State the slope and the y-intercept of the graph of each equation. $y=x+9$

List the minor matrix $M_{ij}$, and calculate the cofactor $A_{i j}=(-1)^{i+j} \operatorname{det}\left(M_{i j}\right)$ for the matrix A given by

$$A=\left[\begin{array}{rrrr} 2 & -1 & 3 & 1 \\ 4 & 1 & 3 & -1 \\ 6 & 2 & 4 & 1 \\ 2 & 2 & 0 & -2 \end{array}\right]$$

$M_{11}$

(a) You may have observed that water waves advancing toward shore have their wavefronts almost always parallel to the shoreline independent of the direction of the wind? Noting the fact that the velocity of waves in water decreases as the depth of the water decreases, use Huygens' Principle to explain this phenomenon. (b) To make the analysis of (a) more specific, assume that waves, initially traveling in the x direction, enter a region in which their speed v has a systematic variation with the distance y perpendicular to the direction of travel. (For example, x could be the direction parallel to the shore, and y would then be the direction perpendicular to the shore.) Show that the direction of the waves will begin to follow the arc of a circle of radius R, such that $R=\frac{v}{d v / d y}$

Find the trigonometric Fourier series for a periodic function f(t) that is equal to $t^{2}$ over the period from t=0 to t=2.

Combing your hair leads to excess electrons on the comb. How fast would you have to move the comb up and down to produce red light?

Identify whether each of the following factors will affect the rate of a reaction: (a) $K_{\mathrm{eq}}$ (b) $\Delta G$ (e) $E_{\mathrm{a}}$ (c) Temperature (f) $\Delta S$

Determine $\cos \theta$ where $\theta$ is the angle between u and v. $u = 2i - 3j + k, \ v = i - 2j + 3k$

Suppose you measure the terminal voltage of a 3.200-V lithium cell having an internal resistance of $5.00\ \Omega$ by placing a $1.00-\mathrm{k}\ \Omega$ voltmeter across its terminals. (a) What current flows? (b) Find the terminal voltage. (c) To see how close the measured terminal voltage is to the emf, calculate their ratio.

What would be the maximum cost of a CFL such that the total cost (investment plus operating) would be the same for both CFL and incandescent 60-W bulbs? Assume the cost of the incandescent bulb is 25 cents and that electricity costs 10 cents/kWh. Calculate the cost for 1000 hours, as in the cost effectiveness of CFL example.

(a) The velocity of sound in a gas is proportional to the square root of the absolute temperature T. Use this fact, and the result of the previous problem, to show that when a thermal gradient exists in the vertical direction (z) sound waves will be turned initially with a radius of curvature $R=\frac{2 T}{d T / d z}$ (b) On a still day, the temperature of the atmosphere is found to decrease more or less linearly with height. Sketch the paths of "rays" of sound emitted from a source suspended high in the atmosphere. Assuming that the velocity of sound at ground level is 1110 ft/sec, estimate the horizontal distance at which an airplane flying at 15,000 ft firs becomes audible to an observer on the ground, if the temperature decreases by $1^{\circ} \mathrm{C}$ per 500-ft increase in altitude.

The area of a triangle is 96 square inches. Its height is 12 inches. Find the base.

$$A=\left[\begin{array}{ll} 1 & 2 \\ 3 & 4 \end{array}\right], \ B=\left[\begin{array}{ll} 1 & 2 \\ 2 & 4 \end{array}\right], \ C=\left[\begin{array}{ll} 1 & 3 \\ 2 & 4 \end{array}\right]$$

$$D=\left[\begin{array}{lll} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 1 & 0 \end{array}\right], \ E=\left[\begin{array}{lll} 0 & 1 & 0 \\ 0 & 0 & 2 \\ 0 & 1 & 3 \end{array}\right]$$

,

$$F=\left[\begin{array}{lll} 1 & 2 & 1 \\ 0 & 3 & 2 \\ 0 & 0 & 1 \end{array}\right]$$

Determine whether the given matrix is singular or nonsingular. If a matrix M is singular, give all solutions of $M \mathbf{x}=\theta$. AB.

Let A = $\mathbb{Z}/4\mathbb{Z}$ and let G be the cyclic group of order 2 acting trivially on A. (a) Prove that $\left|C^{2}(G, A)\right|=2^{8}$. (b) Use the coboundary condition to show that $\left|B^{2}(G, A)\right|=2^{3}$. (c) Use the cocycle condition to show that $\left|Z^{2}(G, A)\right| \leq 2^{4}$. (d) Show that $\left|H^{2}(G, A)\right|=2$ by exhibiting two inequivalent extensions of G by A and their corresponding cocycles.

In the following exercise, use a calculator to draw the graph of the piecewise-defined function and study the graph to evaluate the given limits.

$$g(x)=\left\{\begin{array}{ll} x^{3}-1, & x \leq 0 \\ 1, & x>0 \end{array}\right.$$

(a) $\lim _{x \rightarrow 0^{-}} g(x)$ (b) $\lim _{x \rightarrow 0^{+}} g(x)$