Day 9 in MAT229

Today:

• Announcements
  • I've fixed some of the entry problems for the on-line exercises, I hope....
  • How about handing in your written homework as well (provided that you do it by hand), so that I can have a look at your work if I'm wondering how you did a problem? That would help me. Do so if you wish.
  • Section 7.3 homework is due Thursday.

• Section 7.3:
  • We ended on a linearization yesterday, #108, p. 366:
    
    Did that make sense? We estimated the change in the function using the linearization as
    
    The actual change is
    
    How did we do?;)

• Section 7.4: Exponential Growth and Decay
  • Assignment: pp. 371-373, #4, 5, 12, 13, 19, 20 (on-line: due Tuesday, 9/16); hand in 25 and 33 separately, the same day
  • To motivate this section, I'd like to remind you of a few facts about exponentials: For example, they satisfy a very important differential equation:
    

    The proof presented in the text (p. 368) is interesting: we could just show that

    • is a solution (by differentiation);
    • but the proof shows that it's the only solution: The only function whose derivative is identically zero is a constant function, so or Then we set .

    The upshot is that "...a process obeys an exponential law precisely when its rate of change is proportional to the amount present at time t." (p. 368)

  • When working with exponentials, we always define a few quantities of interest:
    • Doubling time: the time it takes for a growing exponential function to double in size.
      

      This we can easily deduce, starting from a model of the form

      

    • Half life: the time it takes for a dying exponential function to halve in size.
      

      That's funny! Exactly the same formula.... Why is that? Hint:
      

      
      

  • Exercises for section 7.4:
    • #7, p. 372
    • #9, p. 372
    • #16, p. 372
    • Carbon Dating: The ratio of C¹⁴ to C¹² after t years is given by
      
      #18, p. 372
    • #21, p. 372