Day 4 in MAT229

Today:

• Announcements:
  • Keep an eye on the assignment page (there's a new assignment!)
  • We'll have another "quiz opportunity" at our next class meeting, because there's no class on Monday (MLK, Jr. Day). Be ready! The day will come!
  • Calculus lab (MEP 402) hours: coming soon! I'll pass around the sheet....

• quiz opportunity

• Review of Section 6.2 (so far)
  • First of all, exponential functions have a characteristic shape:
    

    This pair of graph should remind of us two important concepts that you will have heard about in science classes at some point: doubling time, and half-life. How so?

  • Exponential functions are useful in all kinds of applications: whenever the rate of change of a function is proportional to the function itself, you'll see exponential functions putting in appearances.
    • Financial (bank accounts, loans, etc.)
    • Biology (population growth)
    • Forensics (Newton's Law of Cooling)

    The concept is often misused in common parlance, however: you'll often hear someone say that something's "growing exponentially!" -- by which they mean what?

  • Laws of Exponents:

    Of course the negative exponents law causes us trouble when we talk about inverses of functions: may people will make the mistake of thinking that the inverse function is the same as because of this law.

    The real problem is that we mathematicians are too lazy: we keep reusing notation, even to the confusion of our students.... We need to be attentive, and interpret the notation in context. Mathematicians are not alone, however: the same is found in language, right? What is "lead"? Did I lead a squad to victory, or were we mowed down in a shower of lead?

  • Demonstration of the derivative property of exponential functions (that an exponential function is its own derivative, up to a constant multiple).
    

  • Shape as a function of base:

• Section 6.2: the exponential function (continued)
  1. The following two properties are two sides of the same coin: but they're the most important reasons for the choice of the irrational base e:
     
     

     This is almost a magical property: the rate of change of an exponential function is proportional to its value. In one case, the rate of change is the function value. The slope of the tangent line is the value of the function. Curious!

  2. Do you think that this physical quantity (Total Global CO₂ emissions) shows signs of exponential growth?
     

     The data is represented by the blue graph -- what do you suppose the dashed graph represents?

  3. Examples:
     1. #13, p. 401 (review of translations, reflections, etc., using )
     2. #17
     3. #36

  4. Now I'd like to turn you loose on a lab which is based on an issue that's near and dear to my heart: global climate change.
     • Exponential Lab