Humanizing Calculus
Humanizing Calculus
Humanizing Calculus
T
he student quoted above raises a con-
cern about the way in which some
mathematics topics are still being
presented today. As teachers, we
sometimes present formulas and rules
but do not take the time to talk about the evolu-
tion of the mathematics as a human invention.
The history of mathematics can supply the why,
where, and how for many concepts that are stud-
ied (Swetz 1995).
Classroom teachers can help their students
develop an appreciation of the invention of calcu-
lus in many ways. To address the concerns raised
by the student in the opening quotation, I sprinkle
quotations and anecdotes throughout a calculus
course, some of which I have included in this
article. Stories about the invention of calculus by
Sir Isaac Newton (1642–1727) and Gottfried Leib-
niz (1646–1716) can add a zesty backdrop to your
calculus lessons and thus help students come to see
mathematics as a body of knowledge developed by
human beings.
d
d d d d f f
acknowledged the correspondence between himself fg = is perfectly
bar sign), Leibniz’s example f gaccessible
= dx .
and Newton, as Newton had done in the first edi-
dx dx dx
to calculus students. What is surprising about this dx g d
g
tion of the Principia. example is Leibniz’s next statement: dx
Although Newton developed his calculus six d
f
years before Leibniz even began his serious study d f
But you get the same thing
= if you. work out d xy
dx
of mathematics, Newton failed to publish his work. dx manner
in a straightforward g d. . . and it is the
g
Newton, however, believed that a scientist’s prior- same thing in the case of divisors.
dx (Child 1920, x
ity derives from having done the work and not from p. 100) d ;
y
the publication of the discovery (Hellman 1998). d xy
This belief produced much conflict and was the Leibniz did not work out the details of this claim.
source of consternation for many years to come. Instead, he considered results from the “inverse dy = 2 z + bβ .
There are various opportunities throughout a x
method of tangents”dand; discovered his error
calculus course to bring in Leibniz and Newton. In y
(Cupillari 2004). The mathematics involved here is = 2 z + bcβ 2 .
dx dy
the next section, I will provide examples that can be not as accessible to beginning calculus students, but
used when teaching the product and quotient rules through that investigation,
dy = 2 z Leibniz
+ bβ . realized:
and implicit differentiation. dv v
=d ;
Hence it appears that it is incorrect d
to say thatψ ψ
dx dy = 2 z + bcβ 2 .
CALCULUS HISTORY FOR SPECIFIC TOPICS dv dy is the same thing as dvy, or that
Derivatives in Early Calculus—The Product and
Quotient Rules dv v
=d ;
After students learn the basic formulas for differen- dψ ψ
tiation of sums and differences
although just above I stated that this was the
d d d case, and it appeared to be proved. This is a dif-
( f ± g) = f± g ,
dx dx dx ficult point. (Child 1920, p. 101)
they should be asked to think about what a product Rather than going back to his original example,
d d d
or quotient rule fg d = look
might f like d (Cupillari
g d 2004). It Leibniz provided a counterexample, using v = x
dx
is likely that students (df dx ±
willg ) suggest
= dx d f ±product gd , and
quo- and y = x. Ten days later, in a manuscript dated
dx
( f ± g) = dx f dx
± g ,
tient rules of the forms: dx dx dx November 21, 1675, Leibniz provided the correct
d product and quotient rules. After stating the correct
d d f fg dx df d product rule, Leibniz wrote, “Now this is a really
= d f g
dx dxd g fgd=dx . d f dx dd g
d =
noteworthy theorem and a general one for
( f ± g )
=g f
dx gd ,
±
dx ddx d all curves” (Child 1920, p. 107).
dd dx dx dd dx(f ± d gd)dx= f ± g ,
and ( f ±
( f ± g ) = dx g ) = ff± ± g ,
gdx
, dx Child (1920) points out that, as a logician, Leib-
dx
dx d
dx
dx f dx dx
d f dx d f niz should have known better than to believe he
dd xy d
d f = .d proved the product rule by providing a single exam-
dxfg g= d=df dx dx. gd d
dd dx
dx
dx ddg f dx
fgdd= f
g g ple. A discussion about what counts as “proof”
x fg == dx
fg f ggdx dx could follow, and students could use the correct
dx
d ; dx dx
dx dx d
dx x
y d product rule to determine that the true value of
d xy
Leibniz, himself, dx f made dthis assumption
d incorrectly
df xy
d(xy) in Leibniz’s original problem should have
d f
before correctly =β
discovering d df the .
. df f =product dx . and quo- been (3cz2 + 2(bc + d)z + bd)dz. Correcting Leibniz’s
ddy
d =ff2zg+ bdx
dx
tient rules (Cupillari x = 2004). dx g
. In a manuscript dated mistake is a fun way to involve students in the
dx gg; x= dd dx
d
dx 1675, dx . g d
g
November 11, yd Leibniz g wrote: human invention. This example shows students
dx dy = y2 z;dx + bcgβ 2 .
dx
dx
that “calculus was not created in one sequential,
Let us nowdexamine xy whether dx dy is the same correct, and ordered way, as it is presented in text-
thing as dddv dy + dbβxy
= 2 z whether .
xy, and
xy dy = 2; z + bβ . dx/dy is the same
v books” and that “even a mathematician as brilliant
thing as dψ x ψ = d as Leibniz made mistakes when he did not check
d ;
xxdx y dy = 2 zd+xbc ;
β2. 2 his work correctly” (Cupillari 2004, p. 195).
dd ;; dx dy = 2yz + bcβ .
yy
When does (fg)′ = f′g′?
dydv = 2=z d+ vbβ;. 2
it may be seen that dv ifdy yv= z + bz, and x = cz + d; While there are an infinite number of counterex-
d ψ ψ =; 2 z + bβ .
dy==22zzd+ψ
then . . . dy + bbβ=β..dIn ψ the same way dx = + cb, amples to the equation ( fg)′ = f ′g′, there are also an
and hence dx dy = 2 z + bcβ 2 . (Child 1920, p. 100) infinite number of pairs of functions f and g that
2 = 2 z + bcβ .
2
dx dy satisfy the equation. For the trivial cases where f or
dx
dxdy dy==22zz ++ bc bcββ 2..
With an explanation dv ofv the notation (we would g is the zero function, or if both f and g are con-
use dz in place of b= and d ;parentheses
dv v in place of the stants, then, of course, ( fg)′ = f ′g + fg′ = f ′g′. As
dvdψ
dv vψ
== dd v ;; dψ
=d ;
ψ
ddψψ ψψ Vol. 101, No. 1 • August 2007 | Mathematics Teacher 25
k2
x
f ( x ) = Ce k−1
k2
x
f ( x ) = Ce k−1
n
x
f (x) = K
Maharam and Shaughnessy (1976) pointed out, When learning x − n ximplicit
n
differentiation, students
f (x) = K
( fg)′ = f ′g′ is true for any functions f and g, when can implicitly − n
xdifferentiate a problem that was actu-
f(x) = C(n – x)–n and g(x) = xn, where C is an arbi- ally posed xand solved by Newton: the problem
2
of
trary nonzero constant and n ≠ 0. One example of ( x)= 5
fcalculating the tangent to the cubic curve x3 – ax2 +
x − 2 x 2
two such functions is f(x) = 3(2 – x)–2 and g(x) = x2. axyf (–xy)3==5 0. In a tract known as the Methodus flux-
Maharam and Shaughnessy (1976) provide a list ionum et serierum x − 2 infinitorum (The Method of
of other, more interesting pairs of functions for xFluxions and Infinite Series), which was written
which the “incorrect product rule” produces cor- in 1671
x but not published until 1736, Newton pre-
x
rect answers. Cupillarie (2004) csc x extends this work, sented this equation simply as an example, ascrib-
y
showing that an infinite e x csc x
number of function pairs ing no significant meaning to this particular curve.
can be found to satisfy the incorrect quotient rule y
However, a curve can be thought of as a path traced
xby a moving point; and, in this case, the curve New-
f ′ f ′ tonxpresented contains the factors (x – y)(x2 + xy –
′ =
fg fg′′ yax + y ), a line and an ellipse. Surely, Newton’s
2
g =
g′ interests in cubic curves of this nature stemmed
y
by letting from his interest in science. For example, Newton
k2
x xo
f ( x ) = Cekk2−1 showed that a planet orbits the sun under an
f ( x ) = Ce k−1
x
square law of attraction moving, not in a cir-
xo
inverse
n yocle, but in an ellipse (Hellman 1998). Additionally,
x
with k ≠ 1 and g(x) =f (ekx x.) Students
= K can n generate hisyo work on the tangent line problem stemmed
xx− n
pairs of functions and f ( xcheck
) = K that they satisfy both x from
+
xo his interest in optics and light refraction
the correct and the incorrectproduct x − n and quotient (Larson, Hostetler, and Edwards 1998).
x + xo
rules. The fact that these are recent 2publications To differentiate x3 – ax2 + axy – y3 = 0, Newton
x y +
yo
also demonstrates that )= 5
f ( xmathematics 2 is an evolving, substituted x + ẋo for x and y + ẏo for y to get
xx− 2
( x)= 5
rather than a static, factivity. y + yo
x − 2
( x + xo )3 − a( x + xo )2 + a( x + xo )( y + yo ) − ( y + yo )3 = 0.
Differing Approaches x and Implicit ( x + xo )3 − a( x + xo )2 + a( x + xo )( y + yo ) − ( y + yo )3 = 0.
Differentiation in xEarly Calculus The expansion
2of2 this3equation
2 gives us
x + 3x xo + 3xx o + x o − ax − 2axxo − ax o
3 2 3 2 2