UT Mathematics, 2015
M 408N Class Activities Exponential functions Unless otherwise noted (by ”You:” in front), all activities are designed for you to tell the students to do, working together, with you and the TA and LAs walking around and helping. Do not put this sheet on the projector, or post online, or otherwise give these notes to the students. (1) (This is a descriptive example of what exponential growth means.) Suppose a piece of paper is .004 inches thick, and very large. Now suppose you fold the paper, so then you have .008 inches (two sheets). Now fold again so that you have .016 inches of folded paper. Continue folding until you have folded a total of 50 times. How thick is your folded paper now? (Hint: as you go begin changing your units to feet, then miles. For reference, the sun is nearly 93 million miles from earth. (.004 ∗ 250 = .450... ∗ 1012 inches, /12 = 3.75... ∗ 1011 feet. /5280 = 71, 079, 539.57 miles) (2) Rewrite or simplify: 27(27x ), e, ex−2 ,32x ∗ 3, others. (3) Sketch and find any asymptotes of the following. (I would put all 4 on the board at once for them to work on.) (a) f (x) = 9x−3 (b) g(x) = 9x − 3 (c) h(x) = −9x (d) s(x) = 9−1 (4) You: put a sketch like the following on the board.
If f (x) = ax and g(x) = bx for real numbers a and b, is a > b or is b > a? (5) Consider an exponential function f (x) = c · ax with y-intercept of y = 2, passing through the point (3, 54). Sketch the graph of f , then find c and a. (6) (If you have time:) Suppose there is a bacterial culture in the lab, and the number of bacteria doubles every hour. There are 500 bacteria when the culture is created. (a) How many bacteria are there after 1 hour? After 2 hours? 3 hours? 4 hours? Write out your work (without simplifying), so you can find the pattern. (b) Use the pattern you found to create a function B(t) that gives the number of bacteria in the culture after t hours. (c) Use your function to determine how long it will be before there are one million bacteria. (This requires using a logarithm!)
