12 DIFFERENTIATION IN SEVERAL VARIABLES 4. Q.y; z/ D y 2 C y sin z , .y; z/ D 2; 2 ; 2; 6 SOLUTION We have Q 2; 2 D 2 2 C 2 sin 2 D 4 C 2 1 D 6 Q 2; 6 D . 2/ 2 2 sin 6 D 4 2 1 2 D 3 In Exercises 5–12, sketch the domain of the function.

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