a. Sketch a phase portrait for the dynamical system $\dot x(t+1)=A \dot x(t)$ where

$$A=\left[\begin{array}{ll} 2 & 1 \\ 3 & 2 \end{array}\right].$$

b. In his paper "On the Measurement of the Circle", the great Greek mathematician Archimedes (c. 280-210 B.C.) uses the approximation $\frac{265}{153} < \sqrt{3} < \frac{1351}{780}$ to estimate $\cos \left(30^{\circ}\right).$ He does not explain how he arrived at these estimates. Explain how we can obtain these approximations from the dynamical system in part a. c. Without using technology, explain why $\frac{1351}{780}-\sqrt{3} < 10^{-6}.$ d. Based on the data in part b, give an underestimate of the form p/q of $\sqrt{3}$ that is better than the one given by Archimedes.

Test the Spearman rank correlation coefficient. The table shows the average hours worked per week and the numbers of on-the-job injuries for a random sample of U.S. construction companies in a recent year. At $\alpha=0.05$, can you conclude that there is a significant correlation between average hours worked and the number of on-the-job injuries?

$$\begin{matrix} \text{Hours worked} & \text{38} & \text{38} & \text{37} & \text{38} & \text{38} & \text{40} & \text{39} & \text{39} & \text{39} & \text{40} & \text{39} & \text{41}\\ \text{Injuries} & \text{11} & \text{11} & \text{9} & \text{10} & \text{10} & \text{17} & \text{15} & \text{14} & \text{14} & \text{16} & \text{15} & \text{17}\\ \text{Hours worked} & \text{41} & \text{42} & \text{41} & \text{41} & \text{41} & \text{42} & \text{42} & \text{42} & \text{42} & \text{41} & \text{41} & \text{39}\\ \text{Injuries} & \text{17} & \text{21} & \text{18} & \text{18} & \text{18} & \text{22} & \text{21} & \text{19} & \text{21} & \text{18} & \text{17} & \text{12}\\ \text{Hours worked} & \text{38} & \text{38} & \text{39} & \text{39} & \text{36} & \text{37} & \text{36} & \text{37} & \text{37} & \text{37} & \text{37} & \text{ }\\ \text{Injuries} & \text{12} & \text{11} & \text{13} & \text{12} & \text{6} & \text{6} & \text{6} & \text{6} & \text{7} & \text{8} & \text{7} & \text{ }\\ \end{matrix}$$

Let R be a ring with 1. Show that $(-1)^{2}=1$ in R.

Fill in the blanks. a. A function f has an absolute maximum at c if ____ for all x in the domains D of f. The number f(c) is called the ___ ___ ___ of f on D. b. A function f has a relative minimum at c if _____ for all values of x is some ____ ___ containing c.

A drawer of socks contains seven black socks, eight blue socks, and nine green socks. Two socks are chosen in the dark. a. What is the probability that they match? b. What is the probability that a black pair is chosen?

Find the Taylor series about the given point for each of the functions. In each case find the radius of convergence. $f(x)=\sin x^{2}$ about $x_{0}=0$

Either use an appropriate theorem to show that the given set, W, is a vector space, or find a specific example to the contrary.

$$\left\{\left[\begin{array}{l} a \\ b \\ c \\ d \end{array}\right]: \begin{array}{l} 3 a+b=c \\ a+b+2 c=2 d \end{array}\right\}$$

Find The critical value(s) and rejection region(s) for the type of t-test with level of significance $\alpha$ and sample size n. Left-tailed test, $\alpha=0.05, n=48$

Find the critical value(s) and rejection region(s) for the type of chi-square test with sample size n and level of significance $\alpha$. Two-tailed test, $n=31, \alpha=0.05$

When every collision between reactants leads to a reaction, what determines the rate at which the reaction occurs?

Fill in the details of the proof that the symmetric groups $S_{\Delta}$ and $S_{\Omega}$ are isomorphic if $|\Delta|=|\Omega|$ as follows: let $\theta: \Delta \rightarrow \Omega$ be a bijection. Define

$$\varphi: S_{\Delta} \rightarrow S_{\Omega} \quad \text { by } \quad \varphi(\sigma)=\theta \circ \sigma \circ \theta^{-1} \quad \text { for all } \sigma \in S_{\Delta}$$

and prove the following: (a) $\varphi$ is well defined, that is, if $\sigma$ is a permutation of $\Delta$ then $\theta \circ \sigma \circ \theta^{-1}$ is a permutation of $\Omega$. (b) $\varphi$ is a bijection from $S_{\Delta}$ onto $S_{\Omega}$. [Find a 2-sided inverse for $\varphi$. (c) $\varphi$ is a homomorphism, that is,

$$\varphi(\sigma \circ \tau)=\varphi(\sigma) \circ \varphi(\tau)$$

How many Boolean functions are there from $Z^n _2$ into $Z_2$?

Take this test as you would take a test in class. Researchers surveyed 19,183 U.S. physicians, asking for the information below. location (region of the U.S.) income (dollars) employment status (private practice or an employee) benefits received (health insurance. liability coverage, etc.) specialty (cardiology, family medicine, radiology. etc.) time spent seeing patients per week (hours) (a) Identify the population and the sample. (h) Is the data collected qualitative, quantitative, or both? Explain your reasoning. (c) Determine the level of measurement for each item above. (d) Determine whether the study is an observational study or an experiment. Explain.

Find the relative maxima and relative minima, if any, of each function. $f(x)=x^{5 / 3}$