Exponential Functions and Their Modeling

Section 3 Exponential Functions and Their Modeling

Task 3.1.

If we start with a single yeast cell under favorable growth conditions, then it will divide in one hour to form two identical "daughter cells". In turn, after another hour, each of these daughter cells will divide to produce two identical cells; we now have four identical "granddaughter cells" of the original parent cell. Under ideal conditions, this "doubling effect" will continue.

Create a table that contains the number of yeast cells $N(t)$ at any given hours between $t=0$ and $t=8\text{.}$
How many yeast cells are there when $t=100\text{?}$ (If you are stuck with this question or it takes too long, move on to the next question and then come back to this one later.)
Write a function equation of the number of yeast cells $N(t)$ in terms of the hours $t\text{.}$
If you were not able to answer Question 2, can you answer that question now using the function equation from Question 3?

The function that models the number of yeast cells in terms of time is called an exponential function. In general, an exponential function is a function of the form $f(x)=b^x$ where $b\gt0\text{.}$ Here $b$ is called the base and $x$ is called the exponent. Let us first explore some properties of exponentials.

Task 3.2.

Consider three expressions that involve exponential functions $2^x \times 3^y\text{,}$ $(2 \times 3)^{xy}\text{,}$ $(2 \times 3)^{x+y}\text{.}$ By plugging in pairs of values for $x$ and $y\text{,}$ decide whether any of these three expressions are the same or not.

Task 3.3.

Consider three expressions that involve exponential functions $2^x \times 2^y\text{,}$ $2^{xy}\text{,}$ $2^{x+y}\text{.}$ By plugging in pairs of values for $x$ and $y\text{,}$ decide whether any of these three expressions are the same or not.

Task 3.4.

Consider three expressions that involve exponential functions $2^x \times 3^x\text{,}$ $(2 \times 3)^{x}\text{,}$ $(2 + 3)^{x+x}\text{.}$ By plugging in value for $x\text{,}$ decide whether any of these three expressions are the same or not.

Task 3.5.

Consider three expressions that involve exponential functions $(2^x)^y\text{,}$ $2^{x+y}\text{,}$ $2^{xy}\text{.}$ By plugging in pairs of values for $x$ and $y\text{,}$ decide whether any of these three expressions are the same or not.

Let us recall that for any $b \gt 0\text{,}$ we have $b^0=1\text{,}$ $b^{-1} = \frac{1}{b}\text{.}$ Let $\frac{p}{q}$ be a rational number where $q \gt 0\text{,}$ we have

\begin{equation*} b^{\frac{p}{q}} = \sqrt[q]{b^p}. \end{equation*}

Task 3.6.

Calculate the following exponentials without using calculators:

$\displaystyle 3^{-2}$
$\displaystyle 8^{2/3}$
$\displaystyle 4^{-3/2}$
$\displaystyle 27^{-4/3}$

In application, exponential functions have a lot of limitations in many cases due to its simple nature. Instead, we use functions of exponential type. A function of exponential type is of the form $f(x) = a \times b^x$ where $a \neq 0$ and $b \gt 0 \text{.}$

Task 3.7.

Let us re-consider the yeast cell growth problem. This time, we will start with three yeast cells instead of one.

Create a table that contains the number of yeast cells $M(t)$ at any given hours between $t=0$ and $t=8\text{.}$
Can you use an exponential function to model this yeast cell growth problem?
Now model this problem using a function of exponential type. Write a function equation of the number of yeast cells $M(t)$ in terms of the hours $t\text{.}$
What is the domain and the range of the function $M(t)\text{.}$
After how many hours will the number of yeast cells be exactly $98304\text{?}$

Task 3.8.

Pam opens an investment account at Vanguard with $$10,000\text{.}$ The account earns approximately $6.5\%$ compounded annually. Let $t$ be the number of years the account has been open, and $B(t)$ the balance in the account.

Make a table showing the balance, $B(t)\text{,}$ in the account at $t=0,1,2,3,$ and $4$ years.
Write a function equation for the function $B(t)\text{.}$
What is the domain and the range of the function $B(t)\text{?}$
Use your function equation to determine the balance in $20$ years.
For the balance of the account to reach $$100,000\text{,}$ at least how many years should Pam wait?

Task 3.9.

The population of a small town has been growing. The population is shown in the following table.

Table 3.10. Population of a small town

$t$ (years since 1980)	$0$	$10$	$20$	$30$
Population $P(t)$	$2,354$	$3,164$	$4,252$	$5,714$

What is the average number of people added to the town per year between 1980 and 2010?
Assuming that the population continues to grow exponentially, write a function equation for $P(t)\text{.}$
If the population continues to grow according to the model, what is the population in 2015?
When will the population reach 10,000?
The gross domestic product (GDP) per capita of this small town in the year 1980 is $$83,145$ and has been increased at the rate of $3.4\%$ every year since then. Write a function equation for the GDP per capita $G(t)$ in terms of time $t$ in years.
Write a function equation for the total GDP of this small town $H(t)$ in terms of time $t$ in years. Simplify the function equation as much as you can.
When will the total GDP of this small town reach one billion dollars?
Suppose that $3\%$ of the annual GDP is spent on infrastructure. How much will be spent on infrastructure in this town in the year 2019?

The Newton’s Law of Cooling (Warming) describes the principle of the heat loss of a body to its surroundings provided the temperature difference is small and the nature of radiating surface remains same. The temperature $T$ of an object at time $t$ is given by the formula

\begin{equation*} T(t)=T_a+(T_0 −T_a)e^{−kt} \end{equation*}

where $T(0) = T_0$ is the initial temperature of the object, $T_a$ is the ambient temperature and $k \gt 0$ is a constant that depends on the object. The letter $e$ represents a number that is often used as the base in exponential functions because it has a number of special properties (you will learn more about this in calculus). The value of the number $e$ is approximately $2.718\text{.}$

Task 3.11.

A $40^{\circ}$F roast is cooked in a $350^{\circ}$F oven. After $2$ hours, the temperature of the roast is $125^{\circ}$F.

Assuming the temperature of the roast follows Newton’s Law of Warming, find a formula for the temperature of the roast $T(t)$ as a function of its time in the oven, $t\text{,}$ in hours.
The roast is done when the internal temperature reaches $165^{\circ}$F. When will the roast be done?

\(t\) (years since 1980)	\(0\)	\(10\)	\(20\)	\(30\)
Population \(P(t)\)	\(2,354\)	\(3,164\)	\(4,252\)	\(5,714\)