The lecture
Here is a writeup of what I discussed in class after the homework problems. In fact, there are two proofs of Taylor's Theorem, and some further discussion. No one has read this yet other than me, so there may be errors. Please send comments, corrections, etc. Thanks.
I have revised (11/13/2005) the essay and corrected some typos and added some interesting material (the last section) on the history of this result.

Who was Taylor?
Here is a biography of Taylor. Please also look at the last section of the essay on Taylor's Theorem for more information about the non-Western history of the result.

Graphical "evidence"
I think Taylor's Theorem is probably the most important result both for theory and practice in the second half of the course (integration by parts was the result in the first half). I not only want to provide a detailed discussion but I want to give supporting evidence. Since people may be convinced in different ways, first I want to present some graphs. The graphs display sin(x) and various Taylor polynomials. The Taylor polynomial of degree 15 is
T15(x)=x-(1/6)x3+(1/120)x5-(1/5,040)x7+(1/362,880)x9-(1/39,916,800)x11+(1/6,227,020,800)x13-(1/1,307,674,368,000)x15
Notice how big the denominators are: factorials certainly grow fast. T7(x) is just the part of T15(x) up to degree 7. These are the partial sums of the whole Taylor series for sin(x).
sin(x) and T1(x)
sin(x) and T3(x)
sin(x) and T5(x)
sin(x) and T7(x)
sin(x) and T7(x)
sin(x) and T9(x)
sin(x) and T11(x)
sin(x) and T13(x)
The polynomials can't approximate sin(x) closely on all numbers, because sin(x) is always between -1 and 1, and any non-constant polynomial will go to +infinity or -infinity as x gets large positive or negative.

Numerical "evidence"
Below are some tables of values of sine and values of Taylor polynomials at various x's. The Taylor polynomial values are colored and in two fonts (for people may be weak on color discrimination). A number like 0.8416666667 has digits 841 which agree with the true value of sine, while the digits 6666667 disagree. Just the colors and the fonts should show you what's going on numerically. For x's close to 0, even low-degree Taylor polynomials are wonderful approximations, while when x gets larged, higher degree Taylor polynomials are needed for the same accuracy.

x=.2
sin(.2)=0.1986693308
T1(.2)=0.2000000000
T3(.2)=0.1986666667
T5(.2)=0.1986693333
T7(.2)=0.1986693308
T9(.2)=0.1986693308
T11(.2)=0.1986693308
T13(.2)=0.1986693308
T15(.2)=0.1986693308
x=.5
sin(.5)=0.4794255386
T1(.5)=0.5000000000
T3(.5)=0.4791666667
T5(.5)=0.4794270833
T7(.5)=0.4794255332
T9(.5)=0.4794255386
T11(.5)=0.4794255386
T13(.5)=0.4794255386
T15(.5)=0.4794255386
x=1
sin(1)=0.8414709848
T1(1)=1.0000000000
T3(1)=0.8333333333
T5(1)=0.8416666667
T7(1)=0.8414682540
T9(1)=0.8414710097
T11(1)=0.8414709846
T13(1)=0.8414709848
T15(1)=0.8414709848
x=2
sin(2)=0.9092974268
T1(2)=2.0000000000
T3(2)=0.6666666667
T5(2)=0.9333333333
T7(2)=0.9079365079
T9(2)=0.9093474427
T11(2)=0.9092961360
T13(2)=0.9092974515
T15(2)=0.9092974265
x=4
sin(4)=-0.7568024953
T1(4)=4.000000000
T3(4)=-6.666666667
T5(4)=1.866666667
T7(4)=-1.384126984
T9(4)=-0.6617283951
T11(4)=-0.7668045535
T13(4)=-0.7560275116
T15(4)=-0.7568486195
x=6
sin(6)=-0.2794154982
T1(6)=6.
T3(6)=-30.
T5(6)=34.80000000
T7(6)=-20.74285714
T9(6)=7.028571429
T11(6)=-2.060259740
T13(6)=0.03716283716
T15(6)=-0.3223953190