Day 28, Mat227

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Announcements:
- Your 6.4 homework is returned.
  - #10 -- just a little ugly.
  - #20 -- use properties of logs.
  - #38 -- I hope that you're feeling confident about tangent lines by now!
- Your 6.6 homework is due Friday
- On a lighter note: "Please join us this Friday for the Major/Minor Lunch. It is a great way to meet faculty and fellow math/stat majors and minors."
  WHERE: MEP 4th floor Center Atrium;
  WHEN: Friday, April 3rd, 11:30-1:30.
section 6.6: now we wrap up our study of the inverse trig functions.
- We're just finishing #48, p. 460 -- this really interesting but algebraically challenging problem
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 ${P(t)={P_0\left(1+\frac{r}{n}\right)^{nt}}={2{P_0}}}$ ?
  
  Solving for t in ${P_0\left(1+\frac{r}{n}\right)^{nt}=2P_0}$ , 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 is provided by 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) in the case where ${f(a)=0}$ and ${g(a)=0}$ .
  Consider ${\lim_{x\to{a}}\frac{f(x)}{g(x)}}$
- 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 9, 11, and 15 from p. 477
  - I asked my son Thad "What's ${0^0}$ ?" and he said "0." I tried to convince him that it should be 1 ("Any number to the 0th power is 1."). He tried to convince me that 0 to any power should be 0. His calculator said "undefined". Mathematica says "Indeterminate".
    
    What's ${\lim_{x\to{0}}{x^x}}$ ?
  - #70, p. 478
  - #86, p. 478
  - #93, p. 479

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Updated on 04/01/2015 17:45:36