Chapter 5

Dissolving salt in water If 10 grams of salt is added to a quantity of water, then the amount \(q(t)\) that is undissolved after \(t\) minutes is given by \(q(t)=10\left(\frac{4}{5}\right)^{t}\). Sketch a graph that shows the value \(q(t)\) at any time from \(t=0\) to \(t=10\).

Sketch the graph of \(f\). $$ f(x)=\log _{2}\left(x^{3}\right) $$

Exer. 39-42: Use natural logarithms to solve for \(x\) in terms of \(y\). $$ y=\frac{e^{x}+e^{-x}}{e^{x}-e^{-x}} $$

Exer. 25-42: Find the inverse function of \(f\). $$ f(x)=x^{2}-6 x, x \geq 3 $$

Compound interest If \(\$ 1000\) is invested at a rate of \(7 \%\) per year compounded monthly, find the principal after (a) 1 month (b) 6 months (c) l year (d) 20 years

Sketch the graph of \(f\). $$ f(x)=\log _{3}\left(x^{3}\right) $$

Exer. 39-42: Use natural logarithms to solve for \(x\) in terms of \(y\). $$ y=\frac{e^{x}-e^{-x}}{e^{x}+e^{-x}} $$

Exer. 37-42: Sketch the graph of \(f\). $$ f(x)=\ln (e+x) $$

Exer. 25-42: Find the inverse function of \(f\). $$ f(x)=x^{2}-4 x+3, x \leq 2 $$

Compound interest If a savings fund pays interest at a rate of \(6 \%\) per year compounded semiannually, how much money invested now will amount to \(\$ 5000\) after I year?

Sketch the graph of \(f\). $$ f(x)=\log _{2} \sqrt{x} $$

Exer. 43-44: Sketch the graph of \(f\), and use the change of base formula to approximate the \(y\)-intercept. $$ f(x)=\log _{2}(x+3) $$

Sketch the graph of \(f\). $$ f(x)=\log _{2} \sqrt[3]{x} $$

Exer. 43-44: Sketch the graph of \(f\), and use the change of base formula to approximate the \(y\)-intercept. $$ f(x)=\log _{3}(x+5) $$

Sketch the graph of \(f\). $$ f(x)=\log _{3}\left(\frac{1}{x}\right) $$

Exer. 45-46: Sketch the graph of \(f\), and use the change of base formula to approximate the \(x\)-intercept. $$ f(x)=4^{x}-3 $$

Manhattan Island The Island of Manhattan was sold for \(\$ 24\) in 1626 . How much would this amount have grown to by 2006 if it had been invested at \(6 \%\) per year compounded quarterly?

Sketch the graph of \(f\). $$ f(x)=\log _{2}\left(\frac{1}{x}\right) $$

Exer. 45-46: Sketch the graph of \(f\), and use the change of base formula to approximate the \(x\)-intercept. $$ f(x)=3^{x}-6 $$

Credit-card interest A certain department store requires its credit-card customers to pay interest on unpaid bills at the rate of \(18 \%\) per year compounded monthly. If a customer buys a television set for \(\$ 500\) on credit and makes no payments for one year, how much is owed at the end of the year?

Exer. 47-50: Chemists use a number denoted by \(\mathrm{pH}\) to describe quantitatively the acidity or basicity of solutions. By definition, \(\mathrm{pH}=-\log \left[\mathrm{H}^{+}\right]\), where \(\left[\mathrm{H}^{+}\right]\)is the hydrogen ion concentration in moles per liter. Approximate the \(\mathrm{pH}\) of each substance. (a) vinegar: \(\left[\mathrm{H}^{+}\right] \approx 6.3 \times 10^{-3}\) (b) carrots: \(\left[\mathrm{H}^{+}\right] \approx 1.0 \times 10^{-5}\) (c) sea water: \(\left[\mathrm{H}^{+}\right] \approx 5.0 \times 10^{-9}\)

Depreciation The declining balance method is an accounting method in which the amount of depreciation taken each year is a fixed percentage of the present value of the item. If \(y\) is the value of the item in a given year, the depreciation taken is \(a y\) for some depreciation rate \(a\) with \(0

Exer. 47-50: Chemists use a number denoted by \(\mathrm{pH}\) to describe quantitatively the acidity or basicity of solutions. By definition, \(\mathrm{pH}=-\log \left[\mathrm{H}^{+}\right]\), where \(\left[\mathrm{H}^{+}\right]\)is the hydrogen ion concentration in moles per liter. Approximate the hydrogen ion concentration \(\left[\mathrm{H}^{+}\right]\)of each substance. (a) apples: \(\mathrm{pH} \approx 3.0\) (b) beer: \(\mathrm{pH} \approx 4.2\) (c) milk: \(\mathrm{pH} \approx 6.6\)

Language dating Glottochronology is a method of dating a language at a particular stage, based on the theory that over a long period of time linguistic changes take place at a fairly constant rate. Suppose that a language originally had \(N_{0}\) basic words and that at time \(t\), measured in millennia ( 1 millennium \(=1000\) years), the number \(N(t)\) of basic words that remain in common use is given by \(N(t)=N_{0}(0.805)^{t}\) (a) Approximate the percentage of basic words lost every 100 years. (b) If \(N_{0}=200\), sketch the graph of \(N\) for \(0 \leq t \leq 5\).

Exer. 47-50: Chemists use a number denoted by \(\mathrm{pH}\) to describe quantitatively the acidity or basicity of solutions. By definition, \(\mathrm{pH}=-\log \left[\mathrm{H}^{+}\right]\), where \(\left[\mathrm{H}^{+}\right]\)is the hydrogen ion concentration in moles per liter. A solution is considered basic if \(\left[\mathrm{H}^{+}\right]<10^{-7}\) or acidic if \(\left[\mathrm{H}^{+}\right]>10^{-7}\). Find the corresponding inequalities involving pH.

Some lending institutions calculate the monthly payment \(M\) on a loan of \(L\) dollars at an interest rate \(r\) (expressed as a decimal) by using the formula $$ M=\frac{L r k}{12(k-1)} $$ where \(k=[1+(r / 12)]^{12 t}\) and \(t\) is the number of years that the loan is in effect. Home mortgage (a) Find the monthly payment on a 30 -year \(\$ 250,000\) home mortgage if the interest rate is \(8 \%\). (b) Find the total interest paid on the loan in part (a).

(a) Prove that the function defined by \(f(x)=a x+b\) (a linear function) for \(a \neq 0\) has an inverse function, and find \(f^{-1}(x)\). (b) Does a constant function have an inverse? Explain.

Exer. 47-50: Chemists use a number denoted by \(\mathrm{pH}\) to describe quantitatively the acidity or basicity of solutions. By definition, \(\mathrm{pH}=-\log \left[\mathrm{H}^{+}\right]\), where \(\left[\mathrm{H}^{+}\right]\)is the hydrogen ion concentration in moles per liter. Many solutions have a \(\mathrm{pH}\) between 1 and 14. Find the corresponding range of \(\left[\mathrm{H}^{+}\right]\).

Some lending institutions calculate the monthly payment \(M\) on a loan of \(L\) dollars at an interest rate \(r\) (expressed as a decimal) by using the formula $$ M=\frac{L r k}{12(k-1)} $$ where \(k=[1+(r / 12)]^{12 t}\) and \(t\) is the number of years that the loan is in effect. Home mortgage Find the largest 25 -year home mortgage that can be obtained at an interest rate of \(7 \%\) if the monthly payment is to be \(\$ 1500\).

Show that the graph of \(f^{-1}\) is the reflection of the graph of \(f\) through the line \(y=x\) by verifying the following conditions: (1) If \(P(a, b)\) is on the graph of \(f\), then \(Q(b, a)\) is on the graph of \(f^{-1}\). (2) The midpoint of line segment \(P Q\) is on the line \(y=x\). (3) The line \(P Q\) is perpendicular to the line \(y=x\).

When the volume control on a stereo system is increased, the voltage across a loudspeaker changes from \(V_{1}\) to \(V_{2}\), and the decibel increase in gain is given by $$ \mathrm{db}=20 \log \frac{V_{2}}{V_{1}} . $$ Find the decibel increase if the voltage changes from 2 volts to \(4.5\) volts.

Use the compound interest formula to determine how long it will take for a sum of money to double if it is invested at a rate of \(6 \%\) per year compounded monthly.

Exer. 51-52: Approximate \(x\) to three significant figures. (a) \(\log x=3.6274\) (b) \(\log x=0.9469\) (c) \(\log x=-1.6253\) (d) \(\ln x=2.3\) (e) \(\ln x=0.05\) (f) \(\ln x=-1.6\)

Some lending institutions calculate the monthly payment \(M\) on a loan of \(L\) dollars at an interest rate \(r\) (expressed as a decimal) by using the formula $$ M=\frac{L r k}{12(k-1)} $$ where \(k=[1+(r / 12)]^{12 t}\) and \(t\) is the number of years that the loan is in effect. Car loan An automobile dealer offers customers no-downpayment 3 -year loans at an interest rate of \(10 \%\). If a customer can afford to pay \(\$ 500\) per month, find the price of the most expensive car that can be purchased.

Solve the compound interest formula $$ A=P\left(1+\frac{r}{n}\right)^{n t} $$ for \(t\) by using natural logarithms.

Exer. 51-52: Approximate \(x\) to three significant figures. (a) \(\log x=1.8965\) (b) \(\log x=4.9680\) (c) \(\log x=-2.2118\) (d) \(\ln x=3.7\) (e) \(\ln x=0.95\) (f) \(\ln x=-5\)

Some lending institutions calculate the monthly payment \(M\) on a loan of \(L\) dollars at an interest rate \(r\) (expressed as a decimal) by using the formula $$ M=\frac{L r k}{12(k-1)} $$ where \(k=[1+(r / 12)]^{12 t}\) and \(t\) is the number of years that the loan is in effect. Business loan The owner of a small business decides to finance a new computer by borrowing \(\$ 3000\) for 2 years at an interest rate of \(7.5 \%\). (a) Find the monthly payment. (b) Find the total interest paid on the loan.

Let \(n\) be any positive integer. Find the inverse function of \(f\) if (a) \(f(x)=x^{n}\) for \(x \geq 0\) (b) \(f(x)=x^{m / n}\) for \(x \geq 0\) and \(m\) any positive integer

Pareto's law for capitalist countries states that the relationship between annual income \(x\) and the number \(y\) of individuals whose income exceeds \(x\) is $$ \log y=\log b-k \log x, $$ where \(b\) and \(k\) are positive constants. Solve this equation for \(y\).

Change \(f(x)=1000(1.05)^{x}\) to an exponential function with base \(e\) and approximate the growth rate of \(f\).

Approximate the function at the value of \(x\) to four decimal places. (a) \(f(x)=13^{\sqrt{x+1.1}}\), \(x=3\) (b) \(g(x)=\left(\frac{5}{42}\right)^{-x}\), \(x=1.43\) (c) \(h(x)=\left(2^{x}+2^{-x}\right)^{2 x}\), \(x=1.06\)

Ventilation is an effective way to improve indoor air quality. In nonsmoking restaurants, air circulation requirements (in \(\mathrm{ft}^{3} / \mathrm{min}\) ) are given by the function \(V(x)=35 x\), where \(x\) is the number of people in the dining area. (a) Determine the ventilation requirements for 23 people. (b) Find \(V^{-1}(x)\). Explain the significance of \(V^{-1}\). (c) Use \(V^{-1}\) to determine the maximum number of people that should be in a restaurant having a ventilation capability of \(2350 \mathrm{ft}^{3} / \mathrm{min}\).

If \(p\) denotes the selling price (in dollars) of a commodity and \(x\) is the corresponding demand (in number sold per day), then the relationship between \(p\) and \(x\) is sometimes given by \(p=p_{0} e^{-a x}\), where \(p_{0}\) and \(a\) are positive constants. Express \(x\) as a function of \(p\).

Change \(f(x)=100\left(\frac{1}{2}\right)^{x}\) to an exponential function with base \(e\) and approximate the decay rate of \(f\).

Approximate the function at the value of \(x\) to four decimal places. (a) \(f(x)=2^{\sqrt[3]{1-x}}\), \(x=2.5\) (b) \(g(x)=\left(\frac{2}{25}+x\right)^{-3 x}, \quad x=2.1\) (c) \(h(x)=\frac{3^{-x}+5}{3^{x}-16}, \quad x=\sqrt{2}\)

The table lists the total numbers of radio stations in the United States for certain years. $$ \begin{array}{|c|c|} \hline \text { Year } & \text { Number } \\ \hline 1950 & 2773 \\ \hline 1960 & 4133 \\ \hline 1970 & 6760 \\ \hline 1980 & 8566 \\ \hline 1990 & 10,770 \\ \hline 2000 & 12,717 \\ \hline \end{array} $$ (a) Determine a linear function \(f(x)=a x+b\) that models these data, where \(x\) is the year. (b) Find \(f^{-1}(x)\). Explain the significance of \(f^{-1}\). (c) Use \(f^{-1}\) to predict the year in which there were 11,987 radio stations. Compare it with the true value, which is 1995 .

If \(v\) denotes the wind velocity (in \(\mathrm{m} / \mathrm{sec}\) ) at a height of \(z\) meters above the ground, then under certain conditions \(v=c \ln \left(z / z_{0}\right)\), where \(c\) is a positive constant and \(z_{0}\) is the height at which the velocity is zero. Sketch the graph of this equation on a \(z v\)-plane for \(c=0.5\) and \(z_{0}=0.1 \mathrm{~m}\).

If a 100 -milligram tablet of an asthma drug is taken orally and if none of the drug is present in the body when the tablet is first taken, the total amount \(A\) in the bloodstream after \(t\) minutes is predicted to be $$ A=100\left[1-(0.9)^{t}\right] \quad \text { for } \quad 0 \leq t \leq 10 $$ (a) Sketch the graph of the equation. (b) Determine the number of minutes needed for \(50 \mathrm{mil}-\) ligrams of the drug to have entered the bloodstream.

If we start with \(q_{0}\) milligrams of radium, the amount \(q\) remaining after \(t\) years is given by the formula \(q=q_{0}(2)^{-t / 600}\). Express \(t\) in terms of \(q\) and \(q_{0}\).

If the pollution of Lake Erie were stopped suddenly, it has been estimated that the level y of pollutants would decrease according to the formula \(y=y_{0} e^{-0.3821 t}\), where \(t\) is the time in years and \(y_{0}\) is the pollutant level at which further pollution ceased. How many years would it take to clear \(50 \%\) of the pollutants?