Chapter 1

Use f(x) to determine verbal, graphical and numerical representations. For the numerical representation use a table wish \(x=-2,-1,0,1,2\) Evaluate \(f(2).\) $$ f(x)=8 $$

Complete the following. (a) Find the domain and range of the relation. (b) Determine the maximum and minimum of the \(x\) -values and then of the y-values. (c) Label appropriate scales on the xy-axes. (d) Plot the relation. $$ ((-1.2,1.5),(1.0,0.5),(-0.3,1.1),(-0.8,-1.3)] $$

Evaluate the expression by hand. Write your result in scientific notation and standard form. $$ \left(8 \times 10^{-3}\right)\left(7 \times 10^{1}\right) $$

Compute the average rate of change of \(f\) from \(x_{1}\) to \(x_{2}\). Round your answer to two decimal places when appropriate. Interpret your result graphically. $$ f(x)=\sqrt{2 x-1}, x_{1}=1, \text { and } x_{2}=3 $$

Analyzing Real Data For the given data set complete the following. (a) Make a line graph of the data. Let this graph represent a function \(f\) (b) Decide whether \(f\) is linear or nonlinear. Median incomes of full-time female workers $$ \begin{array}{|c|c|c|c|c|}\hline \hline \text { Year } & 1970 & 1980 & 1990 & 2000 \\ \hline \text { Income } & \$ 5,440 & \$ 11,591 & \$ 20,591 & \$ 32,442 \end{array} $$

Use f(x) to determine verbal, graphical and numerical representations. For the numerical representation use a table wish \(x=-2,-1,0,1,2\) Evaluate \(f(2).\) $$ f(x)=5-x $$

Evaluate the expression by hand. Write your result in scientific notation and standard form. $$ \frac{6.3 \times 10^{-2}}{3 \times 10^{1}} $$

Compute the average rate of change of \(f\) from \(x_{1}\) to \(x_{2}\). Round your answer to two decimal places when appropriate. Interpret your result graphically. $$ f(x)=\sqrt[3]{x+1}, x_{1}=7, \text { and } x_{2}=26 $$

Analyzing Real Data For the given data set complete the following. (a) Make a line graph of the data. Let this graph represent a function \(f\) (b) Decide whether \(f\) is linear or nonlinear. Median incomes of full-time male workers $$ \begin{array}{|c|c|c|c|c|}\hline \hline \text { Year } & 1970 & 1980 & 1990 & 2000 \\ \hline \text { Income } & \$ 9,184 & \$ 19,173 & \$ 28,979 & \$ 37,435 \end{array} $$

Use f(x) to determine verbal, graphical and numerical representations. For the numerical representation use a table wish \(x=-2,-1,0,1,2\) Evaluate \(f(2).\) $$ f(x)=|x| $$

Evaluate the expression by hand. Write your result in scientific notation and standard form. $$ \frac{8.2 \times 10^{2}}{2 \times 10^{-2}} $$

Consumption The following table lists the number of cigarettes in billions consumed in the United States for selected years. $$ \begin{array}{|ccccc} \hline \text { Year } & 1900 & 1940 & 1980 & 2006 \\ \hline \text { Clgarettes } & 3 & 182 & 632 & 371 \\ \hline \end{array} $$ (a) Find the average rate of change during each time period. (b) Interpret the results.

The table lists the average wind speed in miles per hour at Myrtle Beach, South Carolina. The months are assigned the standard numbers. $$ \begin{array}{|r|c|c|c|c|c|c|}\hline \hline \text { Month } & 1 & 2 & 3 & 4 & 5 & 6 \\ \hline \text { Wind (mph) } & 7 & 8 & 8 & 8 & 7 & 7 \end{array} $$ $$ \begin{array}{|c|c|c|c|c|c|c|}\hline \hline \text { Month } & 7 & 8 & 9 & 10 & 11 & 12 \\ \hline \text { Wind (mph) } & 7 & 7 & 7 & 6 & 6 & 6 \end{array} $$ (a) Could these data be modeled exactly by a constant function? (b) Determine a continuous, constant function \(f\) that models these data approximately. (c) Graph \(f\) and the data.

Use f(x) to determine verbal, graphical and numerical representations. For the numerical representation use a table wish \(x=-2,-1,0,1,2\) Evaluate \(f(2).\) $$ f(x)=\sqrt{x+1} $$

Evaluate the expression by hand. Write your result in scientific notation and standard form. $$ \frac{4 \times 10^{-3}}{8 \times 10^{-1}} $$

Find the center and radius of the circle. $$ x^{2}+y^{2}=25 $$

The following table lists the annual average number of gallons of pure alcohol consumed by each person age 15 and older in the United States for selected years. $$ \begin{array}{rrrrr} \text { Year } & 1940 & 1960 & 1980 & 2000 \\ \hline \text { Alcohol } & 1.56 & 2.07 & 2.76 & 2.18 \end{array} $$ (a) Find the average rate of change during each \(20-\) year period. (b) Interpret the results.

The table lists the marriages per 1000 residents in Kentucky for selected years. $$ \begin{array}{|c|c|c|c|}\hline\hline \text { Year } & 2001 & 2002 & 2003 & 2004 \\ \hline \text { Rate } & 9.0 & 9.0 & 9.1 & 8.9 \end{array} $$ (a) Could these data be modeled exactly by a constant function? (b) Determine a constant function \(f\) that models these data approximately. (c) Graph \(f\) and the data.

Use f(x) to determine verbal, graphical and numerical representations. For the numerical representation use a table wish \(x=-2,-1,0,1,2\) Evaluate \(f(2).\) $$ f(x)=x^{2}-1 $$

Find the center and radius of the circle. $$ x^{2}+y^{2}=100 $$

Evaluate the expression by hand. Write your result in scientific notation and standard form. $$ \frac{2.4 \times 10^{-5}}{4.8 \times 10^{-7}} $$

The distance \(D\) in feet that an object has fallen after \(t\) seconds is given by \(D(t)=16 t^{2}\) (a) Bvaluate \(D(2)\) and \(D(4)\) (b) Calculate the average rate of change of \(D\) from 2 to \(4 .\) Interpret the result.

Sketch a graph that illustrates the motion of the person described. Let the \(x\) -axis represent time and the \(y\) axis represent distance from home. Be sure to label each axis. A person drives a car away from home for 2 hours at 50 miles per hour and then stops for 1 hour.

Use a calculator to approximate the expression. Write your result in scientific notation. $$ \frac{8.947 \times 10^{7}}{0.00095}\left(4.5 \times 10^{8}\right) $$

Find the center and radius of the circle. $$ x^{2}+y^{2}=7 $$

A cylindrical tank contains 100 gallons of water. A plug is pulled from the bottom of the tank and the amount of water in gallons remaining in the tank after \(x\) minutes is given by $$A(x)=100\left(1-\frac{x}{5}\right)^{2}$$ (a) Calculate the average rate of change of \(A\) from 1 to 1.5 and from 2 to \(2.5 .\) Interpret your results. (b) Are the two average rates of change the same or different? Explain why.

Sketch a graph that illustrates the motion of the person described. Let the \(x\) -axis represent time and the \(y\) axis represent distance from home. Be sure to label each axis. A person drives to a nearby park at 25 miles per hour for 1 hour, rests at the park for 2 hours, and then drives home at 50 miles per hour.

Use a calculator to approximate the expression. Write your result in scientific notation. $$ \left(9.87 \times 10^{6}\right)\left(34 \times 10^{11}\right) $$

Find the center and radius of the circle. $$ x^{2}+y^{2}=20 $$

Assume that each function is continuous. Do not use a graphing calculator. On the same coordinate axes, sketch the graphs of a constant function \(f\) and a nonlinear function \(g\) that intersect exactly twice.

Sketch a graph that illustrates the motion of the person described. Let the \(x\) -axis represent time and the \(y\) axis represent distance from home. Be sure to label each axis. A person walks away from home at 4 miles per hour for 1 hour and then turns around and walks home at the same speed.

Express a function \(f\) with the specified representation. Cost of Driving In 2008 the average cost of driving a new car was about 50 cents per mile. Give symbolic, graphical, and numerical representations of the cost in dollars of driving x miles. For the numerical representation use a table with x = 1, 2, 3, 4, 5, 6.

Use a calculator to approximate the expression. Write your result in scientific notation. $$ \left(\frac{101+23}{0.42}\right)^{2}+\sqrt{3.4 \times 10^{-2}} $$

Find the center and radius of the circle. $$ (x-2)^{2}+(y+3)^{2}=9 $$

Assume that each function is continuous. Do not use a graphing calculator. Sketch a graph of a linear function \(f\) that intersects a constant function \(g\) exactly once.

Express a function \(f\) with the specified representation. Counterfeit Money. It is estimated that nine out of every one million bills are counterfeit Give a numerical representation (table) of the predicted number of counterfeit bills in a sample of x million bills where x = 0, 1, 2, ... , 6.

Use a calculator to approximate the expression. Write your result in scientific notation. $$ \sqrt[3]{\left(2.5 \times 10^{-8}\right)+10^{-7}} $$

Find the center and radius of the circle. $$ (x+1)^{2}+(y-1)^{2}=16 $$

Assume that each function is continuous. Do not use a graphing calculator. Sketch a graph of a nonlinear function \(f\) that has only positive average rates of change.

Use a calculator to approximate the expression. Write your result in scientific notation. $$ \left(8.5 \times 10^{-5}\right)\left(-9.5 \times 10^{7}\right)^{2} $$

Find the center and radius of the circle. $$ x^{2}+(y+1)^{2}=100 $$

Assume that each function is continuous. Do not use a graphing calculator. Sketch a graph of a nonlinear function \(f\) that has only negative average rates of change.

Suppose you are given a graphical representation of a function \(f .\) Explain how you would determine whether \(f\) is constant, linear, or nonlinear. How would you determine the type if you were given a numerical or symbolic representation? Give examples.

Find the center and radius of the circle. $$ (x-5)^{2}+y^{2}=19 $$

Use a calculator to approximate the expression. Write your result in scientific notation. $$ \sqrt{\pi\left(4.56 \times 10^{4}\right)+\left(3.1 \times 10^{-2}\right)} $$

Assume that each function is continuous. Do not use a graphing calculator. Sketch a graph of a function that has only positive average rates of change for \(x \leq 0\) and only negative average rates of change for \(x \geq 0\)

Use a calculator to evaluate the expression. Round your result to the nearest thousandth. $$ \sqrt[3]{192} $$

Assume that each function is continuous. Do not use a graphing calculator. Sketch a graph of a function that has only positive average rates of change for \(x \geq 1\) and only negative average rates of change for \(x \leq 1\)

Use a calculator to evaluate the expression. Round your result to the nearest thousandth. $$ \sqrt{\left(32+\pi^{3}\right)} $$

Complete the following for the given \(f(x)\) (a) Find \(f(x+h)\) (b) Find the difference quotient of \(f\) and simplify. $$ f(x)=3 $$