Day 4, Mat128: Calculus A

Last time: Functions

Next time: Section 1.4

Today:

Announcements
- Reminder: turn off your rectangles.
- Quizzes returned
  - Each problems was worth 1 point. You received 0, 1/2, or 1 point on each problem.
  - For the most part, as a group, you didn't know the definition of a function. This must change. Definitions matter, and you need to have them in your brain.
    This course is concerned with doing things to functions. If you don't know what functions are, what chance do you have?
  - You also struggled with classes of functions. I'm going to continue to ask you about classes of functions and their properties.
    You are the Keeper at the Function Zoo: you need to know the characteristics and value of each animal.
A Haiku, by A. Long.

A Function
rule -- associates
with each x in domain D
just one y in range.
This course is really about functions and modeling. We speak of derivatives and integrals in calculus, and both concepts concern functions.
Often functions are not given by God, but rather created by humans (perhaps from data), so we often first model the functions based on data. Let's look at a few examples now, and ask what kinds of questions we might attempt to answer using them. We are studying calculus to be able to create and study functions like these.
Why is it important? Because of news stories like this (2012):
New Law in North Carolina Bans Latest Scientific Predictions of Sea-Level Rise.
Scientists (and mathematicians) make predictions based on data and reason, not on laws passed in State Houses; but those in the State Houses don't have to pay any attention. The people probably should, however -- especially those who own property along the coast.
Any questions about sections 1.1-1.3 before we move along? Last time we talked about function composition, and I emphasized how important this idea is when we create "new functions from old" (which we need to do when we create models).
Today we begin talking about the tangent and velocity problem (section 1.4):
- Gil Strang is one of the most important modern living mathematicians, and here's his "Central Idea of Calculus":
  
  The velocity part is the derivative; the total distance part is the integral. In this course, we're focused on derivatives, hence the velocity. In particular, we speak of "instantaneous velocity" -- the velocity at a single point in time.
  That's what your speedometer is showing.
- Why are we concerned with "limits"? This slide show illustrates the approach of the secant lines to the tangent line for a smooth curve. It's one of the first important problems we'll want to address in calculus. It's why we're interested in limits of things at the outset.
  Notice the focus on linear functions: linear functions are the most important functions in calculus.
- Let's look at some examples from 1.4, using Mathematica. We'll be using Mathematica extensively in this class.
- Next time we'll continue with 1.4, but perhaps begin 1.5.
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