Exam 1: A Library of Functions

Coulomb's law says that the electrical force between two charged objects is directly proportional to the product of the quantity of charge on the objects and inversely proportional to the square of the distance between the objects.Let $q_1$ and $q_2$ be the charge on the two objects.Let d be the distance between the objects and F be the electrical force between them.Translate Coulomb's Law into mathematical symbols.

Given the function $q(x)=x^{3}$ , which of the following is equivalent to $\left.q^{(} x^{5}\right)+q(x+a)$ ?

Given the function $q(x)=x^{3}$ , which of the following is equivalent to $q(2 x+a)+q(x)$ ?

Use the following figure to give the value of $\lim _{x \rightarrow 0^{-}} f(x)$ .If the limit does not exist, enter "DNE".

Find, $\lim _{x \rightarrow \infty} \frac{4 x^{7}-3 x+2}{11 x^{7}+100 x-5}$ if it exists.

In nature, the population of two animals, one of which preys on the other (such as foxes and rabbits)are observed to oscillate with time, and are found to be well approximated by trigonometric functions.The population of foxes is given by the graph below.What is the amplitude?

Solve $6^{x}=2$ for x.Round to 2 decimal places.

Consider the functions $f(x)=5+\sin 3 x$ and $g(x)=3 \sin x$ .Which has a larger vertical intercept?

Write an equation for the graph obtained by shifting the graph of $y=x^{3}$ vertically upward by 3 units, followed by vertically stretching the graph by a factor of 2 and reflecting the graph across the $x -$ axis.

The illumination, I, of a candle is inversely proportional to the square of its distance, d, from the object it illuminates.Write a formula that expresses this relationship.

Given the function $m(z)=z^{2}$ , find and simplify $m(z-h)-m(z)$ .

A one-to-one function is described in words as follows.Take x and multiply by 4.Then add 3 and cube the result.Describe the inverse function in words.

The cost of shipping r kilograms of material is given by the function C = f(r)= 175+ 6r.Find a formula for the inverse function.

One of the graphs below shows the rate of flow, R, of blood from the heart in a man who bicycles for twenty minutes, starting at t = 0 minutes.The other graph shows the pressure, p, in the artery leading to a man's lungs as a function of the rate of flow of blood from the heart.Estimate $p(R(15))$ .

In 1909, the Danish biochemist Sören Peter Lauritz Sörensen (1868-1939)introduced the pH function as a measure of the acidity of a chemical substance: pH = f([H+])=

Tornados are classified in several ways.A tornado's classification on the Fujita Scale as F1 through F5 is most commonly cited.Another classification of tornados is by path length, given by the formula $P l=2 \log (L)+1$ where L is the length of the tornado's path in miles.The Binger, Oklahoma tornado of 1981 was an F4 (originally thought to be an F5)whose path was 16 miles in length.What was its Pl classification?

Consider the graph of the function f(x)given below.What is $\lim _{x \rightarrow-2} f(x)$ ?

A population of bacteria is introduced into a nutrient solution in a dish.The nutrient solution is replenished by the exact same amount each day.At first the bacteria population grows very quickly.Soon, population growth must slow down, as the nutrients become more scarce given the number of bacteria.Eventually the bacteria population stabilizes.A model for the bacteria population growth is given by $P(t)=\frac{5000}{6+e^{-5 t}}$ thousand bacteria.What is $\lim _{t \rightarrow \infty} P(t)$ ?

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 budget is doubled, but prices remain the same, what is the $x_{2}-$ intercept of the new budget constraint?

Given the information in the table and the following conditions, find $h(1)$ . • f(x)is even. • g(x)is odd. • h(x)= $g(x)^{2}-f(x)$ x f(x) g(x) -3 7 22 -2 2 10 -1 -1 1 0 -2 2