Answer true or false to each statement. Explain your answers. If a line has a positive slope, y-values on the line decrease as the x-values decrease.

Consider the SIS epidemic model with vital dynamics $S^{\prime}=\mu N-\beta S \frac{I}{N}+\alpha I-\mu S$, $I^{\prime}=\beta S \frac{I}{N}-\alpha I-\mu I$ where $S, I \geq 0, \beta, \alpha, \mu>0$ are parameters, and $N = S + I.$ (a) Give a biological interpretation of the model and an example of diseases that it might reasonably model. (b) Show that the population remains constant for all t. (c) Find all equilibria and classify their stability. (d) Find the basic reproductive number and interpret it. (e) Eliminate the S equation (because of the constant population) and solve I on the phase line. Do the results differ from the above analysis?

Solve the following systems of equations.

$$\left\{\begin{array}{l} x^{\prime}=x+y+3 \\ y^{\prime}=3 x-y+1 \end{array}\right.$$

In an article on Flurry Blog, a gaming marketing gap for men between the ages of 30 and 40 is identified. You are researching a startup game targeted at the 35-year-old demographic. Your idea is to develop a strategy game that can be played by men from their late 20s through their late 30s. Based on the article’s data, industry research shows that the average strategy player is 28 years old with a standard deviation of 4.8 years. You take a sample of 100 randomly selected gamers. If your target market is 29- to 35-year-olds, should you continue with your development strategy?

Determine whether the given set of vectors forms a basis for the specified vector space.

$$\left\{\left[\begin{array}{l} 1 \\ 1 \\ 0 \end{array}\right],\left[\begin{array}{c} 3 \\ 0 \\ -1 \end{array}\right],\left[\begin{array}{c} 1 \\ -2 \\ -1 \end{array}\right],\left[\begin{array}{c} 1 \\ 3 \\ -1 \end{array}\right]\right\}, \mathbb{R}^{3}$$

Find the eigenvalues and eigenvector (all are real-valued, some repeated roots) of the coefficient matrix either by hand or using MATLAB, Maple, or Mathematica. Obtain three linearly independent solutions and write the general solution.

$$\left\{\begin{array}{l} x^{\prime}=4 x-8 y-10 z \\ y^{\prime}=-x+6 y+5 z \\ z^{\prime}=x-8 y-7 z \end{array}\right.$$

(i) find the eigenvalues and eigenvector of the coefficient matrix by hand; (ii) find two linearly independent solutions; (iii) then use Theorem to find the general solution. Problem have complex (non-real) roots.

$$\left\{\begin{array}{l} x^{\prime}=-y \\ y^{\prime}=16 x \end{array}\right.$$

Suppose that a certain harmful bacteria, once introduced into the body, always persists. The body's immune system is usually able to keep the levels of the bacteria low unless the level of bacteria introduced is initially too large. We thus consider the equation $x^{\prime}=x(x-a)(x-5)$, where $x \geq 0$ represents the population of this harmful bacteria and $0 < a < 5$ is a parameter. (a) Determine the equilibria and their stability. (b) Describe what happens to the bacteria for any initial condition. (You will have two cases to consider.) (c) Give a biological interpretation of the parameter a. In particular, determine whether a healthy person is likely to have a larger or smaller a-value than an unhealthy person. Are there other factors that may change the a-value?

Evaluate

$$\iint_{[0,1] \times[0,1]} \sin ^{2} x \sin ^{2} y d x d y$$

Find an example that meets the given specifications. A subspace S of $\mathbf{R}^{4}$ such that $\operatorname{dim}(S)=\operatorname{dim}\left(S^{\perp}\right)=2$.

Suppose V is a complex vector space, $T \in \mathcal{L}(V),$ and the matrix of T with respect to some basis of V contains only real entries. Show that if $\lambda$ is an eigenvalue of T, then so is $\bar{\lambda}.$

Consider the following theorem: There do not exist three consecutive odd integers a, b, and c such that $a^{2}+b^{2}=c^{2}$. (a) Complete the following restatement of the theorem: For every three consecutive odd integers a, b, and c, _________. (b) Change the sentence in part (a) into an implication $p \Rightarrow q$: If a, b, and c are consecutive odd integers, then _________. (c) Fill in the blanks in the following proof of the theorem. Let a, b, and c be consecutive odd integers. Then a = 2k + 1, b = _______, and c = 2k + 5 for some integer k. Suppose $a^{2}+b^{2}=c^{2}$. Then $(2 k+1)^{2}+( )^{2}=(2 k+5)^{2}$. It follows that $8 k^{2}+16 k+10=4 k^{2}+20 k+25$ and $4 k^{2}-4 k-\quad=0$. Thus k = 5/2 or k = _____. This contradicts k being an _______________. Therefore, there do not exist three consecutive odd integers a, b, and c such that $a^{2}+b^{2}=c^{2}$. (d) Which of the tautologies in Example 3.12 best describes the structure of the proof?

Predict the major products of the following reactions. (a) propane $+\mathrm{BH}_{3} \cdot \mathrm{THF}$. (b) the product from part $(\mathrm{a})+\mathrm{H}_{2} \mathrm{O}_{2} / \mathrm{OH}^{-}$. (c) 2-methylpent-2-ene $+\mathrm{BH}_{3} \cdot \mathrm{THF}$. (d) the product from part $(\mathrm{c})+\mathrm{H}_{2} \mathrm{O}_{2} / \mathrm{OH}^{-}$. (e) 1-methylcyclohexene$+\mathrm{BH}_{3} \cdot \mathrm{THF}$. (f) the product from part $(\mathrm{e})+\mathrm{H}_{2} \mathrm{O}_{2} / \mathrm{OH}^{-}$

Let S = {a, b,c,d,e} and define the equivalence relation R = {(a, a), (b,b), (c,c), (d,d), (e,e), (a,b), (b,a), (a,d), (d,a), (b,d), (d,b)}. Describe the partition $\mathscr{P}$ determined by R by Listing the pieces in $\mathscr{P}$.