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Elements of Analysis in Russian Schools
217
The next chapter of the textbook is called “Antiderivatives and
Integrals.” The problem that motivates the introduction of the concept
of the antiderivative is taken from mechanics: “Given the acceleration
of an object, find its velocity and coordinates at each moment in time.”
The subsequent presentation is sufficiently traditional. The antideriva-
tive F(x) of a function f(x) is defined on an interval by the equality
F
(x) = f(x). The theorem that all antiderivatives of the function f(x)
on an interval have the form F(x) + C is explicitly formulated and
proven using Lagrange’s theorem. A table of antiderivatives is obtained
by means of an inversion of the table of derivatives. Three rules for
finding antiderivatives are formulated and proven by differentiation:
the sum of antiderivatives is the antiderivative of a sum; if F is the
antiderivative of f, then kF is the antiderivative of kf ; and if F is
the antiderivative of f, k = 0, then
1
k
F(kx + b) is the antiderivative of
f(kx + b). In this way, the formula for the substitution of a variable
is introduced only in the linear case. Note that students are not
introduced to the concept of an indefinite integral as the set of all
antiderivatives or to the notation
f(x)dx.
The concept of the integral is introduced in the textbook in an
interesting way. First, the following theorem about the area of a
curvilinear trapezoid is proven (the existence of this area is considered
intuitively obvious and thus not discussed).
Theorem. If f is a function that is continuous and nonnegative on the
interval
[a; b], and F is its antiderivative on this interval, then the area
S of the corresponding curvilinear trapezoid is equal to the change in the
antiderivative over the interval
[a; b], i.e. S = F(b) − F(a). (p. 180)
The theorem is proven using the definition of the derivative, while
the change in area S — the area of a “narrow strip” between two
straight lines with x coordinates x and x + x — is replaced with the
area of the rectangle f(c)x, which is equal to it (the existence of such
a rectangle is justified by citing the continuity of the function). Hence
S
x
= f(c) → f(x) for x → 0 (here, the continuity of the function is
used once again). In this way, the Newton–Leibniz formula is proven
(using geometric language) even before the formal introduction of the
concept of the integral.
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Russian Mathematics Education: Programs and Practices
The integral is then introduced as the limit of integral sums of
a particular kind. The interval
[a, b] is divided into n equal parts,
and the value of the function is taken at the left endpoint of
each of the intervals thus formed. It is claimed that the sequence
S
n
=
b−a
n
(f(x
0
) + · · · + f(x
n−1
)) approaches the area of a curvilinear
trapezoid. Students are then informed that precisely this limit (which
exists for any continuous function) is called the integral. Applications
of integrals in geometry and physics are examined. To compute the
volume of objects, the textbook introduces the formula V =
b
a
S(x)dx,
where S(x) is the cross-section of an object with x-coordinate x,
continuously dependent on x. Let us note, by the way, that in the course
in geometry, the volumes of all studied objects, beginning with the
pyramid, are usually computed using integrals (Atanasyan et al., 2006).
Among the physical problems solved using integrals is the problem of
work done by a variable force, the problem of the force of the water
pressure, and the problem of the centers of masses.
Finally, we should note that in contrast to the textbook by Alimov
et al., examined above, exponential and logarithmic functions are stud-
ied in this textbook after derivatives and integrals. The differentiation of
the exponential function is initially carried out on the function y = e
x
.
The number e is introduced in the following way:
Examining the graphs of the functions y = a
x
for different a
between 2 and 3, we notice that the slopes of the tangents to these
functions at the point (0, 1) increase, passing through, as be might
supposed from geometric considerations, the value 45
◦
(whose tangent
is equal to 1). The textbook concludes:
It appears evident that as a increases from 2 to 3, we will find a value
of a such that the slope will be…equal to 1. (p. 241)
After which the corresponding value of a is called the number e. In
other words, e is defined as a number such that
e
x
−1
x
→ 1 for x → 0.
From this equality, the formulas for the derivatives e
x
and a
x
, and also
for the antiderivatives of these functions, are easily deduced. Then the
derivative of the function y = log x is derived by differentiating the
basic logarithmic identity x = e
log x
, and the derivative of the power
function with an arbitrary real exponent is obtained as the derivative of
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Elements of Analysis in Russian Schools
219
a composite function. Subsequent study of the properties of elementary
functions can be conducted using derivatives.
Let us note that the textbook of Kolmogorov et al. (1990) also
touches on differential equations: it examines equations of exponential
growth and decay, which lead to a function such as f(x) = Ce
kx
,
and the equation of harmonic oscillations, which leads to the function
f(x) = A cos (ωx + φ).
In the opinion of the author of this chapter, the textbook of
Kolmogorov et al. solved an extremely difficult methodological prob-
lem with considerable success: it presented elementary calculus in a way
that is understandable and sufficiently rigorous. No doubt, there is little
reason to suppose that references to the passage to the limit are always
comprehensible to all students, but many topics are presented in a clear
way and with great methodological and mathematical inventiveness.
Very critical judgments of this textbook, however, have also been
expressed (see also Abramov, 2010). This textbook has remained
in print (with certain changes) to this day and plays a role in the
educational process along with other textbooks.
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