Determine whether each of the statements is true or false, and explain why. In a fair game, the elements of the payoff matrix must sum to 0.

Classify the following as acid-base reactions or oxidation-reduction reactions: (a) $\mathrm{Na}_{2} \mathrm{S}(a q)+2 \mathrm{HCl}(a q) \longrightarrow 2 \mathrm{NaCl}(a q)+\mathrm{H}_{2} \mathrm{S}(g)$ (b) $2 \mathrm{Na}(s)+2 \mathrm{HCl}(a q) \longrightarrow 2 \mathrm{NaCl}(a q)+\mathrm{H}_{2}(g)$ (c) $\mathrm{Mg}(s)+\mathrm{Cl}_{2}(g) \longrightarrow \mathrm{MgCl}_{2}(s)$ (d) $\operatorname{Mg} \mathrm{O}(s)+2 \mathrm{HCl}(a q) \longrightarrow \mathrm{MgCl}_{2}(a q)+\mathrm{H}_{2} \mathrm{O}(l)$ (e) $\mathrm{K}_{3} \mathrm{P}(s)+2 \mathrm{O}_{2}(g) \longrightarrow \mathrm{K}_{3} \mathrm{PO}_{4}(s)$ (f) $3 \mathrm{KOH}(a q)+\mathrm{H}_{3} \mathrm{PO}_{4}(a q) \longrightarrow \mathrm{K}_{3} \mathrm{PO}_{4}(a q)+3 \mathrm{H}_{2} \mathrm{O}(l)$

Find all solutions to the system by eliminating one of the variables.

$$\begin{array}{r} x_{1}-4 x_{2}=1 \\ -2 x_{1}+8 x_{2}=-2 \end{array}$$

Marie Revak and Jihan Williams performed an experiment to determine whether Oreo Double Stuf cookies contain twice as much filling as traditional Oreo cookies. The table below gives the results in grams of the amount of filling inside 49 traditional cookies and 52 Double Stuf cookies. (a) Find the mean, maximum, minimum, and standard deviation of the weights for traditional Oreo cookies. (b) Find the mean, maximum, minimum, and standard deviation of the weights for Oreo Double Stuf cookies. (c) What percent of the data of traditional Oreo cookies is within 2 standard deviations of the Double Stuf Oreo mean? (d) What percent of the data of traditional Oreo cookies, when multiplied by 2, is within 2 standard deviations of the Double Stuf Oreo mean? (e) Is there evidence that Double Stuf Oreos have twice as much filling as the traditional Oreo cookie? Explain.

Let g be the function defined by

$$g(x)=\left\{\begin{array}{ll} -\frac{1}{2} x+1 & \text { if } x<2 \\ \sqrt{x-2} & \text { if } x \geq 2 \end{array}\right.$$

Find g(-2), g(0), g(2), and g(4).

Find a particular solution to the given second-order differential equation. $y^{\prime \prime}+4 y=\sec 2 t$

Plot the graph of the function in the standard viewing window.

$$f(x)=x^{3}-20 x^{2}+8 x-10$$

A large but sparsely populated county has two small hospitals, one at the south end of the county and the other at the north end. The south hospital’s emergency room has four beds, whereas the north hospital’s emergency room has only three beds. Let X denote the number of south beds occupied at a particular time on a given day, and let Y denote the number of north beds occupied at the same time on the same day. Suppose that these two rv’s are independent; that the pmf of X puts probability masses .1, .2, .3, .2, and .2 on the x values 0, 1, 2, 3, and 4, respectively; and that the pmf of Y distributes probabilities .1, .3, .4, and .2 on the y values 0, 1, 2, and 3, respectively. Display the joint pmf of X and Y in a joint probability table.

Write an equation for the given situation. One second of video on your digital camera uses the same amount of memory as five pictures. Your phone can hold 300 pictures. I thought the answer would be y=5x. Why is that wrong?

Apply the theorem to calculate the matrix exponential $e^{\mathbf{A} t}$ for each of the matrices in the following problem.

$$\mathbf{A}=\left[\begin{array}{ll} 3 & 4 \\ 0 & 3 \end{array}\right]$$

Open water oil spills can wreak terrible consequences on the environment and be expensive to clean up. Many physical and biological methods have been developed to recover oil from water surfaces. The article “Capacity of Straw for Repeated Binding of Crude Oil from Salt Water and Its Effect on Biodegradation” (J. of Hazardous Toxic and Radioactive Waste, 2012: 75–78) discussed how wheat straw could be used to extract crude oil from a water surface. An experiment was conducted in which crude oil (0 to 16.9g) was added to 100mL of saltwater in separate Petri dishes. Wheat straw (2g) was then added to each dish and all dishes were shaken at 70 rpm overnight. The following data, read from a graph, is based on the x = amount of oil added (in g) and y = the corresponding amount of oil recovered (in g) from wheat straw.

$$\begin{matrix} \text{x} & \text{1.0} & \text{1.5} & \text{2.1} & \text{2.8} & \text{3.6} & \text{4.5} & \text{5.5} & \text{ }\\ \text{y} & \text{0.610} & \text{0.840} & \text{1.512} & \text{1.792} & \text{2.952} & \text{2.880} & \text{4.400} & \text{ }\\ \text{x} & \text{6.6} & \text{7.8} & \text{9.1} & \text{10.5} & \text{12.0} & \text{13.6} & \text{15.2} & \text{16.9}\\ \text{y} & \text{5.346} & \text{6.396} & \text{7.189} & \text{8.085} & \text{9.840} & \text{11.696} & \text{13.224} & \text{14.365}\\ \end{matrix}$$

Construct a scatterplot of the data. Does it appear that recovered oil could be very well predicted by the value of added oil? Explain your reasoning.

A hand-held calculator will suffice for the following problem. In the problem an initial value problem and its exact solution are given. Approximate the values of x(0.2) and y(0.2) in three ways: (a) by the Euler method with two steps of size h=0.1; (b) by the improved Euler method with a single step of size h=0.2; and (c) by the Runge-Kutta method with a single step of size h=0.2. Compare the approximate values with the actual values x(0.2) and y(0.2).

$$\begin{array}{l} x^{\prime}=x+2 y, x(0)=0, y^{\prime}=2 x+y, y(0)=2; x(t)=e^{3 t}-e^{-t}, y(t)=e^{3 t}+e^{-t} \end{array}$$

Perform the operation and write the result in standard form. (15 + 10i) - (2 + 10i)

Use the quadratic formula to solve the quadratic equation. $2 x^{2}+8 x+7=0$