Chapter 1

Find the doubling time of a quantity that is increasing by \(7 \%\) per year.

Solve for \(t\) using natural logarithms. $$10=e^{t}$$

A demand curve is given by \(75 p+50 q=300\), where \(p\) is the price of the product, in dollars, and \(q\) is the quantity demanded at that price. Find \(p\) - and \(q\) -intercepts and interpret them in terms of consumer demand.

An air-freshener starts with 30 grams and evaporates. In each of the following cases, write a formula for the quantity, \(Q\) grams, of air-freshener remaining \(t\) days after the start and sketch a graph of the function. The decrease is: (a) 2 grams a day (b) \(12 \%\) a day

Find the relative, or percent, change. \(R\) changes from 50 to 47

For the functions, find \(f(5)\). $$ f(x)=2 x+3 $$

Find an equation for the line that passes through the given points. $$ (-2,1) \text { and }(2,3) $$

Sketch graphs of the functions. What are their amplitudes and periods? $$y=4 \cos \left(\frac{1}{2} t\right)$$

Determine whether or not the function is a power function. If it is a power function, write it in the form \(y=k x^{p}\) and give the values of \(k\) and \(p\). $$y=\frac{2 x^{2}}{10}$$

Find the following: (a) \(f(g(x))\) (b) \(g(f(x))\) (c) \(f(f(x))\) $$f(x)=2 x^{2}\( and \)g(x)=x+3$$

If $$\$ 12,000$$ is deposited in an account paying \(8 \%\) interest per year, compounded continuously, how long will it take for the balance to reach \(\$ 20,000\) ?

Solve for \(t\) using natural logarithms. $$5=2 e^{t}$$

An amusement park charges an admission fee of $$\$ 7$$ per person as well as an additional $$\$ 1.50$$ for each ride. (a) For one visitor, find the park's total revenue \(R(n)\) as a function of the number of rides, \(n\), taken. (b) Find \(R(2)\) and \(R(8)\) and interpret your answers in terms of amusement park fees.

World population is approximately \(P=6.4(1.0126)^{t}\), with \(P\) in billions and \(t\) in years since 2004 . (a) What is the yearly percent rate of growth of the world population? (b) What was the world population in \(2004 ?\) What does this model predict for the world population in 2010 ? (c) Use part (b) to find the average rate of change of the world population between 2004 and 2010 .

Find the relative, or percent, change. \(W\) changes from \(0.3\) to \(0.05\)

For the functions, find \(f(5)\). $$ f(x)=10 x-x^{2} $$

Find an equation for the line that passes through the given points. $$ (4,5) \text { and }(2,-1) $$

Sketch graphs of the functions. What are their amplitudes and periods? $$y=5-\sin 2 t$$

Determine whether or not the function is a power function. If it is a power function, write it in the form \(y=k x^{p}\) and give the values of \(k\) and \(p\). $$y=\frac{8}{x}$$

Find the following: (a) \(f(g(x))\) (b) \(g(f(x))\) (c) \(f(f(x))\) $$f(x)=2 x+3\( and \)g(x)=5 x^{2}$$

You want to invest money for your child's education in a certificate of deposit (CD). You want it to be worth $$\$ 12,000$$ in 10 years. How much should you invest if the CD pays interest at a \(9 \%\) annual rate compounded (a) Annually? (b) Continuously?

Solve for \(t\) using natural logarithms. $$e^{3 t}=100$$

A company that makes Adirondack chairs has fixed costs of $$\$ 5000$$ and variable costs of $$\$ 30$$ per chair. The company sells the chairs for $$\$ 50$$ each. (a) Find formulas for the cost and revenue functions. (b) Find the marginal cost and marginal revenue. (c) Graph the cost and the revenue functions on the same axes. (d) Find the break-even point.

A \(50 \mathrm{mg}\) dose of quinine is given to a patient to prevent malaria. Quinine leaves the body at a rate of \(6 \%\) per hour. (a) Find a formula for the amount, \(A\) (in \(\mathrm{mg}\) ), of quinine in the body \(t\) hours after the dose is given. (b) How much quinine is in the body after 24 hours? (c) Graph \(A\) as a function of \(t\). (d) Use the graph to estimate when \(5 \mathrm{mg}\) of quinine remains.

Table \(1.10\) gives values of a function \(w=f(t) .\) Is this function increasing or decreasing? Is the graph of this function concave up or concave down? $$ \begin{array}{l} \text { Table } 1.10\\\ \begin{array}{c|c|c|c|c|c|c|c} \hline t & 0 & 4 & 8 & 12 & 16 & 20 & 24 \\ \hline w & 100 & 58 & 32 & 24 & 20 & 18 & 17 \\ \hline \end{array} \end{array} $$

Determine whether or not the function is a power function. If it is a power function, write it in the form \(y=k x^{p}\) and give the values of \(k\) and \(p\). $$y=(5 x)^{3}$$

Find the following: (a) \(f(g(x))\) (b) \(g(f(x))\) (c) \(f(f(x))\) $$f(x)=x^{2}+1\( and \)g(x)=\ln x$$

From October 2002 to October 2006 the number \(N(t)\) of Wikipedia articles was approximated by \(N(t)=\) \(N_{0} e^{t / 500}\), where \(t\) is the number of days after October 1,2002 . Find the doubling time for the number of Wikipedia articles during this period.

Solve for \(t\) using natural logarithms. $$10=6 e^{0.5 t}$$

A photocopying company has two different price lists. The first price list is $$\$ 100$$ plus 3 cents per copy; the second price list is $$\$ 200$$ plus 2 cents per copy. (a) For each price list, find the total cost as a function of the number of copies needed. (b) Determine which price list is cheaper for 5000 copies. (c) For what number of copies do both price lists charge the same amount?

The Hershey Company is the largest US producer of chocolate. In 2008 , annual net sales were \(5.1\) billion dollars and were increasing at a continuous rate of \(4.3 \%\) per year. \({ }^{52}\) (a) Write a formula for annual net sales, \(S\), as a function of time, \(t\), in years since 2008 . (b) Estimate annual net sales in 2015 . (c) Use a graph or trial and error to estimate the year in which annual net sales are expected to pass 8 billion dollars.

A cup of coffee contains \(100 \mathrm{mg}\) of caffeine, which leaves the body at a continuous rate of \(17 \%\) per hour. (a) Write a formula for the amount, \(A \mathrm{mg}\), of caffeine in the body \(t\) hours after drinking a cup of coffee. (b) Graph the function from part (a). Use the graph to estimate the half-life of caffeine. (c) Use logarithms to find the half-life of caffeine.

Solve for \(t\) using natural logarithms. $$40=100 e^{-0.03 t}$$

A company has cost function \(C(q)=4000+2 q\) dollars and revenue function \(R(q)=10 q\) dollars. (a) What are the fixed costs for the company? (b) What is the marginal cost? (c) What price is the company charging for its product? (d) Graph \(C(q)\) and \(R(q)\) on the same axes and label the break-even point, \(q_{0}\). Explain how you know the company makes a profit if the quantity produced is greater than \(q_{0}\). (e) Find the break-even point \(q_{0}\).

The consumer price index (CPI) for a given year is the amount of money in that year that has the same purchasing power as $$\$ 100$$ in 1983 . At the start of 2009 , the CPI was 211 . Write a formula for the CPI as a function of \(t\), years after 2009 , assuming that the CPI increases by \(2.8 \%\) every year.

Graph a function \(f(x)\) which is increasing everywhere and concave up for negative \(x\) and concave down for positive \(x\).

A city's population was 30,700 in the year 2000 and is growing by 850 people a year. (a) Give a formula for the city's population, \(P\), as a function of the number of years, \(t\), since 2000 . (b) What is the population predicted to be in 2010 ? (c) When is the population expected to reach 45,000 ?

For the functions, find \(f(5)\). $$ \begin{array}{c|c|c|c|c|c|c|c|c} \hline x & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 \\ \hline f(x) & 2.3 & 2.8 & 3.2 & 3.7 & 4.1 & 5.0 & 5.6 & 6.2 \\ \hline \end{array} $$

Determine whether or not the function is a power function. If it is a power function, write it in the form \(y=k x^{p}\) and give the values of \(k\) and \(p\). $$y=\frac{x}{5}$$

A population, currently 200 , is growing at \(5 \%\) per year. (a) Write a formula for the population, \(P\), as a function of time, \(t\), years in the future. (b) Graph \(P\) against \(\bar{t}\). (c) Estimate the population 10 years from now. (d) Use the graph to estimate the doubling time of the population.

Solve for \(t\) using natural logarithms. $$a=b^{t}$$

Values of a linear cost function are in Table \(1.25 .\) What are the fixed costs and the marginal cost? Find a formula for the cost function. $$ \begin{array}{l} \text { Table } 1.25\\\ \begin{array}{c|c|c|c|c|c} \hline q & 0 & 5 & 10 & 15 & 20 \\ \hline C(q) & 5000 & 5020 & 5040 & 5060 & 5080 \\ \hline \end{array} \end{array} $$

During the 1980 s, Costa Rica had the highest deforestation rate in the world, at \(2.9 \%\) per year. (This is the rate at which land covered by forests is shrinking.) Assuming the rate continues, what percent of the land in Costa Rica covered by forests in 1980 will be forested in 2015 ?

Find the average rate of change of \(f(x)=2 x^{2}\) between \(x=1\) and \(x=3\).

A cell phone company charges a monthly fee of $$\$ 25$$ plus $$\$ 0.05$$ per minute. Find a formula for the monthly charge, \(C\), in dollars, as a function of the number of minutes, \(m .\) the phone is used during the month.

Let \(y=f(x)=x^{2}+2\). (a) Find the value of \(y\) when \(x\) is zero. (b) What is \(f(3)\) ? (c) What values of \(x\) give \(y\) a value of 11 ? (d) Are there any values of \(x\) that give \(y\) a value of 1 ?

Average daily high temperatures in Ottawa, the capital of Canada, range from a low of \(-6^{\circ}\) Celsius on January 1 to a high of \(26^{\circ}\) Celsius on July 1 six months later. See Figure \(1.102 .\) Find a formula for \(H\), the average daily high temperature in Ottawa in, \({ }^{\circ} \mathrm{C}\), as a function of \(t\), the number of months since January \(1 .\)

Write a formula representing the function. The strength, \(S\), of a beam is proportional to the square of its thickness, \(h\).

Use the variable \(u\) for the inside function to express eac of the following as a composite function: (a) \(y=2^{3 x-1}\) (b) \(P=\sqrt{5 t^{2}+10}\) (c) \(w=2 \ln (3 r+4)\)

A movie theater has fixed costs of $$\$ 5000$$ per day and variable costs averaging $$\$ 2$$ per customer. The theater charges $$\$ 7$$ per ticket. (a) How many customers per day does the theater need in order to make a profit? (b) Find the cost and revenue functions and graph them on the same axes. Mark the break-even point.