Find the specific function values. Complete parts (a) through (d) below. f (x,y,z) = x y y 2 + z = (Simplify your answer.) ID: 14.1.

Transcription

1 . Find the specific function values. Complete parts (a) through (d) below. f (x,y,z) = x y y 2 + z 2 (a) f(2, 4,5) = (b) f 2,, 3 9 = (c) f 0,,0 2 (d) f(4,4,00) = = ID: 4..3

2 2. Given the function f(x,y) = 3xy, answer the following questions. a. Find the function's domain. b. Find the function's range. c. Describe the function's level curves. d. Find the boundary of the function's domain. e. Determine if the domain is an open region, a closed region, both, or neither. f. Decide if the domain is bounded or unbounded. a. Choose the correct domain of the function f(x,y) = 3xy. All points in the first quadrant B. All points in the xy plane All points in the xy plane except the origin D. y 3x b. Choose the correct range of the function f(x,y) = 3xy. All non negative integers B. All real numbers All integers D. All non negative real numbers c. Choose the correct description(s) of the level curves of f(x,y) = 3xy. Select all that apply. Straight lines, when f(x,y) 0 B. Circles, when f(x,y) 0 Hyperbolas, when f(x,y) 0 D. The x and y axes, when f(x,y) = 0 d. Does the function's domain have a boundary? Select the correct choice and if necessary, fill in the answer box below to complete your choice. Yes, at (Type an ordered pair. Use a comma to separate answers as needed.) B. Yes, at = 0 No (Type an expression using x and y as the variables.) e. Choose the correct description of the domain of f(x,y) = 3xy. Closed region Open region Both open and closed Neither open nor closed f. Is the domain of f(x,y) = 3xy bounded or unbounded? Bounded Unbounded ID: 4..2

3 3. Given the function f(x,y) =, answer the following questions. 25 x 2 y 2 a. Find the function's domain. b. Find the function's range. c. Describe the function's level curves. d. Find the boundary of the function's domain. e. Determine if the domain is an open region, a closed region, both, or neither. f. Decide if the domain is bounded or unbounded. a. Choose the correct domain. The set of all points in the xy plane that satisfy x + y <. B. All points in the xy plane. The set of all points in the xy plane that satisfy x + y. D. All points in the xy plane except those that lie on the circle x + y =. b. Choose the correct range. z 5 B. 0 < z 5 z 5 D. 0 < z 5 c. Choose the correct description of the level curves. 5 Circles with radii >. 6 B. Circles with no restrictions on the radii. Circles with radii 5. 6 D. 5 Circles with radii <. 6 d. Choose the correct description of the boundary of the function's domain. B. D. All points (x,y) that satisfy x + y < All points (x,y) that satisfy x + y All points (x,y) that satisfy x + y The circle x + y = e. Choose the correct description of the domain. Neither open nor closed B. Both open and closed Closed Region D. Open Region f. Is the domain of the function bounded or unbounded? Unbounded Bounded ID: 4..23

4 4. Match the set of curves with the appropriate function. y x Choose the correct function below. B. z = 2 (x2 + y 2 ) (4x 2 + y 2 ) z = x 2 x 4 y 2 z = 2x 2 + y 2 2 ID: Match the set of curves with the appropriate function. y x Choose the correct function below. B. z = ( sin y) e x2 / 9 z = ( sin x) e y2 / 9 z = sin x + 2 sin y ID: 4..33

5 6. Match the set of curves with the appropriate function. y x Choose the correct function below. B. z = x 2 + y 2 z = 4x 2 + y 2 z = x y 2 4 ID: Find an equation for the level curve of the function f(x,y) that passes through the given point. 2 2 f(x,y) = 8 4x 4y, 3 3, 3 An equation for the level curve is. (Type an equation.) ID: Find an equation for the level surface of the function f(x,y,z) = x + y + ln z through the given point ( 2,,). An equation for the level surface is x + y + ln z =. ID: Find an equation for the level surface of the function f(x,y,z) = x 2 + y 2 + z 2 through the given point 3,, 2. An equation for the level surface is x 2 + y 2 + z 2 =. ID: 4..63

6 0. Find lim 4x 2 + 3y (x,y) ( 4, 5) lim 4x 2 + 3y 2 39 = (x,y) ( 4, 5) (Type an exact answer, using radicals as needed.) ID: Find the limit. (x,y) lim (ln 6,0) e x y (x,y) lim (ln 6,0) ex y = (Simplify your answer. Type an integer or a simplified fraction.) ID: Find lim 0 (x,y) (, ) 0x y 00x 2 20xy + y 2 0x y by rewriting the fraction first. lim 00x 2 20xy + y 2 (x,y) (, 0 ) = 0x y 0x y ID: Find lim P (4,, 5) + + x 3 y 2 z 4. lim + + P (4,, 5) x 3 y 2 z 4 (Type a simplified fraction.) = ID:

7 4. At what point (x,y) in the plane are the functions below continuous? 2 2 a. f(x,y) = sin (x + y 2) b. f(x,y) = ln (x + y 9) a. Choose the correct answer for points where the function sin (x + y 2) is continuous. for every (x,y) such that x 0 B. for every (x,y) such that x + y 2 > 0 for every (x,y) D. for every (x,y) such that y b. Choose the correct answer for points where the function ln (x + y 9) is continuous. for every (x,y) such that x > 9 and y > 9 B. for every (x,y) such that x < 9 and y < for every (x,y) such that x + y > 9 D. 2 2 for every (x,y) such that x + y < 9 ID: At what points (x,y,z) in space are the functions continuous? a. f(x,y,z) = x + 4y 3z b. f(x,y,z) = x 2 + y a. At which points is f(x,y,z) = x + 4y 3z continuous? Choose the correct answer below. All points satisfying x = y = z B All points satisfying x < y < z All points satisfying x y z D. All points except (0,0,0) E. 3 2 All points satisfying y x z F. All points G. No points b. At which points is f(x,y,z) = x 2 + y 2 4 continuous? Choose the correct answer below. 2 2 All points satisfying x + y 4 B. All points satisfying x y z 2 2 All points satisfying x + y 4 D. 2 2 All points satisfying x + y 4 E. All points except (0,0,0) F. All points G. No points ID:

8 6. By considering different paths of approach, show that the function below has no limit as (x,y) (0,0). x 4 f(x,y) = x 4 + y 2 Examine the values of f along curves that end at (0,0). Along which set of curves is f a constant value? y = kx, x 0 B. y = k x 3, x 0 y = k x 2, x 0 D. y = kx + k x 2, x 0 If (x,y) approaches (0,0) along the curve when k = used in the set of curves found above, what is the limit? If (x,y) approaches (0,0) along the curve when k = 0 used in the set of curves found above, what is the limit? What can you conclude? Since f has the same limit along two different paths to (0,0), in cannot be determined whether or not f has a limit as (x,y) approaches (0,0). B. Since f has two different limits along two different paths to (0,0), in cannot be determined whether or not f has a limit as (x,y) approaches (0,0). Since f has the same limit along two different paths to (0,0), by the two path test, f has no limit as (x,y) approaches (0,0). D. Since f has two different limits along two different paths to (0,0), by the two path test, f has no limit as (x,y) approaches (0,0). ID: