CalGeo: Teaching Calculus using dynamic geometric tools
Outcome 2.1.2 “Historical elements of teaching calculus”
HISTORICAL ELEMENTS OF TEACHING CALCULUS
(I) Introduction to infinite processes
The Greeks geometricians assumed on infinite grounds that simple curvilinear figures such as circles or ellipses, have areas that are geometric magnitudes of the same type as areas of polygonal figures. For their calculation they used two natural properties
i) (monotonicity):
If A is contained in B , then E(A)< E(B), where E(T) denotes the area of T,.
ii) (additivity):
if and then E(Γ) = E(A)+ E(B)
These properties played an essential role, as due to them Eudoxus was led later in a method of calculation of areas curvilinear figures that is today known as “method of exhaustion”. The principle of exhaustion is formulated in the proposition I in X book of the elements of Euclid :
“Two unequal magnitudes being set out, if from the greater there be subtracted a magnitude greater than its half, and from that which is left a magnitude greater than its half , and if this process be repeated continually there will be left some magnitude which will be less than lesser magnitude set out.”
This method is of great importance since it is origin of infinite procedures.
If we want to describe the method in modern terminology, we consider a magnitude α We abstract from α, a magnitude α1 and from the remainder we substract a magnitude α2 greater than the half of the former and so on.
We consider the sequence with e.t.c., then the remainder corresponding to the n-nth step is a-sn. In virtue of the exhaustion principle for any ε>0 there is a n0 N such that
for every which is equivalent to
Application: Organization of activities leading to the introduction
i) in the concept of limit, via measurement of circle’ s area, and
ii) in the integral, with the quadrature of the parabola.
(II) The fundamental theorem of calculus
Medieval investigations, and the subsequent work of Galileo suggested that the motion of a point, along a straight line with varying velocity be represented by means of a graph of its velocity versus time. Indivisibles considerations then indicated that the total distance travelled by the point would equal the area under the velocity-time curve. This led to an embryonic formulation of the fundamental theorem of calculus (Torricelli). Barrow gave a geometric proof of the fundamental theorem of calculus for monotonous and positive functions. Newton in 1666 discusses the computation of areas by means of andidifferention. This is the first Historical appearance of the fundamental theorem in the explicit form
where E denotes the area under curve y=f (x), providing basis for a algorithmic approach to the computation of areas.
Application: Organization of activities leading to the introduction to the fundamental theorem of calculus.
(III) The problem of tangent line, the derivative and the local extreme values.
One of the main mathematical problem at the beginning of 17th century was the determination of the tangent line. Interest in this problem stemmed from more than one source. It was a problem of pure geometry, and it was of great importance for scientific application. Optics was one of the major scientific pursuits of the seventeenth century; the design of lenses was a direct interest to Fermat, Descartes, Huygens and Newton. To study the passage of light through a lens, one must know the angle at which the ray strikes the lens in order to apply the law of refraction. The significant angle is that between the ray and the normal to the curve, the normal being the perpendicular to the tangent. Hence the problem was to find either the normal or the tangent.
Roberval thought a curve as the locus of a point moving under the action of two velocities. While the notion of a tangent as a line having the direction of the resultant velocity was more complicated than the Greek definition of a line touching a curve, this newer concept applied to many curves for which the older one failed. It was also valuable because it linked pure geometry and dynamics, which before Galileo’s work had been regarded as essential distinct. On the other hand, this notion of a tangent was objectionable on mathematical grounds, because it had the definition of tangents on physical concepts. Many curves arose in situations having nothing to do with notion and the definition of tangent was accordingly inapplicable hence other methods of finding tangents grained favor.
In Geometry the tangents lines of two curves, in their common point, determined the angle of the curves. Descartes devised a method of constructing tangent lines that was algebraic rather than infinitesimal in charachter. His method of finding the tangent line to the curve y=f(x) at the point P(x, f(x)) involved list locating the point Q(x0,0) of intersection with the perpendicular through Q to the normal line. P. Fermat, circa 1637, developed a new method in which , for the first time in the history of calculus, the idea of the small change of a variable was presented. He allowed a minimal increase to the variable and after the calculation he left increase to disappear. Fermat observed that the method of drawing the tangent line could be applied to finding of the extreme values which are searched in points x such that (in modern terminology) f’(x)=0
Application: Organization of activities leading to the introduction of concepts of tangent, derivative and extreme values.
(IV) The continuity concept
During 18th century the approximate solution of equations used the theorem of Intermediate values as a basic tool. Many mathematicians considered it (erroneously) as characterization of continuity. In 1817 Bolzano claimed that the theorem required analytic proof. Bolzano and Cauchy gave the strict definition of continuity, while Weierstrass in 1869 in his courses in Berlin, after clarified the distinction between continuous and differentiable fluxion function, first proved that every continuous function is bounded and takes extreme values.
Application: Organization of activities about the continuity concept.
ii