Day 3, Mat128: Calculus A

Last time: Finding roots

Next time: Section 1.4

Today:

Announcements
- Reminder: turn off your rectangles.
- Did anyone go to the breakfast yesterday?
- You have a new assignment.
- No quiz today, but you should have reviewed 1.1-1.3 by now. How's that going? Any questions so far?
Today we continue talking about "the function zoo", and its most important character: the linear function.
1. Here's a practical example of the use of linear functions. Today I reviewed a news report: 'It's a lit fuse:' Release of ancient carbon from melting permafrost measured:
  - A paper published this week in Nature Geoscience has released the first measurements of greenhouse gases from permafrost under Arctic lakes. But while the study confirms those gases locked away in ice for thousands of years are seeping free, it concludes the amounts are not yet large. "It's a lit fuse, but the length of that fuse is very long," said lead author Katey Walter Anthony of the University of Alaska. "According to the model projections, we're getting ready for the part where it starts to explode. But it hasn't happened yet."
  - They found that, across the Arctic, the amount of gas being released from a lake was directly related to its expansion. The more permafrost was melted around the water's edge, the bigger the lake became, and the more greenhouse gases were released.
  What can the mathematician do with this information? Because there's a linear ("direct") relationship between expansion and emissions, we can measure the expansion of lakes to measure the amount of emissions. Measuring lake size is easy -- satellite data is freely available to do that.
2. Last time we talked about how to find the window in which to plot a function.
  - One of the aspects of a function that is "interesting" is the collection of roots of the function.
  - Another is the collection of peaks and pits of a function (maxes and mins).
  I told you that differential calculus -- that which we study this semester -- will help us to find both.
3. The tool we were using last time is simple: linear functions
  I called them the most important functions we use. Let's return to that problem that you must recall from high school or other times in your mathematical life:
  
  How do you find the "right" graph window for a given function, $f(x)$?
  I've updated our Mathematica example, which we will use to figure out
  1. How to find a function's roots (places where $f(x)=0$)?
  2. How to find places where a function hits high and low points -- "peaks and pits"?
  It would be nice if we had an "automated" way of finding these -- then we'd know how we might pick an automated window.
  In particular, we are looking for the method attributed to Sir Isaac Newton, one creator of calculus, who used linear functions to solve both of these problem. Both of them boiled down to the same problem:
  How can we use linear functions to find roots of $f(x)$?
At the end of this session I hope that we'll be asking ourselves the question "How can we find this thing called the derivative function?".

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A function is a rule that assigns to each element x in a set D exactly one element (called f(x)) in a set E.