The average new car cost (in dollars) for selected years from 1980 to 2005 is given in the table.

$$\begin{array}{|l|llllll|} \hline \text { Year } & 1980 & 1985 & 1990 & 1995 & 2000 & 2005 \\ \hline \text { cost } & 7500 & 12,000 & 16,000 & 20,450 & 24,900 & 28,400 \\ \hline \end{array}$$

a. Find a linear equation for the average new car cost in terms of x, the number of years since 1980, using the data for 1980 and 2005. b. Repeat part a, using the data for 1995 and 2005. c. Find the equation of the least squares line using all the data. d. Use a graphing calculator to plot the data and the three lines from parts a-c. e. Discuss which of the three lines found in parts a-c best describes the data, as well as to what extent a linear model accurately describes the data. f. Calculate the correlation coefficient.

Determine whether the statement is true or false. If $T: \mathbb{R}^{2} \rightarrow \mathbb{R}^{2}$ is defined by

$$T\left(\left[\begin{array}{l} x \\ y \end{array}\right]\right)=\left[\begin{array}{ll} 2 & -4 \\ 1 & -2 \end{array}\right]\left[\begin{array}{l} x \\ y \end{array}\right]$$

then T is an isomorphism.

Apollo manufactures smartphones at a variable cost of $V(x)=0.000003 x^{3}-0.03 x^{2}+200 x$ dollars, where x denotes the number of units manufactured per month. The monthly fixed cost attributable to the division that produces them is $100,000. The total revenue realized by Apollo from the sale of x smartphones is given by the total revenue function$R(x)=-0.1 x^{2}+500 x \ (0 \leq x \leq 5000)$ where R(x) is measured in dollars. a. Find a function C that gives the total cost incurred by the manufacture of x smartphones. b. Find the total profit function. c. What is the profit when 1500 units are produced and sold each month?

The rise of digital music and the improvement to the DVD format are some of the reasons why the average selling price of stand-alone DVD recorders has dropped in recent years. The function $A(t)=\frac{699}{(t+1)^{0.94}} \ (0 \leq t \leq 13)$ gives the projected average selling price (in dollars) of stand-alone DVD recorders in year t, where t = 0 corresponds to the beginning of 2002. How fast was the average selling price of stand-alone DVD recorders falling at the beginning of 2002? How fast was it falling at the beginning of 2012?

Suppose that Kendra Gallegos has appointments with 9 potential customers. Kendra will be ecstatic if all 9 of these potential customers decide to make a purchase from her. Of course, in sales there are no guarantees. How many different sets of customers may place an order with Kendra?

Consider a Poisson process on the real line, and denote by $N\left(t_{1}, t_{2}\right)$ the number of events in the interval $\left(t_{1}, t_{2}\right)$. If $t_{0}<t_{1}<t_{2}$, find the conditional distribution of $N\left(t_{0}, t_{1}\right)$ given that $N\left(t_{0}, t_{2}\right)=n$. (Hint: Use the fact that the numbers of events in disjoint subsets are independent.)

(a) identify the claim and state $H_0$ and $H_a$, (b) find the critical values, (c) find the test statistic, (d) decide whether to reject or fail to reject the null hypothesis, and (e) interpret the decision in the context of the original claim. Use $\alpha=0.05$. A coach records the results of the coin toss at the beginning of each football game for a season. The results are shown, where H represents heads and T represents tails. The coach claimed the tosses were not random. Test the coach’s claim. H T T T H T H H T T T T H T H H

Empirical data suggest that the surface area of a 180-cm-tall human body changes at the rate of $S^{\prime}(W)=0.131773 W^{-0.575}$ square meters/kilogram, where W is the mass of the body in kilograms. If the surface area of a 180-cm-tall human body weighing 70 kg is 1.886277 $m^2$, what is the surface area of a human body of the same height weighing 75 kg?

Sketch the graph of the function, using the curve-sketching guide. $g(x)=\frac{x}{x^{2}-4}$

For the following exercise, find the domain, range, and all zeros/intercepts, if any, of the function. $h(x)=\frac{3}{x^{2}+4}$

Find the average value of the function f over the indicated interval [a, b]. $f(x)=2 x^{2}-3 ;[1,3]$

Evaluate the definite integral. $\int_{1}^{4}\left(\sqrt{x}-\frac{1}{\sqrt{x}}\right) d x$

Find the Fourier series for the indicated function on the indicated interval. Plot the function and two partial sums of your choice over the interval.

$$f(x)=\left\{\begin{array}{ll} 2+x, & \mathrm { for } \ -2 \leq x \leq 0, \\ -2+x, & \mathrm { for } \ 0<x \leq 2, \end{array}\right.$$

on [-2, 2]

For the following exercise, for each pair of points, a. find the slope of the line passing through the points and b. indicate whether the line is increasing, decreasing, horizontal, or vertical. (-1, 4) and (3, -1)