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 happens as the viewing window is expanded?

The squirrel population in a certain area is estimated each year and recorded in the table below. Suppose the squirrel population in a given year is a function of the number of years since 1998. Fit a power function to the data and use it to estimate the squirrel population (rounded to the nearest whole number) in 2018 .

A $13 \mathrm{~kg}$ sample of a certain alloy (mixture of metals) contains $3 \mathrm{~kg}$ of tin and $10 \mathrm{~kg}$ of copper. A chemist decides to study the properties of the alloy as its percentage of tin is varied. Suppose $x$ represents the quantity of tin, in $\mathrm{kg}$ , the chemist adds to the sample. Let $f(x)$ represent the fraction of the mixture's mass composed of tin--that is, the ratio of the tin's mass to the mixture's total mass. A negative value of $x$ represents a quantity of tin removed from the original $13 \mathrm{~kg}$ sample. $f^{-1}(0.4)=$ ------------ kg. Round to 2 decimal places.

Let $f(x)=a x^{b}$ and $g(x)=c x^{d}$ be graphed in the following figure. Which of the following is true?

The following table represents the amount of monthly data handled by an Internet server, sampled at six month intervals for the years 1993-1995. Let $d$ be the number of data packets handled monthly (in billions), and $t$ be time in years since December, 1992. The best exponential model for this data is of the form $d(t)=a b^{t}$ , where $a=\ldots$ and $b=\ldots$ . Give both answers to 6 decimal places.

The horizontal asymptote of $y=\frac{2 x+1}{x^{2}}-\frac{5 x+8}{x-5}$ is $\mathrm{y}=\ldots$ . Enter "none" if there is no horizontal asymptote.

Which of the following statements are true for $f(x)=\frac{x}{x^{2}-5 x+6}$ ? Mark all that apply.

Given the following graph, what would the equation of the line be if the horizontal axis is labeled $\ln x$ and the vertical axis is labeled $\ln y$ ?

The function $f(x)=\sin x+0.5$ can be approximated by the function $g(x)=0.5+x-\frac{x^{3}}{6}+\frac{x^{5}}{120}$ . On what interval do the two graphs look similar?

The volume of pollutants (in millions of cubic feet) in a certain reservoir is given by $P(t)=400+35 t$ , where $t$ is time in years. The volume of the reservoir (including pollutants) is gradually increasing and is given by $R(t)=12,000+110 t$ . Let $C(t)$ be the fraction of the reservoir's volume that consists of pollutants. If these trends continue for many years, approximately what percent of the reservoir's total volume would eventually be pollutants? Round to 1 decimal place.

Find the equation of the horizontal line through the $y$ -intercept of the graph $y=7 x^{3}-4 x^{2}+4$ .

Suppose that $a, b$ , and $c$ are integers with $a$ negative and odd, $b$ positive and even, $c$ positive and odd, and $d$ positive and odd. Which of the following functions corresponds to the graph below?

George measures the force exerted by a certain spring when it is stretched various distances beyond its natural length. The data he gathers is compiled in the following table. He wishes to describe the force as a function of the stretched distance, and decides to use an exponential function model. After fitting an exponential function to the data, he uses it to estimate the force when the spring is stretched 16 inches. What is his approximation (to three decimal places) of this force?

The infant mortality, $I$ , in a country is related to the country's GNP (gross national product), $g$ . Some authors (Weld and Helms, 1971) have argued that the relationship is of the form $I=I_{0}+\frac{k}{g+a}$ , where $I_{0}, k$ , and $a$ are positive constants and $g \geq 0$ . For $I_{0}=3, k=8$ , and $a=4$ , the vertical asymptote of the graph of $I$ against $g$ is at $g=$ ------------ If there is no vertical asymptote, enter "none"

Which of the following statements are true for $f(x)=\frac{-11(x-6)(x+5)}{(x+6)(x-2)}$ ? Mark all that apply.

George measures the force exerted by a certain spring when it is stretched various distances beyond its natural length. The data he gathers is compiled in the following table. He wishes to describe the force as a function of the stretched distance, and decides to use a cubic function model. After fitting a cubic function to the data, he uses it to estimate the force when the spring is stretched 22 inches. What is his approximation (to three decimal places) of this force? Stretched Distance (feet) 0.17 0.56 0.73 0.81 1.13 Force (pounds) 16.74 499.82 1,180.23 1,645.11 4,102.94

Write a possible formula for a rational function, $f(x)$ , with zeros at $x=-6, x=2$ , vertical asymptotes at $x=9, x=-11$ , and a horizontal asymptote at $y=3$ .

As $x \rightarrow-\infty, f(x)=\frac{4^{x}-6}{5^{x}+2} \rightarrow \ldots$ . Enter "infinity " or "- infinity" for $\infty$ or $-\infty$ .

A $16 \mathrm{~kg}$ sample of a certain alloy (mixture of metals) contains $5 \mathrm{~kg}$ of tin and $11 \mathrm{~kg}$ of copper. A chemist decides to study the properties of the alloy as its percentage of tin is varied. Suppose $x$ represents the quantity of tin, in $\mathrm{kg}$ , the chemist adds to the sample. Let $f(x)$ represent the fraction of the mixture's mass composed of tin--that is, the ratio of the tin's mass to the mixture's total mass. A negative value of $x$ represents a quantity of tin removed from the original $16 \mathrm{~kg}$ sample. f(0.6)= ----------- (kg tin) (kg mixture) Round to 2 decimal places.

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 power function to the data and use it to estimate the number of childbirths (rounded to the nearest whole number) in 2012 .