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Topic: [ap-calculus] graphing calculator activities
Replies: 1   Last Post: May 21, 2009 11:44 AM

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Dave L. Renfro

Posts: 2,165
Registered: 11/18/05
re:[ap-calculus] graphing calculator activities
Posted: May 21, 2009 11:44 AM
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Martha Martin wrote:

> Does anyone have any activities to help the students
> better understand and learn how to use their graphing
> calculators?

There's a neat idea, which I first saw in this list
(Richard Sisley, 22 November 2000), that you can do
in either an AB class or a BC class. If it's in an
AB class, you can get more mileage out of the idea by
introducing the "quadratic approximation of a function
at a point", which I've found to be fairly easily
understood and worked with by students who have already
worked with the idea of the linear approximation of
a function a point.

While I was looking up my post about this topic
(copied in full below), I came across another post
of mine that might also be something to try, although
in this case, if it's an AB class, you'll first want
to explain the idea of higher order polynomial
approximations to a function at a point. You don't
need to explain how to get them -- although an easy
mechanical method is to assume sin(x) = a + bx + cx^2 + ...,
plug in x = 0 to get a = 0, differentiate both sides
and then plug in x = 0 to get b = 1, differentiate
both sides again and then plug in x = 0 to get c = 0,
differentiate both sides yet again and then plug in
x = 0 to get d = -1/6, etc. -- and you don't have
to deal with convergence theory and the like) and
how to manipulate them algebraically.]

Neat Taylor series application (19 September 2007)


Carol Hart wrote:

> My BC students are just beginning Power Series .. and
> I have to say how much fun is to teach the concept
> with the Foerster text. I used to sort of dread this
> relatively difficult concept .. I'm not sure I ever
> really understood it completely before I started
> using this text. I love the way it is approached.
> We also use the "Explorations" text - Advanced
> Placement Calculus with the TI-89 (Chapter 9 Infinite
> Sequences and Series) from Texas Instruments for
> very concise instructions on how to calculate and
> graph the problems.

I don't know if this is in Foerster's text, but here's
something neat that you can do with Taylor expansions.
I first saw the idea on this list, back in Fall 2000.
I tried it out the next day in a college calculus 2
class (we were in the middle of Taylor series at the
time) where each student had a computer equipped with
Mathematica, and it went over very well, but even with
graphing calculators I've found it's nice (but not as
neat as with CAS's).

If you graph e^x - (1 + x), the behavior near x=0
will look like a parabola.

If you graph e^x - (1 + x + (x^2)/2), the behavior
near x=0 will look like a cubic.

If you graph sin(x) - x, the behavior near x=0
will look like a cubic.

Moreover, these aren't just any parabolas or cubics,
they appear to have the form (constant)*x^2 or

In general, the behavior of f(x) - P_n(x), where
P_n is the n'th order Taylor polynomial about x=0,
will look like the graph of (constant)*x^m near x=0,
where x^m is the first nonzero term in the Taylor
expansion of f(x) about x=0 that isn't in P_n.

This provides a nice graphical illustration for
the remainder term of the Taylor series.

You can also do this for expansions about other
points besides x=0. Moreover, you can do this
near the beginning of a calculus 1 course (and I
have done so many times) to illustrate that the
tangent line is the *best* linear approximation
in this way: Let L be the equation of a line that
isn't the tangent line and let TL be the equation
of the tangent line. Then the graph of y = f(x) - L
and the graph of y = f(x) - TL will differ near
the tangency point in a way that is readily
visible on a graphing calculator. When you zoom
in, the former will look like a line through the
x-axis (i.e. has a positive angle of intersection
with the x-axis) and the latter will look like
a parabola tangent to the x-axis (i.e. has a zero
angle of intersection with the x-axis). The idea
is that BOTH L and TL can be used to approximate
f(x) arbitrarily closely near the point in question
(since both lines intersect the graph of y = f(x)
at that point), but the error involved with TL is
*qualitatively* better than the errors associated
with any of the L's.

This is such a neat idea that I did some searching
to see who brought it up. Here are the posts in
this list where I learned about this graphical

Richard Sisley (November 22, 2000)

Mark Howell (November 26, 2000)

Incidentally, "angles zero measure" are called
horn angles, and they were apparently even studied
by Euclid:

"The result on horn angles in proposition III.16
excludes ratios between horn angles and rectilinear
angles. That proposition states that a horn angle
is less than any rectilinear angle, hence no
multiple of a horn angle is greater than a
rectilinear angle."


Dave L. Renfro
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