Day 10 in MAT229

Last time: Finished Section 7.4

Next time: Finish Section 7.7/Begin 7.8

Today:

Announcements
- Departmental picnic coming up! (You thought I'd forgotten, didn't you?)
  Reserve Sunday, September 21, starting at 4:30 pm for our fall picnic. This year it will be at Highlands Park in Fort Thomas.
  Here's the info.
  You're free to just show up, but if you'd like to make a reservation, email me, or sign up on Brooke Buckley's door (across from the Student Lounge upstairs).
  This is for majors, minors, and anyone who might be interested in becoming one of either. Don't be shy! Come on out and meet us. We only rarely bite.
- Your homework 7.3 is due today. If you'd like to hand in your written work so that I can use it to better grade your on-line work, please do so at this time.
- Section 7.7 homework will be done in two parts:
  - pp. 390-391, #2, 4, 5, 13, 16, 32, 45 will be done on-line;
  - #56 and 62 will be handed in.
  I'll combine the scores. Due Thursday, 9/18.
Section 7.7: L'Hopital's Rule
- In financial mathematics (e.g. compound interest problems, subject of section 7.5, which we're not covering in depth), we tend to use an unusual base for an exponential:
  
  ${P(t)=P_0\left(1+\frac{r}{n}\right)^{nt}}$
- You may have encountered a few laws such as The rule of 72 (70, 71, 69.3,...) to calculate doubling time: forget them! Just do the math:
  
  ${\frac{\ln 2}{n\ln(1+\frac{r}{n})}}$
  where n is the number of compoundings per year, and r is given as a decimal (e.g. 9% is represented by .09).
  When compounding is continuous (i.e. ${n \to \infty}$ ), this reduces to the very lovely rule
  ${\frac{\ln 2}{r}}$ which is found in your book on p. 369.
- Now, how do we know that ${\lim_{n \to \infty}\frac{\ln 2}{n\ln(1+\frac{r}{n})}=\frac{\ln 2}{r}}$ ? That is, how do we know that
  
  ${\lim_{n \to \infty} \ {n\ln(1+\frac{r}{n})}={r}}$ ?
  The answer, of course, is L'Hopital's Rule, which is useful in solving certain indeterminate limits:
  
  Let's rewrite it a little: we want to show that
  
  ${\lim_{n \to \infty}\frac{\ln(1+\frac{r}{n})}{\frac{1}{n}}={r}}$
- Here's the general situation:
- And if the limit of the quotient of the derivatives is indeterminate, then iterate -- that is, do it again! Then work with the ratio of the second (or higher) derivatives.
- Examples:
  - Demonstration that ${\lim_{h \to 0} \frac{e^h-1}{h} = 1}$
  - #2, p. 390
  - #6, p. 390
  - #20, p. 390
  - #42, p. 390
  - A better proof of the fact that
    
    (take logs! Better than induction....)
  - Proof of L'Hopital's Rule

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