It costs a company $30,000 to begin production of a good, plus$3 for every unit of the good produced. Let x be the number of units produced by the company.

(a) Find a formula for $C(x)$, the total cost for the production of $x$ units of the good.

(b) Find a formula for the company's average cost per unit, $a(x)$.

(c) Graph $y=a(x)$ for $0<x \leq 50,000,0 \leq y \leq 10$. Label the horizontal asymptote.

(d) Explain in economic terms why the graph of $a$ has the long-run behavior that it does.

(e) Explain in economic terms why the graph of $a$ has the vertical asymptote that it does.

(f) Find a formula for $a^{-1}(y)$. Give an economic interpretation of $a^{-1}(y)$.

(g) The company makes a profit if the average cost of its good is less than $\$ 5$ per unit. Find the minimum number of units the company can produce and make a profit.

For the polynomials in the following exercise, state the degree and the number of terms, and describe the long-run behavior.

$$y=1-2 x^4+x^3$$

What is the long-run behavior of the functions in the following problem?

$$y=\frac{2^x+3}{x^2+5}$$

An alcohol solution consists of 5 gallons of pure water and x gallons of alcohol, x>0. Let f(x) be the ratio of the volume of alcohol to the total volume of liquid. [Note that $f(x)$ is the concentration of the alcohol in the solution.]

(a) Find a possible formula for $f(x)$.

(b) Evaluate and interpret $f(7)$ in the context of the mixture.

(c) What is the zero of $f$ ? Interpret your result in the context of the mixture.

(d) Find an equation for the horizontal asymptote of $f$. Explain its significance in the context of the mixture.

Compare and discuss the long-run behaviors of the following functions:

$$f(x)=\frac{x^2+1}{x^2+5}, \quad g(x)=\frac{x^3+1}{x^2+5}, \quad h(x)=\frac{x+1}{x^2+5} .$$

For the following problem, find the real zeros (if any) of the polynomials.

$$y=x^4+6 x^2+9$$

The town of Smallsville was founded in 1900. Its population y (in hundreds) is given by the equation

$$y=-0.1 x^4+1.7 x^3-9 x^2+14.4 \bar{x}+5,$$

where $x$ is the number of years since 1900 . Use a the graph in the window $0 \leq x \leq 10,-2 \leq y \leq 13$.

(a) What was the population of Smallsville when it was founded?

(b) When did Smallsville become a ghost town (nobody lived there anymore)? Give the year and the month.

(c) What was the largest population of Smallsville after 1905? When did Smallsville reach that population? Again, include the month and year. Explain your method.

The table below gives approximate values for three functions, f, g, and h. One is exponential, one is trigonometric, and one is a power function. Determine which is which and find possible formulas for each.

For the following problem, find the real zeros (if any) of the polynomials.

$y=a x^2\left(x^2+4\right)(x+3)$, where $a$ is a nonzero constant.

For the following exercise, perform the operations. Express answers in reduced form.

$$\dfrac{1}{2 x}-\dfrac{2}{3}$$

At a constant speed, is the time to drive from Long Island to Albany, NY, proportional, or inversely proportional, to the speed? Explain. At 55 mph, it takes 3.5 hours to make the trip. Write a formula for the relationship and find how fast you would have to drive to get to Albany in 3 hours.

Anthropologists suggest that the relationship between the body weight and brain weight of primates can be modeled with a power function. The table below lists various body weights and the corresponding brain weights of different primates.

(a) Using the table below, find a power function that gives the brain weight, $Q$ (in $\mathrm{mg}$ ), as a function of the body weight, $b$ (in gm).

(b) The erythrocebus (Patas monkey) has a body weight of $7800 \mathrm{gm}$. Estimate its brain weight.

The given figure shows the cost function, C(n), from the previous problem, and a line, $l$, that passes through the origin.

(a) What is the slope of line $l$?

(b) How does line $l$ relate to $a\left(n_0\right)$, the average cost of producing $n_0$ units?

For the following problem, find the real zeros (if any) of the polynomials.

$$y=4 x^2-1$$