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 nonWestern 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
T_{15}(x)=x(1/6)x^{3}+(1/120)x^{5}(1/5,040)x^{7}+(1/362,880)x^{9}(1/39,916,800)x^{11}+(1/6,227,020,800)x^{13}(1/1,307,674,368,000)x^{15}
Notice how big the denominators are: factorials certainly grow
fast. T_{7}(x) is just the part of T_{15}(x) up to
degree 7. These are the partial sums of the whole Taylor series for sin(x).

sin(x) and T_{1}(x) 
 
sin(x) and
T_{3}(x) 


sin(x) and T_{5}(x) 
 
sin(x) and
T_{7}(x) 


sin(x) and T_{7}(x) 
 
sin(x) and
T_{9}(x) 


sin(x) and T_{11}(x) 
 
sin(x) and
T_{13}(x) 


The polynomials can't approximate sin(x) closely on all
numbers, because sin(x) is always between 1 and 1, and any
nonconstant 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 lowdegree 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 
T_{1}(.2)=  0.2000000000 
T_{3}(.2)=  0.1986666667 
T_{5}(.2)=  0.1986693333 
T_{7}(.2)=  0.1986693308 
T_{9}(.2)=  0.1986693308 
T_{11}(.2)=  0.1986693308 
T_{13}(.2)=  0.1986693308 
T_{15}(.2)=  0.1986693308 


x=.5 
sin(.5)=  0.4794255386 
T_{1}(.5)=  0.5000000000 
T_{3}(.5)=  0.4791666667 
T_{5}(.5)=  0.4794270833 
T_{7}(.5)=  0.4794255332 
T_{9}(.5)=  0.4794255386 
T_{11}(.5)=  0.4794255386 
T_{13}(.5)=  0.4794255386 
T_{15}(.5)=  0.4794255386 


x=1 
sin(1)=  0.8414709848 
T_{1}(1)=  1.0000000000 
T_{3}(1)=  0.8333333333 
T_{5}(1)=  0.8416666667 
T_{7}(1)=  0.8414682540 
T_{9}(1)=  0.8414710097 
T_{11}(1)=  0.8414709846 
T_{13}(1)=  0.8414709848 
T_{15}(1)=  0.8414709848 


x=2 
sin(2)=  0.9092974268 
T_{1}(2)=  2.0000000000 
T_{3}(2)=  0.6666666667 
T_{5}(2)=  0.9333333333 
T_{7}(2)=  0.9079365079 
T_{9}(2)=  0.9093474427 
T_{11}(2)=  0.9092961360 
T_{13}(2)=  0.9092974515 
T_{15}(2)=  0.9092974265 

x=4 
sin(4)=  0.7568024953 
T_{1}(4)=  4.000000000 
T_{3}(4)=  6.666666667 
T_{5}(4)=  1.866666667 
T_{7}(4)=  1.384126984 
T_{9}(4)=  0.6617283951 
T_{11}(4)=  0.7668045535 
T_{13}(4)=  0.7560275116 
T_{15}(4)=  0.7568486195 

x=6 
sin(6)=  0.2794154982 
T_{1}(6)=  6. 
T_{3}(6)=  30. 
T_{5}(6)=  34.80000000 
T_{7}(6)=  20.74285714 
T_{9}(6)=  7.028571429 
T_{11}(6)=  2.060259740 
T_{13}(6)=  0.03716283716 
T_{15}(6)=  0.3223953190 
