Chapter 10

In Exercises, find the derivative of the function. $$ y=\left(\frac{1}{4}\right)^{x} $$

In Exercises, use the properties of logarithms to write the expression as a sum, difference, or multiple of logarithms. $$ \ln \frac{2 x}{\sqrt{x^{2}-1}} $$

In Exercises, solve the equation for \(x\). $$ e^{-1 / x}=e^{1 / 2} $$

Find the future value of a \(\$ 6500\) investment if the interest rate is \(6.25 \%\) compounded monthly for 3 years.

The revenues for Sonic Corporation were \(\$ 151.1\) million in 1996 and \(\$ 693.3\) million in 2006. (Source: Sonic Corporation) (a) Use an exponential growth model to estimate the revenue in 2011 . (b) Use a linear model to estimate the 2011 revenue. (c) Use a graphing utility to graph the models from parts (a) and (b). Which model is more accurate?

In Exercises, find the derivative of the function. $$ f(x)=\log _{2} x $$

In Exercises, write the expression as the logarithm of a single quantity. $$ \ln (x-2)-\ln (x+2) $$

The demand function for a product is modeled by \(p=5000\left(1-\frac{4}{4+e^{-0.002 x}}\right)\) Find the price of the product if the quantity demanded is (a) \(x=100\) units and (b) \(x=500\) units. What is the limit of the price as \(x\) increases without bound?

The sales for exercise equipment in the United States were \(\$ 1824\) million in 1990 and \(\$ 5112\) million in 2005. (a) Use the regression feature of a graphing utility to find an exponential growth model and a linear model for the data. (b) Use the exponential growth model to estimate the sales in 2011 . (c) Use the linear model to estimate the sales in 2011 . (d) Use a graphing utility to graph the models from part (a). Which model is more accurate?

In Exercises, find the derivative of the function. $$ g(x)=\log _{5} x $$

In Exercises, write the expression as the logarithm of a single quantity. $$ \ln (2 x+1)+\ln (2 x-1) $$

In Exercises 41 and 42 , the value \(V\) (in dollars) of an item is a function of the time \(t\) (in years). (a) Sketch the function over the interval \([0,10] .\) Use a graphing utility to verify your graph. (b) Find the rate of change of \(V\) when \(t=1\). (c) Find the rate of change of \(V\) when \(t=5\). (d) Use the values \((0, V(0))\) and \((10, V(10))\) to find the linear depreciation model for the item. (e) Compare the exponential function and the model from part (d). What are the advantages of each? $$ V=500,000 e^{-0.2231 t} $$

The demand function for a product is modeled by \(p=10,000\left(1-\frac{3}{3+e^{-0.501 x}}\right)\) Find the price of the product if the quantity demanded is (a) \(x=1000\) units and (b) \(x=1500\) units. What is the limit of the price as \(x\) increases without bound?

The cumulative sales \(S\) (in thousands of units) of a new product after it has been on the market for \(t\) years are modeled by \(S=C e^{k / t}\) During the first year, 5000 units were sold. The saturation point for the market is 30,000 units. That is, the limit of \(S\) as \(t \rightarrow \infty\) is 30,000 . (a) Solve for \(C\) and \(k\) in the model. (b) How many units will be sold after 5 years? (c) Use a graphing utility to graph the sales function.

In Exercises, find the derivative of the function. $$ h(x)=4^{2 x-3} $$

In Exercises, write the expression as the logarithm of a single quantity. $$ 3 \ln x+2 \ln y-4 \ln z $$

The average time between incoming calls at a switchboard is 3 minutes. If a call has just come in, the probability that the next call will come within the next \(t\) minutes is \(P(t)=1-e^{-t / 3} .\) Find the probability of each situation. (a) A call comes in within \(\frac{1}{2}\) minute. (b) A call comes in within 2 minutes. (c) A call comes in within 5 minutes.

In Exercises, find the derivative of the function. $$ y=6^{5 x} $$

In Exercises, write the expression as the logarithm of a single quantity. $$ 2 \ln 3-\frac{1}{2} \ln \left(x^{2}+1\right) $$

The balance \(A\) (in dollars) in a savings account is given by \(A=5000 e^{0.08 t}\), where \(t\) is measured in years. Find the rates at which the balance is changing when (a) \(t=1\) year, (b) \(t=10\) years, and (c) \(t=50\) years.

In Exercises, find the derivative of the function. $$ y=\log _{10}\left(x^{2}+6 x\right) $$

In Exercises, write the expression as the logarithm of a single quantity. $$ 3[\ln x+\ln (x+3)-\ln (x+4)] $$

In Exercises, find the derivative of the function. $$ f(x)=10^{x^{2}} $$

In Exercises, write the expression as the logarithm of a single quantity. $$ \frac{1}{3}\left[2 \ln (x+3)+\ln x-\ln \left(x^{2}-1\right)\right] $$

The population \(P\) (in thousands) of Las Vegas, Nevada from 1960 through 2005 can be modeled by \(P=68.4 e^{0.0467 t}\), where \(t\) is the time in years, with \(t=0\) corresponding to 1960. (Source: U.S. Census Bureau) (a) Find the populations in \(1960,1970,1980,1990,2000\), and 2005 . (b) Explain why the data do not fit a linear model. (c) Use the model to estimate when the population will exceed 900,000 .

Because of a slump in the economy, a company finds that its annual profits have dropped from \(\$ 742,000\) in 1998 to \(\$ 632,000\) in 2000 . If the profit follows an exponential pattern of decline, what is the expected profit for 2003 ? (Let \(t=0\) correspond to \(1998 .\) )

In Exercises, find the derivative of the function. $$ y=x 2^{x} $$

In Exercises, write the expression as the logarithm of a single quantity. $$ \frac{3}{2}\left[\ln x\left(x^{2}+1\right)-\ln (x+1)\right] $$

From 1996 through 2005, the numbers \(y\) (in millions) of employed people in the United States can be modeled by \(y=98.020+6.2472 t-0.24964 t^{2}+0.000002 e^{t}\) where \(t\) represents the year, with \(t=6\) corresponding to 1996. (a) Use a graphing utility to graph the model. (b) Use the graph to estimate the rates of change in the number of employed people in 1996,2000 , and 2005 . (c) Confirm the results of part (b) analytically.

The population \(y\) of a bacterial culture is modeled by the logistic growth function \(y=925 /\left(1+e^{-0.3 t}\right)\), where \(t\) is the time in days. (a) Use a graphing utility to graph the model. (b) Does the population have a limit as \(t\) increases without bound? Explain your answer. (c) How would the limit change if the model were \(y=1000 /\left(1+e^{-0.3 z}\right) ?\) Explain your answer. Draw some conclusions about this type of model.

A small business assumes that the demand function for one of its new products can be modeled by \(p=C e^{k x} .\) When \(p=\$ 45, x=1000\) units, and when \(p=\$ 40, x=1200\) units. (a) Solve for \(C\) and \(k\). (b) Find the values of \(x\) and \(p\) that will maximize the revenue for this product.

In Exercises, find the derivative of the function. $$ y=x 3^{x+1} $$

In Exercises, write the expression as the logarithm of a single quantity. $$ 2\left[\ln x+\frac{1}{4} \ln (x+1)\right] $$

A cell site is a site where electronic communications equipment is placed in a cellular network for the use of mobile phones. From 1985 through 2006 , the numbers \(y\) of cell sites can be modeled by \(y=\frac{222,827}{1+2677 e^{-0.377 t}}\) where \(t\) represents the year, with \(t=5\) corresponding to 1985. (a) Use a graphing utility to graph the model. (b) Use the graph to estimate when the rate of change in the number of cell cites began to decrease. (c) Confirm the result of part (b) analytically.

Suppose that you have a single imaginary bacterium able to divide to form two new cells every 30 seconds. Make a table of values for the number of individuals in the population over 30 -second intervals up to 5 minutes. Graph the points and use a graphing utility to fit an exponential model to the data.

In Exercises, determine an equation of the tangent line to the function at the given point. $$ y=x \ln x $$ $$ (1,0) $$

In Exercises, write the expression as the logarithm of a single quantity. $$ \frac{1}{3} \ln (x+1)-\frac{2}{3} \ln (x-1) $$

A survey of high school seniors from a certain school district who took the SAT has determined that the mean score on the mathematics portion was 650 with a standard deviation of \(12.5\). (a) Assuming the data can be modeled by a normal probability density function, find a model for these data. (b) Use a graphing utility to graph the model. Be sure to choose an appropriate viewing window. (c) Find the derivative of the model. (d) Show that \(f^{\prime}>0\) for \(x<\mu\) and \(f^{\prime}<0\) for \(x>\mu\).

In a learning theory project, the proportion \(P\) of correct responses after \(n\) trials can be modeled by \(P=\frac{0.83}{1+e^{-0.2 n}}\) (a) Use a graphing utility to estimate the proportion of correct responses after 10 trials. Verify your result analytically. (b) Use a graphing utility to estimate the number of trials required to have a proportion of correct responses of \(0.75 .\) (c) Does the proportion of correct responses have a limit as \(n\) increases without bound? Explain your answer.

In Exercises, determine an equation of the tangent line to the function at the given point. $$ y=\frac{\ln x}{x} $$ $$ \left(e, \frac{1}{e}\right) $$

In Exercises, write the expression as the logarithm of a single quantity. $$ \frac{1}{2} \ln (x-2)+\frac{3}{2} \ln (x+2) $$

A survey of a college freshman class has determined that the mean height of females in the class is 64 inches with a standard deviation of \(3.2\) inches. (a) Assuming the data can be modeled by a normal probability density function, find a model for these data. (b) Use a graphing utility to graph the model. Be sure to choose an appropriate viewing window. (c) Find the derivative of the model. (d) Show that \(f^{\prime}>0\) for \(x<\mu\) and \(f^{\prime}<0\) for \(x>\mu\).

In a typing class, the average number \(N\) of words per minute typed after \(t\) weeks of lessons can be modeled by \(N=\frac{95}{1+8.5 e^{-0.12 t}}\) (a) Use a graphing utility to estimate the average number of words per minute typed after 10 weeks. Verify your result analytically. (b) Use a graphing utility to estimate the number of weeks required to achieve an average of 70 words per minute. (c) Does the number of words per minute have a limit as \(t\) increases without bound? Explain your answer.

In Exercises, determine an equation of the tangent line to the function at the given point. $$ y=\log _{3} x $$ $$ (27,3) $$

In Exercises, solve for \(x\) or \(t\). $$ e^{\ln x}=4 $$

Use a graphing utility to graph the normal probability density function with \(\mu=0\) and \(\sigma=2,3\), and 4 in the same viewing window. What effect does the standard deviation \(\sigma\) have on the function? Explain your reasoning.

You want to invest \(\$ 5000\) in a certificate of deposit for 12 months. You are given the options below. Which would you choose? Explain. (a) \(r=5.25 \%\), quarterly compounding (b) \(r=5 \%\), monthly compounding (c) \(r=4.75 \%\), continuous compounding

In Exercises, determine an equation of the tangent line to the function at the given point. $$ g(x)=\log _{10} 2 x $$ $$ (5,1) $$

In Exercises, solve for \(x\) or \(t\). $$ e^{\ln x^{2}}-9=0 $$

Use a graphing utility to graph the normal probability density function with \(\sigma=1\) and \(\mu=-2,1\), and 3 in the same viewing window. What effect does the mean \(\mu\) have on the function? Explain your reasoning.