Transcribed Text
Question 1
25 pts
Which of the answer choices best describes the domain of
f(x, y) = 
All points below and on the parabola y = x2 + 49.
All points above (but not on) the parabola y = x2 + 49.
All points above and on the parabola y = x2 + 49.
All points in the xyplane.
All points below (but not on) the parabola y = x2 + 49.
Question 2
25 pts
Consider f(x,y) = 1/39  x2y2,  and let Z = f(x,y). What is the Z = 4
level curve?
A circle of radius
23.
A circle of radius
29.
A circle of radius
41
A circle of radius
35.
Nothing.
Question 3
25 pts
Each level curve of the function Z = f(x,y) can be obtained by which of the
following?
A slice through the graph, holding the Z variable constant.
A slice through the graph, holding the X variable constant.
A slice through the graph, holding the y variable constant.
A slice through the graph, holding one of the variables constant at zero.
Question 4
25 pts
In general, the graph of Z = f(x,y) is a surface in threedimensional space.
True
False
Question 1
25 pts
Find the partial derivative with respect to Z of the function
1
( 
2(y1)5el2073)
2x3

4el22+3)
4
5 (2x3)(y1)
5
el22t3)108(223)
4
5 (y1) 5
Question 2
25 pts
Find the partial derivative with respect to y of the function
1
1
x 
1
a
(3 y+1)
3
2 (2 x3) 4
3
1
of
3 (223) Tel5215)
emailable
3
4 (3 y+1) 4
of  
Question 3
25 pts
How many second partial derivatives will a function of three variables have,
assuming all the second partial derivatives exist?
12
3
9
4
Question 4
25 pts
Let f be a function of several variables. To find the partial derivative of f with
respect to some variable, we consider the other variables as
fixed constants.
dependent variables.
independent variables.
Question 1
25 pts
Determine which of the following is true about f(x,y) =  4 x²  10 2²  3.
f has a relative minimum at (0,0,3).
f has a saddle point at (0,0,3).
f has a relative maximum at (0,0,3).
f has no critical points.
Question 2
25 pts
Determine which of the following is true about f(x,y) =  7x2 + 7 y  7.
f has a relative minimum at (0,0,7).
f has a relative maximum at (0,0,7).
f has a saddle point at (0,0,7).
f has no critical points.
Question 3
25 pts
How many extrema does the function f(x,y) = 9 x2  7 have?
(Hint: Notice that this is a function of two variables, but one of the variables
does not affect the outcome. What does this mean about the graph? Can we
sketch the graph when y = 0? When y = 5? Does the graph have extrema at
these values of y? What about other y values?)
Infinitely many extrema
One extrema
Two extrema
No extrema
Question 4
25 pts
If fxx (a, b) is positive, then f(x,y) has a relative minimum at (a,b).
Yes, as long as fyy (a,b) is positive as well.
Not necessarily, more information is required to make this conclusion.
Yes, as long as f has a critical point at (a,b) as well.
No, f has a relative maximum at (a,b).
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