Exam 1: A Library of Functions

Suppose there is an initial population of 100 rabbits on Prosperity Island.Assuming that the rabbits have more than enough of everything they need to live prosperously, we might expect the population to grow exponentially.If so, find a formula for P(t), the number of rabbits on Prosperity Island at time t, given that after one year there are 140 rabbits on the island.(Assume t is in years.)Use this formula to determine how many years it will be before there are 400 rabbits on Prosperity Island.Round to 1 decimal place.

A pond has a population of 500 frogs.Over a ten-year period of time the number of frogs drops quickly by 20%, then increases slowly for 5 years before dropping to almost zero.Does the following graph accurately represent the number of frogs in the pond over the ten-year period of time?

You have $500 invested in a bank account earning 8.5% compounded annually.How many years will it take to triple your money? Round to the nearest whole year.

Is the function $g ( x ) = \left\{ \begin{array} { r } | x + 1 | , - 10 < x \leq 2 \\- x ^ { 2 } + 3 x , 2 < x < 10\end{array} \right.$ continuous at x = 2?

If $f(x)$ and $g(x)$ are both continuous functions, what can you say about $f(x)+g(x)$ ?

Could the function described by the following table of values be exponential? x 3.2 3.3 3.4 3.5 3.6 f(x) 26 28.60 31.46 34.61 38.07

Use algebra to evaluate the right-hand limit of $f(x)=\frac{|x-8|}{2 x-16}$ as $x \rightarrow 8$ .

Find, to two decimal places, $\lim _{\theta \rightarrow 0} \frac{8 \sin (6 \theta)}{\theta}$

A school library opened in 1980.In January, 2000 they had 40,000 books.One year later, they had 40,410 books.Assume that they acquire the same number of books at the start of each month.If you graph the function with domain 1980-2010, describe the y-intercept of the graph in the context of the problem.

A school library opened in 1980.In January, 2000 they had 25,000 books.One year later, they had 25,420 books.Assuming they acquire the same number of books at the start of each month, how many books did they have in January, 2003?

In 1992, the Population Crisis Committee wrote: Large cities in developing countries are growing much faster than cities in the industrialized world ever have.London, which in 1810 became the first industrial city to top 1 million, now has a population of 11 million.By contrast, Mexico City's population stood at only a million just 50 years ago and now is 20 million. Assume that the instantaneous percentage growth rates of London and Mexico City were constant over the last two centuries.How many times greater is Mexico City's percentage growth rate than London's (to one decimal place)?

The following describes a function that must be invertible: R(t)is the revenue earned by a shoe shop on the $t ^ { t h }$ th day of the year.

Suppose $\lim _{x \rightarrow 0} f(x)=7$ and $\lim _{x \rightarrow 0} g(x)=11$ .Find $\lim _{x \rightarrow 0} \frac{7 g(x)}{f(x)+7}$ .Round to two decimal places.

Suppose we buy quantities $x_1$ and $x_2$ , respectively, of two goods.The following graph shows the budget constraint $p_{1} x_{1}+p_{2} x_{2}=k$ , where $p_{1}$ and $p_{2}$ are the prices of the two goods and k is the available budget.If the price of the first good is doubled, but the other values are unchanged, what is the $x_{2}-$ intercept of the new budget constraint?

Is the function $f(x)=x^{5}+x^{8}+x^{9}$ odd, even , or neither?

Using the standard viewing rectangle (-10 $\le$ x $\le$ 10, - 10 $\le$ y $\le$ 10), I graphed a cubic polynomial and saw two vertical lines, as shown.There must be another root outside of the window. x

Here are some data from a Scientific American article on Old World monkeys.From the data presented, give an approximate formula for C = cranial capacity (in cm³)as a function of A = arc length of skull (in cm).Round to one decimal place. Cranial Capacity of contemporary Old World monkeys is related to arc length of skull as shown.

The graph of y = f(x)is shown in the first figure.What graph is shown in the second figure?

Is the function $\frac{1}{\sin x}$ continuous on the interval $\frac{3 \pi}{4} \leq x \leq \frac{5 \pi}{4}$ ?

Sketch a graph of a function on the interval [-5, 5] with exactly 2 zeros and at least 2 places where the function is not continuous.