Exam 11: Polynomial and Rational Functions

A right circular cylinder has a volume of 128 cubic meters. The least possible surface area for the cylinder is approximately:

Find the equation of the vertical line through the $x$ -intercept of the graph $y=6(x+7)^{3}$ .

What is the long-run behavior of the function $y=\frac{7^{t}}{\ln |t+2|}$

Let $f(x)=(x-2)^{2}(x+4)$ . The following figure shows the graph of $f(x+a)$ , with $a$ $=$ ------------

Is the function $y=\frac{\left(12 x^{-14}\right)\left(2 x^{5}\right)}{4 x^{-7}}$ a power function? If so, write the function in the form $y=k x^{p}$ .

What is the long-run behavior of the function $y=\frac{e^{7 x}}{\ln |4 x+3|}$

Does the figure below show an accurate graph of $f(x)=\frac{27}{x-3}$ ?

Write the equation for the following graph: The graph of $y=\frac{4}{x^{2}}$ shifted to the left 4 units and up 2 units.

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 power function model. After fitting a power function to the data, he uses it to estimate the force when the spring is stretched 25 inches. What is his approximation (to three decimal places) of this force?

Poiseuille's law says that the rate at which blood is flowing through a blood vessel of radius $R$ is proportional to $R^{4}$ . For medical reasons, we want to know how a reduction in the radius of a blood vessel affects the blood flow. If the radius of the blood vessel decreases by $10 \%$ , by what percentage does the blood flow decrease? Round to the nearest whole percent.

Is $y=\frac{2}{x^{2}-9}+0.25$ a possible formula for the graph below?

The following table contains the names of the nine known planets that orbit about the sun, along with their distance $d$ from the sun in millions of miles and the period $P$ of the revolution about the sun in standard earth years, that is, the time it takes the planet to go around the sun. (For this problem, we will assume that Pluto is a planet). Let $P=f(d)$ . Suppose a tenth planet is discovered at a distance of 5.2 billion (i.e. 5200 million) miles from the sun. Use the best-fit power model for the data to estimate the number of years in the period of the new planet to the nearest year.

The following figure gives the graphs of $f(x)=a x^{p}$ and $g(x)=b x^{q}$ . If $g(f(1))=(a b)^{q}$ , what is $b ?$

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 quadratic function to the data and use it to estimate the number of childbirths (rounded to the nearest whole number) in 2013.

Which of the following formulas has a graph with vertical asymptotes at $x=6$ and $x=8$ , a horizontal asymptote at $y=0$ , and crosses the $x$ -axis at $x=1$ ?

What is the domain of the function $\sqrt{x^{2}-3 x-28}$ ?

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 quadratic function to the data and use it to estimate the squirrel population (rounded to the nearest whole number) in 2017.

As $x \rightarrow \infty, y=\frac{x^{6}+2}{e^{3 x}+9} \rightarrow \ldots$ . Enter "infinity " or "- infinity" for $\infty$ or $-\infty$ .

What is the degree of the polynomial function $y=4 x^{3}-4 x+1$ ?

The sum of two even functions is always an odd function.