chap14-et-student-solutions(2)

chap14-et-student-solutions(2) - DIFFERENTIATION IN 15 S E...

Info iconThis preview shows pages 1–3. Sign up to view the full content.

View Full Document Right Arrow Icon
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. 1. f ( x , y ) = x + yx 3 , ( 2 , 2 ) , ( 1 , 4 ) , ( 6 , 1 2 ) 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 ³ 6 , 1 2 ´ = 6 + 1 2 · 6 3 = 114 g ( x , y ) = y x 2 + y 2 , ( 1 , 3 ) , ( 3 , 2 ) 3. h ( x , y , z ) = xyz 2 , ( 3 , 8 , 2 ) , ( 3 , 2 , 6 ) Substituting ( x , y , z ) = ( 3 , 8 , 2 ) and ( x , y , z ) = ( 3 , 2 , 6 ) in the function, we obtain h ( 3 , 8 , 2 ) = 3 · 8 · 2 2 = 3 · 8 · 1 4 = 6 h ( 3 , 2 , 6 ) = 3 · ( 2 ) · ( 6 ) 2 6 · 1 36 1 6 Q ( y , z ) = y 2 + y sin z , ( y , z ) = ( 2 , π 2 ), ( 2 , 6 ) In Exercises 5–16, sketch the domain of the function. 5. f ( x , y ) = 4 x 7 y The function is deFned for all x and y , hence the domain is the entire -plane. f ( x , y ) = p 9 x 2 7. f ( x , y ) = ln ( y 2 x )
Background image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
324 CHAPTER 15 DIFFERENTIATION IN SEVERAL VARIABLES (ET CHAPTER 14) SOLUTION The function is deFned if y 2 x > 0or y > 2 x . This is the region in the xy -plane that is above the line y = 2 x .
Background image of page 2
Image of page 3
This is the end of the preview. Sign up to access the rest of the document.

This note was uploaded on 02/17/2009 for the course MATH 2057 taught by Professor Estrada during the Fall '08 term at LSU.

Page1 / 127

chap14-et-student-solutions(2) - DIFFERENTIATION IN 15 S E...

This preview shows document pages 1 - 3. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online