Creation and Manipulation of Taylor Series

1
Creation and Manipulation of Taylor Series
Dan KennedyBaylor SchoolChattanooga,
TN dkennedy_at_baylorschool.org
2
The recommended impetus for this session is
free-response question 6 from the 2007 BC exam
3
The BC students in 2007 did not exactly ace this
one.
The mean score was 2.28 out of 9. A mere 0.3 of
the students earned 9s. Approximately 40.6 of
the students earned no points at all.
4
Bad idea Find the first three derivatives of f
and use them to build the series. Good idea
Plug into the Maclaurin series for
5
Plug in

which can be simplified to
6
Other popular simplifications of
7
Bad idea Forget part (a). Lets use
LHopitals rule to find
Dissing the AP Committee is a no-no!
8
Good idea Do what you are told.
9
Thus
All this was worth 1 point out of 9.
10
Baaaad idea Start by antidifferentiating
Good idea Import your series from part (a).
11
By the FTC, is the
antiderivative of that equals 0 at x 0.
From part (a)
So
12
Using the first two terms of this series, we
estimate
Notice that the next term of the series
approximation would have been
This number plays a big role in (d).
13
Quel imbecile!
Baaad idea Try arguing your case using the
Lagrange error bound. Good idea Use the bound
associated with the Alternating Series Test. Be
sure to justify that it applies here!
14
The series in (c) for is
an alternating series of terms that decrease in
absolute value with a limit of zero. Thus, the
truncation error after two terms is less than the
absolute value of the third term
15
So lets talk about series manipulation. In the
old days (pre-1989), the approach to series in a
calculus class was quite different from what it
is today (as with many other topics). 1. Convergen
ce tests for series of constants
2. Constructing Taylor series3. Intervals of
convergence
For example, consider BC-4 from 1979
16
Notice that the series is geometric.
17
Here is the actual grading standard for part (a),
which was worth 5 points out of 15. Notice that
it was assumed that students would build the
series using nth derivatives. Since it was 1979,
they all did.
18
The grading standard for part (b) assumed the
standard Ratio Test with endpoint analysis. There
is no visible acknowledgment that
justifies the answer in one step!
19
Todays calculus students would (I hope)
recognize as a variation of
,but the students in 1979 apparently did
not. How did our students get better???? We need
to reflect on these beneficial changes
occasionally if only to remind ourselves that
the good old days were actually not all that
great.
20
What happened was a new emphasis on series as
functions the real reason for having them in
the course. In fact, many teachers are talking
about series as functions long before they talk
about convergence tests for series of constants.
What follow are some of my favorite student
explorations
21
When they find that this function is its own
derivative, most students will guess that it is
. You hope someone will realize why it must be
.
22
The third task is particularly rich. Students
might forget the constant of integration. When
reminded, theyll cheerfully add it. Then remind
them that they can use tan 0 to find it!
23
This began with a series that was valid for -1 lt
x lt 1. An interesting postscript is that we also
get convergence at -1 and 1.
24
1
Limit tan 1
25
Students can generally do this. They will
discover that the polynomial is
26
Now theyre ready for the nicest little
exploration in the course!
27
One of the most powerful visualizations in
mathematics is the spectacle of the convergence
of Taylor series. Here are the Taylor polynomials
for sin x about x 0
28
Here are their graphs, superimposed on the graph
of y sin x
29
Coming next year from the College Board
The 2008-2009 AP Calculus Focus Materials The
topic Infinite Series!
30
dkennedy_at_baylorschool.org