Question 1 25 pts Which of the answer choices best describes the ...

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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 xy-plane. All points below (but not on) the parabola y = x2 + 49. Question 2 25 pts Consider f(x,y) = 1/39 - x2-y2, - 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 three-dimensional space. True False Question 1 25 pts Find the partial derivative with respect to Z of the function 1 ( - 2(-y-1)5el-2073) 2x-3 - 4el-22+3) 4 5 (2x-3)(y-1) 5 el-22t3)108(22-3) 4 5 (-y-1) 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 x-3) 4 3 1 of 3 (22-3) Tel-5215) 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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