Chapter 10

In Exercises, find the second derivative. $$ f(x)=(1+2 x) e^{4 x} $$

In Exercises, use a graphing utility to graph the function. $$ y=2^{-x^{2}} $$

In Exercises, use the properties of logarithms and the fact that \(\ln 2 \approx 0.6931\) and \(\ln 3 \approx 1.0986\) to approximate the logarithm. Then use a calculator to confirm your approximation. (a) \(\ln 6\) (b) \(\ln \frac{3}{2}\) (c) \(\ln 81\) (d) \(\ln \sqrt{3}\)

In Exercises, find the second derivative. $$ f(x)=5 e^{-x}-2 e^{-5 x} $$

In Exercises, use the properties of logarithms and the fact that \(\ln 2 \approx 0.6931\) and \(\ln 3 \approx 1.0986\) to approximate the logarithm. Then use a calculator to confirm your approximation. (a) \(\ln 0.25\) (b) \(\ln 24\) (c) \(\ln \sqrt[3]{12}\) (d) \(\ln \frac{1}{72}\)

In Exercises, find the second derivative. $$ f(x)=(3+2 x) e^{-3 x} $$

In Exercises, use a graphing utility to graph the function. $$ s(t)=2^{-t}+3 $$

In Exercises, use a calculator to evaluate the logarithm. Round to three decimal places. $$ \log _{4} 7 $$

In Exercises, use the properties of logarithms to write the expression as a sum, difference, or multiple of logarithms. $$ \ln \frac{2}{3} $$

In Exercises, graph and analyze the function. Include extrema, points of inflection, and asymptotes in your analysis. $$ f(x)=\frac{1}{2-e^{-x}} $$

The population \(P\) (in millions) of the United States from 1992 through 2005 can be modeled by the exponential function \(P(t)=252.12(1.011)^{\prime}\), where \(t\) is the time in years, with \(t=2\) corresponding to 1992 . Use the model to estimate the population in the years (a) 2008 and (b) 2012. (Source: U.S. Census Bureau)

In Exercises, use a calculator to evaluate the logarithm. Round to three decimal places. $$ \log _{6} 10 $$

In Exercises, use the properties of logarithms to write the expression as a sum, difference, or multiple of logarithms. $$ \ln \frac{1}{5} $$

In Exercises, graph and analyze the function. Include extrema, points of inflection, and asymptotes in your analysis. $$ f(x)=\frac{e^{x}-e^{-x}}{2} $$

The sales \(S\) (in millions of dollars) for Starbucks from 1996 through 2005 can be modeled by the exponential function \(S(t)=182.34(1.272)^{2}\), where \(t\) is the time in years, with \(t=6\) corresponding to 1996 . Use the model to estimate the sales in the years (a) 2008 and (b) 2014 .

In Exercises, determine the principal \(P\) that must be invested at interest rate \(r\), compounded continuously, so that \(\$ 1,000,000\) will be available for retirement in \(t\) years. $$ r=7.5 \%, t=40 $$

In Exercises, use a calculator to evaluate the logarithm. Round to three decimal places. $$ \log _{2} 48 $$

In Exercises, use the properties of logarithms to write the expression as a sum, difference, or multiple of logarithms. $$ \ln 2 x y $$

In Exercises, graph and analyze the function. Include extrema, points of inflection, and asymptotes in your analysis. $$ f(x)=x^{2} e^{-x} $$

On the day of a child's birth, a deposit of \(\$ 20,000\) is made in a trust fund that pays \(8 \%\) interest, compounded continuously. Determine the balance in this account on the child's 21 st birthday.

Suppose that the value of a piece of property doubles every 15 years. If you buy the property for \(\$ 64,000\), its value \(t\) years after the date of purchase should be \(V(t)=64,000(2)^{2 / 15} .\) Use the model to approximate the value of the property (a) 5 years and (b) 20 years after it is purchased.

In Exercises, determine the principal \(P\) that must be invested at interest rate \(r\), compounded continuously, so that \(\$ 1,000,000\) will be available for retirement in \(t\) years. $$ r=10 \%, t=25 $$

In Exercises, use a calculator to evaluate the logarithm. Round to three decimal places. $$ \log _{5} 12 $$

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

In Exercises, graph and analyze the function. Include extrema, points of inflection, and asymptotes in your analysis. $$ f(x)=x e^{-x} $$

A deposit of \(\$ 10,000\) is made in a trust fund that pays \(7 \%\) interest, compounded continuously. It is specified that the balance will be given to the college from which the donor graduated after the money has earned interest for 50 years. How much will the college receive?

After \(t\) years, the value of a car that originally cost \(\$ 16,000\) depreciates so that each year it is worth \(\frac{3}{4}\) of its value for the previous year. Find a model for \(V(t)\), the value of the car after \(t\) years. Sketch a graph of the model and determine the value of the car 4 years after it was purchased.

The effective yield is the annual rate \(i\) that will produce the same interest per year as the nominal rate \(r\) compounded \(n\) times per year. (a) For a rate \(r\) that is compounded \(n\) times per year, show that the effective yield is \(i=\left(1+\frac{r}{n}\right)^{n}-1 .\) (b) Find the effective yield for a nominal rate of \(6 \%\), compounded monthly.

In Exercises, use a calculator to evaluate the logarithm. Round to three decimal places. $$ \log _{3} \frac{1}{2} $$

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

In Exercises, use a graphing utility to graph the function. Determine any asymptotes of the graph. $$ f(x)=\frac{8}{1+e^{-0.5 x}} $$

Find the effective rate of interest corresponding to a nominal rate of \(9 \%\) per year compounded (a) annually, (b) semiannually, (c) quarterly, and (d) monthly.

Suppose that the annual rate of inflation averages \(4 \%\) over the next 10 years. With this rate of inflation, the approximate cost \(C\) of goods or services during any year in that decade will be given by \(C(t)=P(1.04)^{r}, \quad 0 \leq t \leq 10\) where \(t\) is time in years and \(P\) is the present cost. If the price of an oil change for your car is presently \(\$ 24.95\), estimate the price 10 years from now.

The effective yield is the annual rate \(i\) that will produce the same interest per year as the nominal rate \(r\). (a) For a rate \(r\) that is compounded continuously, show that the effective yield is \(i=e^{r}-1\). (b) Find the effective yield for a nominal rate of \(6 \%\), compounded continuously.

In Exercises, use a calculator to evaluate the logarithm. Round to three decimal places. $$ \log _{7} \frac{2}{9} $$

In Exercises, use the properties of logarithms to write the expression as a sum, difference, or multiple of logarithms. $$ \ln \sqrt{\frac{x^{3}}{x+1}} $$

In Exercises, use a graphing utility to graph the function. Determine any asymptotes of the graph. $$ g(x)=\frac{8}{1+e^{-0.5 / x}} $$

Find the effective rate of interest corresponding to a nominal rate of \(7.5 \%\) per year compounded (a) annually, (b) semiannually, (c) quarterly, and (d) monthly.

In Exercises, use a calculator to evaluate the logarithm. Round to three decimal places. $$ \log _{1 / 5} 31 $$

In Exercises, use the properties of logarithms to write the expression as a sum, difference, or multiple of logarithms. $$ \ln \left[z(z-1)^{2}\right] $$

In Exercises, solve the equation for \(x\). $$ e^{-3 x}=e $$

How much should be deposited in an account paying \(7.2 \%\) interest compounded monthly in order to have a balance of \(\$ 15,503.77\) three years from now?

In Exercises, use a calculator to evaluate the logarithm. Round to three decimal places. $$ \log _{2 / 3} 32 $$

In Exercises, use the properties of logarithms to write the expression as a sum, difference, or multiple of logarithms. $$ \ln \left(x \sqrt[3]{x^{2}+1}\right) $$

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

How much should be deposited in an account paying \(7.8 \%\) interest compounded monthly in order to have a balance of \(\$ 21,154.03\) four years from now?

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

In Exercises, use the properties of logarithms to write the expression as a sum, difference, or multiple of logarithms. $$ \ln \frac{3 x(x+1)}{(2 x+1)^{2}} $$

In Exercises, solve the equation for \(x\). $$ e^{\sqrt{x}}=e^{3} $$

Find the future value of an \(\$ 8000\) investment if the interest rate is \(4.5 \%\) compounded monthly for 2 years.