15.1%20Prel.%20Q%20and%20Ex%201-%2022

# 15.1%20Prel.%20Q%20and%20Ex%201-%2022 - DIFFERENTIATION IN...

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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 xy -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 -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 xz -plane. The intersection of the graph of f ( x , y ) = x 2 sin ( x 3 y ) with the -plane is obtained by setting y = 0in 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 -plane. 3. Is it possible for two different level curves of a function to intersect? Explain. Two different level curves of f ( x , y ) are the curves in the -plane deFned by equations f ( x , y ) = c 1 and f ( x , y ) = c 2 for c 1 6= c 2 . If the curves intersect at a point ( x 0 , y 0 ) ,then f ( x 0 , y 0 ) = c 1 and f ( x 0 , y 0 ) = c 2 ,wh ich 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. 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? 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 1–4, evaluate the function at the speciFed points.

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15.1%20Prel.%20Q%20and%20Ex%201-%2022 - DIFFERENTIATION IN...

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