Day 11 in MAT229

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Today:

Announcements:
- Your quiz opportunity is over section 6.8 -- L'Hopital's rule.
- Your assignment over 6.4 will be due next time (Monday)
- Your assignment over 6.3 is returned:
  - 14 -- Tamra
  - 56 -- Sam
- I found some study materials from our textbook's author, and have made a sample available on-line. You might try the chapter 6 materials.
Last time we began to study the inverse trig functions (section 6.6).
- Let's look at a few more problems involving these curious inverse functions:
  1. 6, p. 459
  2. 14
  3. 22
  4. 67
  5. 49 -- there's always a ladder somewhere in calculus....
Section 6.8: L'Hopital's Rule (Sometimes limits are kind of tricky....)
- In financial mathematics 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: the question is
  
  For what value of t is ${P(t)=P_0\left(1+\frac{r}{n}\right)^{nt}=2P_0}$ ?
  
  Solving t is ${P_0\left(1+\frac{r}{n}\right)^{nt}=2P_0}$ for t, we get
  ${t=\frac{\ln 2}{n\ln(1+\frac{r}{n})}}$
  
  where n is the number of compoundings per year, and r is given as a decimal (e.g. 9% is represented by .09). This is the doubling time.
  When compounding is continuous (i.e. ${n \to \infty}$ ), this reduces to the very lovely rule
  ${\frac{\ln 2}{r}}$
- 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.
- Motivation of L'Hopital's Rule (requires the limit definition of the derivative -- p. 471)
- Examples:
  - Demonstration that ${\lim_{h \to 0} \frac{e^h-1}{h} = 1}$ (remember that this limit figured into the definition of $f'(x),$ where $f(x)=e^x$ )
  - Let's do a bunch of odds, #7-33, p. 477
  - #70, p. 478
  - #86, p. 478
  - #93, p. 479

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