- I'm collecting 11.2 homework tomorrow. I'll have it available on my door on Wednesday, if you'd like to pick it up before our exam next Thursday.
- The exam will cover material from chapter 8 to 11.3 (remember that
one problem of those I've assigned for 11.3 will figure prominently in
the exam).
- I've finished grading your 9.4 homework
- New things (in addition to the old things):
- When the problem says "Graph P, f, and the linear approximation L(x)=1 on a common screen." I'm expecting... graphs!
- When the problem says "Comment on how well the functions P and L approximate f." I'm expecting... comments!
- Problem 3: "To approximate a function f...." In this problem we're no longer assuming that f(x)=cos(x). Don't impose elements of previous problems, when they're not appropriate.
- Problem 4: Your graph (and there should be a graph!) should be focused on the region around x=1, since that's the point about which we're doing our expansion.
- Problem 5: a fair number of you reached the conclusion that
: that's what we're looking for! Once we have the values of
we'll know
.
- If you don't have a graphing tool, then you might try this one: it's a web-based function plotter.
- Your "polynomials" have to be polynomials!;) You can't have square root functions, etc. playing a role. All those derivatives are evaluated at the abscissa about which you're doing your expansion (x=a).
- To find out where the two curves intersect (#2), you could use Newton's method (using the "solve" command in your calculator is equivalent):
- Once you've got a polynomial expanded about a, in terms of (x-a), don't expand those out! You're wasting time, and losing sight of the fact that we want to know what's going on right around a -- when (x-a) is small.
- New things (in addition to the old things):
- Last time I previewed the major results, and we did a few
examples. I'll propose a few more examples, and let you take a crack at
asking that we treat a few problems.
- Again, this section concerns several different means of testing
for convergence of series with all positive terms (obviously, if we're
dealing with series of all negative terms, similar results are going to
apply).
- We're not so concerned with the actual limits (that is the values
of the series) as we are with the convergence (or divergence) of the
series.
- Theorems:
Theorem 3 is just a corollary of Theorem 2, where the integrals are the obvious power functions:
This theorem says that, in the long run, one series has terms which are simply a constant times the terms in the other series.
- Examples:
- Last time we did these:
- Integral Test: #9, 10, p. 563
- Comparison Test: #20, p. 563
- Limit Comparison Test: #34, p. 563
- Would you like to pick some more?
- #84, p. 565: Divergence of the harmonic series (and dominoes!)
- Last time we did these:
- Assignment: To be handed in (due 10/27): pp. 570-571, #7, 10, 11, 14, 17, 20-26, 28
-
Again, our mission here is not necessarily the value of the
series, but may be simply knowledge of whether it converges or
not.
That being said, Theorem 2 below does give us some help in determining the value of (or at least bounds on the value of) the series.
- Series that converge absolutely, and
- Series that don't.
- Examples:
- #2, p. 570
- #18, p. 570:
- #27, p. 570