Table of Contents

• Pre-quiz
• Part I: Exponential Growth
• 1: Building the basic growth equation
• 2: What does exponential growth look like?
• 3: Why does the semi-log plot unbend the plot of exponential growth?
• 4: Do ALL curves get 'unbent' by taking the log?
• Part II: Finding Generation Time
• 5: But how long is a generation?
• 6: Finding generation time on a graph
• 7: Is this growth exponential?
• Part III: The Numbers get messy
• 8: In search of ... the exact doubling time
• 9: Finding doubling time with messy numbers
• 10: Doubling time practice for fun and profit
• 11: In Search of ... a more convenient equation
• 12: Down and dirty with e-to-the-t
• 13: Comparing apples and oranges
• Part IV: Extended Problems
• 14: Fun with Rhizobia
• 15: Extended problem #1: E. coli ate my lab report
• 16: Extended problem #2: Exponential growth ends (with a whimper)
• 17: More practice with exponential growth

In search of ... the exact doubling time

So far, we've been guesstimating doubling time from a graph of population over time, or we've calculated it directly from a table. But there are a few problems...

First, guesstimating a parameter from a graph is not really approved scientific procedure. It would be hard to imagine publishing a report in the Journal of the American Medical Association that started out "we looked at the graph, squinted a little, and decided that the doubing time was 23 minutes...".

Secondly, when we calculated the doubling time from the table, that only worked because the table happened to include an exactly doubled population. If the table does NOT contain a doubled population entry, then you're in trouble. In other words, we can figure out the exact doubling time for this E. nice below, but not for E. naughty:

time	E. nice	E. naughty
10:00	10 million	10 million
10:20	20 million	15 million

At first glance, it appears that the population got "halfway" to doubling, so doubling time should be 40 minutes. Let's see if that's true -- if the population keeps growing at the same rate, will it double in 40 minutes?

time	E. messy
10:00	10 million
	... multiply by 1.5 to get
10:20	15 million
	... multiply by 1.5 to get
10:40	22.5 million

 

Oops, we got to 22.5, not 20 million. Granted, this is a small difference, but it's still not the right answer. Like the old joke goes, a million here, a million there, pretty soon you're talking about a real epidemic. What we need is a foolproof way to determine exactly what the doubling time is.