Day 4 in MAT229

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Today:
- Monday is a holiday, so we'll not drop any new material on you that day. We'll "see you" next on lab day, Tuesday, with more new material to come on Wednesday.
- Your first labs are due Saturday, by midnight. Let me know if that's a problem. You can finish up by hand, if you get frustrated with Mathematica! Remember to submit as a pdf, unless you need me to work out Mathematica issues, in which case you might submit the Mathematica notebook (.nb) instead.
- Matthew asked for a quick recap on implicit differentiation.
- We quickly review some notions from exponential functions.
- We introduce logarithmic functions: another really important new member of the function zoo.
  One really cool property that logarithmic functions have is that derivatives are related to powers of $x$. More on that later.
Quick review of exponentials:
- Exponential functions are frequently referenced because of their rapid growth -- "The virus is out of control! It's taking off exponentially!"
  Frequently there's no truth to that -- the person really just means that things are growing rapidly.
  We have a very particular understanding of exponential growth: the rate of growth of an exponential function is proportional to the function value itself: i.e. \[ \frac{d}{dx}b^x= C b^x \]
  We assume that base $b>0$; then the function $f(x)=b^x$ is always positive, and monotone increasing or monotone decreasing (unless $b=1$ -- in which case the function is constant -- not very exponential at all!).
- Exponentials may decay rapidly, too: it depends on the value of $b$: if \[ b > 1 \longrightarrow \textrm{ the exponential is increasing;}\\ \] \[ 0 < b < 1 \longrightarrow \textrm{ the exponential is decreasing.} \]
  
  $0 < b=1/e < 1$ $b=e>1$
- Exponential functions are generally expressed in terms of the base $e \approx 2.718$.
  This is because that is the base of an exponential function that possesses the amazing property \[ \frac{d}{dx}e^x=e^x \] The only (really interesting) function that is its own derivative (although any multiple of this exponential function also satisfies that property).
  That means it's its own anti-derivative, too (up to a constant): \[ \int e^x \ dx = e^x+C \]
- Since I frequently teach STA205, I think a lot about normal distributions of things (bell-shaped curves). The standard normal probability density is given by \[ \rho(x)=\frac{1}{\sqrt{2 \sigma^2 \pi}}e^{-\frac{(x-\mu)^2}{2\sigma^2}} \]
  It's defined in terms of an exponential function, composed with a quadratic. (Isn't it beautiful?)
  
  You statistician wannabes are going to be especially interested in exponentials.... But exponentials are important all over the sciences. Nature loves the exponential -- and also its inverse, of course: the logarithm.
- So let's get into 6.3: Logarithmic functions: the inverse function we've all been waiting for!

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$0 < b=1/e < 1$	$b=e>1$