Use properties of determinants to evaluate the given determinant by inspection. Explain your reasoning.

$$\left| \begin{array} { r r r } { 2 } & { 3 } & { - 4 } \\ { 1 } & { - 3 } & { - 2 } \\ { - 1 } & { 5 } & { 2 } \end{array} \right|$$

Let $A$ and $B$ be $3 \times 3$ matrices, with $\operatorname{det} A=-2$ and det $B=3$. Use properties of determinants (in the text and in the preceding exercises) to compute:

$$\operatorname{det} A B$$

The National Sporting Goods Association conducted a survey of persons 7 years of age or older about participation in sports activities (Statistical Abstract of the United States, 2002). The total population in this age group was reported at 248.5 million, with 120.9 million male and 127.6 million female. The number of participants for the top five sports activities appears here.

$$\begin{matrix} & \text{Perticipants (millions)}\\ \text{Activity} & \text{Male} & \text{Female}\\ \text{Bicycle riding} & \text{22.2} & \text{21}\\ \text{Camping} & \text{25.6} & \text{24.3}\\ \text{Exercise walking} & \text{28.7} & \text{57.7}\\ \text{Exercising with equipment} & \text{20.4} & \text{24.4}\\ \text{Swimming} & \text{26.4} & \text{34.4}\\ \end{matrix}$$

a. For a randomly selected female, estimate the probability of participation in each of the sports activities. b. For a randomly selected male, estimate the probability of participation in each of the sports activities. c. For a randomly selected person, what is the probability the person participates in exercise walking? d. Suppose you just happen to see an exercise walker going by. What is the probability the walker is a woman? What is the probability the walker is a man?

$$\begin{array} { l } { \text { Let } I = \langle 2 \rangle . \text { Prove that } I [ x ] \text { is not a maximal ideal of } Z [ x ] \text { even } } \\ { \text { though } I \text { is a maximal ideal of } Z } \end{array}$$

Compute the arc length exactly. $y=\frac{1}{4}\left(e^{2 x}+e^{-2 x}\right), 0 \leq x \leq 1$

Find the mode for each group of data items. If there is no mode, so state.

$$100,40,70,40,60$$

Identify u = g(x) and du = g’(x) dx for the integral. ∫ (8x²+1)²(16x) dx

What power did the Tonkin Gulf Resolution give to the president?

Experience shows that wooden homes are more resistant to earthquake damage than homes made from other materials. Why is this true? (1) Wood has a high hardness on Mohs' scale. (2) Wood is more dense than stone or brick. (3) Wood frames can bend and still hold together. (4) Wood does not conduct heat energy as well as stone or brick.

The following problem deal with the Vandermonde determinant

$$V\left(x_{1}, x_{2}, \ldots, x_{n}\right)=\left|\begin{array}{ccccc} 1 & x_{1} & x_{1}^{2} & \dots & x_{1}^{n-1} \\ 1 & x_{2} & x_{2}^{2} & \dots & x_{2}^{n-1} \\ \vdots & \vdots & \vdots & \ddots & \vdots \\ 1 & x_{n} & x_{n}^{2} & \dots & x_{n}^{n-1} \end{array}\right|$$

that will play an important role in the section. Show by direct computation that V(a, b) = b - a and that

$$V(a, b, c)=\left|\begin{array}{ccc} 1 & a & a^{2} \\ 1 & b & b^{2} \\ 1 & c & c^{2} \end{array}\right|=(b-a)(c-a)(c-b)$$

.

Explain the difference between a population and a community.

Prove that the determinant of the Vandermonde matrix

$$\left[\begin{array}{lll}{1} & {a} & {a^{2}} \\ {1} & {b} & {b^{2}} \\ {1} & {c} & {c^{2}}\end{array}\right]$$

is (b - a)(c - a)(c - b).

Explain why the function

$$f(x) = 4x^2 + 8x$$

is neither even nor odd.

Explain the steps you would take to create a circle graph if the data are given as percents.