Chapter 1

For each function, find the largest possible domain and determine the range. $$ \text { 36. } f(x)=\frac{1}{1-x^{2}} $$

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 worm is the earthworm \(M i\) crochaetus 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 \((-2,3)\) 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 \(\quad\) 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 \((-1,-2)\)

The Etruscan shrew (Suncus etruscus) is by weight the smallest mammal in the world, weighing \(1.8 \mathrm{~g}\) on average. By comparison, the largest mammal in the world, the blue whale \((B a l-\) aenoptera musculus), may weigh as much as 190 tonnes \(\left(190 \times 10^{3}\right.\) \(\mathrm{kg}\) ). A blue whale is order(s) of magnitude heavier than an Etruscan shrew.

Let $$ f(x)=\frac{x}{x+1}, \quad x>-1 $$ (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 \(f(x)=3 / 4\) ? (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 $$ f(x)=\frac{2}{3+x}, \quad x>-3 $$ (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 in the range of \(f(x)\), you can find exactly one value \(x \geq 0\) such that \(f(x)=a\). Determine \(x\) for general \(a\) 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 \((1,3)\) and perpendicular to the horizontal line passing through \((3,2)\)

Let $$ f(x)=\frac{2 x+1}{1+x}, \quad x \geq 0 $$ (a) Use a graphing calculator to graph \(f(x)\). (b) Find the range of \(f(x)\). (c) For which values of \(x\) is \(f(x)=5 / 4\) ? (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 \((-1,-7)\)

he 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 \(\quad\) order(s) of magnitude smaller than that of a small eukaryotic cell. (Hint: Approximate the shapes of both cells he coberes

In Problems 42-44, we discuss the Monod growth function, which was introduced in Example 6 of this section. In Example 6 we met the Monod growth function. The most general form of this function has two constants in it: $$ r(N)=\frac{a N}{k+N}. $$ (Compare with Example 6, where we took \(k=1 .\) ) In this question we will consider how the function changes if the constants \(a\) or \(b\) are changed. (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) By comparing the graphs of (a)(i) and (a)(iii), describe in words what happens when you change \(a\). (c) By comparing the graphs of (a)(i) and (a)(ii), 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 \((-1,4)\)

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 rela\mathrm{\\{} t i o n s h i p . ~ ( T h e ~ o r i g i n a l ~ \(x-y\) coordinates are given.) $$ \left(x_{1}, y_{1}\right)=(0,5),\left(x_{2}, y_{2}\right)=(3,1) $$

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 $$ r(N)=\frac{5 N}{1+N}, \quad N \geq 0 $$ 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 \(N=20\) to \(N=40 .\) This is an example of diminishing return.

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 rela\mathrm{\\{} t i o n s h i p . ~ ( T h e ~ o r i g i n a l ~ \(x-y\) 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. In Example 6 we met the Monod growth function. The most general form of this function has two constants in it: $$ r(N)=\frac{a N}{k+N}. $$ (Compare with Example 6 where we took \(k=1 .\) ) In this question we will consider how, given some experimental data, we can determine values for \(a\) and \(k\) to fit the Monod growth function to the data. First, we measure growth rate for three values of \(N\) : $$ \begin{array}{cl} \hline \boldsymbol{N} & \boldsymbol{r}(\boldsymbol{N}) \\ \hline 0 & 0 \\ 2 & 1.5 \\ 4 & 2 \\ \hline \end{array} $$ We want to find the values of \(a\) and \(k\) that would fit the Monod growth function to this data. Write out the equations for \(r(0)\), \(r(2)\), and \(r(4)\); $$ \begin{array}{l} r(0): 0=0\\\ r(2): \frac{2 a}{k+2}=1.5\\\ r(4): \frac{4 a}{k+4}=2 \end{array} $$ Equation (1.5) is automatically satisfied. We need to pick values of \(a\) and \(k\) that satisfy \((1.6)\) and \((1.7) .\) To do this, we need to eliminate one variable so that we have one equation in one unknown. (a) To eliminate \(a\), divide \((1.6)\) into \((1.7)\) (i.e., divide the left-hand side of \((1.7)\) by the left-hand side of \((1.6)\) and the right-hand side of \((1.7)\) by the right-hand side of \((1.6))\) : $$ \frac{2(k+2)}{k+4}=\frac{2}{1.5}=\frac{4}{3}. $$ (i) Solve this equation for \(k\). (ii) Substitute your value for \(k\) back into (1.6) and solve for \(a\). (iii) What if you instead substitute your value for \(k\) from (i) into (1.7), and solve for \(a\) ? Do you get a different answer? (b) Suppose in a different experiment you measured the following data: $$ \begin{array}{ll} \hline \boldsymbol{N} & \boldsymbol{r}(\boldsymbol{N}) \\ \hline 0 & 0 \\ 1 & 1 \\ 3 & 2.25 \\ \hline \end{array} $$ Calculate values for \(a\) and \(k\) to fit the Monod growth function to this data. (c) Suppose in a different experiment you measured the following data: $$ \begin{array}{cl} \hline \boldsymbol{N} & \boldsymbol{r}(\boldsymbol{N}) \\ \hline 0 & 0.5 \\ 1 & 1 \\ 3 & 1.5 \\ \hline \end{array} $$ Are there any values for \(a\) and \(k\) that would fit the Monod growth function to this data?

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 rela\mathrm{\\{} t i o n s h i p . ~ ( T h e ~ o r i g i n a l ~ \(x-y\) coordinates are given.) $$ \left(x_{1}, y_{1}\right)=(-1,1),\left(x_{2}, y_{2}\right)=(1,3) $$

Let $$ f(x)=\frac{x^{2}}{4+x^{2}}, \quad x \geq 0 $$ (a) Use a graphing calculator to graph \(f(x)\). (b) On the basis of your graph in (a), find the range of \(f(x)\). (c) What happens to \(f(x)\) as \(x\) gets larger?

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 rela\mathrm{\\{} t i o n s h i p . ~ ( T h e ~ o r i g i n a l ~ \(x-y\) coordinates are given.) $$ \left(x_{1}, y_{1}\right)=(1,4),\left(x_{2}, y_{2}\right)=(4,1) $$

We sometimes talk about functions increasing at accelerating (or decelerating) rates. To clarify what we mean by these terms, we will consider some specific examples. (a) First, the Monod growth function: $$ r(N)=\frac{N}{2+N} \quad N \geq 0 $$ (i) Calculate \(r(0.1)\) and \(r(0.2) .\) How much does \(r(N)\) change when \(N\) is increased by \(0.1\), from \(N=0.1 ?\) (ii) Now calculate \(r(2)\) and \(r(2.1) .\) Show that \(r\) increases less from when \(N\) is initially 2 and is increased by \(0.1\) than when \(N\) is initially \(0.1\) and is increased by \(0.1\). (iii) Calculate \(r(4)\) and \(r(4.1) .\) Show that \(r\) increases less when \(N\) is initially 4 than it does when \(N\) is 2 initially. \(r\) is increasing with \(N\), but the effect of increasing \(N\) by the same amount is smaller for \(N=4\) than \(N=2\), and smaller for \(N=2\) than \(N=0.1 .\) So we say that \(r(N)\) is increasing at a decelerating rate. (b) Now consider the growth function $$ s(N)=\frac{N^{2}}{4+N^{2}} \quad N \geq 0 $$ This is an example of a sigmoidal function. (i) Plot \(s(N)\) and \(r(N)\) on the same axes. Show that \(s(N)\) and \(r(N)\) have the same range. (ii) Compare the increases in \(s(N)\) when \(N\) is increased by \(0.1\), from $$ \begin{array}{llll} 0 \text { to } 0.1,2 & \text { to } 2.1, & \text { and } & 4 \text { to } 4.1 . \end{array} $$ Since \(s\) increases more from 2 to \(2.1\) than from 0 to \(0.1\), we say that s increases at an accelerating rate between \(N=0\) and \(N=2\). (iii) Show that the increase of \(s(N)\) is decelerating for large \(N\).

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 \(213 \mathrm{~g}\) has 13 seeds.

Use a logarithmic transformation to find \(a\) linear relationship between the given quantities and graph the \mathrm{\\{} r e s u l t i n g ~ l i n e a r ~ r e l a t i o n s h i p ~ o n ~ a ~ l o g - l i n e a r ~ p l o t . ~ $$ y=3 \times 10^{-2 x} $$

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 \mathrm{\\{} r e s u l t i n g ~ l i n e a r ~ r e l a t i o n s h i p ~ o n ~ a ~ l o g - l i n e a r ~ p l o t . ~ $$ y=10^{1.5 x} $$

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 \mathrm{\\{} r e s u l t i n g ~ l i n e a r ~ r e l a t i o n s h i p ~ o n ~ a ~ l o g - l i n e a r ~ p l o t . ~ $$ v=2 e^{-1.2 x} $$

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

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 \mathrm{\\{} r e s u l t i n g ~ l i n e a r ~ r e l a t i o n s h i p ~ o n ~ a ~ l o g - l i n e a r ~ p l o t . ~ $$ y=1.5 e^{2 x} $$

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 Los Angeles and Las Vegas is \(434 \mathrm{~km}\). How many miles is this?

Use a logarithmic transformation to find \(a\) linear relationship between the given quantities and graph the \mathrm{\\{} r e s u l t i n g ~ l i n e a r ~ r e l a t i o n s h i p ~ o n ~ a ~ l o g - l i n e a r ~ p l o t . ~ $$ y=5 \times 2^{4 x} $$

(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

In the United States, measurements in recipes are usually given in cups. In Europe, measurements are usually given in grams. One cup of flour weighs \(120 \mathrm{~g}\). (a) A U.S. recipe requires two and a half cups of flour. How many grams is this? (b) A British recipe calls for \(225 \mathrm{~g}\) of flour. How many cups is this? (c) Write a formula to convert flour measurements in grams to measurements in cups.

Use a logarithmic transformation to find \(a\) linear relationship between the given quantities and graph the \mathrm{\\{} r e s u l t i n g ~ l i n e a r ~ r e l a t i o n s h i p ~ o n ~ a ~ l o g - l i n e a r ~ p l o t . ~ $$ y=3 \times 5^{-1.3 x} $$

(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

(a) To measure temperature, three scales are commonly used: Fahrenheit, Celsius, and Kelvin. These scales are linearly related. The Celsius scale is devised so that \(0^{\circ} \mathrm{C}\) is the freezing point of water and \(100^{\circ} \mathrm{C}\) is the boiling point of water. 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. (c) Is there any temperature that reads the same in Celsius and Fahrenheit?

Use a logarithmic transformation to find \(a\) linear relationship between the given quantities and graph the \mathrm{\\{} r e s u l t i n g ~ l i n e a r ~ r e l a t i o n s h i p ~ o n ~ a ~ l o g - l i n e a r ~ p l o t . ~ $$ y=4 \times 3^{2 x} $$

Measuring Brain Activity fMRI is a method for inferring brain activity by measuring changes in blood flow to different parts of the brain (blood flow changes can be measured noninvasively using an MRI scanner). The technique works because of a correlation between blood flow rate (which can be measured using the scanner) and brain activity. Logothetis et al. (2001) showed in experiments on macaque monkeys that blood flow \((y)\) is linearly related to brain activity \((x) .\) Both \(x\) and \(y\) are measured on scales from 0 to 1 . (a) Here are two data points from Logothetis et al. (2001): \begin{tabular}{ll} \hline \(\boldsymbol{x}\) & \(\boldsymbol{y}\) \\ \hline \(0.16\) & \(0.52\) \\ \(1.0\) & \(1.0\) \\ \hline \end{tabular} Find a formula for \(y\) in terms of \(x\). (b) Find the blood flow rate \((y)\) corresponding to each of the following brain activity measurements. (i) \(x=0.5\) (ii) \(x=0.9\) (iii) \(x=0\) (c) It is most useful to have a formula for brain activity \((x)\) in terms of blood flow \((y)\), since blood flow can be measured. Derive this formula from your answer to part \((a)\).

Use a logarithmic transformation to find \(a\) linear relationship between the given quantities and graph the \mathrm{\\{} r e s u l t i n g ~ l i n e a r ~ r e l a t i o n s h i p ~ o n ~ a ~ l o g - l i n e a r ~ p l o t . ~ $$ y=2^{x+1} $$