Exam 11: Polynomial and Rational Functions

The graphs of $y_{1}=0.25 x^{4}$ and $y_{2}=0.25\left(x^{4}+x^{3}-23 x^{2}+3 x+90\right)$ are shown in the figure below as viewed on the window $[-10,10]$ by $[-25,25]$ . What is the double zero of $y_{2}$ ?

Find the long-run behavior of the function $y=\frac{x^{2}-3}{x^{3}}-\frac{1}{4 x^{2}}$ .

Let $f(x)=\frac{8}{x+3}$ . As $x \rightarrow-3$ from the right, $f(x) \rightarrow \ldots$ -------- .Enter "infinity" for $\infty$ and "-infinity" for $-\infty$ .

As $x \rightarrow \infty, g(x)=\frac{6 x+4}{5 x+5} \rightarrow \ldots$ . Enter "infinity " or "- infinity" for $\infty$ or $-\infty$ .

Let $f(x)=k x^{p}$ satisfy the conditions $f(-x)=f(x), \lim _{x \rightarrow \infty} f(x)=0$ , and $f(2)=-\frac{5}{4}$ . Which of the following must be true?

Let $f(x)=2 x^{3}-4 x^{2}+5$ . Which of the following statements are true?

The graph of $y=\frac{1}{(x-4)^{8}}-4$ is a transformation of the graph of $y=\frac{1}{x^{p}}$ for some $p$ . Which of the following correctly describes the transformation?

The profit earned by a producer to manufacture and sell $n$ units of a good is given by $P(n)=11 n-2365$ . The average profit for $n$ units is given by $A(n)=\frac{P(n)}{n}$ . A) Compute $A(1), A(215), A(282)$ . B) Interpret the economic significance of the values in part (A). C) What trend do you notice in the values of $A(n)$ as $n$ gets large? Explain this trend in economic terms.

Perform the operations and then express the answer in reduced form: $\frac{2}{5 x^{2}}-\frac{1}{6 x+5}$

If the function $f(x)=\frac{6}{x^{2}}+\frac{x+3}{5}$ is written in the form $f(x)=\frac{p(x)}{q(x)}$ , a ratio of polynomials, which of the following could be $p(x)$ ?

Does $f(x)=(x-7)^{4}+11$ have an inverse that is a function?

Suppose $y$ is directly proportional to $x$ . If $y=1$ when $x=5$ , what is the value of $x$ when $y$ is 2 ?

List the zeros of the function $y=x^{3}-3 x^{2}-18 x$ in ascending order, separated by commas.

A right circular cylinder has a volume of 122 cubic meters. a) What is the formula for surface area in terms of the radius, $r$ , of the base of the cylinder? b) Approximate the least possible surface area for the cylinder. c) Approximate the radius necessary to achieve the minimum surface area. Round answers to 3 decimal places if necessary.

The functions $f$ and $F$ are given by $f(x)=k x^{p}$ and $F(x)=\sqrt{\frac{k x^{p-1}}{p}}$ . Suppose $f(x)=\frac{x^{9}}{8}$ . What must $F$ be?

The following figure shows the graph of a power function, $f(x)$ , whose formula has the form $f(x)=k x^{p}$ . Which of the following statements are true? Mark all that apply.

Which of the following statements describe the graph of $f(x)=\frac{x+9}{x+3}$ ? Mark all that apply.

The following graphs are of functions of the form $f(x)=M x+\frac{N}{x}$ , where $M$ and $N$ are constants. Which graph has $M>0$ and $N<0$ ?

Can $f(x)=\frac{2}{3 x^{4}}$ be written in the form of an exponential function, a power function, or neither?

The following table represents the amount of monthly data handled by MichNet, the University of Michigan's gateway to the Internet, sampled at six month intervals for the years 1993-1995. (The data is from the Michigan Daily, March 30, 1996, p.4.) The article also stated that the data handling capacity of the MichNet system was about 40 billion data packets per month. Date June '93 Dec. '93 June '94 Dec. '94 June '95 Dec. '95 Data packets 1.8 2.2 4.1 5.0 7.8 10.5 (millions) Let $d$ be the number of data packets handled monthly (in billions), and $t$ be time in years since December, 1992. Find an equation for the best fit power function that models these data (you do not have to enter your equation). Use this equation to determine in what year the network became overloaded.