Let $g(x)=\sqrt{x}$ for $x \geq 0$

a. Find the average rate of change of $g(x)$ with respect to $x$ owerthe intervals $[1,2],[1,1.5]$ and $[1,1+h]$.

b. Make a table of values of the average rate of change of $g$ with respect to $x$ over the interval $[1,1+h]$ for some values of $h$ approaching zero, say $h=0.1,0.01,0.001,0.0001,0.00001$ and $0.000001 .$

c. What does your table indicate is the rate of change of $g(x)$ with respect to $x$ at $x=1 ?$

d. Calculate the limit as $h$ approaches zero of the average rate of change of $g(x)$ with respect to $x$ over the interval $[1,1+h]$ .

## Discussion

## Video Transcript

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## Recommended Questions

The profits of a small company for each of the first five years of its operation are given in the following table: $$\begin{array}{lc} \hline \text { Year } & \text { Profit in } \$ 1000 \mathrm{s} \\\hline 2010 & 6 \\2011 & 27 \\2012 & 62 \\2013 & 111 \\2014 & 174 \\\hline\end{array}$$

a. Plot points representing the profit as a function of year, and join them by as smooth a curve as you can.

b. What is the average rate of increase of the profits between 2012 and $2014 ?$

c. Use your graph to estimate the rate at which the profits were changing in 2012

The profits of a small company for each of the first five years of its operation are given in the following table:

\begin{equation}

\begin{array}{ll}{\text { Year }} & {\text { Profit in } \$ 1000 \text { s }} \\ {2010} & {6} \\ {2011} & {6} \\ {2012} & {62} \\ {2013} & {62} \\ {2014} & {174} \\ \hline\end{array}

\end{equation}

\begin{equation}

\begin{array}{l}{\text { a. Plot points representing the profit as a function of year, and }} \\ {\text { join them by as smooth a curve as you can. }}\end{array}

\end{equation}

\begin{equation}

\begin{array}{l}{\text { b. What is the average rate of increase of the profits between }} \\ {2012 \text { and } 2014 ?} \\ {\text { c. Use your graph to estimate the rate at which the profits were }} \\ {\text { changing in } 2012 .}\end{array}

\end{equation}

A manufacturer determines that his profit and cost functions over one year are represented by the following graphs.

(GRAPH CAN'T COPY)

Business. Find the intervals on which profit is increasing. decreasing, and constant.

For the following exercises, use the graph in Figure 2.59, showing the profit, $y$ , in thousands of dollars, of a company in a given year, $x$ , where $x$ represents years since 1980.

In 2004, a school population was 1250. By 2012 the population had dropped to 875. Assume the population is changing linearly.

a. How much did the population drop between the year 2004 and 2012?

b. What is the average population decline per year?

c. Find an equation for the population, $P$, of the school $t$ years after 2004.

The scatterplot diagram above shows the profits of a start-up

company during its first year of business. If the company's profits continue to grow at the same rate as predicted by the line of best fit, which has been drawn in, which of the following will be closest to the company's monthly profit after it has been in business for a year and a half?

$$\begin{array}{l}{\text { (A) } \$ 40,000} \\ {\text { (B) } \$ 60,000} \\ {\text { (C) } \$ 80,000} \\ {\text { (D) } \$ 85,000}\end{array}$$

During an economic downturn the annual profits of a company dropped from $\$ 850,000$ in 2008 to $\$ 525,000$ in $2010 .$ Assume the exponential model $P(t)=P_{0} e^{t t}$ for the annual profit where $P$ is profit in thousands of dollars, and $t$ is time in years.

a. Find the exponential model for the annual profit.

b. Assuming the exponential model was applicable in the year 2012 , estimate the profit (to the nearest thousand dollars) for the year 2012

A manufacturer determines that his profit and cost functions over one year are represented by the following graphs.

(GRAPH CAN'T COPY)

Business. Find the intervals on which cost is increasing. decreasing, and constant.

Business Suppose you start a lawn-mowing business and make a profit of 400 dollars in the first year. Each year, your profit increases $5 \% .$ i.

a. Write a function that models your annual profit.

b. If you continue your business for $10 \mathrm{yr},$ what will your total profit be?

Use a graphing calculator and the following information.

A software company’s net profit for each year from 1993 to 1998 lead a financial analyst to model the company’s net profit by

$$P=6.84 t^{2}-3.76 t+9.29$$

where $P$ is the profit in millions of dollars and $t$ is the number of years since $1993 .$ In 1993 the net profit was approximately 9.29 million dollars $(t=0)$.

Use the graph to predict whether the net profit will reach 650 million dollars.

Del Monte Fruit Company recently released a new applesauce. By the end of its first year, profits on this product amounted to $\$ 30.000 .$ The anticipated profit for the end of the fourth year is $\$ 66,000$. The ratio of change in time to change in profit is constant. Let $x$ be years and $P$ be profit.

a. Write a linear function $P(x)$ that expresses profit as a function of time.

b. Use this function to predict the company's profit at the end of the seventh year.

c. Predict when the profit should reach $\$ 126,000$.

Profit Karla Harby Communications, a small company of science writers, found that its rate of profit (in thousands of dollars) after $t$ years of operation is given by

$$P^{\prime}(t)=(3 t+3)\left(t^{2}+2 t+2\right)^{1 / 3}$$

(a) Find the total profit in the first three years.

(b) Find the profit in the fourth year of operation.

(c) What is happening to the annual profit over the long run?

For the following exercises, use the graph in Figure 2.56 showing the profit, $y$, in thousands of dollars, of a company in a given year, $x$, where $x$ represents years since 1980.

Find the linear function $y,$ where $y$ depends on $x,$ the number of years since 1980 .

The graph given shows the revenue $R(t)$ and operating costs $C(t)$ of Space Travel Resources (STR), for the years 2000 to $2010 .$ Use the graph to find the

a. revenue in $2002: R(2)$

b. costs in $2008: C(8)$

c. years STR broke even: $R(t)=C(t)$

d. years costs exceeded revenue: $C(t) > R(t)$

e. years STR made a profit: $R(t) > C(t)$

f. For the year $2005,$ use function notation to write the profit equation for STR. What was their profit?

(GRAPH CAN'T COPY)

DESCRIBING PROFITS Management originally predicted that the profits from the sales of a new product would be approximated by the graph of the function $f$ shown. The actual profits are shown by the function $g$ along with a verbal description. Use the concepts of transformations of graphs to write $g$ in terms of $f$.

(a) The profits were only three-fourths as large as expected.

(b) The profits were consistently $\$10,000$ greater than predicted.

(c) There was a two-year delay in the introduction of the product. After sales began, profits grew as expected.

For the following exercises, use the graph in Figure 2.59, showing the profit, $y$ , in thousands of dollars, of a company in a given year, $x$ , where $x$ represents years since 1980.

Find the linear function $y,$ where $y$ depends on $x,$ the number of years since 1980 .

Use a graphing calculator and the following information.

A software company’s net profit for each year from 1993 to 1998 lead a financial analyst to model the company’s net profit by

$$P=6.84 t^{2}-3.76 t+9.29$$

where $P$ is the profit in millions of dollars and $t$ is the number of years since $1993 .$ In 1993 the net profit was approximately 9.29 million dollars $(t=0)$.

Use a graphing calculator to estimate how many years it will take for the company’s net profit to reach 475 million dollars according to the model.

Use the following graph, which shows the average price for regular gasoline at the beginning of January of each year.

CAN'T COPY THE GRAPH

What was the percent of decrease in the average price of a gallon of gasoline from January 2001 to January $2002 ?$

T-shirt profits A clothing company makes a profit of $\$ 10$ on its long-sleeved T-shirts and $\$ 5$ on its short-sleeved T-shirts. Assuming there is a $\$ 200$ setup cost, the profit on T-shirt sales is $z=10 x+5 y-200,$ where $x$ is the number of long-sleeved T-shirts sold and $y$ is the number of short-sleeved T-shirts sold. Assume $x$ and $y$ are nonnegative.

a. Graph the plane that gives the profit using the window $$[0,40] \times[0,40] \times[-400,400]$$

b. If $x=20$ and $y=10,$ is the profit positive or negative?

c. Describe the values of $x$ and $y$ for which the company breaks even (for which the profit is zero). Mark this set on your graph.

Examine the graph of expense and revenue.

a. What is the breakeven point?

b. If quantity $C$ is sold and $C < A,$ is there a profit or a loss? Explain.

c. If quantity $D$ is sold and $D > A,$ is there a profit or a loss? Explain.

d. The $y$ -intercept of the expense function is $Z$ . Interpret what the company is doing if it operates at the point $(0, Z)$ .

Estimating Profit An appliance manufacturer estimates that the profit $y$ (in dollars) generated by producing $x$ cooktops per month is given by the equation

$$

y=10 x+0.5 x^{2}-0.001 x^{3}-5000

$$

where $0 \leq x \leq 450$.

(a) Graph the equation.

(b) How many cooktops must be produced to begin generating a profit?

(c) For what range of values of $x$ is the company's profit greater than $\$ 15,000 ?$

Use a graphing calculator and the following information.

A software company’s net profit for each year from 1993 to 1998 lead a financial analyst to model the company’s net profit by

$$P=6.84 t^{2}-3.76 t+9.29$$

where $P$ is the profit in millions of dollars and $t$ is the number of years since $1993 .$ In 1993 the net profit was approximately 9.29 million dollars $(t=0)$.

Give the domain and range of the function for 1993 through 1998.

The graph given shows a large corporation's investment in research and development $R(t)$ over time, and the amount paid to investors as dividends $D(t),$ in billions of dollars. Use the graph to find thea. dividend payments in $2002: D(2)$

b. investment in $2006: R(6)$

c. years where $R(t)=D(t)$

d. years where $R(t) > D(t)$

e. years where $R(t) < D(t)$

f. Use function notation to write an equation for the total expenditures of the corporation in year $t$ What was the total for $2010 ?$

(GRAPH CAN'T COPY)

For the following exercises, use the graph in Figure $2.45,$ which shows the profit, $y,$ in thousands of dollars, of a company in a given year, $t,$ where $t$ represents the number of years since $1980 .$

Find the linear function $y,$ where $y$ depends on $t,$ the number of years since 1980 .

Profit Suppose that the total profit in hundreds of dollars from

selling $x$ items is given by

$$P(x)=2 x^{2}-5 x+6$$

Find the average rate of change of profit for the following changes in $x$ .

$$ \text { a. }4 \quad \text { b. } 2 \text { to } 3$$

c. Find and interpret the instantaneous rate of change of profit with respect to the number of items produced when $x=2$ . (This number is called the marginal profit at $x=2 . )$

d. Find the marginal profit at $x=4$

Use the following graph to answer the questions below. Let $Y_{1}=\operatorname{cost}$ function and $Y_{2}=$ revenue function.

a. Explain the significance of point $(D, B)$

b. Explain the significance of point $(E, A)$

c. Explain the significance of point $(F, C)$

d. Explain the significance of point $(G, 0)$

e. Explain the significance of point $(H, 0)$

f. Where do you think the maximum profit might occur?

Use the following graph, which shows the average price for regular gasoline at the beginning of January of each year.

CAN'T COPY THE GRAPH

What was the percent of increase in the average price of a gallon of gasoline from January 2002 to January $2008 ?$

For the following exercises, use the graph in Figure 2.59, showing the profit, $y$ , in thousands of dollars, of a company in a given year, $x$ , where $x$ represents years since 1980.

Draw a best-fit line for the plotted data.

Karla Harby Communications, a small company of science writers, found that its rate of profit (in thousands of dollars) after t years of operation is given by

$$P^{\prime}(t)=(3 t+3)\left(t^{2}+2 t+2\right)^{1 / 3}$$

a. Find the total profit in the first three years.

b. Find the profit in the fourth year of operation.

c. What is happening to the annual profit over the long run?

For the following exercises, consider this scenario: The profit of a company decreased steadily over a ten-year span. The following ordered pairs shows dollars and the number of units sold in hundreds and the profit in thousands of over the tenyear span, (number of units sold, profit) for specific recorded years:

$$(46,1,600), \quad(48,1,550), \quad(50,1,505), \quad(52,1,540), \quad(54,1,495).$$

Find to the nearest tenth and interpret the $y$-intercept.

The profit function, in thousands of dollars, for a company that makes graphing calculators is $\mathrm{P}(x)=-5 x^{2}+5,400 x-106,000$ where $x$ is the number of calculators sold in the millions.

a. Graph the profit function $\mathrm{P}(x)$

b. How many calculators must the company sell in order to make a profit?

Write an exponential growth model for the profit. A business had a $10,000 profit in 1990. Then the profit increased by 25% per year for the next 10 years.

For the following exercises, consider this scenario The profit of a company decreased steadily over a ten-year span. The following ordered pairs shows dollars and the number of units sold in hundreds and the profit in thousands of over the ten-year span, (number of units sold, profit) for specific recorded years: $(46,600),(48,550),(50,505),(52,540),(54,495)$

Find to the nearest tenth and interpret the $y$ -intercept.

Write an exponential growth model for the profit. A business had a $15,000 profit in 1990. Then the profit increased by 30% per year for the next 15 years.

Use the following graph, which shows the average price for regular gasoline at the beginning of January of each year.

CAN'T COPY THE GRAPH

What was the percent of increase in the average price of a gallon of gasoline from January 2000 to January $2010 ?$

For the following exercises, consider this scenario: The profit of a company decreased steadily over a ten-year span. The following ordered pairs show dollars and the number of units sold in hundreds and the profit in thousands of over the tenyear span (number of units sold, profit) for specific recorded years:

$$(46,250),(48,225),(50,205),(52,180),(54,165).$$

Use linear regression to determine a function $y$, where the profit in thousands of dollars depends on the number of units sold in hundreds .

The figure shows the graphs of the cost and revenue functions for a company that manufactures and sells small radios. Use the information in the figure to solve Exercises 67–72.

a. Use the formulas shown in the voice balloons to write the company's profit function, $P$ , from producing and selling radios.

b. Find the company's profit if $10,000$ radios are produced and sold.

SALES From 2003 through 2008, the sales $R_1$ (in thousands of dollars) for one of two restaurants owned by the same parent company can be modeled by

$R_1 = 480 - 8t - 0.8t^2$, $t = 3, 4, 5, 6, 7, 8$

where $t = 3$ represents 2003. During the same six-year period, the sales $R_2$ (in thousands of dollars) for the second restaurant can be modeled by

$R_2 = 254 - 0.78t$, $t = 3, 4, 5, 6, 7, 8$.

(a) Write a function $R_3$ that represents the total sales of the two restaurants owned by the same parent company.

(b) Use a graphing utility to graph $R_1$, $R_2$, and $R_3$ in the same viewing window.

For the following exercises, use the graph in Figure 2.56 showing the profit, $y$, in thousands of dollars, of a company in a given year, $x$, where $x$ represents years since 1980.

Find and interpret the $y$ -intercept.

A small business expects an income stream of $\$ 5000$ per year for a four-year period.

(a) Find the present value of the business if the annual interest rate, compounded continuously, is

(i) $3 \%$

(ii) $10 \%$

(b) In each case, find the value of the business at the end of the four-year period.

For the following exercises, consider this scenario The profit of a company decreased steadily over a ten-year span. The following ordered pairs shows dollars and the number of units sold in hundreds and the profit in thousands of over the ten-year span, (number of units sold, profit) for specific recorded years: $(46,600),(48,550),(50,505),(52,540),(54,495)$

Use linear regression to determine a function $P$ where the profit in thousands of dollars depends on

the number of units sold in hundreds.

The company you work for has been giving a $5 \%$ increase in salary every year. Your salary $S$ can be modeled by $S=38,000(1.05)^{t}$ where $t=0$ represents the year $2000 .$ Make a table showing your salary in $1995,2000,2005,$ and $2010 .$ Then graph the points given by this table and draw a smooth curve through these points.

AVERAGE SALARY The graph shows the average salaries for senior high school principals from 1996 through 2008. (Source: Educational Research Service)

(a) Use the slopes of the line segments to determine the time periods in which the average salary increased the greatest and the least.

(b) Find the slope of the line segment connecting the points for the years 1996 and 2008.

(c) Interpret the meaning of the slope in part (b) in the context of the problem.

Use the given linear equation to answer the questions. The equation $p=0.24 r-45,000$ describes the profit for a company, where $r$ represents revenue in dollars.

a. Find the profit if the revenue is $\$ 250,000$.

b. Find the revenue required to break even (the point at which profit is $\$ 0$ ).

c. Graph the equation with $r$ on the horizontal axis and $p$ on the vertical axis.

For the following exercises, use the graph in Figure $2.45,$ which shows the profit, $y,$ in thousands of dollars, of a company in a given year, $t,$ where $t$ represents the number of years since $1980 .$

Find and interpret the slope.

For the following exercises, consider this scenario: The profit of a company increased steadily over a ten-year span. The following ordered pairs show the number of units sold in hundreds and the profit in thousands of over the ten year span, (number of units sold, profit) for specific recorded years:

$$(46,250), \quad(48,305), \quad(50,350), \quad(52,390), \quad(54,410).$$

Use linear regression to determine a function $y$, where the profit in thousands of dollars depends on the number of units sold in hundreds .

Solve.

At the end of the first year of operation for a business, the business recorded a profit of $\$ 15,280 .$ Two years later the business recorded a profit of $\$ 80,450$. What was the percent of increase in profit?

A movie theater has fixed costs of $$ 5000$ per day and variable costs averaging $$ 6 per customer. The theater charges $$ 11 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.

HOW DO YOU SEE IT? The graph shows the profit $P$ (in thousands of dollars) of a company in terms of its advertising cost $x$ (in thousands of dollars).

\begin{equation}

\begin{array}{l}{\text { (a) Estimate the interval on which the profit is }} \\ {\text { increasing. }} \\ {\text { (b) Estimate the interval on which the profit is }} \\ {\text { decreasing. }}\end{array}

\end{equation}

\begin{equation}

\begin{array}{l}{\text { (c) Estimate the amount of money the company }} \\ {\text { should spend on advertising in order to yield a }} \\ {\text { maximum profit. }} \\ {\text { (d) The point of diminishing returns is the point at which }} \\ {\text { the rate of growth of the profit function begins to }} \\ {\text { decline. Estimate the point of diminishing returns. }}\end{array}

\end{equation}

Using the cost and revenue graphs in Figure $4.50,$ sketch the following functions. Label the points $q_{1}$ and $q_{2}$

(a) Total profit

(b) Marginal cost

(c) Marginal revenue

FIGURE CAN'T COPY

Suppose demand for a monopoly’s product falls so that its profit-maximizing price is below average variable cost. How much output should the firm supply? Hint: Draw the graph.

For the following exercises, use the graph in Figure $2.44,$ which shows the profit, $y,$ in thousands of dollars, of a company in a given year, $t,$ where $t$ represents the number of years since $1980 .$

Find the linear function $y,$ where $y$ depends on $t,$ the number of years since 1980 .

The figure shows the graphs of the cost and revenue functions for a company that manufactures and sells small radios. Use the information in the figure to solve Exercises 67–72.

a. Use the formulas shown in the voice balloons to write the company's profit function, $P$ , from producing and selling $x$ radios.

b. Find the company's profit if $20,000$ radios are produced and sold.

For the following exercises, use the graph in Figure $7,$ which shows the profit, $y$ , in thousands of dollars, of a company in a given year, $t,$ where t represents the number of years since 1980 .

Find and interpret the slope.

The business manager of a small manufacturing company finds that she can model the company’s annual growth as roughly exponential, but with cyclical fluctuations. She uses the function $S(t)=75(1.04)^{t}+4 \sin (\pi t / 3)$ to estimate sales (in millions of dollars), $t$ years after 2005.

(a) What are the company’s sales in 2005?

(b) What is the approximate annual growth rate?

(c) What does the model predict for sales in 2013?

(d) How many years are in each economic cycle for this company?

For the following exercises, use the graph in Figure 2.59, showing the profit, $y$ , in thousands of dollars, of a company in a given year, $x$ , where $x$ represents years since 1980.

Draw a scatter plot for the data provided in Table 2.23. Then determine whether the data appears to be linearly related.

$$\begin{array}{|c|c|c|c|c|c|}\hline 0 & {2} & {4} & {6} & {8} & {10} \\ \hline-450 & {-200} & {10} & {265} & {500} & {755} \\ \hline\end{array}$$

A company produces and sells shirts. The fixed costs are 7000 dollars and the variable costs are 5 dollars per shirt.

(a) Shirts are sold for 12 dollars each. Find cost and revenue as functions of the quantity of shirts, $q$

(b) The company is considering changing the selling price of the shirts. Demand is $q=2000-40 p$ where $p$ is price in dollars and $q$ is the number of shirts. What quantity is sold at the current price of $$ 12 ?$ What profit is realized at this price?

(c) Use the demand equation to write cost and revenue as functions of the price, $p .$ Then write profit as a function of price.

(d) Graph profit against price. Find the price that maximizes profits. What is this profit?

For the following exercises, consider this scenario: The profit of a company decreased steadily over a ten-year span. The following ordered pairs show dollars and the number of units sold in hundreds and the profit in thousands of over the tenyear span (number of units sold, profit) for specific recorded years:

$$(46,250),(48,225),(50,205),(52,180),(54,165).$$

Predict when the profit will dip below the $\$ 25,000$ threshold.

For the following exercises, use the graph in Figure 2.59, showing the profit, $y$ , in thousands of dollars, of a company in a given year, $x$ , where $x$ represents years since 1980.

Find and interpret the $y$-intercept.

A company produces two different tables, a round top and a square top. If

$r$ represents the number of round-top tables and $s$ represents the number of square-top tables, then $145 r+215 s+100$ describes the revenue from the sales of the two types of table. The polynomial $110 r+140 s+345$ describes the cost of producing the two types of table.

a. Write an expression in simplest form for the net.

b. In one month, the company sells 120 round-top tables and 106 square-top tables. Find the net profit or loss.

SALES The graph shows the sales (in billions of dollars) for Apple Inc. for the years 2001 through 2007. (Source: Apple Inc.)

(a) Use the slopes of the line segments to determine the years in which the sales showed the greatest increase and the least increase.

(b) Find the slope of the line segment connecting the points for the years 2001 and 2007.

(c) Interpret the meaning of the slope in part (b) in the context of the problem.

Examine each of the graphs in Exercises $2-5 .$ In each case, the blue graph represents the expense function and the black graph represents the revenue function. Describe the profit situation in terms of the expense and revenue functions.

(Graph Cant Copy)

Examine each of the graphs in Exercises $2-5 .$ In each case, the blue graph represents the expense function and the black graph represents the revenue function. Describe the profit situation in terms of the expense and revenue functions.

(Graph Cant Copy)

Examine each of the graphs in Exercises $2-5 .$ In each case, the blue graph represents the expense function and the black graph represents the revenue function. Describe the profit situation in terms of the expense and revenue functions.

(Graph Cant Copy)

(Graph Cant Copy)

[T] A profit is earned when revenue exceeds cost. Suppose the profit function for a skateboard manufacturer is given by $P(x)=30 x-0.3 x^{2}-250,$ where $x$ is the number of skateboards sold.

a. Find the exact profit from the sale of the thirtieth skateboard.

b. Find the marginal profit function and use it to estimate the profit from the sale of the thirtieth skateboard.

Based on your answers to the WipeOut Skic company in Exercise $7.3,$ now imagine a situation where the firm produces a quantity of 5 units that it sells for a price of $\$ 25$ each.

a. What will be the company's profits or losses?

b. How can you tell at a glance whether the company is making or losing money at this price by looking at average cost?

c. At the given quantity and price, is the marginal unit produced adding to profits?

Figure 1.55 shows cost and revenue for a company.

(a) Approximately what quantity does this company have to produce to make a profit?

(b) Estimate the profit generated by 600 units.

(GRAPH CANNOT COPY)

A company producing jigsaw puzzles has fixed costs of 6000 dollars and variable costs of 2 dollars per puzzle. The company sells the puzzles for 5 dollars each.

(a) Find formulas for the cost function, the revenue function, and the profit function.

(b) Sketch a graph of $R(q)$ and $C(q)$ on the same axes. What is the break-even point, $q_{0},$ for the company?