Day 13 in MAT227

Last time: Section 6.3 (volumes)

Next time: Section 7.2

Today:

Announcements:
- Homework Section 6.2 due today.
Today we're looking at section 7.1: Derivative of and the number e.
So: did you visit Gil Strang's website, and see what he has to say about "the magic number e" (watch the video)? What do you think?
- ${e^x}$ : "the great function of calculus", that only calculus could create.
- Amazing properties:
  1. ${e\approx{2.718281828459045}}$ (irrational! the decimal representation doesn't repeat....)
  2. ${\frac{dy}{dx}=y}$ (a differential equation, relating y and dy/dx)
  3. ${e^xe^X=e^{x+X}}$
  4. ${e^x=1+x+\frac{x^2}{2}+\frac{x^3}{3!}+\ldots+\frac{x^n}{n!}+\ldots}$ (this is called an infinite power series -- you can think of an exponential function as a sum of polynomial functions of higher and higher degree, going on forever! No wonder exponentials grow faster than any polynomial....)
  5. We can use the power series to write ${e=1+1+\frac{1}{2}+\frac{1}{3!}+\ldots+\frac{1}{n!}+\ldots}$ That's exactly what e is, and we can calculate e to any desired accuracy using it.
- RE differential equations: "Fantastic description of nature is by differential equations."
- Gil mentions the binomial theorem (related to Pascal's triangle) -- "a big deal"
- I hope that you might consider visiting Gil for other classics, such as Integrals: the Big Picture
Before I give you all the floor, I'd like to emphasize one thing: the limit definition of the derivative, and the derivative of ${b^x}$ .
${\lim_{\Delta{x}\rightarrow{0}}\frac{b^{x+\Delta{x}}-b^{x}}{\Delta{x}}}=\lim_{\Delta{x}\rightarrow{0}}\frac{b^{x}b^{\Delta{x}}-b^{x}}{\Delta{x}}=b^{x}\lim_{\Delta{x}\rightarrow{0}}\frac{b^{\Delta{x}}-1}{\Delta{x}}=b^{x}\lim_{\Delta{x}\rightarrow{0}}\frac{b^{\Delta{x}}-b^0}{\Delta{x}}=b^{x}[b^x]'_{x=0}$
Now, there's got to be one base ( ${b>0}$ ) such that the slope of the graph at 0 is exactly 1 -- and that base is e! That's the miracle....
What questions do you have about this topic, or any of our previous topics?
- The magic number e
- Setting up Volumes by Disks and Washers: ${V=\int{dV}}$
- Setting up integrals: ${I=\int{dI}}$
- The difference in area between two curves
- The last lab
- Substitution
- Net change as the integral of a rate
- The Fundamental Theorem of Calculus, Part II
- The Fundamental Theorem of Calculus, Part I (it's really that easy!)
- anti-derivatives
- Definition of the Riemann sum (a function of three variables! See p. 262).
- Definition of the definite integral.
- Properties of the definite integral.
- Summation formulas
- Limits of summations (power sums)
- Putting these formulas into your calculator.
- ???
Let's begin by having a look at the mass of the atmosphere activity that I gave you last time. It actually includes an exponential function, so it's a perfect lead in!
For the rest of class:
You will undertake the following activity: modeling total emissions as an exponential.
It was inspired by a graph like this one:
which I saw in D.C. last week.
For the final 20 minutes, we'll switch to a class discussion of the proposed solution(s).

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