Chapter 10

The demand function for a product is given by \(p=5000\left(1-\frac{4}{4+e^{-0.002 x}}\right)\) where \(p\) is the price per unit and \(x\) is the number of units sold. Find the numbers of units sold for prices of (a) \(p=\$ 200\) and (b) \(p=\$ 800\).

In Exercises, find \(d x / d p\) for the demand function. Interpret this rate of change when the price is \(\$ 10\). $$ x=\frac{500}{\ln \left(p^{2}+1\right)} $$

The demand function for a product is given by \(p=10,000\left(1-\frac{3}{3+e^{-0.001 x}}\right)\) where \(p\) is the price per unit and \(x\) is the number of units sold. Find the numbers of units sold for prices of (a) \(p=\$ 500\) and (b) \(p=\$ 1500\).

The population \(P\) (in thousands) of Orlando, Florida from 1980 through 2005 can be modeled by \(P=131 e^{0.019 r}\) where \(t=0\) corresponds to \(1980 .\) (Source: U.S. Census Bureau) (a) According to this model, what was the population of Orlando in 2005 ? (b) According to this model, in what year will Orlando have a population of \(300,000 ?\)

The population \(P\) (in thousands) of Houston, Texas from 1980 through 2005 can be modeled by \(P=1576 e^{0.01 t}\), where \(t=0\) corresponds to 1980 . (a) According to this model, what was the population of Houston in 2005 ? (b) According to this model, in what year will Houston have a population of \(2,500,000 ?\)

The cost of producing \(x\) units of a product is modeled by \(C=500+300 x-300 \ln x, \quad x \geq 1\) (a) Find the average cost function \(\bar{C}\). (b) Analytically find the minimum average cost. Use a graphing utility to confirm your result.

In Exercises \(83-86\), you are given the ratio of carbon atoms in a fossil. Use the information to estimate the age of the fossil. In living organic material, the ratio of radioactive carbon isotopes to the total number of carbon atoms is about 1 to \(10^{12}\). (See Example 2 in Section 10.1.) When organic material dies, its radioactive carbon isotopes begin to decay, with a half- life of about 5715 years. So, the ratio \(R\) of carbon isotopes to carbon- 14 atoms is modeled by \(R=10^{-12}\left(\frac{1}{2}\right)^{t / 5715}\), where \(t\) is the time (in years) and \(t=0\) represents the time when the organic material died. $$ R=0.32 \times 10^{-12} $$

The cost of producing \(x\) units of a product is modeled by \(C=100+25 x-120 \ln x, \quad x \geq 1\) (a) Find the average cost function \(\bar{C}\). (b) Analytically find the minimum average cost. Use a graphing utility to confirm your result.

In Exercises \(83-86\), you are given the ratio of carbon atoms in a fossil. Use the information to estimate the age of the fossil. In living organic material, the ratio of radioactive carbon isotopes to the total number of carbon atoms is about 1 to \(10^{12}\). (See Example 2 in Section 10.1.) When organic material dies, its radioactive carbon isotopes begin to decay, with a half- life of about 5715 years. So, the ratio \(R\) of carbon isotopes to carbon- 14 atoms is modeled by \(R=10^{-12}\left(\frac{1}{2}\right)^{t / 5715}\), where \(t\) is the time (in years) and \(t=0\) represents the time when the organic material died. $$ R=0.27 \times 10^{-12} $$

The retail sales \(S\) (in billions of dollars per year) of e-commerce companies in the United States from 1999 through 2004 are shown in the table. $$ \begin{array}{|l|l|l|l|l|l|l|} \hline t & 9 & 10 & 11 & 12 & 13 & 14 \\ \hline S & 14.5 & 27.8 & 34.5 & 45.0 & 56.6 & 70.9 \\ \hline \end{array} $$ The data can be modeled by \(S=-254.9+121.95 \ln t\), where \(t=9\) corresponds to 1999.

In Exercises \(83-86\), you are given the ratio of carbon atoms in a fossil. Use the information to estimate the age of the fossil. In living organic material, the ratio of radioactive carbon isotopes to the total number of carbon atoms is about 1 to \(10^{12}\). (See Example 2 in Section 10.1.) When organic material dies, its radioactive carbon isotopes begin to decay, with a half- life of about 5715 years. So, the ratio \(R\) of carbon isotopes to carbon- 14 atoms is modeled by \(R=10^{-12}\left(\frac{1}{2}\right)^{t / 5715}\), where \(t\) is the time (in years) and \(t=0\) represents the time when the organic material died. $$ R=0.22 \times 10^{-12} $$

The term \(t\) (in years) of a \(\$ 200,000\) home mortgage at \(7.5 \%\) interest can be approximated by \(t=-13.375 \ln \frac{x-1250}{x}, x>1250\) where \(x\) is the monthly payment in dollars. (a) Use a graphing utility to graph the model. (b) Use the model to approximate the term of a home mortgage for which the monthly payment is \(\$ 1398.43 .\) What is the total amount paid? (c) Use the model to approximate the term of a home mortgage for which the monthly payment is \(\$ 1611.19 .\) What is the total amount paid? (d) Find the instantaneous rate of change of \(t\) with respect to \(x\) when \(x=\$ 1398.43\) and \(x=\$ 1611.19\). (e) Write a short paragraph describing the benefit of the higher monthly payment.

In Exercises \(83-86\), you are given the ratio of carbon atoms in a fossil. Use the information to estimate the age of the fossil. In living organic material, the ratio of radioactive carbon isotopes to the total number of carbon atoms is about 1 to \(10^{12}\). (See Example 2 in Section 10.1.) When organic material dies, its radioactive carbon isotopes begin to decay, with a half- life of about 5715 years. So, the ratio \(R\) of carbon isotopes to carbon- 14 atoms is modeled by \(R=10^{-12}\left(\frac{1}{2}\right)^{t / 5715}\), where \(t\) is the time (in years) and \(t=0\) represents the time when the organic material died. $$ R=0.13 \times 10^{-12} $$

On the Richter scale, the magnitude \(R\) of an earthquake of intensity \(I\) is given by \(R=\frac{\ln I-\ln I_{0}}{\ln 10}\) where \(I_{0}\) is the minimum intensity used for comparison. Assume \(I_{0}=1\). (a) Find the intensity of the 1906 San Francisco earthquake for which \(R=8.3\). (b) Find the intensity of the May 26, 2006 earthquake in Java, Indonesia for which \(R=6.3\). (c) Find the factor by which the intensity is increased when the value of \(R\) is doubled. (d) Find \(d R / d I\)

Students in a mathematics class were given an exam and then retested monthly with equivalent exams. The average scores \(S\) (on a 100 -point scale) for the class can be modeled by \(S=80-14 \ln (t+1)\) \(0 \leq t \leq 12\), where \(t\) is the time in months. (a) What was the average score on the original exam? (b) What was the average score after 4 months? (c) After how many months was the average score 46 ?

Students in a learning theory study were given an exam and then retested monthly for 6 months with an equivalent exam. The data obtained in the study are shown in the table, where \(t\) is the time in months after the initial exam and \(s\) is the average score for the class. $$ \begin{array}{|l|l|l|l|l|l|l|} \hline t & 1 & 2 & 3 & 4 & 5 & 6 \\ \hline s & 84.2 & 78.4 & 72.1 & 68.5 & 67.1 & 65.3 \\ \hline \end{array} $$ (a) Use these data to find a logarithmic equation that relates \(t\) and \(s\). (b) Use a graphing utility to plot the data and graph the model. How well does the model fit the data? (c) Find the rate of change of \(s\) with respect to \(t\) when \(t=2\). Interpret the meaning in the context of the problem.

In a group project in learning theory, a mathematical model for the proportion \(P\) of correct responses after \(n\) trials was found to be \(P=\frac{0.83}{1+e^{-0.2 n}}\) (a) Use a graphing utility to graph the function. (b) Use the graph to determine any horizontal asymptotes of the graph of the function. Interpret the meaning of the upper asymptote in the context of the problem. (c) After how many trials will \(60 \%\) of the responses be correct?

You are investing \(P\) dollars at an annual interest rate of \(r\), compounded continuously, for \(t\) years, Which of the following options would you choose to get the highest value of the investment? Explain your reasoning. (a) Double the amount you invest. (b) Double your interest rate. (c) Double the number of years.

Demonstrate that \(\frac{\ln x}{\ln y} \neq \ln \frac{x}{y}=\ln x-\ln y\) by using a spreadsheet to complete the table. $$ \begin{array}{|c|c|c|c|c|} \hline x & y & \frac{\ln x}{\ln y} & \ln \frac{x}{y} & \ln x-\ln y \\ \hline 1 & 2 & & & \\ \hline 3 & 4 & & & \\ \hline 10 & 5 & & & \\ \hline 4 & 0.5 & & & \\ \hline \end{array} $$

Use a spreadsheet to complete the table using \(f(x)=\frac{\ln x}{x}\). $$ \begin{array}{|l|l|l|l|l|l|l|} \hline x & 1 & 5 & 10 & 10^{2} & 10^{4} & 10^{6} \\ \hline f(x) & & & & & & \\ \hline \end{array} $$ (a) Use the table to estimate the limit: \(\lim _{x \rightarrow \infty} f(x)\). (b) Use a graphing utility to estimate the relative extrema of \(f\)

In Exercises, use a graphing utility to verify that the functions are equivalent for \(x>0\). $$ \begin{aligned} &f(x)=\ln \frac{x^{2}}{4} \\ &g(x)=2 \ln x-\ln 4 \end{aligned} $$

In Exercises, use a graphing utility to verify that the functions are equivalent for \(x>0\). $$ \begin{aligned} &f(x)=\ln \sqrt{x\left(x^{2}+1\right)} \\ &g(x)=\frac{1}{2}\left[\ln x+\ln \left(x^{2}+1\right)\right] \end{aligned} $$

In Exercises, determine whether the statement is true or false given that \(f(x)=\ln x .\) If it is false, explain why or give an example that shows it is false. $$ f(0)=0 $$

In Exercises, determine whether the statement is true or false given that \(f(x)=\ln x .\) If it is false, explain why or give an example that shows it is false. $$ f(a x)=f(a)+f(x), \quad a>0, x>0 $$

In Exercises, determine whether the statement is true or false given that \(f(x)=\ln x .\) If it is false, explain why or give an example that shows it is false. $$ f(x-2)=f(x)-f(2), \quad x>2 $$

In Exercises, determine whether the statement is true or false given that \(f(x)=\ln x .\) If it is false, explain why or give an example that shows it is false. $$ \sqrt{f(x)}=\frac{1}{2} f(x) $$

In Exercises, determine whether the statement is true or false given that \(f(x)=\ln x .\) If it is false, explain why or give an example that shows it is false. $$ \text { If } f(u)=2 f(v), \text { then } v=u^{2} $$

In Exercises, determine whether the statement is true or false given that \(f(x)=\ln x .\) If it is false, explain why or give an example that shows it is false. $$ \text { If } f(x)<0, \text { then } 0

Use a graphing utility to graph \(y=10 \ln \left(\frac{10+\sqrt{100-x^{2}}}{10}\right)-\sqrt{100-x^{2}}\) over the interval \((0,10]\). This graph is called a tractrix or pursuit curve. Use your school's library, the Internet, or some other reference source to find information about a tractrix. Explain how such a curve can arise in a real-life setting.