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Unformatted text preview: 15 DIFFERENTIATION IN SEVERAL VARIABLES 15.1 Functions of Two or More Variables (ET Section 14.1) Preliminary Questions 1. What is the difference between a horizontal trace and a level curve? How are they related? SOLUTION A horizontal trace at height c consists of all points ( x , y , c ) such that f ( x , y ) = c . A level curve is the curve f ( x , y ) = c in the x y-plane. The horizontal trace is in the z = c plane. The two curves are related in the sense that the level curve is the projection of the horizontal trace on the x y-plane. The two curves have the same shape but they are located in parallel planes. 2. Describe the trace of f ( x , y ) = x 2- sin ( x 3 y ) in the x z-plane. SOLUTION The intersection of the graph of f ( x , y ) = x 2- sin ( x 3 y ) with the x z-plane is obtained by setting y = 0 in the equation z = x 2- sin ( x 3 y ) . We get the equation z = x 2- sin 0 = x 2 . This is the parabola z = x 2 in the x z-plane. 3. Is it possible for two different level curves of a function to intersect? Explain. SOLUTION Two different level curves of f ( x , y ) are the curves in the x y-plane defined by equations f ( x , y ) = c 1 and f ( x , y ) = c 2 for c 1 y= c 2 . If the curves intersect at a point ( x , y ) , then f ( x , y ) = c 1 and f ( x , y ) = c 2 , which implies that c 1 = c 2 . Therefore, two different level curves of a function do not intersect. 4. Describe the contour map of f ( x , y ) = x with contour interval 1. SOLUTION The level curves of the function f ( x , y ) = x are the vertical lines x = c . Therefore, the contour map of f with contour interval 1 consists of vertical lines so that every two adjacent lines are distanced one unit from another. 5. How will the contour maps of f ( x , y ) = x and g ( x , y ) = 2 x with contour interval 1 look different? SOLUTION The level curves of f ( x , y ) = x are the vertical lines x = c , and the level curves of g ( x , y ) = 2 x are the vertical lines 2 x = c or x = c 2 . Therefore, the contour map of f ( x , y ) = 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 ) = 2 x (with contour interval 1) the distance between two adjacent vertical lines is 1 2 . Exercises In Exercises 14, evaluate the function at the specified points. 1. f ( x , y ) = x + yx 3 , ( 2 , 2 ) , (- 1 , 4 ) , ( 6 , 1 2 ) 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 ) = 2 + 2 2 3 = 18 f (- 1 , 4 ) = - 1 + 4 (- 1 ) 3 = - 5 f 6 , 1 2 = 6 + 1 2 6 3 = 114 2. g ( x , y ) = y x 2 + y 2 , ( 1 , 3 ) , ( 3 ,- 2 ) SOLUTION We substitute ( x , y ) = ( 1 , 3 ) and ( x , y ) = ( 3 ,- 2 ) in the function to obtain g ( 1 , 3 ) = 3 1 2 + 3 2 = 3 10 ; g ( 3 ,- 2 ) =- 2 3 2 + (- 2 ) 2 = - 2 13 3. h ( x , y , z ) = x yz- 2 , ( 3 , 8 , 2 ) , ( 3 ,- 2 ,- 6 ) SOLUTION Substituting ( x , y , z ) = ( 3 , 8 , 2 ) and ( x , y , z ) = ( 3 ,- 2 ,- 6...
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This note was uploaded on 07/15/2009 for the course MATH 210 taught by Professor Hubscher during the Spring '08 term at University of Illinois at Urbana–Champaign.
- Spring '08