OBJECTIVE To become familiar with the rapid rate of growth of exponential functions.

If you want to be a millionaire, you can start by taking Benjamin Franklin's wise advice:

"A penny saved is a penny earned."

But you're going to have to save more and more pennies each day. So suppose you put a penny in your piggy bank today, two pennies tomorrow, four pennies the next day, and so on, doubling the number of pennies you add to the bank each day. Let's call the first day we put a penny in the bank Day 0, the next Day 1, and so on. So on Day 3 you put $8$ pennies in the bank, on Day 5 you put in $32$ pennies, and on Day 10 you put in $1024$ pennies, or about $10$ dollars worth of pennies.

This doesn't seem like an effective way of becoming a millionare; after all, we're just saving pennies. How many years does it take to save a million dollars this way? We'll work it out in this exploration, and in the process we hope to improve our intuition for exponential growth.

I. How to Save a Million Dollars

1. 1. Complete the table for the number of pennies saved each day.

      Day $x$	$0$	$1$	$2$	$3$	$4$	$5$
      Pennies saved $f(x)$	$1$

   2. The number of pennies saved each day grows exponentially. What is the growth factor? What is the initial value? Find an exponential function $f(x) = Ca^x$ that models the number of pennies saved on day $x$. $$f(x) = \color{red}{\fbox{  }} . \color{red}{\fbox{  }}^x$$
2. Let's complete the table for the number of pennies saved on each day for $30$ days, without using a calculator—but we'll make the calculations easier by making some approximations. On Day 10 we must save $2^{10}$, or $1024$ pennies. Since this is about $\$10$, we'll write $\$10$ instead of the $1024$ pennies. On Day 20 we must save $2^{20}$ pennies; check that this is approximately $\$10,000$.

   Day $x$	$0$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$
   Pennies saved $f(x)$	$1$

   Day $x$	$10$	$11$	$12$	$13$	$14$	$15$	$16$	$17$	$18$	$19$
   Dollars saved	$\$10$

   Day $x$	$20$	$21$	$22$	$23$	$24$	$25$	$26$	$27$	$28$	$29$
   Dollars saved	$\$10,000$

3. On which day does the amount we put in first exceed one million dollars? Is your answer surprising? What does this experiment tell us about exponential growth?

II. How to Manage Population Growth

We know that population grows exponentially. Let's see what this means for a type of bacteria that splits every minute. Suppose that at 12:00 noon a single bacterium colonizes a discarded food can. The bacterium and its descendents are all happy, but they fear the time when the can is completely full of bacteria—doomsday.

1. How many bacteria are in the can at 12:05? At 12:10?
2. The can is completely full of bacteria at 1:00 P.M. At what time was the can only half full of bacteria?
3. When the can is exactly half full, the president of the bacteria colony reassures his constituents that doomsday is far away—after all, there is as much room left in the can as has been used in the entire previous history of the colony. Is the president correct? How much time is left before doomsday?
4. When the can is a quarter full, how much time is left till doomsday?