Chapter 7

Each of Exercises \(1-4\) gives a value of \(\sinh x\) or cosh \(x .\) Use the definitions and the identity cosh \(^{2} x-\sinh ^{2} x=1\) to find the values of the remaining five hyperbolic functions. $$ \sinh x=-\frac{3}{4} $$

Find simpler expressions for the quantities in Exercises \(1-4\). $$ \begin{array}{llll}{\text { a. } e^{\ln 7.2}} & {\text { b. } e^{-\ln x^{2}}} & {\text { c. } e^{\ln x-\ln y}}\end{array} $$

Human evolution continues The analysis of tooth shrinkage by C. Loring Brace and colleagues at the University of Michigan's Museum of Anthropology indicates that human tooth size is continuing to decrease and that the evolutionary process did not come to a halt some \(30,000\) years ago as many scientists contend. In northern Europeans, for example, tooth size reduction now has a rate of 1\(\%\) per 1000 years. a. If \(t\) represents time in years and \(y\) represents tooth size, use the condition that \(y=0.99 y_{0}\) when \(t=1000\) to find the value of \(k\) in the equation \(y=y_{0} e^{k t}\) . Then use this value of \(k\) to answer the following questions. b. In about how many years will human teeth be 90\(\%\) of their present size? c. What will be our descendants' tooth size \(20,000\) years from now (as a percentage of our present tooth size)?

Which of the following functions grow faster than \(e^{x}\) as \(x \rightarrow \infty ?\) Which grow at the same rate as \(e^{x} ?\) Which grow slower? $$ \begin{array}{ll}{\text { a. } x+3} & {\text { b. } x^{3}+\sin ^{2} x} \\\ {\text { c. } \sqrt{x}} & {\text { d. } 4^{x}} \\ {\text { e. }(3 / 2)^{x}} & {\text { f. } e^{x / 2}} \\ {\text { g. } e^{x} / 2} & {\text { h. } \log _{10} x}\end{array} $$

Simplify the expressions. a. \(5^{\log _{5} 7}\) b. \(8^{\log _{8} \sqrt{2}}\) c. \(1.3^{\log _{13} 75}\) d. \(\log _{4} 16 \) e. \(\log _{3} \sqrt{3}\) f. \(\log _{4}\left(\frac{1}{4}\right)\)

Express the following logarithms in terms of \(\ln 2\) and \(\ln 3 .\) $$ \begin{array}{lll}{\text { a. } \ln 0.75} & {\text { b. } \ln (4 / 9)} & {\text { c. } \ln (1 / 2)} \\ {\text { d. } \ln \sqrt[3]{9}} & {\text { e. } \ln 3 \sqrt{2}} & {\text { f. } \ln \sqrt{13.5}}\end{array} $$

Each of Exercises \(1-4\) gives a value of \(\sinh x\) or cosh \(x .\) Use the definitions and the identity cosh \(^{2} x-\sinh ^{2} x=1\) to find the values of the remaining five hyperbolic functions. $$ \sinh x=\frac{4}{3} $$

Find simpler expressions for the quantities in Exercises \(1-4\). $$ \text { a. } e^{\ln \left(x^{2}+y^{2}\right)} \quad \text { b. } e^{-\ln 0.3} \quad \text { c. } e^{\ln \pi x-\ln 2} $$

Atmospheric pressure The earth's atmospheric pressure \(p\) is often modeled by assuming that the rate \(d p / d h\) at which \(p\) changes with the altitude \(h\) above sea level is proportional to \(p .\) Suppose that the pressure at sea level is 1013 millibars (about 14.7 pounds per square inch ) and that the pressure at an altitude of 20 \(\mathrm{km}\) is 90 millibars. a. Solve the initial value problem Differential equation: \(\quad d p / d h=k p \quad(k \text { a constant })\) Initial condition: \(\quad p=p_{0}\) when \(h=0\) to express \(p\) in terms of \(h .\) Determine the values of \(p_{0}\) and \(k\) from the given altitude-pressure data. b. What is the atmospheric pressure at \(h=50 \mathrm{km}\) ? c. At what altitude does the pressure equal 900 millibars?

Which of the following functions grow faster than \(e^{x}\) as \(x \rightarrow \infty ?\) Which grow at the same rate as \(e^{x} ?\) Which grow slower? $$ \begin{array}{ll}{\text { a. } 10 x^{4}+30 x+1} & {\text { b. } x \ln x-x} \\\ {\text { c. } \sqrt{1+x^{4}}} & {\text { d. }(5 / 2)^{x}} \\ {\text { e. } e^{-x}} & {\text { f. } x e^{x}} \\ {\text { g. } e^{\cos x}} & {\text { h. } e^{x-1}}\end{array} $$

Simplify the expressions. a. \(2^{\log _{2} 3}\) b. \(10^{\log _{10}(1 / 2)}\) c. \(\pi^{\log _{\pi} 7}\) d. \(\log _{11} 121 \) e. \(\log _{121} 11\) f. \(\log _{3}\left(\frac{1}{9}\right)\)

Express the following logarithms in terms of \(\ln 5\) and \(\ln 7\) $$ \begin{array}{ll}{\text { a. } \ln (1 / 125)} & {\text { b. } \ln 9.8} \\\ {\text { d. } \ln 1225} & {\text { e. } \ln 0.056} \\ {\text { f. }(\ln 35+\ln (1 / 7)) /(\ln 25)}\end{array} $$

Each of Exercises \(1-4\) gives a value of \(\sinh x\) or cosh \(x .\) Use the definitions and the identity cosh \(^{2} x-\sinh ^{2} x=1\) to find the values of the remaining five hyperbolic functions. $$ \cosh x=\frac{17}{15}, \quad x>0 $$

Find simpler expressions for the quantities in Exercises \(1-4\). $$ \text { a. } 2 \ln \sqrt{e} \quad \text { b. } \ln \left(\ln e^{e}\right) \quad \text { c. } \ln \left(e^{-x^{2}-y^{2}}\right) $$

First-order chemical reactions In some chemical reactions, the rate at which the amount of a substance changes with time is proportional to the amount present. For the change of \(\delta\) -glucono lactone into gluconic acid, for example, $$ \frac{d y}{d t}=-0.6 y $$ when \(t\) is measured in hours. If there are 100 grams of \(\delta\) -glucono lactone present when \(t=0,\) how many grams will be left after the first hour?

Which of the following functions grow faster than \(x^{2}\) as \(x \rightarrow \infty ?\) Which grow at the same rate as \(x^{2} ?\) Which grow slower? $$ \begin{array}{ll}{\text { a. } x^{2}+4 x} & {\text { b. } x^{5}-x^{2}} \\\ {\text { c. } \sqrt{x^{4}+x^{3}}} & {\text { d. }(x+3)^{2}} \\ {\text { e. } x \ln x} & {\text { f. } 2^{x}} \\ {\text { g. } x^{3} e^{-x}} & {\text { h. } 8 x^{2}}\end{array} $$

Simplify the expressions. a. \(2^{\log _{4} x} \) b. \(9^{\log _{3} x}\) c. \(\log _{2}\left(e^{(\ln 2)(\sin x)}\right)\)

Use the properties of logarithms to simplify the expressions in Exercises 3 and \(4 .\) $$ \begin{array}{l}{\text { a. } \ln \sin \theta-\ln \left(\frac{\sin \theta}{5}\right) \quad \text { b. } \ln \left(3 x^{2}-9 x\right)+\ln \left(\frac{1}{3 x}\right)} \\ {\text { c. } \frac{1}{2} \ln \left(4 t^{4}\right)-\ln 2}\end{array} $$

Each of Exercises \(1-4\) gives a value of \(\sinh x\) or cosh \(x .\) Use the definitions and the identity cosh \(^{2} x-\sinh ^{2} x=1\) to find the values of the remaining five hyperbolic functions. $$ \cosh x=\frac{13}{5}, \quad x>0 $$

Find simpler expressions for the quantities in Exercises \(1-4\). $$ \text { a. } \ln \left(e^{\sec \theta}\right) \quad \text { b. } \ln \left(e^{\left(e^{7}\right)}\right) \quad \text { c. } \ln \left(e^{2 \ln x}\right) $$

Which of the following functions grow faster than \(x^{2}\) as \(x \rightarrow \infty ?\) Which grow at the same rate as \(x^{2} ?\) Which grow slower? $$ \begin{array}{ll}{\text { a. } x^{2}+\sqrt{x}} & {\text { b. } 10 x^{2}} \\\ {\text { c. } x^{2} e^{-x}} & {\text { d. } \log _{10}\left(x^{2}\right)} \\\ {\text { e. } x^{3}-x^{2}} & {\text { f. }(1 / 10)^{x}} \\ {\text { g. }(1.1)^{x}} & {\text { h. } x^{2}+100 x}\end{array} $$

Simplify the expressions. a. \(25^{\log _{5}\left(3 x^{2}\right)}\) b. \(\log _{e}\left(e^{x}\right)\) c. \(\log _{4}\left(2^{x} \sin x\right)\)

Use the properties of logarithms to simplify the expressions in Exercises 3 and \(4 .\) $$ \begin{array}{l}{\text { a. } \ln \sec \theta+\ln \cos \theta \quad \text { b. } \ln (8 x+4)-2 \ln 2} \\ {\text { c. } 3 \ln \sqrt[3]{t^{2}-1}-\ln (t+1)}\end{array} $$

Rewrite the expressions in Exercises \(5-10\) in terms of exponentials and simplify the results as much as you can. $$ 2 \cosh (\ln x) $$

In Exercises \(5-10,\) solve for \(y\) in terms of \(t\) or \(x,\) as appropriate. $$ \ln y=2 t+4 $$

Working underwater The intensity \(L(x)\) of light \(x\) feet beneath the surface of the ocean satisfies the differential equation $$ \frac{d L}{d x}=-k L $$ As a diver, you know from experience that diving to 18 \(\mathrm{ft}\) in the Caribbean Sea cuts the intensity in half. You cannot work without artificial light when the intensity falls below one-tenth of the surface value. About how deep can you expect to work without artificial light?

Which of the following functions grow faster than \(\ln x\) as \(x \rightarrow \infty ?\) Which grow at the same rate as \(\ln x ?\) Which grow slower? $$ \begin{array}{ll}{\text { a. } \log _{3} x} & {\text { b. } \ln 2 x} \\\ {\text { c. } \ln \sqrt{x}} & {\text { d. } \sqrt{x}} \\ {\text { e. } x} & {\text { f. } 5 \ln x} \\ {\text { g. } 1 / x} & {\text { h. } e^{x}}\end{array} $$

Rewrite the expressions in Exercises \(5-10\) in terms of exponentials and simplify the results as much as you can. $$ \sinh (2 \ln x) $$

In Exercises \(5-10,\) solve for \(y\) in terms of \(t\) or \(x,\) as appropriate. $$ \ln y=-t+5 $$

Voltage in a discharging capacitor Suppose that electricity is draining from a capacitor at a rate that is proportional to the voltage \(V\) across its terminals and that, if \(t\) is measured in seconds, $$ \frac{d V}{d t}=-\frac{1}{40} V $$ Solve this equation for \(V,\) using \(V_{0}\) to denote the value of \(V\) when \(t=0 .\) How long will it take the voltage to drop to 10\(\%\) of its original value?

Which of the following functions grow faster than \(\ln x\) as \(x \rightarrow \infty ?\) Which grow at the same rate as \(\ln x ?\) Which grow slower? $$ \begin{array}{ll}{\text { a. } \log _{2}\left(x^{2}\right)} & {\text { b. } \log _{10} 10 x} \\ {\text { c. } 1 / \sqrt{x}} & {\text { d. } 1 / x^{2}} \\\ {\text { e. } x-2 \ln x} & {\text { f. } e^{-x}} \\ {\text { g. } \ln (\ln x)} & {\text { h. } \ln (2 x+5)}\end{array} $$

In Exercises \(5-36,\) find the derivative of \(y\) with respect to \(x, t,\) or \(\theta,\) as appropriate. \(y=\ln k x, k\) constant

Rewrite the expressions in Exercises \(5-10\) in terms of exponentials and simplify the results as much as you can. $$ \cosh 5 x+\sinh 5 x $$

In Exercises \(5-10,\) solve for \(y\) in terms of \(t\) or \(x,\) as appropriate. $$ \ln (y-40)=5 t $$

Cholera bacteria Suppose that the bacteria in a colony can grow unchecked, by the law of exponential change. The colony starts with 1 bacterium and doubles every half-hour. How many bacteria will the colony contain at the end of 24 hours? (Under favorable laboratory conditions, the number of cholera bacteria can double every 30 \(\mathrm{min.}\) In an infected person, many bacteria are destroyed, but this example helps explain why a person who feels well in the morning may be dangerously ill by evening.)

Order the following functions from slowest growing to fastest growing as \(x \rightarrow \infty\) . $$ \begin{array}{ll}{\text { a. } e^{x}} & {\text { b. } x^{x}} \\ {\text { c. }(\ln x)^{x}} & {\text { d. } e^{x / 2}}\end{array} $$

In Exercises \(5-36,\) find the derivative of \(y\) with respect to \(x, t,\) or \(\theta,\) as appropriate. $$ y=\ln \left(t^{2}\right) $$

Solve the equations. \(3^{\log _{3}(7)}+2^{\log _{2}(5)}=5^{\log _{5}(x)}\)

In Exercises \(5-10,\) solve for \(y\) in terms of \(t\) or \(x,\) as appropriate. $$ \ln (1-2 y)=t $$

Growth of bacteria A colony of bacteria is grown under ideal conditions in a laboratory so that the population increases exponentially with time. At the end of 3 hours there are \(10,000\) bacteria. At the end of 5 hours there are \(40,000\) . How many bacteria were present initially?

Order the following functions from slowest growing to fastest growing as \(x \rightarrow \infty\) . $$ \begin{array}{ll}{\text { a. } 2^{x}} & {\text { b. } x^{2}} \\ {\text { c. }(\ln 2)^{x}} & {\text { d. } e^{x}}\end{array} $$

In Exercises \(5-36,\) find the derivative of \(y\) with respect to \(x, t,\) or \(\theta,\) as appropriate. $$ y=\ln \left(t^{3 / 2}\right) $$

Solve the equations. \(8^{\log _{8}(3)}-e^{\ln 5}=x^{2}-7^{\log _{7}(3 x)}\)

Rewrite the expressions in Exercises \(5-10\) in terms of exponentials and simplify the results as much as you can. $$ (\sinh x+\cosh x)^{4} $$

In Exercises \(5-10,\) solve for \(y\) in terms of \(t\) or \(x,\) as appropriate. $$ \ln (y-1)-\ln 2=x+\ln x $$

True, or false? As \(x \rightarrow \infty\) $$ \begin{array}{ll}{\text { a. } x=o(x)} & {\text { b. } x=o(x+5)} \\ {\text { c. } x=O(x+5)} & {\text { d. } x=O(2 x)} \\ {\text { e. } e^{x}=o\left(e^{2 x}\right)} & {\text { f. } x+\ln x=O(x)} \\ {\text { g. } \ln x=o(\ln 2 x)} & {\text { h. } \sqrt{x^{2}+5}=O(x)}\end{array} $$

In Exercises \(5-36,\) find the derivative of \(y\) with respect to \(x, t,\) or \(\theta,\) as appropriate. $$ y=\ln \frac{3}{x} $$

Solve the equations. \(3^{\log _{3}\left(x^{2}\right)}=5 e^{\ln x}-3 \cdot 10^{\log _{10}(2)}\)

In Exercises \(5-10,\) solve for \(y\) in terms of \(t\) or \(x,\) as appropriate. $$ \ln \left(y^{2}-1\right)-\ln (y+1)=\ln (\sin x) $$

The U.S. population The Museum of Science in Boston displays a running total of the U.S. population. On May \(11,1993\) , the total was increasing at the rate of 1 person every 14 sec. The dis- played population figure for \(3 : 45\) P.M. that day was \(257,313,431\) . a. Assuming exponential growth at a constant rate, find the rate constant for the population's growth (people per 365 -day year). b. At this rate, what will the U.S. population be at \(3 : 45 \mathrm{PM}\) . Boston time on May \(11,2008 ?\)