Day 3 in MAT229

Today:

• Announcements:
  • Visit from the STEM FORCE crew
  • Keep an eye on the assignment page (there's a new assignment!)
  • Remember that we'll have another "quiz opportunity" tomorrow.

• Review of Section 6.1: inverses
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    Proof: using the definition of the inverse, and the chain rule.

• Section 6.2: the exponential function
  1. 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?

  2. 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?

     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. 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?

  4. Shape as a function of base:

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

  6. 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!

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