Looking for calc problems screwed up by calculators/computers
sjmil...@cc.memphis.edu
> I know this topic has come up before but I couldn't find any saved posts
> about it. We're ready to start teaching calculus again and the discussion
> began anew about the role of machines in the curriculum for this course.
> Let's just say there are two sides to the issue, and individual instructors
> will tailor the course to suit themselves -- I really don't want to
> strike up that argument again here in sci.math.
>
> What I _do_ want is a collection of examples which _appear_ appropriate for
> calculator use but in which the calculator will err. I have in mind some
> examples to bring up in a lecture to warn the students of pitfalls and
> limitations of technology; I also want to be ready with test questions which
> will trap students who try to get the machine to think for them.
> Of course, there are lots of questions we address throughout the course
> for which the students will see the calculator is not helpful. I was
> thinking instead of more borderline examples.
>
> For instance, the machine will graph y= 3 x^2; it won't graph
> y=(a^2+1) x^2. Another favorite of mine is to ask for the graph of
> y=x^(2/3), which most calculators will insist is different from
> y=(x^2)^(1/3). I understand there will be calculators on the market later
> this fall which can handle symbolic derivatives, so my old trick of
> asking for derivatives of exp(a x) rather than exp(3 x) will no longer
> be effective. You can train students to watch out for singularities with
> a few examples like y=sin(1/x) (I find the pixelated graph to be much
> less informative than the hand-drawn one), and for overflow with
> examples like y=10^(-6)(x-1000)^3 (taken from Finney and Thomas).
>
> As you can see most of the examples I can think of right now are related
> to graphing, but I know there are others. For example, someone whose
> name escapes me took the trouble to prepare a list of problems, mostly
> symbolic integrals, for which Mathematica in particular produced answers
> which were flat-out wrong. (As I recall this was done with the WRI people
> present, in a setting which I could hardly imagine to be cordial). I
> probably saw it presented in this forum a couple of years ago. Has anyone
> got these or similar examples?
Have them compute the derivative of |x| at 0, or 1/x or 1/x^2 at 0. At least
the Ti's use the symmetric derivative - computing the slope of the secant from
x-h to x+h for a small value of h. Hence, the answers come back 0, 88.93...,
and 0. You can use this as a lead in to a good discussion of the limit giving
the symmetric derivative and how it relates to the limit for the standard
derivative. By the way, if you go in this direction, you might have them look
at the symmetric derivative of x^2.
Regards,
Bob
>
> I hope I'm explaining specifically enough what I'm looking for, though I
> know it's a vague idea. If you want to send examples by email I can
> summarize the most on-topic responses in a followup post.
>
> dave
>
>
Dave Rusin
I hope I'm explaining specifically enough what I'm looking for, though I
Mark Janeba
>What I _do_ want is a collection of examples which _appear_ appropriate for
>calculator use but in which the calculator will err. I have in mind some
>examples to bring up in a lecture to warn the students of pitfalls and
>limitations of technology;
Here's two of my favorites.
1) (From Demana or Waits, at the '88 Atlanta meeting)
Find lim (x->0) of (1-cos(x^6))/x^12; do this by examining the graph.
2) (My own) Find all local extrema of f(x)=(x^2)exp(-.004x).
Students who attempt this using only a graphing calculator almost
invariably get it wrong. Try it to see why. It shows up in my examples
and on exams.
For the record, I strongly support graphing calculator use - thoughtful use.
Cheers,
--
Mark Janeba (503)370-6123
Dept. of Mathematics e-mail: mja...@willamette.edu
Willamette University "Speaking for myself, not the university"
Salem, Oregon 97301-3922 USA
Dave Rusin
>What I _do_ want is a collection of examples which _appear_ appropriate for
>calculator use but in which the calculator will err.
I received a number of helpful replies, and I even succeeded in finding an
old post on the subject. I will summarize the examples below. They're sorted
according to the error or trap rather than the application, since the
mathematicians in this group are likely to be able to think of novel ways
of re-using or re-presenting them.
The examples are all pretty obvious once you think to look for them, but
it's not me who thought to look for them. For the most part they're
adapted from examples mailed, posted, or published in calc books already
(including those of Stewart, Finney/Thomas, and Hughes-Hallett/Gleason).
Thanks to those who contributed. If I get any more I'll add them to this
file and leave the collection out at
http://www.math.niu.edu/~rusin/known-math/calc.errors
(FTP and gopher to math.niu.edu/pub/papers/Rusin/... work too.)
Have fun,
dave
I. Limitations to numeric data and common functions
A. Machines can't graph y = A x^2 (not even y=(A^2+1) x^2, whose graph is
arguably "the same" for all real A.)
B. Machines can't find e.g. extrema of y=A x^2 + B x + C. Warning: computers
with symbolic algebra programs _can_ do this; calculators with such
capabilities will be on the market shortly.
C. Some machines cannot graph or compute functions defined in parts (e.g.
f(x)=f1(x) for x< 0, =f2(x) for x > 0.) In particular, some cannot
handle the greatest integer function.
Of course, there is no difficulty creating more conceptual questions which
will forever be just beyond the machine's reach, e.g. "Suppose f is a
function whose graph is sketched below..."
II. Underflow errors
Sample: Find lim (x->0) of (1-cos(x^6))/x^12; do this by examining the graph.
(x^6 quickly underflows to zero)
III. Overflow errors
1) Find lim (x->0) 1/(1-cos (x^4)) - 2/x^8 graphically (the x^(-8) terms
quickly swamp the O(1) terms which you really need).
2) Calculate the derivative of sqrt(x) at x=10^6 numerically from the
definition. If you try to push (sqrt(10^6+h)-1000)/h for h around 10^-6
the h's get lost; of course this can better be computed as
1/(sqrt(10^6+h)+1000).
IV. Discretization errors
A. In singularities like y=sin(1/x) I find the pixelated graph to be much
less informative than the hand-drawn one. Likewise, the graph of
y=sin(10^8 * x) shows no apparent pattern.
(Interestingly, my TI-82 with the standard window shows a very
definite pattern for the graph of y=sin(10^7 * x) !)
B. Of course there are plenty of numerical examples, fooling students who
think the calculator is secretly carrying infinitely many places of accuracy.
They may be convinced that e^(pi*sqrt(43)) is an integer (if they use a
computer, have them check e^(pi*sqrt(163)) instead). If you succeed in
convincing them pi is not 22/7, try 355/113; or ask them for the positive
root of 242 x^4 + 113 x - 23928, and see how many make a leap of faith.
(I'd like to find a natural-looking limit which evaluates to, say,
0.333340 -- any examples?)
V. Graphing-window traps
Students, naturally, begin with the default window on the graph of a function
(typically showing the behaviour on [-10,10], say). It's easy to make
examples in which this
--- shows none of the graph (e.g. y=40+ln(x) )
--- fails to show large-scale behaviour (e.g. y=10^(-6)(x-1000)^3 )
--- misses extrema (e.g. f(x)=(x^2)exp(-.004x) )
--- suggests false limits (e.g. lim x-->-oo x^3/(x^3+10 x^2) is
a particularly good trap on the TI default window [-10,10]! )
These problems can be fixed by zooming out far enough. Others require
zooming in, especially
--- failing to show small-scale variation (e.g. y=x^4-x^2+0.2 has 3 roots)
Of course you can combine these into examples for which both zooming in
and zooming out are necessary to get a complete picture. Here's a polynomial
I once cooked up for students who were studying different techniques of
root-finding:
f(x)=4 x^7-10 x^6-5 x^5+(53/2)x^4 -22 x^3 + (59/2)x^2 -(69/2)x + 9
(its roots are -1.8794, 0.3473, 1.5, 1.5, 1.5321, and -.25 +- sqrt(-15)/4 ;
the successive extrema between the roots are +191.37, -16.22, and -.0003 --
not at all pleasant for a graphing calculator to display!)
VI. Faulty or misleading algorithms
A. Fractional powers (e.g. y=x^(2/3) has a different domain from the
calculator's y=exp((2/3)*ln(x)).)
B. TI's compute numerical derivative with _symmetric_ displacements of
the input; thus |x| has a derivative of 0 at 0. (At least my TI-82
does not inform me 1/x^2 is differentiable at zero!)
C. I'm not sure how to classify integration algorithms. My TI-82 does
not mind me asking for the integral of x^(-1) from 0 to 1; it wisely
informs me after a while that the "tolerance was not met". But it
doesn't succeed in integrating x^(-.99) from 0 to 1 either (gives
the same error message).
Incidentally, I received no pointers to
> a list of problems, mostly
>symbolic integrals, for which Mathematica in particular produced answers
>which were flat-out wrong,
a post I'm sure I saw once, but of course there are bug reports for all
the packages from time to time. I guess for the most part these fall under
the "bad algorithm" category, with the occasional "insufficient attention
paid to error estimates" complaint.
One howler I received does indeed check out on my copy of
Maple: given
integrate(sqrt((cos(x))^2),x=0..Pi);
it says the answer is 0 .
Collected (not invented) by dave ru...@math.niu.edu
Mark Janeba
>II. Underflow errors
>
>Sample: Find lim (x->0) of (1-cos(x^6))/x^12; do this by examining the graph.
>(x^6 quickly underflows to zero)
No, that's not the problem. For example, the TI-85 reports the value of
the expression (1-cos(x^6))/x^12 at x=.06 as 0, but x^6 is 4.6656E-8.
Underflow is a long way off. However, cos(x^6) is, to 13+ significant
digits, evaluated as "1" for this x, so the TI-85 thinks the expression
equals zero.
I think numerical analysts have some term like "loss of precision through
subtraction" [of nearly equal quantities, i.e. "1" and "cos(x^6)"].
>VI. Faulty or misleading algorithms
>C. I'm not sure how to classify integration algorithms. My TI-82 does
>not mind me asking for the integral of x^(-1) from 0 to 1; it wisely
>informs me after a while that the "tolerance was not met". But it
>doesn't succeed in integrating x^(-.99) from 0 to 1 either (gives
>the same error message).
The TI-82 and -85 use adaptive Gauss-Kronrod integration (check out
http://archive.ppp.ti.com/pub/graph-ti/calc-apps/info/numinter.txt ).
Some experimentation shows that the '85 uses 7 nodes while the '82 uses
(if I remember correctly) 15.
So an error you can catch the '85 at:
Integrate ( x(5x^2-3)(891x^4-990x^2+155) )^2 on [-1,1] and get 0.
The HP-48GX seems to use a change of variables and a Romberg method
that dates back to some much earlier HP calculators. ("Handheld
Calculator Evaluates Integrals", William M. Kahan, Hewlett-Packard
Journal, August 1980) The handheld referred to in the title is the
HP-34C; it's a detailed article and tests lead me to believe the same
integrator is still in place in HP's latest machines.
An error you can catch the HP-48G at:
Integrate (cos(512 Pi x))^2 on [0,1] and get 1.
[Just try to graph the integrand on any calculator and have some fun].
As the HP Journal article points out, any numerical integration routine
has an Achilles' heel. This fact is pointed out in most numerical
integration texts.
For the record, I have never worked for either HP or TI, but I personally
like TI's better.
Regards to all,
D. J. Bernstein
> I think numerical analysts have some term like "loss of precision through
> subtraction" [of nearly equal quantities, i.e. "1" and "cos(x^6)"].
And Kahan keeps pointing out that the subtraction is not at fault. If
there's any cancellation in the subtraction of two floating-point
numbers, then _all_ digits of the difference are calculated correctly.
Cancellation merely reveals errors from previous operations.
---Dan