Day 12 in MAT229

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Today:

Announcements:
- Your quiz opportunity tomorrow is over section 7.1.
- Your assignment over 6.4 is due today.
- Your assignment of the labs is due tomorrow
Today we cover 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:
  
  ${P(t)=P_0\left(1+\frac{r}{n}\right)^{nt}}$
- Let's see where this model comes from: it's the compound interest model -- that is, interest on top of interest. If you receive simple interest at a rate ${r}$ on an initial investment ${P_0}$ , then at the end of the year you'd have
  
  ${P(1)=P_0+r\cdot{P_0}=P_0(1+r)}$
  And if you did it for ${t}$ years, you'd have
  ${P(t)=P_0\left(1+r\right)^t}$
- Now, in compound interest, you take your interest rate, split it up several (say 12) ways, and then do the computation several (e.g. 12) times:
  
  ${P(1/12)=P_0+(r/12)\cdot{P_0}=P_0(1+r/12)}$
  So that
  
  ${P(1)=P_0\left(1+r/12\right)^{12}}$
  And if you did it for ${t}$ years, you'd have
  ${P(t)=P_0\left(1+r/12\right)^{12t}}$
  Hence, more generally,
  
  ${P(t)=P_0\left(1+\frac{r}{n}\right)^{nt}}$
- 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
    - #7, p. 477 (don't forget your roots!;)
    - 11
    - 17
    - 29
    - 25 (sometimes you have to do this more than once!)
  - #70, p. 478
  - #86, p. 478
  - #93, p. 479

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