Chapter 1

(a) Show that \(f(x)=x^{2}-x, x \geq \frac{1}{2}\), is one to one, and find its inverse together with its domain. (b) Graph \(f(x)\) and \(f^{-1}(x)\) in one coordinate system, together with the line \(y=x\), and convince yourself that the graph of \(f^{-1}(x)\) can be obtained by reflecting the graph of \(f(x)\) about the line \(y=x\).

Solve \(\sec ^{2} x=\sqrt{3} \tan x+1\) on \([0, \pi)\).

Evaluate the following exponential expressions: (a) \(2^{4} 8^{-2 / 3}\) (b) \(\frac{3^{3} 3^{-1 / 2}}{3^{1 / 2}}\) (c) \(\frac{5^{k}(25)^{k-1}}{5^{2-k}}\)

The following table is based on a functional relationship be tween \(x\) and \(y\) that is either an exponential or a power function: \begin{tabular}{lc} \hline \(\boldsymbol{x}\) & \(\boldsymbol{y}\) \\ \hline \(0.5\) & \(1.21\) \\ 1 & \(0.74\) \\ \(1.5\) & \(0.45\) \\ 2 & \(0.27\) \\ \(2.5\) & \(0.16\) \\ \hline \end{tabular} Use an appropriate logarithmic transformation and a graph to decide whether the table comes from a power function or an exponential function, and find the functional relationship between \(x\) and \(y\).

The reciprocal of a function \(f(x)\) can be written as either \(1 / f(x)\) or \([f(x)]^{-1} .\) The point of this problem is to make clear that a reciprocal of a function has nothing to do with the inverse of a function. As an example, let \(f(x)=x+1, x \in \mathbf{R}\). Find both \([f(x)]^{-1}\) and \(f^{-1}(x)\), and compare the two functions. Graph all three functions together.

Evaluate the following exponential expressions: (a) \(\left(2^{4} 2^{-3 / 2}\right)^{2}\) (b) \(\left(\frac{6^{5 / 2} 6^{2 / 3}}{6^{1 / 3}}\right)^{3}\) (c) \(\left(\frac{3^{-2 k+3}}{3^{4+k}}\right)^{3}\)

The following table is based on a functional relationship between \(x\) and \(y\) that is either an exponential or a power function: \begin{tabular}{cc} \hline \(\boldsymbol{x}\) & \(\boldsymbol{y}\) \\ \hline\(-1\) & \(0.398\) \\ \(-0.5\) & \(1.26\) \\ 0 & 4 \\ \(0.5\) & \(12.68\) \\ 1 & \(40.18\) \\ \hline \end{tabular} Use an appropriate logarithmic transformation and a graph to decide whether the table comes from a power function or an exponential function, and find the functional relationship between \(x\) and \(y\).

Find the inverse of \(f(x)=3^{x}, x \in \mathbf{R}\), together with its domain, and graph both functions in the same coordinate system.

Which real number \(x\) satisfies (a) \(\log _{4} x=-2\) ? (b) \(\log _{1 / 3} x=-3 ?\) (c) \(\log _{10} x=-2\) ?

Find the inverse of \(f(x)=4^{x}, x \in \mathbf{R}\), together with its domain, and graph both functions in the same coordinate system.

Which real number \(x\) satisfies (a) \(\log _{2} x=-3\) ? (b) \(\log _{1 / 4} x=-\frac{1}{2} ?\) (c) \(\log _{3} x=0 ?\)

The following table is based on a functional relationship between \(x\) and \(y\) that is either an exponential or a power function: \begin{tabular}{ll} \hline \(\boldsymbol{x}\) & \(\boldsymbol{y}\) \\ \hline \(0.1\) & \(0.045\) \\ \(0.5\) & \(1.33\) \\ 1 & \(5.7\) \\ \(1.5\) & \(13.36\) \\ 2 & \(24.44\) \\ \hline \end{tabular} Use an appropriate logarithmic transformation and a graph to decide whether the table comes from a power function or an exponential function, and find the functional relationship between \(x\) and \(y\).

Find the inverse of \(f(x)=\left(\frac{1}{4}\right)^{x}, x \in \mathbf{R}\), together with its domain, and graph both functions in the same coordinate system.

Which real number \(x\) satisfies (a) \(\log _{1 / 2} 32=x\) ? (b) \(\log _{1 / 3} 81=x\) ? (c) \(\log _{10} 0.001=x ?\)

The following table is based on a functional relationship between \(x\) and \(y\) that is either an exponential or a power function: \begin{tabular}{ll} \hline \(\boldsymbol{x}\) & \(\boldsymbol{y}\) \\ \hline \(0.1\) & \(0.067\) \\ \(0.5\) & \(0.22\) \\ 1 & \(1.00\) \\ \(1.5\) & \(4.48\) \\ 2 & \(20.09\) \\ \hline \end{tabular} Use an appropriate logarithmic transformation and a graph to decide whether the table comes from a power function or an exponential function, and find the functional relationship between and \(y\).

Find the inverse of \(f(x)=\left(\frac{1}{3}\right)^{x}, x \in \mathbf{R}\), together with its domain, and graph both functions in the same coordinate system.

Which real number \(x\) satisfies (a) \(\log _{3} 81=x\) ? (b) \(\log _{5} \frac{1}{25}=x\) ? (c) \(\log _{10} 1000=x ?\)

Use the indicated base to logarithmically transform each exponential relationship so that a linear relationship results. Then use the indicated base to graph each relationship either in log or semilog transformed coordinates so that a straight line results. $$ y=2^{x} ; \text { base } 2 $$

Simplify the following expressions: (a) \(2^{5 \log _{2} x}\) (b) \(4^{3 \log _{4} x}\) (c) \(5^{5 \log _{1 / 5} x}\) (d) \(4^{3 \log _{2} x}\) (e) \(2^{3 \log _{1 / 2} x}\) (f) \(8^{\log _{1 / 2} x}\)

Simplify the following expressions: (a) \(-\ln \frac{1}{3}\) (b) \(\log _{4}\left(x^{2}-4\right)\) (c) \(\log _{2} 4^{3 x-1}\)

Use the indicated base to logarithmically transform each exponential relationship so that a linear relationship results. Then use the indicated base to graph each relationship either in log or semilog transformed coordinates so that a straight line results. $$ y=5^{x} ; \text { base } 5 $$

Simplify the following expressions: (a) \(\log _{4} 16^{x}\) (b) \(\log _{2} 16^{x}\) (c) \(\log _{3} 27^{x}\) (d) \(\log _{1 / 2} 4^{x}\) (e) \(\log _{1 / 2} 8^{-x}\) (f) \(\log _{3} 9^{-x}\)

Simplify the following expressions: (a) \(-\log _{3} \frac{1}{4}\) (b) \(\log \left(\frac{x^{3}-x}{x-1}\right)\) (c) \(\ln \left(e^{x-2}\right)\)

Use the indicated base to logarithmically transform each exponential relationship so that a linear relationship results. Then use the indicated base to graph each relationship either in log or semilog transformed coordinates so that a straight line results. $$ y=2^{-x} ; \text { base } 2 $$

Simplify the following expressions: (a) \(\ln x^{2}+\ln x^{-1}\) (b) \(\ln x^{4}-\frac{1}{3} \ln x^{-2}\) (c) \(\ln \left(x^{2}-1\right)-\ln (x+1)\) (d) \(\frac{1}{2} \ln x^{-1}+\ln x^{-3}\)

Solve for \(x\). (a) \(e^{3 x-1}=2\) (b) \(e^{-2 x}=10\) (c) \(e^{x^{2}-1}=10\)

Use the indicated base to logarithmically transform each exponential relationship so that a linear relationship results. Then use the indicated base to graph each relationship either in log or semilog transformed coordinates so that a straight line results. $$ y=3 e^{-2 x} ; \text { base } 3 $$

Simplify the following expressions: (a) \(e^{3 \ln x}\) (b) \(e^{-\ln \left(x^{2}+1\right)}\) (c) \(e^{-2 \ln (1 / x)}\) (d) \(e^{-2 \ln x}\)

Solve for \(x\). (a) \(5^{x}=625\) (b) \(4^{4 x}=256\) (c) \(10^{2 x}=0.0001\)

Suppose that \(N(t)\) denotes a population size at time \(t\) and satisfies the equation $$ N(t)=2 e^{3 t} \quad \text { for } t \geq 0 $$ (a) If you graph \(N(t)\) as a function of \(t\) on a semilog plot, a straight line results. Explain why. (b) Graph \(N(t)\) as a function of \(t\) on a semilog plot, and determine the slope of the resulting straight line.

Write the following expressions in terms of base \(e\), and simplify: (a) \(3^{x}\) (b) \(4^{x^{2}-1}\) (c) \(2^{-x-1}\) (d) \(3^{-4 x+1}\)

Solve for \(x\). (a) \(\ln (x-3)=5\) (b) \(\ln (x+2)+\ln (x-2)=1\) (c) \(\log _{3} x^{2}-\log _{3} 2 x=2\)

Suppose that you follow a population over time. When you plot your data on a semilog plot, a straight line with slope \(0.03\) results. Furthermore, assume that the population size at time 0 was 20\. If \(N(t)\) denotes the population size at time \(t\), what function best describes the population size at time \(t\) ?

Write the following expressions in terms of base \(e\) : (a) \(\log _{2}\left(x^{2}-1\right)\) (b) \(\log _{3}(5 x+1)\) (c) \(\log (x+2)\) (d) \(\log _{2}\left(2 x^{2}-1\right)\)

Solve for \(x\). (a) \(\log _{3}(2 x-1)=2\) (b) \(\ln (2-3 x)=0\) (c) \(\log (x)-\log (x+1)=\log \left(\frac{2}{3}\right)\)

Many studies have shown that the number of species on an island increases with the area of the island. Frequently, the functional relationship between the number of species, \(S\), and the area, \(A\), is approximated by \(S=C A^{z}\), where \(z\) is a constant that depends on the particular species and habitat in the study. (Actual values of \(z\) range from about \(0.2\) to \(0.35 .\) ) Suppose that the best fit to your data points on a log-log scale is a straight line. Is your model \(S=C A^{z}\) an appropriate description of your data? If yes, how would you find \(z\) ?

Show that the function \(y=(1 / 2)^{x}\) can be written in the form \(y=e^{-\mu x}\), where \(\mu\) is a positive constant. Determine \(\mu .\)

Simplify each expression and write it in the standard form \(a+b i\). \((3-2 i)-(-5+2 i)\)

Michaelis-Menten Equation Enzymes serve as catalysts in many chemical reactions in living systems. The simplest such reactions transform a single substrate into a product with the help of an enzyme. The Michaelis-Menten equation describes the rate of such enzymatically controlled reactions. The equation, which gives the relationship between the initial rate of the reaction, \(v\), and the concentration of the substrate, \(s\), is $$ v(s)=\frac{v_{\max } s}{s+K} $$ where \(v_{\max }\) is the maximum rate at which the product may be formed and \(K\) is called the Michaelis-Menten constant. Note that this equation has the same form as the Monod growth function. Given some data on the reaction rate \(v\), for different substrate concentrations \(s\), we would like to infer the parameters \(K\) and (a) The graph of \(v\) against \(s\) is nonlinear, so it is hard to determine \(K\) and \(v_{\max }\) directly from a graph of the function \(v(s) .\) In the remaining parts of this question you will be guided to transform your plot into one in which the dependent variable depends linearly on the independent variable. First plot, using a graphing calculator, or by hand, \(v(s)\) for the following values of \(K\) and \(v_{\max }\) : $$ \left(K, v_{\max }\right)=(1,1), \quad(2,1), \quad(1,2) $$ (b) Show that the Michaelis-Menten equation can be written in the form $$ \frac{1}{v}=\frac{K}{v_{\max }} \frac{1}{s}+\frac{1}{v_{\max }} $$

Show that if \(a>1\), then the function \(y=a^{x}\) can be written in the form \(y=e^{\mu x}\), where \(\mu\) is a positive constant. Write \(\mu\) in terms of \(a\).

Simplify each expression and write it in the standard form \(a+b i\). \((6+i)-4 i\)

The Richter magnitude scale is used to measure the strength of earthquakes. The magnitude \(m\) of an earthquake is calculated from the amplitude of shaking, \(A\) (measured in \(\mu \mathrm{m}\), where \(1 \mu \mathrm{m}=10^{-6} \mathrm{~m}\) ), measured by a seismometer, and from the distance of the seismometer to the epicenter of the earthquake, \(D\) (measured in \(\mathrm{km}\) ), using the following formula. $$ m=\log A-2.48+2.76 \log D $$ (a) A seismometer distance \(100 \mathrm{~km}\) from the earthquake epicenter measures shaking with an amplitude of \(100 \mu \mathrm{m} .\) Calculate \(m .\) (b) The smallest amplitude of shaking that most people can feel is \(1 \mathrm{~mm}\left(1 \mathrm{~mm}=10^{3} \mu \mathrm{m}\right) .\) Calculate the smallest magnitude of earthquake a person might feel if they were \(10 \mathrm{~km}\) away from the earthquake epicenter. (c) An earthquake is measured to have magnitude \(m=7.2\). Calculate the amplitude of shaking if (i) \(D=10 \mathrm{~km}\) from the epicenter. (ii) \(D=100 \mathrm{~km}\) from the epicenter. (d) Measured at the same distance from the epicenter, an increase of 1 in the Richter magnitude of an earthquake (e.g., from \(m=3\) to \(m=4\) ) corresponds to what factor increase in the amplitude of shaking?

Simplify each expression and write it in the standard form \(a+b i\). \((4-2 i)+(9+4 i)\)

In a case study in which the maximal rates of oxygen consumption (in \(\mathrm{ml} / \mathrm{s}\) ) of nine species of wild African mammals were plotted against body mass (in \(\mathrm{kg}\) ) on a log-log plot, it was found that the data points fell on a straight line with slope approximately equal to \(0.8\) and vertical-axis intercept approximately equal to \(0.105 .\) Find an equation that relates maximal oxygen consumption and body mass. (Adapted from Reiss, 1989).

Species Diversity In Example 4 we introduced the GiniSimpson index for measuring the diversity of a region containing two different types of individuals. Another index of diversity that is very commonly used is the Shannon diversity index. For a region that contains two different types of organisms, and in which a proportion \(p\) of organisms are of type 1, and a proportion \(1-p\) are of type 2, the Shannon diversity index is given by the formula: $$ H(p)=-p \ln p-(1-p) \ln (1-p) $$ (a) What is the domain of the function \(H\) ? (b) Show by plotting \(H\) against \(p\) that the maximum value for the Shannon diversity index occurs when \(p=1 / 2\), that is, when both types of organisms are equally abundant in the population. (c) Show that \(H\) has the following symmetry: \(H(1-p)=H(p)\). Explain why this implies that if we swap the labels on the types of organisms (so that type 1 is now called type 2 , and type 2 is now called type 1), then the Shannon diversity index will not change. (d) If type 1 goes extinct (so that \(p=0\) ), what happens to the Shannon diversity index? (i) First, explain why we cannot directly evaluate the formula for \(H\) if \(p=0\) (ii) Then consider what happens if \(p \neq 0\) but is very small, say \(p=0.001\). Then you should be able to use (1.10) to evaluate \(H(p) .\) (iii) Make a table of values of \(H(p)\) if \(p=0.001,10^{-4}\), \(10^{-5}, 10^{-6}\). Show that \(H\) gets closer to 0 as \(p\) gets smaller. Because of this observation we define \(H(0)=0\) and \(H(1)=1\). We will return to this idea when we discuss continuity in Chapter \(3 .\) (e) If we use \((1.10)\) to evaluate \(H\) when \(0

Simplify each expression and write it in the standard form \(a+b i\). \((6-4 i)+(2+5 i)\)

For each given pair of functions, use a graphing calculator to compare the functions. Describe what you see. \(y=\sin x\) and \(y=2 \sin x\)

Simplify each expression and write it in the standard form \(a+b i\). \(4(5+3 i)\)

Bite Strength The bite strength of finches increases as body mass increases. Van de May and Bout (2004) made the following measurements of bird mass \((B\), measured in \(\mathrm{g}\) ) and wind beat frequency \((f\), measured in \(\mathrm{Hz})\) : \begin{tabular}{lcc} \hline Species & Body Mass \((\boldsymbol{B}\), Measured in g) & Bite Strength \((\boldsymbol{S}, \mathbf{M e a s u r e d ~ i n ~} \mathbf{N})\) \\ \hline Java sparrow & \(30.4\) & \(9.6\) \\ Red-billed firefinch & \(6.9\) & \(1.2\) \\ Double barred finch & \(9.7\) & \(1.9\) \\ \hline \end{tabular} Assume that there is a power-law dependence of \(S\) upon \(B\) : \(S=a B^{c}\), where \(a\) and \(c\) are some constants. By plotting \(\log S\) against log \(B\), estimate the parameters \(a\) and \(c\).

For each given pair of functions, use a graphing calculator to compare the functions. Describe what you see. \(y=\cos x\) and \(y=\cos 2 x\)