Day 29 in MAT229

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Announcements:
- I know that it's getting nicer outside, and the tendency is to look out the window and think "let's see: I could do calculus, or I could go outside!" Mighty tempting.... But stick with me.
- Whenever someone visits with me on Zoom I say "Now that wasn't so bad, was it?" I encourage them to come again, and to tell their friends to come, too! If you're struggling, please stop by at office hours, and ask your questions.
- We're in week 8, covering a little more on improper integrals, and today getting a start into the world of Taylor polynomials.
  We love polynomials because they're so simple; we use polynomials to approximate functions for the same reason that we use rectangles to approximate general areas -- because they're simple!
Today:
1. Taylor polynomials are generalizations of tangent lines: if you're good with tangent lines, you're off to a good start.
  Tangent lines provide a great fit -- but really only near one point, $x=a$. At that point, tangent lines get both the function value and the slope (first derivative) exactly right.
2. Lines are great -- but they can't bend. Why not try fitting quadratics to functions, instead?
  So we'll fit second degree polynomials (whose graphs are parabolas) to functions.
  Parabolas provide an even better fit -- but really only near one point, $x=a$. At that point, tangent parabolas get the function value, and the slope, and the inflection (second derivative) exactly right.
3. And so on. Why stop at quadratics? Look at cubics, and then quartics, and then quintics, and then .... what's the sixth degree polynomial called? I can't recall....
4. The point is that we can keep going forever, and we'll get more and more right about the function -- but only at that one point. But higher derivatives should also mean that we can stay close to the function for values further and further away from the point $x=a$.
5. Materials for today:
Links/Notes:
- Nothing new to report.

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