Chapter 1

(a) Show that \(f(x)=\sqrt{x}, x \geq 0\), 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 \([0, \pi)\).

73\. The following table is based on a functional relationship between \(x\) and \(y\) that is either an exponential or a power function: \begin{tabular}{rl} \hline \(\boldsymbol{x}\) & \(\boldsymbol{y}\) \\ \hline 1 & \(1.8\) \\ 2 & \(2.07\) \\ 4 & \(2.38\) \\ 10 & \(2.85\) \\ 20 & \(3.28\) \\ \hline \end{tabular}

(a) Show that \(f(x)=1 / x^{3}, x>0\), 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 \(y=x\)

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

74\. 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 \(\boldsymbol{x}\) & \multicolumn{1}{c} {\(\boldsymbol{y}\)} \\ \hline \(0.5\) & \(7.81\) \\ 1 & \(3.4\) \\ \(1.5\) & \(2.09\) \\ 2 & \(1.48\) \\ \(2.5\) & \(1.13\) \\ \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 \(y\).

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) \(\log _{10} x=-2 ?\)

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 \(\boldsymbol{x}\) & \multicolumn{1}{c} {\(\boldsymbol{y}\)} \\ \hline 0 & 3 \\ \(0.5\) & \(2.20\) \\ 1 & \(1.61\) \\ \(1.5\) & \(1.18\) \\ 2 & \(0.862\) \\ \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 \(f(x)=5^{x}, x \in \mathbf{R}\), together with its domain, and graph both functions in the same coordinate system.

Which real number \(x\) satisfies (a) \(\log _{1 / 2} x=-4\) ? (b) \(\log _{1 / 4} x=2 ?\) (c) \(\log _{5} x=3\) ?

The following table is based on a functional relationship between \(x\) and \(y\) that is either an exponential or a power function: \begin{tabular}{ll} \hline \(\boldsymbol{x}\) & \multicolumn{1}{c} {\(\boldsymbol{y}\)} \\ \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 \(\boldsymbol{x}\) & \multicolumn{1}{c} {\(\boldsymbol{y}\)} \\ \hline \(0.1\) & \(1.72\) \\ \(0.5\) & \(1.41\) \\ 1 & \(1.11\) \\ \(1.5\) & \(0.872\) \\ 2 & \(0.685\) \\ \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 \(f(x)=\left(\frac{1}{3}\right)^{x}, x \in \mathbf{R}\), together with its domain, and graph both functions in the same coordinate system.

Which real number \(x\) satisfies (a) \(\log _{4} 64=x ?\) (b) \(\log _{1 / 5} 625=x ?\) (c) \(\log _{10} 10,000=x\) ?

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 in a coordinate system whose axes are accordingly transformed so that a straight line results. $$ y=2^{x} ; \text { base } 2 $$

Find the inverse of \(f(x)=2^{x}, x \geq 0\), together with its domain, and graph both functions in the same coordinate system.

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 in a coordinate system whose axes are accordingly transformed so that a straight line results. $$ y=3^{x} ; \text { base } 3 $$

Find the inverse of \(f(x)=\left(\frac{1}{2}\right)^{x}, x \geq 0\), together with its domain, and graph both functions in the same coordinate system.

Simplify the following expressions: (a) \(-\ln \frac{1}{5}\) (b) \(\ln \frac{x^{2}-y^{2}}{-\sqrt{x}}\) (c) \(\log _{3} 3^{2 x+1}\)

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 in a coordinate system whose axes are accordingly transformed so that a straight line results. $$ y=2^{-x} ; \text { base } 2 $$

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

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 in a coordinate system whose axes are accordingly transformed so that a straight line results. $$ y=3^{-x} ; \text { base } 3 $$

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) \(\log _{3} 9^{-x}\)

Solve for \(x\). (a) \(3^{x}=81\) (b) \(9^{2 x+1}=27\) (c) \(10^{5 x}=1000\)

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.

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

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 \(t\) ?

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) \(\ln (2 x-3)=0\) (b) \(\log _{2}(1-x)=3\) (c) \(\ln x^{3}-2 \ln x=1\)

Species-Area Curves 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 \(z\) ?

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

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

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) \(\log _{2}\left(2 x^{2}-1\right)\)

Simplify each expression and write it in the standard form \(a+b i\). \((7+i)-4\)

Continuation of Problem 86) Estimating \(v_{\max }\) and \(K_{m}\) from the Lineweaver-Burk graph as described in Problem 86 is not always satisfactory. A different transformation typically yields better estimates (Dowd and Riggs, 1965 ). Show that the Michaelis-Menten equation can be written as $$ \frac{v_{0}}{s_{0}}=\frac{v_{\max }}{K_{m}}-\frac{1}{K_{m}} v_{0} $$ and explain why this transformation results in a straight line when you graph \(v_{0}\) on the horizontal axis and \(\frac{v_{0}}{s_{0}}\) on the vertical axis. Explain how you can estimate \(v_{\max }\) and \(K_{m}\) from the graph.

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 \(a+b i\). \((4-2 i)+(9+4 i)\)

Adapted from Reiss, 1989\()\) In a case study in which the maximal rates of oxygen consumption (in \(\mathrm{ml} / \mathrm{s}\) ) of nine species of wild African mammals (Taylor et al., 1980 ) 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.

Show that if \(0

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

Adapted from Benton and Harper, 1997) In vertebrates, embryos and juveniles have large heads relative to their overall body size. As the animal grows older, proportions change; for instance, the ratio of skull length to body length diminishes. That this is the case not only for living vertebrates, but also for fossil vertebrates, is shown by the following example: Ichthyosaurs are a group of marine reptiles that appeared in the early Triassic and died out well before the end of the Cretaceous. \({ }^{1}\) They were fish shaped and comparable in size to dolphins. In a study of 20 fossil skeletons, the following allometric relationship between skull length \(S\) (measured in \(\mathrm{cm}\) ) and backbone length \(B\) (measured in \(\mathrm{cm}\) ) was found: $$ S=1.162 B^{0.93} $$ (a) Choose suitable transformations of \(S\) and \(B\) so that the resulting relationship is linear. Plot the transformed relationship, and find the slope and the \(y\) -intercept. (b) Explain why the allometric equation confirms that juveniles had relatively large heads. (Hint: Compute the ratio of \(S\) to \(B\) for a number of different values of \(B-\) say \(, 10 \mathrm{~cm}, 100 \mathrm{~cm}, 500 \mathrm{~cm}-\) and compare.)

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