Exam 11: Polynomial and Rational Functions

Let $f(x)=(x-2)^{2}(x+4)$ . Which of the following is the graph of $-f(x)$ ?

Suppose that $f(x)=4 x^{5}+6 x^{3}+4 x^{2}-14$ . Select all that are true:

The following figure gives the graphs of $f(x)=a x^{p}$ and $g(x)=b x^{q}$ . Which is smaller, $p$ or $q$ ?

Which of the following figures shows the graph of $f(x)=\frac{x}{1+x^{2}}$ ?

The functions $f$ and $F$ are given by $f(x)=k x^{p}$ and $F(x)=\sqrt{\frac{k x^{p-1}}{p}}$ . Suppose $f(x)=\frac{x^{11}}{6}$ . What must $F$ be?

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.) Let $d$ be the number of data packets handled monthly (in billions), and $t$ be time in years since December, 1992. Find equations for the best fit exponential function and the best fit power function that model these data (you do not have to enter your equations). By comparing their correlation coefficients, choose which model gives a better fit.

Suppose that $a, b, c$ , and $d$ are integers and that $a$ is positive and even, $b$ is positive and odd, $c$ is negative and even, and $d$ is negative and odd. Which of the following graphs could correspond to the power function $y=a x^{c}$ ? If none of the graphs correspond to the function, enter "none".

Let $p(x)$ be a polynomial of degree 6 . Let $q(x)$ be a polynomial of degree 7 . Which of the following are true about $r(x)=\frac{p(x)}{q(x)}$ ?

Let $f(x)=\frac{6}{x+8}$ . As $x \rightarrow \infty, f(x) \rightarrow \ldots$ . Enter "infinity" for $\infty$ and "-infinity" for $-\infty$ .

Compute the following limits: a) $\lim _{x \rightarrow \infty}\left(-5 x^{4}+7 x^{3}-114 x^{2}\right)$ b) $\lim _{x \rightarrow-\infty}\left(-5 x^{4}+7 x^{3}-114 x^{2}\right)$

Which of the following statements are true for $f(x)=\frac{x^{2}-1}{x^{2}+5 x}$ ? Mark all that apply.

Describe the graph of $y=\frac{1}{2}(x+2)(x-2)(x+4)$ .

What is the domain of the function $y=\ln \left((2 x-9) x^{2}\right)$ ?

Let $f(x)=x^{1 / 2}$ and $g(x)=\ln x$ . Which function dominates as $x \rightarrow \infty$ ?

The number of yearly childbirths for a growing town are recorded in the table below. Suppose the number of childbirths each year is a function of the number of years since 1996. Fit a cubic function to the data and use it to estimate the number of childbirths (rounded to the nearest whole number) in 2012.

The volume of pollutants (in millions of cubic feet) in a certain reservoir is given by $P(t)=370+25 t$ , where $t$ is time in years. The volume of the reservoir (including pollutants) is gradually increasing and is given by $R(t)=11,000+130 t$ . Let $C(t)$ be the fraction of the reservoir's volume that consists of pollutants. What percent of the reservoir's total volume consisted of pollutants in the year $t=0$ ? Round to 1 decimal place.

Must the sum of two functions with horizontal asymptotes also have a horizontal asymptote?

According to one advertisement, Burger King's all-beef hamburger patties have $85 \%$ more beef than McDonald's all-beef hamburger patties. If both chains serve circular patties of the same thickness, then the diameter of Burger King's patties, $d_{B}$ , will be directly proportional to the diameter of McDonald's patties, $d_{M}$ . Which of the following formulas express $d_{B}$ as a function of $d_{M}$ ?

The figure below shows the graphs of two power functions, $f$ and $g$ . The formula for $g$ is $g(x)=k x^{p}$ , where $k=$ -------- and $p=$ -------------

The volume occupied by a fixed quantity of gas such as oxygen is inversely proportional to its pressure, provided that its temperature is held constant. Suppose that a quantity of oxygen occupies a 110 liter volume at a pressure of 12 atmospheres. If the temperature of the oxygen does not change, how many liters will it occupy if its pressure rises to 17 atmospheres? Round to 1 decimal place.