Chapter 1

Specify the domain of the function $$ f(x)=\frac{2 x-1}{\sqrt{4-x^{2}}} $$

Find the composite function \(g(h(x))\), where \(g(u)=\frac{1}{2 u+1}\) and \(h(x)=\frac{x+2}{2 x+1}\)

Find an equation for each of these lines: a. Through the point \((-1,2)\) with slope \(-\frac{1}{2}\) b. With slope 2 and \(y\) intercept \(-3\)

Sketch the graph of each of these functions. Be sure to show all intercepts and any high or low points. a. \(f(x)=3 x-5\) b. \(f(x)=-x^{2}+3 x+4\)

PRICE OF GASOLINE Since the beginning of the year, the price of unleaded gasoline has been increasing at a constant rate of 2 cents per gallon per month. By June first, the price had reached \(\$ 3.80\) per gallon. a. Express the price of unleaded gasoline as a function of time, and draw the graph. b. What was the price at the beginning of the year? c. What will be the price on October \(1 ?\)

truck is 300 miles due east of a car and is traveling west at the constant speed of 30 miles per hour. Meanwhile, the car is going north at the constant speed of 60 miles per hour. Express the distance between the car and truck as a function of time.

BACTERIAL POPULATION The population (in thousands) of a colony of bacteria \(t\) minutes after the introduction of a toxin is given by the function $$ f(t)= \begin{cases}t^{2}+7 & \text { if } 0 \leq t<5 \\ -8 t+72 & \text { if } t \geq 5\end{cases} $$ a. When does the colony die out? b. Explain why the population must be 10,000 some time between \(t=1\) and \(t=7\).

MUTATION In a study of mutation in fruit flies, researchers radiate flies with X-rays and determine that the mutation percentage \(M\) increases linearly with the X-ray dosage \(D\), measured in kilo- Roentgens (kR). When a dose of \(D=3 \mathrm{kR}\) is used, the percentage of mutations is \(7.7 \%\), while a dose of \(5 \mathrm{kR}\) results in a \(12.7 \%\) mutation percentage. Express \(M\) as a function of \(D\). What percentage of the flies will mutate even if no radiation is used?

Find the slope and \(y\) intercept of the given line and draw the graph. a. \(2 y+3 x=0\) b. \(\frac{x}{3}+\frac{y}{2}=4\)

Find equations for these lines: a. Slope 5 and \(y\) intercept \((0,-4)\) b. Slope \(-2\) and contains \((1,3)\) c. Contains \((5,4)\) and is parallel to \(2 x+y=3\)

Find equations for these lines: a. Passes through the points \((-1,3)\) and \((4,1)\). b. \(x\) intercept \((3,0)\) and \(y\) intercept \(\left(0,-\frac{2}{3}\right)\) c. Contains \((-1,3)\) and is perpendicular to \(5 x-3 y=7\)

Find the points of intersection (if any) of the given pair of curves, and draw the graphs. a. \(y=-3 x+5\) and \(y=2 x-10\) b. \(y=x+7\) and \(y=-2+x\)

Find \(c\) so that the curve \(y=3 x^{2}-2 x+c\) passes through the point \((2,4)\).

Find \(c\) so that the curve \(y=4-x-c x^{2}\) passes through the point \((-2,1)\).

Either find the given limit or show it does not exist. If the limit is infinite, indicate whether it is \(+\infty\) or \(-\infty\). $$ \lim _{x \rightarrow 1} \frac{x^{2}+x-2}{x^{2}-1} $$

Either find the given limit or show it does not exist. If the limit is infinite, indicate whether it is \(+\infty\) or \(-\infty\). $$ \lim _{x \rightarrow 2} \frac{x^{2}-3 x}{x+1} $$

Either find the given limit or show it does not exist. If the limit is infinite, indicate whether it is \(+\infty\) or \(-\infty\). $$ \lim _{x \rightarrow 2} \frac{x-8}{2-x} $$

Either find the given limit or show it does not exist. If the limit is infinite, indicate whether it is \(+\infty\) or \(-\infty\). $$ \lim _{x \rightarrow 1}\left(\frac{1}{x^{2}}-\frac{1}{x}\right) $$

Either find the given limit or show it does not exist. If the limit is infinite, indicate whether it is \(+\infty\) or \(-\infty\). $$ \lim _{x \rightarrow-\infty}\left(2+\frac{1}{x^{2}}\right) $$

Either find the given limit or show it does not exist. If the limit is infinite, indicate whether it is \(+\infty\) or \(-\infty\). $$ \lim _{x \rightarrow-\infty} \frac{x}{x^{2}+5} $$

Either find the given limit or show it does not exist. If the limit is infinite, indicate whether it is \(+\infty\) or \(-\infty\). $$ \lim _{x \rightarrow 0^{-}}\left(x^{3}-\frac{1}{x^{2}}\right) $$

Either find the given limit or show it does not exist. If the limit is infinite, indicate whether it is \(+\infty\) or \(-\infty\). $$ \lim _{x \rightarrow-\infty} \frac{x^{4}+3 x^{2}-2 x+7}{x^{3}+x+1} $$

Either find the given limit or show it does not exist. If the limit is infinite, indicate whether it is \(+\infty\) or \(-\infty\). $$ \lim _{x \rightarrow-\infty} \frac{x^{3}-3 x+5}{2 x+3} $$

Either find the given limit or show it does not exist. If the limit is infinite, indicate whether it is \(+\infty\) or \(-\infty\). $$ \lim _{x \rightarrow-\infty} \frac{x x^{2}}{x^{2}+3 x-1} $$

Either find the given limit or show it does not exist. If the limit is infinite, indicate whether it is \(+\infty\) or \(-\infty\). $$ \lim _{x \rightarrow-\infty} \frac{x(x-3)}{7-x^{2}} $$

Either find the given limit or show it does not exist. If the limit is infinite, indicate whether it is \(+\infty\) or \(-\infty\). $$ \lim _{x \rightarrow 0^{-}} x \sqrt{1-\frac{1}{x}} $$

Either find the given limit or show it does not exist. If the limit is infinite, indicate whether it is \(+\infty\) or \(-\infty\). $$ \lim _{x \rightarrow 0^{+}} \sqrt{x}\left(1+\frac{1}{x^{2}}\right) $$

List all values of \(x\) for which the given function is not continuous. f(x)=\frac{x^{2}-1}{x+3}

List all values of \(x\) for which the given function is not continuous. $$ f(x)=5 x^{3}-3 x+\sqrt{x} $$

List all values of \(x\) for which the given function is not continuous. $$ g(x)=\frac{x^{3}+5 x}{(x-2)(2 x+3)} $$

PRICE As advances in technology result in the production of increasingly powerful, compact calculators, the price of calculators currently on the market drops. Suppose that \(x\) months from now, the price of a certain model will be \(P\) dollars per unit, where $$ P(x)=40+\frac{30}{x+1} $$ a. What will be the price 5 months from now? b. By how much will the price drop during the fifth month? c. When will the price be \(\$ 43\) ? d. What happens to the price in the long run (as \(x \rightarrow \infty\) )?

ENVIRONMENTAL ANALYSIS An environmental study of a certain community suggests that the average daily level of smog in the air will be \(Q(p)=\sqrt{0.5 p+19.4}\) units when the population is \(p\) thousand. It is estimated that \(t\) years from now, the population will be \(p(t)=8+0.2 t^{2}\) thousand. a. Express the level of smog in the air as a function of time. b. What will the smog level be 3 years from now? c. When will the smog level reach 5 units?

EDUCATIONAL FUNDING A private college in the southwest has launched a fund- raising campaign. Suppose that college officials estimate that it will take \(f(x)=\frac{10 x}{150-x}\) weeks to reach \(x \%\) of their goal. a. Sketch the relevant portion of the graph of this function. b. How long will it take to reach \(50 \%\) of the campaign's goal? c. How long will it take to reach \(100 \%\) of the goal?

CONSUMER EXPENDITURE The demand for a certain commodity is \(D(x)=-50 x+800 ;\) that is, \(x\) units of the commodity will be demanded by consumers when the price is \(p=D(x)\) dollars per unit. Total consumer expenditure \(E(x)\) is the amount of money consumers pay to buy \(x\) units of the commodity. a. Express consumer expenditure as a function of \(x\), and sketch the graph of \(E(x)\). b. Use the graph in part (a) to determine the level of production \(x\) at which consumer expenditure is largest. What price \(p\) corresponds to maximum consumer expenditure?

MICROBIOLOGY A spherical cell of radius \(r\) has volume \(V=\frac{4}{3} \pi r^{3}\) and surface area \(S=4 \pi r^{2}\). Express \(V\) as a function of \(S\). If \(S\) is doubled, what happens to \(V\) ?

MANUFACTURING EFFICIENCY A manufacturing firm has received an order to make 400,000 souvenir silver medals commemorating the anniversary of the landing of Apollo 11 on the moon. The firm owns several machines, each of which can produce 200 medals per hour. The cost of setting up the machines to produce the medals is \(\$ 80\) per machine, and the total operating cost is \(\$ 5.76\) per hour. Express the cost of producing the 400,000 medals as a function of the number of machines used. Draw the graph and estimate the number of machines the firm should use to minimize cost.

OPTIMAL SELLING PRICE A retailer can obtain digital cameras from the manufacturer at a cost of \(\$ 150\) apiece. The retailer has been selling the cameras at the price of \(\$ 340\) apiece, and at this price, consumers have been buying 40 cameras a month. The retailer is planning to lower the price to stimulate sales and estimates that for each \(\$ 5\) reduction in the price, 10 more cameras will be sold each month. Express the retailer's monthly profit from the sale of the cameras as a function of the selling price. Draw the graph, and estimate the optimal selling price.

PROPERTY TAX Khalil is trying to decide between two competing property tax propositions. With Proposition A, he will pay \(\$ 100\) plus \(8 \%\) of the assessed value of his home, while Proposition B requires a payment of \(\$ 1,900\) plus \(2 \%\) of the assessed value. Assuming Khalil's only consideration is to minimize his tax payment,develop a criterion based on the assessed value \(V\) of his home for deciding between the propositions.

INVENTORY ANALYSIS A businessman maintains inventory over a particular 30 -day month as follows: \(\begin{array}{ll}\text { days } 1-9 & 30 \text { units } \\ \text { days } 10-15 & 17 \text { units } \\ \text { days } 16-23 & 12 \text { units } \\\ \text { days } 24-30 & \text { steadily decreasing from } 12 \text { units } \\\ & \text { to } 0 \text { units }\end{array}\) Sketch the graph of the inventory as a function of time \(t\) (days). At what times is the graph discontinuous?

BREAK-EVEN ANALYSIS A manufacturer can sell a certain product for \(\$ 80\) per unit. Total cost consists of a fixed overhead of \(\$ 4,500\) plus production costs of \(\$ 50\) per unit. a. How many units must the manufacturer sell to break even? b. What is the manufacturer's profit or loss if 200 units are sold? c. How many units must the manufacturer sell to realize a profit of \(\$ 900\) ?

PRODUCTION MANAGEMENT During the summer, a group of students builds kayaks in a converted garage. The rental for the garage is \(\$ 1,500\) for the summer, and the materials needed to build a kayak cost \(\$ 125\). The kayaks can be sold for \(\$ 275\) apiece. a. How many kayaks must the students sell to break even? b. How many kayaks must the students sell to make a profit of at least \(\$ 1,000\) ?

LEARNING Some psychologists believe that when a person is asked to recall a set of facts, the rate at which the facts are recalled is proportional to the number of relevant facts in the subject's memory that have not yet been recalled. Express the recall rate as a function of the number of facts that have been recalled.

COST-EFFICIENT DESIGN A cable is to be run from a power plant on one side of a river 900 meters wide to a factory on the other side, 3,000 meters downstream. The cable will be run in a straight line from the power plant to some point \(P\) on the opposite bank and then along the bank to the factory. The cost of running the cable across the water is \(\$ 5\) per meter, while the cost over land is \(\$ 4\) per meter. Let \(x\) be the distance from \(P\) to the point directly across the river from the power plant. Express the cost of installing the cable as a function of \(x\).

MANUFACTURING COST A manufacturer is capable of producing 5,000 units per day. There is a fixed (overhead) cost of \(\$ 1,500\) per day and a variable cost of \(\$ 2\) per unit produced. a. Express the daily cost \(C(x)\) as a function of the number of units produced, and sketch the graph of \(C(x)\). b. Find the average daily \(\operatorname{cost} A C(x)\). What is the average daily cost of producing 3,000 units per day? c. Is \(C(x)\) continuous? If not, where do its discontinuities occur?

At what time between 3 P.M. and 4 P.M. will the minute hand coincide with the hour hand? [Hint: The hour hand moves \(\frac{1}{12}\) as fast as the minute hand.]

The radius of the earth is roughly 4,000 miles, and an object located \(x\) miles from the center of the earth weighs \(w(x)\) lb, where $$ w(x)= \begin{cases}A x & \text { if } x \leq 4,000 \\ \frac{B}{x^{2}} & \text { if } x>4,000\end{cases} $$ and \(A\) and \(B\) are positive constants. Assuming that \(w(x)\) is continuous for all \(x\), what must be true about \(A\) and \(B\) ? Sketch the graph of w(x).

In each of these cases, find the value of the constant \(A\) that makes the given function \(f(x)\) continuous for all \(x\). a. \(f(x)= \begin{cases}2 x+3 & \text { if } x<1 \\ A x-1 & \text { if } x \geq 1\end{cases}\) b. \(f(x)= \begin{cases}\frac{x^{2}-1}{x+1} & \text { if } x<-1 \\ A x^{2}+x-3 & \text { if } x \geq-1\end{cases}\)

The accompanying graph represents a function \(g(x)\) that oscillates more and more frequently as approaches 0 from either the right or the left but with decreasing magnitude. Does \(\lim _{x \rightarrow 0} g(x)\) exist? If so, what is its value? [Note: For students with experience in trigonometry, the function \(g(x)=|x| \sin (1 / x)\) behaves in this way.]

Graph \(f(x)=\frac{3 x^{2}-6 x+9}{x^{2}+x-2} .\) Determine the values of \(x\) where the function is undefined.

Graph \(y=\frac{21}{9} x-\frac{84}{35}\) and \(y=\frac{654}{279} x-\frac{54}{10}\) on the same set of coordinate axes using \([-10,10] 1\) by \([-10,10] 1\). Are the two lines parallel?