If the truck brakes and the crate slides to the right relative to the truck, the horizontal acceleration of the crate is given by $\overline{s}=-g \mu_{k},$ where g is the acceleration of gravity, $\mu_k=0.87.$ is the kinetic friction coefficient, and s is the position of the crate relative to a coordinate system attached to the ground (rather than the truck). Assuming that the crate slides without hitting the right end of the truck bed, determine the time it takes to stop if its velocity at the start of the sliding motion is $v_0 = 55 mph.$

The area of the rectangular solar panel is given by the trinomial $x^2+7x+12$. The height of the solar panel is $x+3$. What is an expression for the length of the panel? Explain your reasoning.

Indicate whether the statement is true or false. The integral $\int_{0}^{3}(1-x) d x$ is the area between the graph of f(x)=1-x and the x-axis between x=0 and x=3.

Determine whether the statement is true or false. If it is true, explain why. If it is false, explain why or give an example that disproves the statement. The following system of differential equations is linear:

$$\begin{aligned} &\frac{d x}{d t}=3 x+y+3\\ &\frac{d y}{d t}=2 x y-x+5 \end{aligned}$$

A circular metal loop is kept entirely within a magnetic field, but moved to a region of higher field strength while not rotated. What could you do to its diameter to prevent an induced current in it? Explain.

Condense the expression to the logarithm of asingle quantity. $\log _{10}(x+2)+2 \log _{10} x-3 \log _{10}(x+4)$

The index of refraction of crown glass is 1.515 for red light and 1.523 for blue light. (a) If light of both colors is incident on crown glass from air, the blue color will be refracted (1) more, (2) less, or (3) the same amount as the red color. Explain. (b) Find the angle separating rays of the two colors in a piece of crown glass if their angle of incidence is $37^{\circ}.$

Write a formula for the derivative of the product of four differentiable functions. That is, find a formula for $\frac{d}{d x}\left[f_{1}(x) f_{2}(x) f_{3}(x) f_{4}\right]$. Also find a formula for $\frac{d}{d x}[f(x)]^{4}$.

You will give a general expression for the n-dimensional volume of the n-dimensional ball $B=\left\{\left(x_{1}, x_{2}, \ldots, x_{n}\right) | x_{1}^{2}+x_{2}^{2}+\cdots x_{n}^{2} \leq a^{2}\right\}.$ Let $V_n(a)$ denote this n-dimensional volume. Let $C_n$ denote the n-dimensional volume of the unit ball U; that is, $C_{n}=V_{n}(1).$ Use the previous two exercises to establish the recursive formula $V_{n}(a)=\left(\frac{2 \pi a^{2}}{n}\right) V_{n-2}(a)$

Use second order Taylor polynomials $P_{2}(x)$ for the given function about the point specified to approximate the indicated value. Estimate the error, and write the smallest interval you can be sure contains the value. $f(x)=x^{1 / 3}$ about 8; approximate $9^{1 / 3}$.

Air inside a rigid tank is heated from 550 to 600 R. Find the entropy increase s2 − s1. What is the entropy increase if the tank is heated from 2300 to 2350 R?

Supplementary Exercises A large used-book store randomly selected 1,000 of its customers and asked them to complete a survey about their satisfaction with the the merchandise in the store. The following data are from the 224 customers who returned the survey. Although the survey requested a considerable amount of information, the store was most interested in the frequency of purchases (number of books purchased in past 3 months) and the customer’s rating of the adequacy of book selection in the store. The data from the 224 surveys are given in the following table.a. Is there evidence of a significant relationship between frequency of purchases and adequacy of selection? Use $\alpha$ = .05.b. Describe the relationship between frequency of purchases and adequacy of selection (if any).

$$\begin{matrix} \text{Frequency} & \text{ } & \text{Adequacy of Selection}\\ \text{of Purchases} & \text{Poor} & \text{Average} & \text{Good} & \text{Excellent}\\ \text{1} & \text{3} & \text{4} & \text{37} & \text{44}\\ \text{2} & \text{2} & \text{6} & \text{30} & \text{28}\\ \text{3} & \text{3} & \text{8} & \text{16} & \text{19}\\ \text{4 or more} & \text{2} & \text{12} & \text{5} & \text{5}\\ \end{matrix}$$

Determine the intervals of constant concavity of the given function, and locate any inflection points. $f(x)=x-2 \sin x$

Determine whether the statement is true or false. If it is true, explain why. If it is false, explain why or give an example that disproves the statement. If y is an eigenvector of A, then Ay = ky for some scalar k.