Exam 11: Polynomial and Rational Functions

Use a graphing calculator to find all the real zeros of $f(x)=0.3 x^{3}+2.7 x^{2}-0.3 x-2.7$ . List them in ascending order, separated by commas.

Estimate the following limits A) $\lim _{x \rightarrow 3+} \frac{6 x^{2}}{(x-3)^{3}(x-4)}$ B) $\lim _{x \rightarrow 3+} \frac{(x-3)^{3}(x-4)}{6 x^{2}}$

The formula for the power function shown in the following table is $f(x)=k x^{p}$ , where $k=$ ----------- and $p=$ ------------ . Round both answers to 1 decimal place.

Which of the following are polynomials:

Is $\frac{e^{x}}{x^{6}+1}-\frac{x-1}{x+2}$ a rational function?

As $x \rightarrow-\infty, g(x)=e^{-x}-8 \rightarrow$ ------------- . Enter "infinity " or "- infinity" for $\infty$ or $-\infty$ .

If the function $f(x)=\frac{1}{x+3}-\frac{x}{x-5}$ is written in the form $f(x)=\frac{p(x)}{q(x)}$ , a ratio of polynomials, which of the following could be $p(x)$ ? A) $-x^{2}-2 x-5$ B) $-x^{2}-3 x+1$ C) $1-x$ D) -8

Which of the following could represent a complete graph of $f(x)=a x+x^{3}$ where $a$ is a constant?

Farmer Brown has 115 feet of fence. He wishes to close in a rectangular field using the barn as one side of the field. Suppose each side of the field perpendicular to the barn has length $x$ feet. What is the area of the fenced in field?

Is $y=x^{6}$ a power function?

The following figure gives the graphs of four power functions. Which one could be the graph of $y=\frac{1}{1000 x^{6}}$ ? If none of the graphs match, enter "none".

The formula for the polynomial function of degree 2 shown in the following table is $f(x)=a x^{2}+b x+c$ , where $a=$ ----------- $b=$ -------------- , and $c=$ ------------

One of the following tables of data comes from a linear function, one from an exponential function, and one from a power function. The formula for the power function is $y=a x^{b}$ with $a=$ ----------- and $b=$ ----------

Let $y=1.4 x^{5}+1.1$ . If $y$ is a polynomial, give its degree. If not, enter "not a polynomial".

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 an exponential function to the data and use it to estimate the number of childbirths (rounded to the nearest whole number) in 2015 .

Find a power function through the two points $(3,729)$ and $(6,11,664)$ .

Farmer Brown has 112 feet of fence. He wishes to close in a rectangular field using the barn as one side of the field. Suppose each side of the field perpendicular to the barn has length $x$ feet. a) What is the area of the fenced in field in terms of $x$ ? b) In the context of this problem, what values of $x$ make sense? c) Approximate the maximum volume of the fenced field.

Suppose that $f(x)$ is a power function and that $f(9)=\frac{1}{9}$ . Must $f(x)=\frac{1}{x}$ ?

A $10 \mathrm{~kg}$ sample of a certain alloy (mixture of metals) contains $3 \mathrm{~kg}$ of tin and $7 \mathrm{~kg}$ 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 $10 \mathrm{~kg}$ sample. There is a zero at $x=$ ---------

The following figure is a graph of $y=\frac{a(x+b)(x-c)}{(x+m)(x-n)}$ , where $P>Q>R>S>0$ and all of the constants appearing in the function are positive. What is the value of $b$ (in terms of $P, Q$ , etc.)?