Application of Computer Algebra (ACA) 2007
Session: Computer Algebra in Education
19-22 July 2007, Rochester, Michigan, USA
( )
Using ln x as an Antiderivative for 1/ x is a Bad Choice!
Michel Beaudin
Service des enseignements généraux,
École de technologie supérieure (ETS),
1100, Notre-Dame street west,
Montréal, Québec, Canada, H3C 1K3
Email: [email protected]
Abstract
All major CAS are using ln(x) for their indefinite integral of 1 x . Some will say because these
systems are “complex oriented”. The symbolic handheld calculator Voyage 200 (or TI-89
Titanium) from Texas Instruments gives the user the choice: when the exponential format is set
to “rectangular”, you get ln(x) for the indefinite integral of 1 x . But, unfortunately, the
( )
exponential format defaults to “real” and we obtain ln x . In single variable calculus, we teach
students that two antiderivatives of a given function over an interval differ by a constant. But
calculus textbooks are still using the formula
∫
dx
= ln ( x ) + C because they want to include
x
both cases: negative and positive x. There is no need for this: in the case of a definite integral,
−2
say
dx
, we lose the chance of recalling that the function 1 x is an odd function … or we lose a
x
−3
∫
chance to introduce students to complex numbers (using log of negative numbers). In their
studies, (engineering) students will also follow a Differential Equations course. They will make
use of a “dsolve” command. And sometimes, in this case, some internal complex values will be
necessary in order to obtain the solution of a given DE. So, here again, choosing the “real
branch” is a bad choice. Finally, as far as complex analysis is concerned, the antiderivative of
1 z is ln(z) and not ln ( z ) .
Using technology becomes an opportunity to question textbooks: this will be the main goal of this
talk.
1. Introduction
Let us first make it clear: the Voyage 200 handheld calculator (and also the TI-89
Titanium and former TI-92) is a very impressive CAS calculator, very easy to use and powerful
enough to teach mathematics at the undergraduate level to future engineers. That is what we have
been doing for the past 8 years with enthusiasm: since September 1999, every student at Ecole de
technologie supérieure (ETS) has to buy this calculator. Along with CAS like Maple and Derive
and also the Matlab program, all available in the computer labs, learning and teaching
mathematics at ETS is done in a “computer algebra environment”.
With the Voyage 200
calculator (latest OS: 3.10), the “Complex Format” defaults to “real”: this is reasonable when we
want “the” cubic root of −1 to be −1 (high school students would be confused if “the” cubic root
of −1 is a complex number!). But this will also simplify the antiderivative of 1/x into ln(|x|).
Take a look at figures 1a and 1b.
Figure 1a
Figure 1b
Unfortunately, the status regarding the complex format appears nowhere on the screen (but this
will change with TI new handheld “Nspire”). Changing the “Complex Format” to “rectangular”,
we see that “the cubic root” of −1 simplifies into a complex number (because of the principal
branch of the cubic root for a negative real number) and that the antiderivative of 1/x becomes
ln(x): figure 2a and 2b show this.
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Figure 2a
Figure 2b
With Derive, the “real branch” affects “the cubic root” but not the integral of 1/x:
Figure 3
So, what is our point? Well, we think that if computer algebra systems are here to stay and if
more and more teachers are going to incorporate it into their teaching, we just can’t avoid using
complex numbers. The “complex format” should default to “rectangular” and not to “real” just
like the default setting of the numeric ODE solver of the Voyage 200 defaults to “RK method”
and not to “Euler’s method”. If, some 20 years ago, computer algebra systems were not written
to meet the needs of beginning calculus students, there is still no contradiction by using a CAS
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and teaching to such students: the teacher has to tell them what should be said: there is no need
at all to write the following formula:
∫
Write down
∫
dx
x
dx
x
= ln ( x ) + C
= ln x + C and recall that two antiderivatives of a continuous function over an
interval differ by a constant. When a student will encounter logs of negative real numbers, he/she
will learn something new, something that is well implemented into modern CAS, especially
Voyage 200 calculators: avoiding it will eventually lead to more serious problems. Moreover,
putting absolute values inside the logs also affects the shape of the answers in certain cases.
Using logarithms properties valid for positive numbers, the device also collects the logs when a
rational function is integrated. The “complex branch” has the advantage to show what partial
fraction expansion was applied in order to perform the integration: see figure 4 where, between
the two integrals, we have switched the “Complex Format” from “real” to “rectangular”.
Figure 4
In the next section, we are going to show some consequences of using computer algebra in
teaching mathematics: it becomes an opportunity to question textbooks! But the converse is also
true: using technology is an opportunity to question choices made by developers or to ask them
to incorporate new features in upcoming versions.
2. Examples
When you are using computer algebra in your everyday teaching, you collect many funny
examples of bugs and strange answers given by the system(s). In some cases, you can use it to
remind the students to be aware of the answers on the screen. In other cases, you rapidly face the
following: computer algebra systems and “classical” textbooks are not incompatible but one
needs to adapt the questions from the textbooks to fit the use of a CAS. The following examples
will try to demonstrate this.
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Example 1: from a private talk with David Jeffrey, corner of St-Urbain and Duluth
streets, Montreal, Canada, summer 2006.
Solve
(
x − 1 x + 1 = c, x
)
where c is any real
constant. Before taking a look at the answers given by Maple 9.5, Derive 6.10 and Voyage 200
OS 3.10, let us first ask the following question. What is the domain of the function on the LHS of
this equation? We teach students that the domain of a real function f is those real numbers x such
that f ( x) ∈ . In calculus, if you define f ( x) =
x − 1 x + 1 , you probably don’t want to deal
with square roots of negative numbers even if the (final) result is real! So, in calculus, the
domain of f should be x ≥ 1 and the equation can only have one solution for every c ≥ 0 , namely
c 2 + 1 . On the computer algebra ground, internal complex values are allowed, so
c 2 + 1 will
be the solution if c ≥ 0 and − c 2 + 1 if c ≤ 0 . Consequently, the exact answer to our equation
should be (according to David Jeffrey):
c 2 + 1sign(c) (if we set sign(0) to be ±1 , this covers
all cases). But take a look at the following Maple session, Derive session and Voyage 200
session.
> solve(sqrt(x-1)*sqrt(x+1)=c,x);
2
1+c ,
2
2
1+c ,- 1+c ,- 1+c
2
Figure 5
So, all of 3 systems are wrong! Here again, the Voyage 200 shows many educational features
when you define this function. The Complex Format shows different graphs, depending on the
branch:
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Figure 6a (“real branch”)
Figure 6b (“complex branch”)
This example shows that it is ridiculous to not incorporate complex numbers into a single variable
calculus course. The amount of complex numbers knowledge is small and the benefit for the
students using computer algebra systems is much bigger. Moreover, using the simple x1 3
function, students already remarked that no curve will be plot in the third quadrant if the
“complex branch” is selected. So, calculus textbooks should include, in their treatment of real
variable functions, examples of internal complex values and authors should talk about the
difference between “real” and “complex” branches.
Example 2: We wrote earlier that the main goal of our talk will be “the opportunity to
question textbooks” while using technology. Here is a nice example of this. In Differential
Equations courses, modern textbooks are very well illustrated if we compare to older books.
Graphics of solutions are often showed and numerical methods are introduced at early stage.
Moreover, many exercises are “projects” or “CAS problems”, so with the aid of technology,
students have the opportunity to explore new avenues. Unfortunately, we don’t see many “pencil
and paper problems” related with technology. This is strange and this is, according to us, one
major reason why “computer algebra has not yet lived up to its promise in the field of
mathematics education”.
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Let us illustrate our point with the following ODE. When my students are asked to solve the IC
problem
dy
dx
=
4y
2
x −9
,
y (0) = 5 ,
they are rapidly concerned (again!) with “complex format”: depending if “real” or “rectangular”
is the chosen one, the get the following “answer” on their Voyage 200 screen: in figure 7a, the
“real branch” was selected and in figure 7b, the “rectangular branch” was selected:
Figure 7a
Figure 7b
Which ideas should textbooks suggest to the students? Well, before solving the ODE, we have to
think about the existence of the solution. The function f ( x, y ) =
4y
2
x −9
and its partial derivative
with respect to y are continuous over a certain rectangle containing the point (0, 5) as long as x
belongs to the open interval ]−3, 3[ . The complex answer given by the TI in figure 7b (Maple
and Derive give the same answer) will become “real” when the system is informed with the
command that x belongs to the interval ]−3, 3[ :
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Figure 8
This is a nice illustration that computer algebra has the power to reunify theory and practice in a
much better way than many textbooks!
Example 3: Curve of intersection of 2 surfaces. In many calculus textbooks, authors
show nice computer graphics and give examples of finding parametric equation for the curve of
intersection of two surfaces.
Here is one example: the plane x + y + z = 3 intersects the
paraboloid z = x 2 + y 2 (see figure 9 where the range is −5 < x, y < 5, −1 < z < 10):
Figure 9
In order to find parametric equations for the curve of intersection, they simply equate both z
values, complete the square and use the first trigonometric identity. This gives, using t as the
parameter:
1
14
1
14
14
14
x=− +
cos t , y = − +
sin t , z = 4 −
cos t −
sin t ,
2
2
2
2
2
2
0 ≤ t ≤ 2π .
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When we plot this curve in the same window as before, we get:
Figure 10
The critic we can address to textbooks is the following: when partial derivatives are presented to
students, why not say a word on the following approach? If someone wants to find parametric
equations for the curve of intersection of two surfaces defined by the equations f = 0 and g = 0,
and if this curve is traced by a particular vector function r (t ) , then one possible function r is
dr
obtained by solving the following ODE system:
= ∇f × ∇g (there exists, of course, other
dt
possible ways for finding parametric equations).
And, later on, using Laplace transforms
techniques (at least for a linear system), students will be able to solve the system (and will find a
different paramétrisation if we consider the former plane and the former paraboloid). In fact,
textbooks never give the example of finding the curve of intersection of a plane and a sphere! Of
course, eigenvalues will be needed but the ODE system approach is again possible. In fact, no
textbooks, even those dealing with ODEs, show examples of using numerical methods for
plotting the curve of intersection! They just don’t want to mix Calculus and ODEs! With
technology, there is no reason at all to do so. Let us conclude this example using Derive in order
to plot the great circle when the sphere
x 2 + y 2 + z 2 = 25 intersects the plane x + y + z = 0 .
Setting z = 0 in both equations gives one point on the curve. The following Derive session does
the job very well:
9
Figure 11
Figure 12
Example 4: Now, here is a critique of technology! When our students, in a multiple
variable calculus course, have to use cylindrical coordinates for the computation of a given triple
integral, they also use the given rectangular coordinates in order to check their answer, because
the TI is doing the calculations! If both answers are the same (at least, numerically), they are
confident that their triple integral was set correctly! Colleagues at ETS found an interesting
example and I found where is the bug in the device! So, this example is something on the “bad
side” of technology: Texas Instruments fixed the bug in the upcoming new version of the
operating system (OS) for the Nspire CAS calculator (and software). Suppose you want to
compute the volume of a solid that is bounded below by the plane z = 0, above by the cone
z = x 2 + y 2 and on the sides by the cylinder x 2 + y 2 = 2 y . You will have to compute the
following triple integral:
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V=
1 1+ 1− x 2
x2 + y2
∫ ∫
∫
−1 1− 1− x 2
0
dz dy dx
Using cylindrical coordinates, we easily get the exact answer:
π 2sin θ r
is simply V = ∫
0
∫ ∫ r dz dr dθ =
0
0
32
≈ 3.55556 .
9
32
≈ 3.55556 because the integral
9
When the students compared the answer
obtained using rectangular coordinates, they started to be convinced of the importance of
cylindrical coordinates! They did not get the answer 3.55556 but the value 3.90789 or the
complex value 3.90789 − 0.523599i! Here again, the rectangular branch has the advantage for
expecting a bug because the imaginary parts is not close to 0.
Figure 13a
Figure 13b
1 1+ 1− x 2
x2 + y2
0 1− 1− x 2
0
But some of them used the fact that the volume must be equal to 2∫
∫
∫
dz dy dx . With this
new triple integral, the TI makes no error (see figure 14). In fact, the problem seems to be caused
not by the numerical integrator of the device, but by the following fact: when the inner double
integral has been computed, the integrand, as a function of x, is not an even function  in fact it
a
is not defined for negative x , so the device could not use the shortcut
∫
−a
a
f ( x) dx = 2 ∫ f ( x) dx ,
0
for an even function f.
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Figure 14
Here is the mistake made by the TI : it concerns the simplification of nested radicals! When
evaluating the integral
1+ 1− x 2
∫
x 2 + y 2 dy ,
1− 1− x 2
the device correctly used the antiderivative
(
x 2 ln y + x 2 + y 2
2
)+ y
x2 + y 2
.
2
Not let us substitute for y, the lower limit of integration, namely 1 − 1 − x 2 . The square root
(
inside the log or in the second term becomes x 2 + 1 − 1 − x 2
)
2
= 2 − 2 1 − x 2 . In order to
simplify this last expression, we will use the rule shown in this Derive show step simplification:
Figure 15
For −1 < x < 1, we have
2 − 2 1 − x2 =
x + 1 − 1 − x . Which is equal to
and only if 0 ≤ x ≤ 1 . But the TI wrongly simplified
2 − 2 1 − x 2 into
x + 1 − 1 − x if
x + 1 − 1 − x as figure
16 shows:
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Figure 16
As we said earlier, this bug is fixed in the new Nspire calculator (and also software).
Unfortunately, TI does not seem to release new OS for the Voyage 200…
3. Conclusion
Let us conclude with a critique of both technology and textbooks! When introduced to
the concept of limit, students are asked, in some exercises, to “compute” limit numerically… In
fact, this is correct as long as you show (classical) examples of the importance of arbitrary
precision.
And this arbitrary precision is impossible on calculator (even handheld CAS
calculator) for obvious reasons. But we always have access to arbitrary precision using a CAS on
a computer. We think that textbooks should, inside their “CAS projects”, give students exercises
about “catastrophic cancellation”. In this way, students will be aware of doing approximate
arithmetic and, when exact arithmetic will not be possible or possible but not useful, they will pay
more attention to the results they get on their screen. We conclude this presentation with some
exercises that teachers could do with their students.


Problem 1: Compute 1 +
n
1
5
10
15
 for n = 10 ,10 ,10 ,… The limit should be the number e…
n
Students who are using a calculator and tables will eventually find 1 as the limit value. This is a
simple but important example to do.
Problem 2: At an higher level, ask your students to find the real solutions of the following
x19
equation, using a computer: ∏ ( x − k ) − 7 = 0 . Some will find integers solutions ... unless
10
k =1
20
they use a sufficient number of digits precision. This equation has 10 real roots, located between
0 and 21.
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Problem 3: Textbooks often use a graph in order to locate zeros of a given function. But few
among it will ask students to use their calculator or a computer for plotting the graph of a rational
function, for example something like
1000( x − 1)
(101x − 100)(100 x − 99)
.
Here, you must practice the
“zooming techniques”!
Problem 4: Let f ( x) = x x
(
x + 1 + x −1 − 2 x
)
( x ≥ 1) . What happens if x goes to
1
infinity? Some students will be able to show that the limit is − . Here again, if we substitute
4
for x the value 106 , we need a sufficient precision.
Problem 5 : From [3] , page 198. Alaina wants to get to the bus stop as quickly as possible. The
bus stop is across a grassy park, 2000 feet west and 600 feet north of her starting position (figure
17). Alaina can walk west along the edge of the park on the sidewalk at a speed of 6 ft/sec. She
can also travel through the grass in the park, but only at a rate of 4 ft/sec. What path will get her
to the bus stop the fastest?
Figure 17
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Despite the massive presence of technology that we have today, we still see many textbooks (but,
fortunately, reference [3] is quite progressive compare to other textbooks) that will start doing
this example by finding the critical point of the function
t=
x
6
+
(2000 − x) 2 + 6002
4
( 0 ≤ x ≤ 2000 )
Why not recall that a continuous function over a bounder closed interval achieve an absolute
minimum value and, then, use the zoomfit command of some device?!
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REFERENCES
[1] Computer Algebra Systems. A practical Guide. Edited by Michael J. Wester. John Wiley &
Sons, Ltd. 1999.
[2] Beaudin, Michel & Picard, Gilles. Using the Voyage 200 (OS 3.10) in the classroom:
surprising results. Lecture at the Dresden International Symposium on Technology and its
Integration into Mathematics Education, 20-23 July 2006, Dresden, Germany.
[3] Hugues-Hallett, Gleason, McCallum et al, Calculus, Single Variable, Third Edition,
Wiley, 2002
[4] Koepf, Wolfram. Keynote lecture at the Second International Derive and TI-92 Conference,
Bonn, Germany, July 1996.
[5] Kutzler, Bernhard & Kokol-Voljic, Vlasta. Introduction to Derive 6, 2003.
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