Chapter 1

For each of the functions, mark and label the amplitude, period, average value, and horizontal shift. \(f(x)=5 \sin (2 x-1)+3\)

Determine whether the given pair of functions can be combined into the required function. If so, then a. draw an input/output diagram for the new function. b. write a statement for the new function, complete with function notation and input and output units and descriptions. Construct a function for the number of business calculus students given functions \(m\) and \(f\) where \(m(t)\) is the number of men taking business calculus in year \(t\) and \(f(t)\) is the number of women taking business calculus in year \(t\).

Simple Interest Calculate the total amount due after two years on a loan of \(\$ 1500\) with a simple interest charge of \(7 \%\).

For Activities 1 through \(6, \quad\) for each linear model a. give the slope of the line defined by the equation. b. write the rate of change of the function in a sentence of interpretation. c. evaluate and give a sentence of interpretation for \(f(0)\). The cost to rent a newly released movie is \(f(x)=0.3 x+5\) dollars, where \(x\) is the number of years since 2010 .

For each of the functions, mark and label the amplitude, period, average value, and horizontal shift. \(f(x)=0.1 \sin (4 x-2)-0.5\)

Determine whether the given pair of functions can be combined into the required function. If so, then a. draw an input/output diagram for the new function. b. write a statement for the new function, complete with function notation and input and output units and descriptions. Construct a function for the total number of household pets given functions \(h\) and \(p\) where \(h(x)\) is the number of households with income \(x\) dollars and \(p(x)\) is the average number of pets per household when the household income is \(x\) dollars.

Simple Interest Calculate the total amount due after four years on a loan of \(\$ 3500\) with \(4 \%\) simple interest.

For Activities 1 through \(6, \quad\) for each linear model a. give the slope of the line defined by the equation. b. write the rate of change of the function in a sentence of interpretation. c. evaluate and give a sentence of interpretation for \(f(0)\). The number of people of age \(x\) years in a certain country is \(f(x)=-0.5 x+3.2\) million people.

For each of the functions, mark and label the amplitude, period, average value, and horizontal shift. \(s(t)=3 \sin t-4\)

Compound Interest To offset college expenses, at the beginning of your freshman year you obtain a nonsubsidized student loan for \(\$ 15,000\). Interest on this loan accrues at a rate of \(4.15 \%\) compounded monthly. However, you do not have to make any payments against either the principal or the interest until after you graduate. a. Write a model giving the total amount you will owe on this loan after \(t\) years in college. b. What is the APR? c. What is the APY?

For Activities 1 through \(6, \quad\) for each linear model a. give the slope of the line defined by the equation. b. write the rate of change of the function in a sentence of interpretation. c. evaluate and give a sentence of interpretation for \(f(0)\). The profit is \(f(x)=2 x-4.5\) thousand dollars when \(x\) hundred units are sold. The profit is \(f(x)=2 x-4.5\) thousand dollars when \(x\) hundred units are sold.

For each of the functions, mark and label the amplitude, period, average value, and horizontal shift. \(j(u)=7 \sin (2 u+\pi)-6\)

For Activities 1 through 4 , a. Identify each graph as either increasing or decreasing. b. Identify the type of concavity for each graph. c. Match each graph with its equation. $$ \begin{array}{l} f(x)=2.4-7 \ln x \\ g(x)=2+2 \ln x \end{array} $$

For Activities 1 through \(6, \quad\) for each linear model a. give the slope of the line defined by the equation. b. write the rate of change of the function in a sentence of interpretation. c. evaluate and give a sentence of interpretation for \(f(0)\). The quantity of tomatoes harvested is \(f(x)=5 x+6\) hundred pounds when \(x\) inches of rain fall.

For each of the functions, state the amplitude, period, average value, and horizontal shift. \(\quad p(x)=\sin (2.2 x+0.4)+0.7\)

The total cost for producing 1000 units of a commodity is \(\$ 4.2\) million, and the revenue generated by the sale of 1000 units is \(\$ 5.3\) million. a. What is the profit on 1000 units of the commodity? b. Assuming \(C(q)\) represents total cost and \(R(q)\) represents revenue for the production and sale of \(q\) units of a commodity, write an expression for profit.

Credit Card Balance Your credit card statement indicates the interest charged is \(12 \%\) compounded monthly on the outstanding balance. a. What is the nominal rate (APR)? b. What is the effective rate of interest (APY)?

a. indicate whether the function describes exponential growth or decay. b. give the constant percentage change. $$ f(x)=72\left(1.05^{x}\right) $$

For Activities 1 through \(6, \quad\) for each linear model a. give the slope of the line defined by the equation. b. write the rate of change of the function in a sentence of interpretation. c. evaluate and give a sentence of interpretation for \(f(0)\). The production of a coated wire is \(f(x)=100 x\) feet, wh \(x\) dollars is the amount spent on raw materials.

Test Scores A student's raw score on a spelling test with 20 evenly weighted questions can be expressed by \(g(n)=5 n\) when she spells \(n\) words correctly.

For each of the functions, state the amplitude, period, average value, and horizontal shift. \(f(x)=6.1 \sin (6.3 x+0.2)+10.4\)

The revenue generated by the sale of 5000 units of a commodity is \(\$ 400\) thousand dollars, and the average cost of producing 5000 units is \(\$ 20\) per unit. a. What is the profit on 5000 units of the commodity? b. Assuming \(R(q)\) represents revenue and \(\bar{C}(q)\) represents the average cost for the production and sale of \(q\) units of a commodity, write an expression for profit.

Certificate of Deposit A CD is bought for \(\$ 2500\) and held 3 years until maturity. What is the furure value of the \(\mathrm{CD}\) at the end of the 3 years if it earns interest compounded quarterly at a nominal rate of \(6.6 \% ?\)

a. indicate whether the function describes exponential growth or decay. b. give the constant percentage change. $$ K(r)=33\left(0.92^{\prime}\right) $$

For Activities 1 through \(6, \quad\) for each linear model a. give the slope of the line defined by the equation. b. write the rate of change of the function in a sentence of interpretation. c. evaluate and give a sentence of interpretation for \(f(0)\). Under overcrowding conditions on an assembly line, productivity is \(f(x)=1700-20 x\) units when labor is \(x\) workers over capacity.

For each of the functions, state the amplitude, period, average value, and horizontal shift. \(g(x)=3.62 \sin (0.22 x+4.81)+7.32\)

A company posted costs of 72 billion euros and a profit of 129 billion euros during the same quarter. a. What was the company's revenue during that quarter? b. Assuming \(C(t)\) represents total cost and \(P(t)\) represents profit during the \(t\) h quarter, write an expression for revenue.

Doubling Time How long would it take an investment to double under each of the following conditions? a. Interest is \(6.3 \%\) compounded monthly. b. Interest is \(8 \%\) compounded continuously.

a. indicate whether the function describes exponential growth or decay. b. give the constant percentage change. $$ y(x)=16.2\left(0.87^{x}\right) $$

For Activities 7 through \(12,\) write a linear model for the given rate of change and initial output value. The cost to produce plastic toys increases by 30 cents per toy produced. The fixed cost is 50 dollars.

For each of the functions, state the amplitude, period, average value, and horizontal shift. \(\quad p(x)=235 \sin (300 x+100)-65\)

A company receives \(\$ 2.9\) million for each ship it sells and can build the ships for \(\$ 0.2\) million each. a. What is the company's revenue from building and selling a ship? b. Assuming \(\bar{C}(x)\) represents average cost and \(\bar{P}(x)\) represents average profit when \(x\) ships are built and sold, write an expression for the revenue from the building and selling of \(x\) ships.

For Activities 7 through \(12,\) write a linear model for the given rate of change and initial output value. The population of a county was 175 thousand in 2008 and has continued to increase by 2.5 thousand people per year.

Doubling Time How long would it take an investment to double under each of the following conditions? a. Interest is \(4.3 \%\) compounded semi-annually. b. Interest is \(5 \%\) compounded daily (use 365 days).

a. indicate whether the function describes exponential growth or decay. b. give the constant percentage change. $$ A(t)=0.57\left(1.035^{t}\right) $$

Geometry The equation \(y=\pm \sqrt{36-x^{2}}\) is the \(y\) -coordinate for a point with \(x\) -coordinate between -6 and 6 on a circle with radius \(6 \mathrm{~cm}\) centered at the origin.

For each of the functions, state the amplitude, period, average value, and horizontal shift. \(f(x)=\sin (\pi x-2)\)

A company posted a net loss of \(\$ 3\) billion during the 3rd quarter. During the same quarter, the company's revenue was \(\$ 5\) billion. a. What was the company's cost in the 3 rd quarter? b. Assuming \(R(t)\) represents revenue and \(P(t)\) represents profit during the \(t\) th quarter, write an expression for cost.

For Activities 7 through \(12,\) write a linear model for the given rate of change and initial output value. During the first snowfall of the year, snow fell at a rate of 0.25 inch per hour. Three inches had fallen by noon. The snow stopped just before \(3: 30 \mathrm{PM}\).

Investment You have \(\$ 1000\) to invest, and you have two options: Option A: \(4.725 \%\) compounded semiannully Option B: \(4.675 \%\) compounded continuously. a. Calculate the annual percentage yield for each option. Which is the better option? b. Calculate the future value of each investment after 2 years and after 5 years. Does your choice of option depend on the number of years you leave the money invested?

In 1990 , there were 3 women among the CEOs of Fortune 500 companies. In \(2009,\) this number had risen to 15 women. (Source: H. Ibarra and Hansen, M., Women CEOs: Why So Few?, The Conversation-Harvard Business Review, Dec 21,2009\()\) a. What was the change in the number of Fortune 500 women CEOs between 1990 and \(2009 ?\) b. What was the percentage change in the number of Fortune 500 women CEOs between 1990 and \(2009 ?\)

The number of U.S. farms with milk cows can be modeled as $$ f(x)=45.183\left(0.831^{x}\right)+60 \text { thousand farms } $$ where \(x\) is the number of years since \(2000,\) based on data for years between 2001 and 2007 . (Source: Based on data from Statistical Abstract, 2007 and 2008\()\). a. Were the number of farms with milk cows increasing or decreasing between 2001 and \(2007 ?\) b. What is the concavity of the function on the interval \(1 \leq x \leq 7 ?\)

For each of the functions, state the amplitude, period, average value, and horizontal shift. \(\quad g(x)=\sin (x-\pi)\)

It costs a company \(\$ 19.50\) to produce 150 glass bottles. a. What is the average cost of production of a glass bottle? b. Assuming \(C(q)\) represents total cost of producing \(q\) bottles, write an expression for average cost.

For Activities 7 through \(12,\) write a linear model for the given rate of change and initial output value. The value of an antique plate increased by \(\$ 10\) per year from an initial value of \(\$ 50\) in 2004.

Investment You have \(\$ 5000\) to invest, and you have two options: Option A: \(4.9 \%\) compounded monthly Option B: \(4.8 \%\) compounded continuously. a. Calculate the annual percentage yield for each option. Which is the better option? b. By how much would the two investments differ after 3.5 years?

The U.S. labor force is considered to be the civilian, noninstitutional population 16 years old and over. In \(2008,59.5 \%\) of women in the labor force were actively employed. The U.S. Bureau of Labor Statistics predicts that by \(2016,\) this participation rate will be \(59.2 \%\). a. What is the predicted change in the percentage of women who are actively employed between 2008 and \(2016 ?\) b. What is the predicted percentage change in the percentage of women who are actively employed between 2008 and \(2016 ?\)

The percentage of people in the United States who earn at least \(t\) thousand dollars, \(25 \leq t \leq 150,\) can be modeled as $$ p(t)=119.931\left(0.982^{t}\right) \text { percent. } $$ a. Is \(p\) increasing or decreasing on the interval \(25 \leq t \leq 150 ?\) b. What is the concavity of \(p\) on the interval \(25 \leq t \leq 150 ?\)

For each of the functions, state the amplitude, period, average value, and horizontal shift. \(\quad y(x)=\sin x\)

Piroxicam Concentration The table gives estimated concentrations (in micrograms per milliliter) of the drug piroxicam taken in \(20 \mathrm{mg}\) doses once a day. Piroxicam Concentration in the Bloodstream $$ \begin{array}{|c|c|} \hline \text { Days } & \begin{array}{c} \text { Concentration } \\ (\mu \mathrm{g} / \mathrm{ml}) \end{array} \\ \hline 1 & 1.5 \\ \hline 3 & 3.2 \\ \hline 5 & 4.5 \\ \hline 7 & 5.5 \\ \hline 9 & 6.2 \\ \hline 11 & 6.5 \\ \hline 13 & 6.9 \\ \hline 15 & 7.3 \\ \hline 17 & 7.5 \\ \hline \end{array} $$ a. Find a log model for the data. b. Express the end behavior of the model by using limits. c. Estimate the concentration of the drug after 2 days of piroxicam doses.