Chapter 1

For each function, find the largest possible domain and determine the range. $$ f(x)=\frac{1}{x^{2}+1} $$

Determine the equation of the line that satisfies the stated requirements. Put the equation in standard form. The line passing through \((-1,5)\) and parallel to the horizontal line passing through \((2,-1)\)

The longest known species of worms is the earthworm Microchaetus rappi of South Africa; in 1937, a 6.7-m-long specimen was collected from the Transvaal. The shortest worm is Chaetogaster annandalei, which measures less than \(0.51 \mathrm{~mm}\) in length. \(M .\) rappi is order(s) of magnitude longer than C. annandalei.

Compare \(y=\frac{1}{x}\) and \(y=\frac{1}{x^{2}}\) for \(x>0\) by graphing the two functions. Where do the curves intersect? Which function is greater for small values of \(x ?\) for large values of \(x\) ?

Determine the equation of the line that satisfies the stated requirements. Put the equation in standard form. The line passing through \((-1,1)\) and parallel to the vertical line passing through \((2,-1)\)

Both the La Plata river dolphin (Pontoporia blainvillei) and the sperm whale ( Physeter macrocephalus) belong to the suborder Odontoceti (individuals that have teeth). A La Plata river dolphin weighs between 30 and \(50 \mathrm{~kg}\), whereas a sperm whale weighs between 35,000 and \(40,000 \mathrm{~kg}\). A sperm whale is order(s) of magnitude heavier than a La Plata river dolphin.

Let \(n\) and \(m\) be two positive integers with \(m \leq n\). Answer the following questions about \(y=x^{-n}\) and \(y=x^{-m}\) for \(x>0\) : Where do the curves intersect? Which function is greater for small values of \(x ?\) for large values of \(x\) ?

Determine the equation of the line that satisfies the stated requirements. Put the equation in standard form. The line passing through \((3,1)\) and parallel to the vertical line passing through \((-1,-2)\)

Compare a ball of radius \(1 \mathrm{~cm}\) against a ball of radius \(10 \mathrm{~cm}\). The radius of the larger ball is \(-\) order(s) of magnitude bigger than the radius of the smaller ball. The volume of the larger ball is order(s) of magnitude bigger than the volume of the smaller ball.

Let $$ f(x)=\frac{1}{x+1}, \quad x>-1 $$ (a) Use a graphing calculator to graph \(f(x)\). (b) On the basis of the graph in (a), determine the range of \(f(x)\). (c) For which values of \(x\) is \(f(x)=2\) ? (d) On the basis of the graph in (a), determine how many solutions \(f(x)=a\) has, where \(a\) is in the range of \(f(x)\)

Determine the equation of the line that satisfies the stated requirements. Put the equation in standard form. The line passing through \((1,-3)\) and perpendicular to the horizontal line passing through \((-1,-1)\)

Compare a square with side length \(1 \mathrm{~m}\) against a square with side length \(100 \mathrm{~m}\). The area of the larger square is order(s) of magnitude larger than the area of the smaller square.

Let $$ f(x)=\frac{2 x}{3+x}, \quad x \geq 0 $$ (a) Use a graphing calculator to graph \(f(x)\). (b) Find the range of \(f(x)\). (c) For which values of \(x\) is \(f(x)=1 ?\) (d) Based on the graph in (a), explain in words why, for any value \(a\) in the range of \(f(x)\), you can find exactly one value \(x \geq 0\) such that \(f(x)=a\). Determine \(x\) by solving \(f(x)=a\).

Determine the equation of the line that satisfies the stated requirements. Put the equation in standard form. The line passing through \((4,2)\) and perpendicular to the horizontal line passing through \((3,1)\)

The diameter of a typical bacterium is about \(0.5\) to \(1 \mu \mathrm{m}\). An exception is the bacterium Epulopiscium fishelsoni, which is about \(600 \mu \mathrm{m}\) long and \(80 \mu \mathrm{m}\) wide. The volume of \(E\). fishelsoni is about order(s) of magnitude larger than that of a typical bacterium. (Hint: Approximate the shape of a typical bacterium by a sphere and the shape of \(E\). fishelsoni by a cylinder.)

Let $$ f(x)=\frac{3 x}{1+x}, \quad x \geq 0 $$ (a) Use a graphing calculator to graph \(f(x)\). (b) Find the range of \(f(x)\). (c) For which values of \(x\) is \(f(x)=2\) ? (d) On the basis of the graph in (a), explain in words why, for any value \(a\) in the range of \(f(x)\), you can find exactly one value \(x \geq 0\) such that \(f(x)=a\). Determine \(x\) by solving \(f(x)=a\).

Determine the equation of the line that satisfies the stated requirements. Put the equation in standard form. The line passing through \((7,3)\) and perpendicular to the vertical line passing through \((-2,4)\)

The length of a typical bacterial cell is about one-tenth that of a small eukaryotic cell. Consequently, the cell volume of a bacterium is about \(-\) order(s) of magnitude smaller than that of a small eukaryotic cell. (Hint: Approximate the shapes of both types of cells by spheres.)

In Problems 42-44, we discuss the Monod growth function, which was introduced in Example 6 of this section. Use a graphing calculator to investigate the Monod growth function $$ r(N)=\frac{a N}{k+N}, \quad N \geq 0 $$ where \(a\) and \(k\) are positive constants. (a) Graph \(r(N)\) for (i) \(a=5\) and \(k=1\), (ii) \(a=5\) and \(k=3\), and (iii) \(a=8\) and \(k=1\). Place all three graphs in one coordinate system. (b) On the basis of the graphs in (a), describe in words what happens when you change \(a\). (c) On the basis of the graphs in (a), describe in words what happens when you change \(k\).

Determine the equation of the line that satisfies the stated requirements. Put the equation in standard form. The line passing through \((-2,5)\) and perpendicular to the vertical line passing through \((1,4)\)

We discuss the Monod growth function, which was introduced in Example 6 of this section. The Monod growth function \(r(N)\) describes growth as a function of nutrient concentration \(N\). Assume that $$ r(N)=5 \frac{N}{1+N}, \quad N \geq 0 $$ Find the percentage increase when the nutrient concentration is doubled from \(N=0.1\) to \(N=0.2\). Compare this result with what you find when you double the nutrient concentration from \(N=10\) to \(N=20\). This is an example of diminishing return.

To convert a length measured in feet to a length measured in centimeters, we use the facts that a length measured in feet is proportional to a length measured in centimeters and that \(1 \mathrm{ft}\) corresponds to \(30.5 \mathrm{~cm}\). If \(x\) denotes the length measured in \(\mathrm{ft}\) and \(y\) denotes the length measured in \(\mathrm{cm}\), then $$ y=30.5 x $$ (a) Explain how to use this relationship. (b) Use the relationship to convert the following measurements into centimeters: (i) \(6 \mathrm{ft}\) (ii) \(3 \mathrm{ft}, 2 \mathrm{in}\) (iii) \(1 \mathrm{ft}, 7\) in (c) Use the relationship to convert the following measurements into ft: (i) \(173 \mathrm{~cm}\) (ii) \(75 \mathrm{~cm}\) (iii) \(48 \mathrm{~cm}\)

When log \(y\) is graphed as a function of \(x, a\) straight line results. Graph straight lines, each given by two points, on a log-linear plot, and determine the functional relationship. (The original \(x-y\) coordinates are given.) $$ \left(x_{1}, y_{1}\right)=(-1,4),\left(x_{2}, y_{2}\right)=(2,8) $$

We discuss the Monod growth function, which was introduced in Example 6 of this section. The Monod growth function \(r(N)\) describes growth as a function of nutrient concentration \(N\). Assume that $$ r(N)=a \frac{N}{k+N}, \quad N \geq 0 $$ where \(a\) and \(k\) are positive constants. (a) What happens to \(r(N)\) as \(N\) increases? Use this relationship to explain why \(a\) is called the saturation level. (b) Show that \(k\) is the half-saturation constant; that is, show that if \(N=k\), then \(r(N)=a / 2\).

(a) To convert the weight of an object from kilograms (kg) to pounds (lb), you use the facts that a weight measured in kilograms is proportional to a weight measured in pounds and that 1 kg corresponds to \(2.20 \mathrm{lb} .\) Find an equation that relates weight measured in kilograms to weight measured in pounds. (b) Use your answer in (a) to convert the following measurements: (i) \(63 \mathrm{lb}\) (ii) \(150 \mathrm{lb}\) (iii) \(2.5 \mathrm{~kg}\) (iv) \(140 \mathrm{~kg}\)

When log \(y\) is graphed as a function of \(x, a\) straight line results. Graph straight lines, each given by two points, on a log-linear plot, and determine the functional relationship. (The original \(x-y\) coordinates are given.) $$ \left(x_{1}, y_{1}\right)=(-2,3),\left(x_{2}, y_{2}\right)=(1,1) $$

Assume that the distance a car travels is proportional to the time it takes to cover the distance. Find an equation that relates distance and time if it takes the car 15 min to travel \(10 \mathrm{mi}\). What is the constant of proportionality if distance is measured in miles and time is measured in hours?

When log \(y\) is graphed as a function of \(x, a\) straight line results. Graph straight lines, each given by two points, on a log-linear plot, and determine the functional relationship. (The original \(x-y\) coordinates are given.) $$ \left(x_{1}, y_{1}\right)=(1,4),\left(x_{2}, y_{2}\right)=(6,1) $$

The function $$ f(x)=\frac{x^{n}}{b^{n}+x^{n}}, \quad x \geq 0 $$ where \(n\) is a positive integer and \(b\) is a positive real number, is used in biochemistry to model reaction rates as a function of the concentration of some reactants. (a) Use a graphing calculator to graph \(f(x)\) for \(n=1,2\), and 3 in one coordinate system when \(b=2\). (b) Where do the three graphs in (a) intersect? (c) What happens to \(f(x)\) as \(x\) gets larger? (d) For an arbitrary positive value of \(b\), show that \(f(b)=1 / 2\). On the basis of this demonstration and your answer in (c), explain why \(b\) is called the half-saturation constant.

Assume that the number of seeds a plant produces is proportional to its aboveground biomass. Find an equation that relates number of seeds and aboveground biomass if a plant that weighs \(217 \mathrm{~g}\) has 17 seeds.

In Problems \(47-54\), use a logarithmic transformation to find a linear relationship between the given quantities and graph the resulting linear relationship on a log-linear plot. $$ y=3 \times 10^{-2 x} $$

Use a graphing calculator to sketch the graphs of the functions. $$ y=x^{3 / 2}, x \geq 0 $$

Experimental study plots are often squares of length \(1 \mathrm{~m}\). If \(1 \mathrm{ft}\) corresponds to \(0.305 \mathrm{~m}\), compute the area of a square plot of length \(1 \mathrm{~m}\) in \(\mathrm{ft}^{2}\).

use a logarithmic transformation to find a linear relationship between the given quantities and graph the resulting linear relationship on a log-linear plot. $$ y=4 \times 10^{5 x} $$

Use a graphing calculator to sketch the graphs of the functions. $$ y=x^{1 / 3}, x \geq 0 $$

Large areas are often measured in hectares (ha) or in acres. If \(1 \mathrm{ha}=10,000 \mathrm{~m}^{2}\) and 1 acre \(=4046.86 \mathrm{~m}^{2}\), how many acres is 1 hectare?

use a logarithmic transformation to find a linear relationship between the given quantities and graph the resulting linear relationship on a log-linear plot. $$ y=2 e^{-1.2 x} $$

Use a graphing calculator to sketch the graphs of the functions. $$ y=x^{-1 / 4}, x>0 $$

To convert the volume of a liquid measured in ounces to a volume measured in liters, we use the fact that 1 liter equals \(33.81\) ounces. Denote by \(x\) the volume measured in ounces and by \(y\) the volume measured in liters. Assume a linear relationship between these two units of measurements. (a) Find the equation relating \(x\) and \(y\). (b) A typical soda can contains 12 ounces of liquid. How many liters is this?

use a logarithmic transformation to find a linear relationship between the given quantities and graph the resulting linear relationship on a log-linear plot. $$ y=7 e^{3 x} $$

Use a graphing calculator to sketch the graphs of the functions. $$ y=2 x^{-7 / 8}, x>0 $$

To convert a distance measured in miles to a distance measured in kilometers, we use the fact that 1 mile equals \(1.609\) kilometers. Denote by \(x\) the distance measured in miles and by \(y\) the distance measured in kilometers. Assume a linear relationship between these two units of measurements. (a) Find an equation relating \(x\) and \(y\). (b) The distance between Minneapolis and Madison is 261 miles. How many kilometers is this?

use a logarithmic transformation to find a linear relationship between the given quantities and graph the resulting linear relationship on a log-linear plot. $$ y=5 \times 2^{4 x} $$

(a) Graph \(y=x^{-1 / 2}, x>0\), and \(y=x^{1 / 2}, x \geq 0\), together, in one coordinate system. (b) Show algebraically that $$ x^{-1 / 2} \geq x^{1 / 2} $$ for \(0

Car speed in many countries is measured in kilometers per hour. In the United States, car speed is measured in miles per hour. To convert between these units, use the fact that 1 mile equals \(1.609\) kilometers. (a) The speed limit on many U.S. highways is 55 miles per hour. Convert this number into kilometers per hour. (b) The recommended speed limit on German highways is 130 kilometers per hour. Convert this number into miles per hour.

use a logarithmic transformation to find a linear relationship between the given quantities and graph the resulting linear relationship on a log-linear plot. $$ v=6 \times 2^{-0.9 x} $$

(a) Graph \(y=x^{5 / 2}, x \geq 0\), and \(y=x^{1 / 2}, x \geq 0\), together, in one coordinate system. (b) Show algebraically that $$ x^{5 / 2} \leq x^{1 / 2} $$ for \(0 \leq x \leq 1\). (Hint: Show that \(x^{1 / 2} / x^{-1 / 2}=x \leq 1\) for \(0

To measure temperature, three scales are commonly used: Fahrenheit, Celsius, and Kelvin. These scales are linearly related. We discuss these scales in Problems 52 and \(53 .\) (a) The Celsius scale is devised so that \(0^{\circ} \mathrm{C}\) is the freezing point of water (at 1 atmosphere of pressure) and \(100^{\circ} \mathrm{C}\) is the boiling point of water (at 1 atmosphere of pressure). If you are more familiar with the Fahrenheit scale, then you know that water freezes at \(32^{\circ} \mathrm{F}\) and boils at \(212^{\circ} \mathrm{F}\). Find a linear equation that relates temperature measured in degrees Celsius and temperature measured in degrees Fahrenheit. (b) The normal body temperature in humans ranges from \(97.6^{\circ} \mathrm{F}\) to \(99.6^{\circ} \mathrm{F}\). Convert this temperature range into degrees Celsius.

use a logarithmic transformation to find a linear relationship between the given quantities and graph the resulting linear relationship on a log-linear plot. $$ y=4 \times 3^{2 x} $$

In Problems 53-56, sketch each scaling relation (Niklas, 1994). In a sample based on 46 species, leaf area was found to be proportional to (stem diameter) \(^{1.84}\). On the basis of your graph, as stem diameter increases, does leaf area increase or decrease?