MATH 211

Midterm 1 | General
1. Evaluate $g(1, 0, -1)$ for $g(x, y, z) = \frac{2xyz}{x^2 + y^2 + z^2}$

A) $1$
B) $-1$
C) $0$
D) $2$
E) $4$

2. Your weekly cost (in dollars) to manufacture $x$ cars and $y$ trucks is $C(x, y) = 280000 + 5000x + 4000y$ What is the marginal cost of a truck?

A) \$ $285000$
B) \$ $5000$
C) \$ $4000$
D) \$ $2000$
E) \$ $284000$

3. Calculate $\frac{\partial f}{\partial x}|_{(5, 8)}$ and $\frac{\partial f}{\partial y}|_{(5, 8)}$ when defined. $f(x, y) = x^2y^3 - x^3y^2 - xy$

A) $\frac{\partial f}{\partial x} |_{(5, 8)} = 312, \frac{\partial f}{\partial y} |_{(5, 8)} = 2795$
B) $\frac{\partial f}{\partial x} |_{(5, 8)} = -312, \frac{\partial f}{\partial y} |_{(5, 8)} = -2800$
C) $\frac{\partial f}{\partial x} |_{(5, 8)} = 2795, \frac{\partial f}{\partial y} |_{(5, 8)} = 312$
D) $\frac{\partial f}{\partial x} |_{(5, 8)} = 320, \frac{\partial f}{\partial y} |_{(5, 8)} = 2800$
E) $\frac{\partial f}{\partial x} |_{(5, 8)} = -312, \frac{\partial f}{\partial y} |_{(5, 8)} = 2800$

4. Calculate $\frac{\partial^2 f}{\partial x^2}$ and $\frac{\partial^2 f}{\partial y}{\partial x}$ when defined. $f(x, y) = 7x^{0.5} y^{0.1}$

A) $\frac{\partial^2 f}{\partial x^2} = -3.5y^{0.1}x^{-1.5}, \frac{\partial^2 f}{\partial y \partial x} = 3.5y^{-0.9}x^{-0.5}$
B) $\frac{\partial^2 f}{\partial x^2} = -3.5y^{0.1}x^{-0.5}, \frac{\partial^2 f}{\partial y \partial x} = 0.7y^{-0.9}x^{0.5}$
C) $\frac{\partial^2 f}{\partial x^2} = -7y^{-0.9}x^{-1.5}, \frac{\partial^2 f}{\partial y \partial x} = 7y^{-1.9}x^{-0.5}$
D) $\frac{\partial^2 f}{\partial x^2} = --1.75y^{0.1}x^{-1.5}, \frac{\partial^2 f}{\partial y \partial x} = 0.35y^{-0.9}x^{-0.5}$
E) $\frac{\partial^2 f}{\partial x^2} = 1.75y^{0.1}x^{-1.5}, \frac{\partial^2 f}{\partial y \partial x} = -0.35y^{-0.9}x^{0.5}$

5. Calculate $\frac{\partial f}{\partial x}, \frac{\partial f}{\partial y}$, and $\frac{\partial f}{\partial z}$ when defined. $f(x, y) = x^{0.9}y^{0.3}z^{0.6}$

A) $\frac{\partial f}{\partial x} = 0.9x^{-0.1}y^{0.3}z^{0.6}, \frac{\partial f}{\partial y} = 0.3x^{0.9} y^{-0.7} z^{0.6}, \frac{\partial f}{\partial z} = 0.6 x^{0.9} y^{0.3} z^{-0.4}$
B) $\frac{\partial f}{\partial x} = 0.3x^{-0.7}y^{0.3}z^{0.6}, \frac{\partial f}{\partial y} = 0.3x^{0.9} y^{-0.1} z^{0.6}, \frac{\partial f}{\partial z} = 0.6 x^{0.3} y^{0.9} z^{-0.4}$
C) $\frac{\partial f}{\partial x} = 0.9x^{-0.1}y^{0.3}z^{0.6}, \frac{\partial f}{\partial y} = 0.3x^{0.9} y^{-0.4} z^{0.3}, \frac{\partial f}{\partial z} = 0.6 x^{0.9} y^{0.6} z^{-0.7}$
D) $\frac{\partial f}{\partial x} = 0.9x^{-0.1}y^{0.7}z^{0.4}, \frac{\partial f}{\partial y} = 0.3x^{0.9} y^{-0.7} z^{0.6}, \frac{\partial f}{\partial z} = 0.6 x^{0.9} y^{0.3} z^{-0.4}$
E) $\frac{\partial f}{\partial x} = 0.9x^{-0.1}y^{0.3}z^{0.6}, \frac{\partial f}{\partial y} = 0.3x^{0.9} y^{-0.3} z^{0.6}, \frac{\partial f}{\partial z} = 0.6 x^{0.9} y^{0.3} z^{-0.6}$

6. A regino is defined by the set of inequalities. Which graph satisfies the solution to these inequalities? $2x + y \le 30 \\ x + y \le 20 \\ x \ge 0, y \ge 0$

A) !!q6-1.png!!
B) !!q6-2.png!!
C) !!q6-3.png!!
D) !!q6-4.png!!
E) !!q6-5.png!!

7. Solve the linear programming problem. $\begin{align} \text{Maximize} \quad & P = 4x + 7y \\ \text{subject to} \quad & x + y \le 4 \\ & 2x + y \le 5 \\ & x \ge 0, y \ge 0 \end{align}$

A) $x = 0, y = 4, P = 28$
B) $x = 1, y = 0, P = 25$
C) $x = 2.5, y = 0, P = 10$
D) $x = 0, y = 5, P = 35$
E) $x = 1, y = 3, P = 25$

8. Draw $3$ level curves (contours) for the following function. Show your work, i.e. show how you obtained the equations of the level curves. Use an appropriate scale. Label each level curve with the corresponding z-value. Show the scale on the x- and y-axes, Look at the sketch of the surface. Use the given spaces. The graph of the surface is given. $z = -\sqrt{9 - x^2 - y^2}$


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9. Find the absolute minimum and maximum of the function $f(x, y) = -5x^2 + 4y^2 + 6x + 6y + 12$ on the region bounded by $3$ lines: the line with equation $x = -3$, the line with equation $y = 3$, and the line with equation: $x - y = 10$.

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10. Miss Persida Buda, the owner of Slatina Jeans Inc., decided to branch out into home décor and opened up a factory where she produces two types of decorative lamps made out of Emu eggshells. Let $x$ represent the number of table lamps and y represent the number of floor lamps produced and sold. Her weekly revenue function is given by $R(x, y) = -x^2 - xy - y^2 + 35x + 40y + 150$. As Miss Budau’s consultant, help her with the following:
Find the values of $x$ and $y$ that maximize the weekly revenue, and give the revenue as well. Assume there are no constraints.

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