Chapter 1

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

Hummingbird Flight Hummingbird wing-beat frequency decreases as bird mass increases. Altshuler et al. (2010) made the following measurements of bird size (measured in mass, \(B\), in \(g\) ) and wind beat frequency (frequency, \(f\), in \(\mathrm{Hz}\) ). \begin{tabular}{lcc} \hline & & Wing Beat \\ Species & Body Mass (B, & Frequency ( \(\boldsymbol{f},\), \\ & Measured in g) & Measured in Hz) \\ \hline Giant hummingbird & \(22.025\) & \(14.99\) \\ Volcano hummingbird & \(2.708\) & \(43.31\) \\ Blue-mantled thornbill & \(6.000\) & \(29.27\) \\ \hline \end{tabular} Assume that there is a power-law dependence of \(f\) upon \(B\) : $$ f=b B^{a} $$ for some constants \(a\) and \(b\). By plotting \(\log f\) against \(\log B\), estimate the parameters \(a\) and \(b\).

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

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

For each given pair of functions, use a graphing calculator to compare the functions. Describe what you see. \(y=\cos x\) and \(y=\cos (x+1)\)

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

Drug Absorption After a patient takes the painkiller acetaminophen (often sold under the brand name Tylenol), the concentration of drug in their blood increases at first, as the painkiller is absorbed into the blood, and then starts to decrease as the drug is metabolized or removed by the liver. In one study, the concentration of drug \((c\), measured in \(\mu \mathrm{g} / \mathrm{ml}\) ) was measured in a patient as a function of time \((t\), measured in hours since the drug was administered). The data in this example is taken from Rowling et al. (1977). \begin{tabular}{lc} \hline \(\boldsymbol{t}\) & \multicolumn{1}{c} {\(\boldsymbol{c}\)} \\ \hline 1 & \(10.61\) \\ \(1.5\) & \(8.73\) \\ 2 & \(7.63\) \\ 3 & \(5.55\) \\ 4 & \(3.97\) \\ 5 & \(3.01\) \\ 6 & \(2.39\) \\ \hline \end{tabular} (a) You want to determine from the data whether the relationship between concentration and time follows a power law $$ c=a t^{b} $$ for some set of constants \(a\) and \(b\), or whether it instead follows an exponential law $$ c=k d^{t} $$ for some constants \(k\) and \(d\). Explain how you could plot the data with transformed horizontal and vertical axes to determine which mathematical model is correct. (b) By plotting \(\log c\) against \(\log t\) in one graph, and \(\log c\) against \(t\) in another, explain why the data supports the second model (exponential decay) better than it supports the first model. (c) From your plot of \(\log c\) against \(t\), estimate the parameter \(d\).

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

Let \(z=1+2 i, u=2-3 i, v=1-5 i\), and \(w=1+i\). Compute the following expressions: $$ \bar{z} $$

For each given pair of functions, use a graphing calculator to compare the functions. Describe what you see. \(y=\tan x\) and \(y=\tan (x+1)\)

Let \(z=1+2 i, u=2-3 i, v=1-5 i\), and \(w=1+i\). Compute the following expressions: $$ z+u $$

Find the amplitude and the period of \(f(x)\) : $$ f(x)=2 \sin \left(\frac{x}{2}\right), \quad x \in \mathbf{R} $$

Let \(z=1+2 i, u=2-3 i, v=1-5 i\), and \(w=1+i\). Compute the following expressions: $$ \overline{z+v} $$

This problem relates to self-thinning of plants (Example 8). If we plot the logarithm of the over age weight of a plant, \(\log w\), against the logarithm of the density of plants, \(\log d\) (base 10 ), a straight line with slope \(-3 / 2\) results. Find the equation that relates \(w\) and \(d\), assuming that \(w=1 \mathrm{~g}\) when \(d=10^{3} \mathrm{~m}^{-2}\).

Find the amplitude and the period of \(f(x)\) : $$ f(x)=3 \cos 4 x, \quad x \in \mathbf{R} $$

Let \(z=1+2 i, u=2-3 i, v=1-5 i\), and \(w=1+i\). Compute the following expressions: $$ \overline{v-w} $$

Find the amplitude and the period of \(f(x)\) : $$ f(x)=-4 \sin (\pi x), \quad x \in \mathbf{R} $$

Let \(z=1+2 i, u=2-3 i, v=1-5 i\), and \(w=1+i\). Compute the following expressions: $$ \overline{v w} $$

Find the amplitude and the period of \(f(x)\) $$ f(x)=-\frac{3}{2} \sin \left(\frac{\pi}{3} x\right), \quad x \in \mathbf{R} $$

Let \(z=1+2 i, u=2-3 i, v=1-5 i\), and \(w=1+i\). Compute the following expressions: $$ \overline{u z} $$

Solve each quadratic equation in the complex number system. \(2 x^{2}-3 x+2=0\)

Solve each quadratic equation in the complex number system. \(x^{2}+x+1=0\)

Use the fact that \(\sec x=\frac{1}{\cos x}\) to explain why the maximum domain of \(y=\sec x\) consists of all real numbers except odd integer multiples of \(\pi / 2\).

Solve each quadratic equation in the complex number system. \(-x^{2}+x+2=0\)

Use the fact that \(\cot x=\frac{1}{\tan x}\) to explain why the maximum domain of \(y=\csc x\) consists of all real numbers except integer multiples of \(\pi\).

Solve each quadratic equation in the complex number system. \(x^{2}+2 x+3=0\)

Solve each quadratic equation in the complex number system. \(x^{2}+x+6=0\)

Logistic Transformation Suppose that $$ f(x)=\frac{1}{1+e^{-(b+m x)}} $$ where \(b\) and \(m\) are constants. A function of the form (1.15) is called a logistic function. The logistic function was introduced by the Dutch mathematical biologist Verhulst around 1840 to describe the growth of populations with limited food resources. (a) Show that $$ \ln \frac{f(x)}{1-f(x)}=b+m x $$ This transformation is called the logistic transformation. (b) Given some data, a table of values of \(x\), and the function \(f(x)\) at each value \(x\), explain how to plot the data to produce a straight line, and to estimate the constant \(b\) and \(m\) from the model.

Solve each quadratic equation in the complex number system. \(-2 x^{2}+4 x-3=0\)

Not every study of species diversity as a function of productivity produces a hump-shaped curve. Owen (1988) studied rodent assemblages in Texas and found that the number of species was a decreasing function of productivity. Sketch a graph that would describe this situation.

First determine whether the solutions of each quadratic equation are real or complex without solving the equation. Then solve the equation. \(3 x^{2}-4 x-7=0\)

Species diversity in a community may be controlled by disturbance frequency. The intermediate disturbance hypothesis states that species diversity is greatest at intermediate disturbance levels. Sketch a graph of species diversity as a function of disturbance frequency that illustrates this hypothesis.

First determine whether the solutions of each quadratic equation are real or complex without solving the equation. Then solve the equation. \(x^{2}-x-1=0\)

Preston (1962) investigated the dependence of number of bird species on island area in the West Indian islands. He found that the number of bird species increased at a decelerating rate as island area increased. Sketch this relationship.

First determine whether the solutions of each quadratic equation are real or complex without solving the equation. Then solve the equation. \(3 x^{2}-x-4=0\)

Phytoplankton converts carbon dioxide to organic com. pounds during photosynthesis. This process requires sunlight. It has been observed that the rate of photosynthesis is a function of light intensity: The rate of photosynthesis increases approx imately linearly with light intensity at low intensities, saturates at intermediate levels, and decreases slightly at high intensities. Sketch a graph of the rate of photosynthesis as a function of light intensity.

First determine whether the solutions of each quadratic equation are real or complex without solving the equation. Then solve the equation. \(4 x^{2}-x+1=0\)

Brown lemming densities in the tundra areas of North America and Eurasia show cyclic behavior: Over three to four years, lemming densities build up, and they typically crash the next year. Sketch a graph that describes this situation.

First determine whether the solutions of each quadratic equation are real or complex without solving the equation. Then solve the equation. \(3 x^{2}-5 x+6=0\)

Yang et al. (2014) measured the time that animals of different sizes spend urinating. For animals larger than \(1 \mathrm{~kg}\), the time spent urinating increases (slowly) with animal size. The smallest animal in their study was a cat (mass \(5 \mathrm{~kg}\), duration of urination 18s) and the largest was an elephant (mass \(5000 \mathrm{~kg}\), duration of urination \(29 \mathrm{~s}\) ). Make a sketch of time spent urinating as a function of animal size.

First determine whether the solutions of each quadratic equation are real or complex without solving the equation. Then solve the equation. \(-3 x^{2}-x-4=0\)

A study of Borcherts (1994) investigated the relationship between stem water storage and wood density in a number of tree species in Costa Rica. The study showed that water storage is inversely related to wood density; that is, higher wood density corresponds to lower water content. Sketch a graph of water content as a function of wood density that illustrates this situation.

If \(z=a+b i\), find \(z+\bar{z}\) and \(z-\bar{z}\).

Species richness can be a hump-shaped function of productivity. In the same coordinate system, sketch two hump-shaped graphs of species richness as a function of productivity, one in which the maximum occurs at low productivity and one in which the maximum occurs at high productivity.

If \(z=a+b i\), find \(\bar{z}\). Use your answer to compute \(\overline{(\bar{z})}\), and compare your answer with \(z\).

The size distribution of zooplankton in a lake is typically a hump-shaped curve; that is, the number of zooplankton of a given size increases with size up to a critical size and then decreases with size for organisms larger than that critical size. Brooks and Dodson (1965) studied the effects of introducing a planktivorous fish in a lake. They found that the composition of zooplankton after the fish was introduced shifted to smaller individuals. In the same coordinate system, sketch the size distribution of zooplankton before and after the introduction of the planktivorous fish.

Show \(\overline{(\bar{z})}=z\)

Daphnia is a genus of zooplankton that comprises a number of species. The body growth rate of Daphnia depends on food concentration. A minimum food concentration is required for growth: Below this level, the growth rate is negative; above, it is positive. In a study by Gliwicz (1990), it was found that growth rate is an increasing function of food concentration and that the minimum food concentration required for growth decreases with increasing size of the animal. Sketch two graphs in the same coordinate system, one for a large and one for a small Daphnia species, that illustrates this situation.

Show \(\overline{z+w}=\bar{z}+\bar{w}\)

Show \(\overline{z w}=\bar{z} \bar{w}\).