9.2: Exponential Growth
This activity is intended to supplement Algebra I, Chapter 8, Lesson 5.
Problem 1
Before beginning this activity, change your window settings to match those to the right.
Enter the function
Use TRACE to observe how the value of
 Write at least three observations about the effect of the value of
b on the graph off(x) .  What value of
b results in a constant function? Explain.  Explain why the value of
b cannot be negative.
Problem 2
Now you are going to graph function
Press
Enter an
The calculator draws the tangent line and displays the equation of the line. Record the

x:−−−−−  slope of tangent:
−−−−−
Now find the value of the function
The calculator displays the

f(x):−−−−−  How does the slope of the tangent line at this point compare to the value of the function,
f(x) ?
Return to the
Record the values of



slope of tangent at 

2  
3 
Return to the
 Write at least two observations about the graph and/or the slope of its tangent at
T .
Problem 3
Slope is a measure of rate of change in a function. In this example, sometimes the slope is less than
When the rate of change of



2  
3 
To begin the search for this value of
Value of
Value of \begin{align*}b\end{align*} that is closest to \begin{align*}1\end{align*} and less than \begin{align*}1\end{align*}: ______
The value of \begin{align*}b\end{align*} we are looking for must be between these two.
Choose some values of \begin{align*}b\end{align*} that are between two numbers and repeat the process of graphing the function, drawing a tangent line, recording the value of the function and the slope of the tangent line at that point, and calculating the ratio. Narrow in on the value of \begin{align*}b\end{align*} that yields a ratio of \begin{align*}1\end{align*} as closely as you can.
\begin{align*}x\end{align*}  \begin{align*}f(x)\end{align*}  slope of tangent at \begin{align*}x\end{align*}  \begin{align*}\frac{\text{slope of tangent at} \ x}{f(x)}\end{align*} 

What is this value of \begin{align*}b? \ b \approx \underline{\;\;\;\;\;\;\;\;\;}\end{align*}
Applications
The number you found is an approximation for the mathematical constant \begin{align*}e\end{align*}. As you discovered, it is unique in that it is the only value of \begin{align*}b\end{align*} such that \begin{align*}y = b^x\end{align*} changes at a rate that is equal to the value of the function itself. Some examples are: (a) the growth of populations of people, animals, and bacteria; (b) the value of a bank account in which interest is compounded continuously; (c) and radioactive decay.
The common feature is that the rate of growth or decay is proportional to the size of the population, account balance, or mass of radioactive material. Growth and decay situations can be modeled by equations of the form \begin{align*}P = P_{0}e^{kt}\end{align*}, where \begin{align*}P\end{align*} is the current amount or population, \begin{align*}P_0\end{align*} is the initial amount, \begin{align*}t\end{align*} is time, and \begin{align*}k\end{align*} is a growth constant. An amount is growing if \begin{align*}k > 0\end{align*} and declining if \begin{align*}k < 0\end{align*}.
The following are examples of exponential growth or decay. For each exercise, write an equation to represent the situation and solve your equation to find the answer.
1. Suppose you invest $1,000 in a CD that is compounded continuously at the rate of \begin{align*}5\%\end{align*} annually. (Compounded continuously means that the investment is always growing rather than increasing in discrete steps.) What is the value of this investment after one year?
Two years? Five years?
2. A colony of bacteria is growing at a rate of \begin{align*}50\%\end{align*} per hour. What is the approximate population of the colony after one day if the initial population was 500?
3. Suppose a glacier is melting proportionately to its volume at the rate of \begin{align*}15\%\end{align*} per year. Approximately what percent of the glacier is left after ten years if the initial volume is one million cubic meters? (This is an example of exponential decay.)
4. A snowball is rolling down a snow covered hill. Suppose that at any time while it is rolling down the hill, its weight is increasing proportionately to its weight at a rate of \begin{align*}10\%\end{align*} per second. What is its weight after \begin{align*}10 \ seconds\end{align*} if its weight initially was \begin{align*}2 \ pounds\end{align*}? After \begin{align*}20 \ seconds\end{align*}? After \begin{align*}45 \ seconds\end{align*}? After \begin{align*}1 \ minute\end{align*}? What limitations might exist on this problem?
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