Download 3 The Introductory Course on Higher Mathematics\ V.B.Zhivetin. The

The Introductory Course on Higher Mathematics\ V.B.Zhivetin.
The Kazan society of mathematicians. Kazan, 2000, 44 p.
This book is intended for a wide range of readers studying higher
mathematics and enables a comprehensive understanding of this discipline. It can be used by teachers in technical and humanitarian high
schools as well as by those who wish to depend their knowledge or to
study this course on their own.
Tabl. – Ill. – 21 Lit. –
Reviewed by: Prof. Dr. A.N.Sherstnev (Kazan state university);
Prof. Dr. D.Kh.Mushtari (NIIMM named after
N.G. Chebotarev).
3

V.B.Zhivetin,
1998
ZHIVETIN Vladimir Borisovich
The Introductory Course on Higher Mathematics
The master copy was prepared by the author
Printed from the original by
«Olitex» Ltd
420094, 2 Chuikov street
Kazan
4
National Academic Institute
«Graph» Publishing House
License 0210 dated 07.10.97
Order 021
5
Elements of mathematical symbolism
We will use the following logical symbols to put down mathematical
reasoning.
1. The symbol ⇒ means logical sequence. Let α and β be some
properties. Notation α⇒β is to be read as: β follows from α.
2. The symbol ⇔ means logical equivalence. Notation α⇔β means
that β follows from α, and α follows from β.
3. The symbol ∀ is called universal quantifier. Notation ∀x corresponds to the statements: for all x, for any x, for whatever x.
4. The symbol ∃ — stands for existential quantifier. Notation ∃x
corresponds to the statement: there exists such an x, which … .
Chapter 1. Limits
1.1. The limit of numerical sequence
We’ll examine sets (aggregates) of different nature, consisting of
either finite or infinite number of elements (finite or infinite sets). A set
is called ordered, if for every two of its elements it is defined, which of
them precedes the other. For example, coordinates (x,y,z) of some point
in space constitute an ordered finite set (the abscissa x is preceding the
ordinate y, which in its turn is proceeding the z-coordinate).
An element of the set is often called the point of set. It is related
to the fact that, for example, there exists one-to-one correspondence
between the set of real numbers x and the points of a number line (as
we call a straight line with selected reference point, scale and positive
direction), i. e. to every real number x corresponds a point on the
straight line with coordinate x, and vice versa, to every point on the
straight line corresponds a certain real number – the coordinate of this
point. The set of points on the straight line (set of real numbers) is denoted by R1.
Further, between the set of ordered pairs of real numbers (x,y) and
the set of points on a plane there also exists one-to-one correspon6
dence, i. e. to every pair of numbers (x,y) corresponds a point on a
plane with coordinates x, y, and, vice versa, to each point of the plane
corresponds a definite pair of numbers (x,y) — its coordinates. The set
of points on a plane (set of ordered pairs of real numbers) is denoted by
R2.
And lastly, there exists one-to-one correspondence both between
the set of triplets of real numbers (x,y,z) and the set of points in space,
i. e. to each triplet of numbers there corresponds a point in space with
coordinates x, y, z, and, vice versa, to each point in space there corresponds a triplet of numbers (x,y,z) — its coordinates. The set of points
in space (the set of ordered triplets of real numbers) is denoted by R3.
To summarize what was said above, we can introduce a notion of
n-dimensional space Rn, which is understood as the set of ordered ntuples of real numbers (x1, x2, … , xn), referred points of n-dimensional
space, where the set of real numbers (the set of points on a straight
line) will be one-dimensional space, the set of ordered pairs of real
numbers (the set of points on a plane) — two-dimensional space, the
set of ordered triplets of real numbers (the set of points in physical
space) — three-dimensional space.
There are quantities of two kinds that have to be distinguished in
mathematics, physics and engineering: scalar quantities, characterized
by numerical value only (temperature, density and etc.) and vector
quantities, characterized by direction (velocity, force and so on) as well
as by numerical value. If we assume that a certain point 0 reflects the
zero vector (reference point), we can geometrically show the vector
quantity as a directed segment (vector) beginning at zero point, its
length corresponding to the numerical value of the vector quantity and
its direction coinciding with the direction of the vector quantity. The
position of a vector on a number line is characterized by one real number x — the coordinate of its endpoint; on a plane – by two numbers
(x,y) — projection of the vector on the coordinate axes or, which is the
same, the coordinates of its endpoint; in physical space — by three coordinates (x,y,z) — also projections of the vector on the coordinate
axes or the coordinates of its endpoint. There is a one-to-one correspondence between the set of real numbers and vectors on a straight
line as well as between the set of ordered pairs of real numbers (x,y)
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and vectors on a plane, or between the set of ordered triplets of real
triplets (x,y,z) and vectors in physical space. For that reason we can
regard the set of real numbers (space R1) not only as the set of points
on a straight line, but also as the set of vectors on a straight line (ending
at points with coordinate x); the set of pairs of real numbers (x,y)
(space R2) not only as the set of points on a plane, but also as the set of
vectors on a plane ending at points with coordinates x, y, or with projections on the coordinate axes equal to x, y; the set of triplets of real
numbers (x,y,z) (space R3) can be regarded not only as the set of points
in physical space, but also as the set of vectors in this space with ends
at points with coordinates x, y, z, or with projections on the coordinate
axes of equal to x, y, z.
Summarizing what was said above, the set of ordered n-tuples of
real numbers x=(x1, … , xn) (space Rn) can be regarded not only as the
set of points of n-dimensional space, but also as the set of vectors in
this space (with the ends at points with coordinates x1, … , xn or, which
is the same, with projections on the coordinate axes equal to x1, … ,
xn).
The distance between the points x, y on the straight line, between
the points (x1,y1) (x2,y2) on the plane and between the points (x1,y1,z1)
(x2,y2,z2) in three-dimensional space is determined from the following
equations
ρ1 = | x − y | = ( x − y) 2 , ρ 2 = ( x 2 − x1 ) 2 + ( y 2 − y1 ) 2 ,
(1)
ρ 3 = ( x 2 − x 1 ) 2 + ( y 2 − y 1 ) 2 + ( z 2 − z1 ) 2
from which it is seen that the first two formulas are particular cases of
the third one. Generalizing the formula (1) for n-dimensional case, let us
determine the distance between the points x=(x1, … , xn), y=(y1, … ,
yn) of n-dimensional space from the formula
ρ n (x , y) = (x 1 − y 1 ) 2 + ... + ( x n − y n ) 2 .
(2)
The set, for elements of which a metric notion of distance has been
introduced (has been introduced), is called metric space. The metric
(2) is called Euclidean metric. The space Rn with Euclidean metric is
called Euclidean space.
In the formula (2) ρ(x,y) has the following three properties:
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1) symmetry ρ(x,y)=p(y,x);
2) positivity ρ(x,y)≥0;
3) triangle inequality (each side of a triangle doesn’t exceed the
sum of the other two sides) ρ(x,z)≤ρ(x,y)+ρ(y,z).
Now, let us discuss the infinite sequence of numbers x1, … , xn, …
The numbers x1, x2, etc. are called members or elements of the sequence; the element xn, which determines elements of the sequence
with any number by means of formula xn=f(n), is called the common
member of the sequence. Discussing a sequence, we usually mention
only its common member xn, as it determines in a unique way members
of the sequence with any numbers (for example, the sequence xn=1/n
goes as following: 1, 1/2, 1/3, … , 1/n, …).
The number x is called the limit of sequence xn, if all members of
the sequence with sufficiently large numbers are unrestrictedly approaching the number x as their number n grows. In other words, if a
sequence xn has the limit x, all members of the sequence beginning with
a certain sufficiently large number N, reach the arbitrarily small vicinity
of the limit point x (reach the segment of arbitrarily small radius ε with
the centre at point x). Hence, the number x is called the limit of sequence xn, if
∀ε>0 ∃N: n ≥N ⇒ ρ 1(x,xn)<ε,
(3)
where ρ 1(x,xn)=|x–xn| (see the first formula (1)). It is obvious that number N, starting from which the inequality ρ 1(x,xn)<ε holds true, is, in
generally dependent on the value ε, i. e. N=N(ε). A sequence which
has a limit is called convergent, without a limit — divergent.
If a sequence xn has the limit x, it is expressed as follows
xn→x when n→∞ or xn→x
or
lim x n = x or lim xn=x.
n→ ∞
It should be marked that there are two evident properties of sequences. If limxn=x, limyn=y then lim(xn+yn)=limxn+lim yn, i. e. the limit
of the sum of sequences is equal to the sum of their limits. Indeed,
when n→∞ the elements xn, yn are unrestrictedly approaching their
limits x, y and, therefore, the sum xn+yn is unrestrictedly approaching
number x+y, and it implies that
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lim(xn+yn) = x+y, i. e. lim(xn+yn) = limxn+lim yn.
Further, if limxn=x and C is constant, then limCxn=Clim xn, i. e. the constant factor could be placed after the limit sign (entered under the limit
sign). Indeed, when n→∞ the elements xn are unrestrictedly approaching their limit x and hence, the elements Cxn are unrestrictedly approaching the number Cx, and it implies that Cxn=Cx, i. e. lim
Cxn=Climxn.
As an example, let us examine a sequence with the common member xn=n/n+1, having the following expanded form
1 2 3
n
, , , ... ,
, ... .
2 3 4
n +1
When n=99 we obtain xn=99/100=0,99 — the number close to 1, when
n=999 we obtain xn=999/1000=0,999 — the number still closer to 1 and
so on, i. e. members of the sequence with sufficiently large number are
unrestrictedly approaching number x=1, being, therefore, the limit of the
sequence xn=n/n+1. Obviously it should be proved, that inequality (3)
which in the case considered takes on the form
1−
n
< ε,
n+1
will hold true beginning from a certain N at indefinitely small given ε>0.
As 1>n/(n+1), 1–n/(n+1)>0 we can leave out the sign of modulus and
sequentially obtain
1−
n
1
1
< ε, n + 1 - n - (n + 1)ε > 0, n + 1 > , n > − 1.
n +1
ε
ε
As n is a natural number (n≥1), its least value N at which the equation
holds true
N>
1
−1
ε
evidently equals (considering the small ε, i. e. ε<<1, 1/ε>>1)
 1
N =  ,
ε
where [1/ε] — is an integer part 1/ε. So, when
 1
n≥ N = 
ε
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for any indefinitely small ε>0 we’ll get
1−
n
< ε,
n+1
and it implies (see definition (3)) that the number x=1 is the limit of the
sequence xn=n/n+1.
Now let’s examine a sequence (x11,x21,x31), … , (x1n,x2n,x3n) of triplets of numbers or, which is the same, a sequence of points
Mn(x1n,x2n,x3n) in three-dimensional space (in two dimensional space, i.
e. on a plane, the reasoning is similar). The point M(x1,x2,x3) is called
the limit of a sequence of points Mn(x1n,x2n,x3n) if points Mn with sufficiently large numbers are unrestrictedly approaching the point M as
their number is grows.
In other words if a sequence of points Mn has the limit point M, all
points Mn beginning with a certain sufficiently large number N reach
indefinitely the arbitrarily small vicinity of the limit point M (reach the
sphere of arbitrarily small radius ε with the center at point M). Hence,
the point M(x1,x2,x3) is called the limit of the sequence of points Mn(x1n,
x2n, x3n), if for
∀ε>0 ∃N: n≥N ⇒ ρ 3(M,M1)<ε,
(4)
where the distance (metric) p3 is determined from the last formula (1),
i. e.
ρ 3 (M , M n ) = ( x 1 − x 1 n ) 2 + (x 2 − x 2 n ) 2 + (x 3 − x 3 n ) 2 .
Now it is not quite easy to give the definition for the limit of a sequence of points Mn(x1n, x2n, … , xkn) in k-dimensional space. Preliminarily, on the analogy with three-dimensional case, the aggregate of
points M∗ satisfying the inequality
ρ k(M, M*) < ε,
∗
where points M, M
have coordinates M(x1,x2,…,xk),
*
*
*
*
M (x 1 , x 2 , K , x k ) and the distance
ρ k (M , M ∗ ) = (x 1 − x ∗1 ) 2 + (x 2 − x ∗2 ) 2 + ... + ( x k − x ∗k )2 ,
(5)
is called a k-dimensional sphere with the centre at point M of radius
ε, and the aggregate of points M∗ satisfying the equation ρ k(M, M∗ )=ε,
— a k-dimensional sphere with the centre at point M of the same
radius ε. Using the notion of a k-dimensional sphere the limit of a se11
quence of points Mn(x1n, … , xkn) in k-dimensional space can be determined at follows: point M(x1, … , xk) is called the limit of a sequence of
points Mn(x1n, … , xkn) if points Mn with sufficiently large number N,
reach indefinitely the arbitrarily small vicinity of the point M (reach the
sphere of an arbitrary small radius ε with the center at point M).
Hence, the point M(x1, … , xk) is called the limit of a sequence of
points Mn(x1n, … , xkn) if for
∀ε>0 ∃N: n≥N ⇒ ρ k(M,Mn)>ε
(6)
1.2. Limit of functional sequence
As is known, the function of one argument y=f(x) establishes a correspondence: number x → number y.
This correspondence is usually established either analytically (by a
formula), or by a table a or graph. The set of values of an argument x,
where the function is studied, forms the domain of the function, and the
set of corresponding values of the function itself forms the ranger
space. It can be said that the function of one argument y=f(x) is mapping x numbers from its domain of the function onto corresponding y
numbers from its range space. It is customary to express the above
mapping (correspondence) as follows: (the short form)
f : R1 → R1,
and it implies: f – function mapping elements of R1 space (points) onto
elements (points) of the same space. In practice, one often deals with
functions of many variables (arguments) y=f(x1, … , xn), establishing a
correspondence between the ordered set of numbers (x1, … , xn) and a
number, i. e. establishing mapping (correspondence) to
vector (x1, … , xn) → y number.
A function of this kind is expressed as:
f : Rn → R1,
which means: f – function mapping elements (points) of Rn space onto
elements (points) of R1 space.
A function of many arguments, as well as a function of one argument, can be expressed as:
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y=f(x),
but it should be noted that
x=(x1, … , xn) or x∈Rn, n≥2.
Function y=f(x), x∈Rn, n≥2 is often called the function of vectorial argument, while function y=f(x), x∈R1 — the function of scalar argument. If function of many arguments is expressed as y=f(x), x∈Rn,
n≥2, then the above definition of the function domain and the range
space of functions will be valid. It should be added that if the domain of
the function y=f(x) for scalar or vectorial argument is not indicated,
then the domain is understood as the aggregate of all values of argument x at which the function has sense (exists).
In conclusion of this paragraph we’ll study the sequence of functions fn(x), x∈R1, defined at interval (a,b). If at any fixed x∈(a,b), the
number sequence fn(x) is converging then it is said that the sequence of
functions fn(x) is converging at (a,b).
Let the sequence fn(x) converge at (a,b). Then a certain y number,
which is the limit of number sequence fn(x) corresponds to each fixed
point x∈(a,b), i. e. establishes a correspondence: number x → to number y, i. e. for the sequence fn(x) converging at (a,b) there exists a
function y=f(x) which at each fixed x∈(a,b) gives the value of a limit of
the number sequence fn(x). The above — mentioned function f(x) is
called the limit of functional sequence f n(x) at intervals (a,b).
As the function f(x) at any fixed x∈(a,b) is the limit of the number
sequence fn(x), than basing on definition (3) for the limit of a number
sequence we can give the following definition for the functional sequence limit: function f(x), defined at (a,b), is called the limit of sequence fn(x), consisting of functions also defined at (a, b), if for
∀ε>0 è ∀ x∈(a,b) ∃N : n≥N ⇒ |f(x) – fn(x)| < ε
(6′)
In general case, number N, starting from which the inequality |f(x)–
fn(x)|<ε holds true, depends on ε and on x, i. e. N=N(ε, x). However, it
can happen that number N is one and the same for all x∈(a,b), i. e.
N=N(ε). In this case, the sequence fn(x) is called uniformly convergent at (a,b). If the sequence has the limit f(x) it will be expressed as:
fn(x)→f(x) when n→∞ or in short fn(x)→f(x)
or
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lim f n (x ) = f (x ) or in short lim fn(x)=f(x).
n→ ∞
We’ll use the above material and introduce the definition of function
limit. For this matter we’ll investigate the general case of function f(x)
with vectorial or scalar argument, i. e. x∈Rn, n≥1. Let’s denote the
domain of function f(x) by Df and let the point x0∈Df. The number f0 is
called the limit of function f(x) at point x0, if the values of function f(x)
(number f(x)) are unrestrictedly approaching number f0 while argument
(point) x is unrestrictedly approaching point x0. In other words, if function f(x) has the limit f0 at x0, it means that the values of function (number f(x)) reach the arbitrary small vicinity of number f0 (reach the segment of arbitrary small radius ε with the centre at point f0), if only the
argument reaches sufficiently small vicinity of point x0 (reaches the
sphere of sufficiently small radius δ with the centre at point x0). Hence,
the number f0 is called the limit of function f(x) at point x0, if for
∀ε>0 ∃δ>0 : ρ(x0,x)<δ ⇒ |f(x) – fn(x)| < ε,
(7)
where x=(x1, … , xn), x0=(x01, … , x0n),
ρ n = (x 01 − x 1 )2 + L + ( x 0 n − x n )2 .
(8)
In one-dimensional case, when x∈R , x0∈R , the inequality ρ 1(x0,x)<δ
could be rearranged in the equivalent form |x0–x|<δ.
If there exists the limit f0 of the function f(x) at point x0, then it is
expressed as
lim f ( x) = f0 or f(x)→f0, with x→x0.
1
1
x →x o
In conclusion of this paragraph we’ll note that if for functions f1(x),
f2(x), x∈Rn, n≥1, it is known that
lim f1(x)=f1o, lim f2(x)=f2,
then
lim [f1 ( x) + f 2 (x )] = lim f1 (x ) + lim f 2 (x ) = f10 + f 20 ,
(9)
x→ x 0
x →x 0
x → x0
i. e. the limit of the sum is equal to the sum of limits. Indeed, with x→x0
the values of function f1(x) are unrestrictedly approaching number f1o,
and the value of function f2(x) — number f2. Therefore, the values of
the sum f1(x)+f2(x) with x→x0 are unrestrictedly approaching number
f1o+f2o. Exactly this fact is expressed by the equation (9).
14
In addition, if the lim f ( x) = f0 , then with C=const
x →x o
lim Cf (x ) = C lim f ( x ) = Cf 0 ,
x →x 0
(10)
x→ x 0
i. e. the constant factor can be placed after the limit sign. Indeed, if the
value f(x) when x→x0 is unrestrictedly approaching number f0, then the
value of the product Cf(x) when x→x0 is unrestrictedly approaching
number Cf0. The equation (10) is stating this particular fact.
1.3. Continuity of function
Now let us refer to the definition of function continuity. Basing on
its geometric representation, it would be natural to describe the function
f: R1→R1 as continuous, if its graph has the form of continuously
smooth curve (Fig. 1) without runout of points (Fig. 2, 3) or discontinuous jumps (Fig. 4, 5).
These simple geometric representations bring us to conclusion that
the function f : R1→R1, should be considered continuous at a certain
point x0:
1) if it is determined at this point, because then there exists a point
(x0, f(x0)) on the function graph, otherwise there would have inevitably
been discontinuity at this point (see Fig. 2 and 3, where f(x0) doesn’t
exist);
2) if there exists lim f (x ), because in this case when approaching
x→ x o
point x0 from any side, the function graph will be arbitrarily closely
(continuously) approaching one and the same value lim f (x ) (see Fig.
x→ x o
1, where lim f (x ) exists and Fig. 2, where lim f (x ) doesn’t exist);
x→ x o
x→ x o
3) if the limit of the function at point x0 is equal to its particular
value at this point, i. e.
lim f ( x) = f ( x 0 ),
x→ x 0
(11)
otherwise when moving along the curve y=f(x) to a point with abscissa
x0 the curve may make a final jump (there will be discontinuity) as on
15
Fig. 5, where the jump is equal to the ordinate difference
f (x 0 ) − lim f ( x) .
x →x 0
So the equation (11) is assumed to be the definition of continuity at
the function point of one or several arguments, silently presuming that
at the considered point x0 there exist the function limit and its particular
value. Using the definition of a limit (7) and assuming for it that
f0=f(x0), the definition (11) of continuity of function at a point can be
stated in a different way: the function f(x), x∈Rn, n≥1 is continuous at
point x0 if for
∀ε>0 ∃δ>0: ρ(x0,x)<δ ⇒ |f(x0)–f(x)|<ε.
(12)
The definition of continuity (12) is fully (⇔) equivalent to the definition (11). At the same time, another definition arises from it: the function f(x), x∈Rn, n≥1 is continuous at point x0, if to a sufficiently small
change of argument (ρ(x0,x)<δ) corresponds arbitrarily small change of
function (|f(x0) – f(x)|<ε). The function f(x), x∈Rn, n≥1, which is continuous at any one point of a certain set D⊂Rn is called continuous on
this D set.
Fig. 1
lim f (x ) = f (x 0 )
x→ x 0
Fig. 2
lim f (x ) exists
x→ x 0
f(x0) doesn’t exist
Fig. 3
f(x0) doesn't exists
and
lim f (x ) doesn’t exist
x→ x 0
16
Fig. 4
lim f (x ) doesn’t exist
x→ x 0
Fig. 5
lim f (x ) ≠ f (x 0 ) , though
x→ x 0
lim f (x ) and f(x0) exist
x→ x 0
Charter 2. Derivatives
2.1. Notion of derivative
When we examine any functional relationship f : Rn→R1, determining of rate the function change is of special interest. Let’s begin with
one-dimensional case when f : R1→R1. It is known that the motion of a
material point M is governed by law the s=f(t), where s — is distance, t
— is time, so the quantity
∆ s f ( t + ∆ t) − f ( t)
=
∆t
∆t
is called the rate of motion of point M at a time interval (t, t+∆t) and
∆s
f (t + ∆t ) − f ( t )
= lim
∆t →0 ∆ t
∆t → 0
∆t
lim
is the velocity of point M at time point t.
By analogy, if we deal with functional relation y=f(x), x∈R1, then,
irrespective of physical meaning of variables x, y the quantity
∆ y f (x + ∆ x) − f ( x)
=
∆x
∆x
— is called the average rate of change of function f(x) at interval
(x,x+∆x), and
17
∆y
f ( x + ∆ x ) − f (x )
= lim
∆x →0 ∆ x
∆ x→ 0
∆x
lim
(13)
is called the rate of change of function f(x) at point x. In the course of
mathematical analysis the limit (13) is also called the derivative of function y=f(x), x∈R1 and it is denoted by y′ or f ′ (x), i. e.
∆y
,
∆x → 0 ∆ x




f (x + ∆x ) − f ( x) 
f ′( x) = lim
.
∆x → 0
∆x

y′ = lim
which is the same as
(14)
Thus, if a certain process is described by functional relationship y=f(x),
x∈R1, the rate of change of this process is determined by the derivative
(14).
Let’s use (14) and introduce f ′ =A, as well as A⋅∆x=dy. The last
relation is called the differential of function f(x) at point x. Change of
argument ∆x is denoted by dx and called the differential of a dependent variable. Then it can be expressed as
dy = f ′′(x)dx
or
f ′( x) = dy / dx,
i. e. the derivative is equal to the relation of differentials dy and dx.
Examples.
1. Calculate the derivative of the a constant, i. e. the derivative of a
function y=C, C=const. It is characteristic for this function that at any
value of argument x, it takes on the same value C. Hence, y(x)=C,
y(x+∆x)=C, ∆y=0 at any ∆x≠0, ∆y/∆x=0 at ∀ ∆x≠0, whence
y′ = lim
∆ x →0
∆y
=0
∆x
as the limit of constant is the constant itself. So
C′=0.
2. Calculate the derivative of a power function y=x3. We have
y=(x+∆x)=(x+∆x)3, ∆y=y(x+∆x) – y(x)=(x+∆x)3 - x3,
∆y=3x2∆x+3x∆x2+∆x3, ∆y/∆x=3x2+(3x+ x2) ∆x.
Hence
18
∆y
= 3x 2 ,
∆x →0 ∆ x
y′= lim
i. e.
x3=3 x2.
Note. Similarly, it is easy to obtain derivatives
(x′)=1, (x2)′=2x.
Comparing the derivatives of functions x, x2, x3 we see that apparently
the derivative of function xn, where n is any natural number, has the
following form
(xn)′=nxn-1.
The validity of this formula will be further strictly determined. More
over, it will be proved that this formula is valid for any real n.
We can present the derivative of y=f(x), x∈R1 geometrically to
show its visual meaning. From Fig. 6 it is seen that
Fig. 6
∆ y ÑB
=
= tgβ,
∆x AC
where β is the angle of inclination of secant ABL with respect to axis
x. When ∆x→0, point B is unrestrictedly approaching point A in the line
of the graph of function y=f(x), the secant ABL rotates and under the
existing limit (14) tends to take a certain extreme position described by
the following slope ratio (see Fig. 6):
tgα = lim tgβ = lim
B→ A
∆ x →0
∆y
= y ′(x ) .
∆x
The limiting value of secant ABL when B→A along the curve y=f(x),
is called a tangential to the curve y=f(x) at point A. Hence, the de19
rivative y′(x) of the function y=f(x) at a certain point x defines the slope
ratio of a tangential with respect to the function graph at points (x,f(x)).
2.2. Partial derivative of function
Let’s define a partial derivative in the case of function f: Rn→R1, i.
e. for a multivariable function u=f(x1, … , xn). When fixing arguments
x1, x2, … , xn, let’s assign to any single argument x change of ∆x and by
analogy with a single variable function we obtain the relation
∆ i u f (x 1 , K, x i -1 , x i + ∆x i , x i +1 ,K , x n ) − f (x 1 , K, x i -1 , x i , x i +1 ,K , x n )
=
.
∆x i
∆x i
This relation can be regarded as the average rate of change of
function f with respect to variable xi at interval (xi, xi+1) when other
variables are fixed. In the obtained relation if we go to a limit at ∆xi→0
and denote this limit as ∂u/∂xi or ∂f/∂xi, we obtain
∂u
∆i u
= lim
∆
x
→
0
∂x i
i
∆x i
or, which is the same
∂f
f ( x1 , K , x i -1 , x i + ∆ x i , x i +1 , K , x n ) − f ( x1 , K , x i -1 , x i , x i + 1 , K, x n )
= lim
.
∂ x i ∆x i →0
∆x i
(15)
The limiting value (15) is called the partial derivative of function
u=f(xi, … , xn) with respect to variable xi. It defines the rate of change
of function f with respect to variable xi at a fixed point (x1, … , xn). It is
essential that when we calculate a partial derivative, we change only
one of the arguments, keeping others at constant (fixed) value.
As the limit of the sum is equal to the sum of limits and the constant
factor can be placed after the limit sign, so the derivative of the sum is
equal to the sum of derivatives and the constant factor can be placed
after the derivative sign, i. e.
(f1 + f 2 )′(x ) = f 1′(x ) + f 2′ (x ), (Cf )′( x) = Cf ′(x ) ,
and similarly for multivariable functions.
∂(u1 + u 2 )
∂u 1 ∂u 2
=
+
,
∂x i
∂x 1 ∂x 2
20
∂ (Cu)
∂u
=C
∂x i
∂x i
.
2.3. Derivative of higher orders
The derivative of function f : R1→R1 is at the same time a function,
i. e. f ′ : R1→R1. It may be necessary to define function acceleration, i.
e. to determine the rate of change of derivative (f ′ )′(x). Using the
definition (14) and introducing the notation f ′′ (x) for the derivative
( f ′ )′(x), we obtain
f ′′(x )= (f ′)′(x ) = lim
∆x → 0
f ′(x + ∆ x) − f ′(x )
.
∆x
(16)
The derivative f ′′ (x), determined by the equality (16), is called a
derivative of the second order or the second derivative, and the
original derivative f ′ (x) — is called a derivative of the first order or
first the derivative.
Similarly we can introduce and examine a derivative of the third
order
f ′′′(x )= (f ′′)′( x) = lim
∆x → 0
f ′ ( x + ∆ x ) − f ′ ( x)
∆x
and, in general, a derivative of k-order
f (k ) ( x)= (f ( k -1) )′( x) = lim
∆x → 0
f ( k -1) (x + ∆ x ) − f (k -1 ) ( x)
∆x
Derivatives with orders higher than the first order are called derivatives of higher orders or senior derivatives.
Example. It is known that derivative of a power function xn is
calculated according to the rule (xn)′=nxn-1. That is why
(xn)″ = (nxn–1)′ = n(n–1)xn–2,
(xn)″′ = (nxn–1)″ = (n(n–1)xn–2)′ = n(n–1)(n–2)xn–3
and in general
(xn)(k) = n(n–1)(n–2) ... [n – (k–1)]xn–k.
Derivatives of higher order can also be introduced for a multivariable function. Let’s assume that the derivative of function u=f(x1, … ,
xn) was calc ulated as ∂f/∂xi. This derivative is at the same time a function of variables (x1, … , xn) and now we can calculate its derivative,
for example, by variable xk. As a result we’ll obtain the second deriva-
21
tive of function f using the variables xi, xk which is denoted by
∂ 2f
.
∂x i ∂x k
We could have obtained the derivative of function ∂f/∂xi using the
same variable xi, and then we should have obtained the second derivative of function f by one and the same argument:
∂ 2f
. The similar
∂x 2i
process of differentiation can be continued and derivatives of higher
order of multivariable functions can be obtained.
Example. To find derivatives of the second order for the function
u=xy2
∂ 2u
∂ 2u
∂ 2u
,
,
∂x 2 ∂x∂y ∂y2
.
Apparently,
∂u
∂2 u
∂2 u
= y2 ,
=
0
,
= 2y ,
∂x
∂x 2
∂x∂y
∂u
= 2 xy,
∂y
∂2 u
= 2x .
∂y 2
Charter 3. Integrals
3.1. Definite integrals
Let’s specify a problem: to compute the area S under the graph of
continuous function y=f(x), y>0 on a certain segment [a,b] (Fig. 7). This
problem will bring us to a new notion of a definite integral of the function f(x). At first we’ll attempt to obtain an approximated formula for
the area S of curvilinear trapezoid aABb, i. e.
n
S ≈ Sn = ∑ f (ξ i )∆x i ,
(17)
i =1
where ∆xi=xi – xi-1.
Now, we’ll be unrestrictedly increasing the number of partitions of
the segment [a,b] so that the lengths |xi–xi-1| of all partial segments will
tend to zero (the lengths of all partial segments will tend to zero if the
sequence λn of biggest lengths of partial segments will tend to zero,
22
what is expected further on). Since for each partition there exists a corresponding sum Sn (and its biggest segment λn), we obtain, therefore,
the sequence of sums Sn of the form (17) (and the sequence of the biggest segments λn). It case there exists a limit of the sequence of sums
Sn as the number of partitions in increased unrestrictedly in the way
that the sequence λn→0 (one and the same number irrespective of the
selection of points xi, ξ i), then the limit it is taken as the precise value of
the area of curvilinear trapezoid aABb. Thus, by definition
S = lim
λn → 0
n
∑ f (ξ i )∆ x i .
(18)
i =1
Fig. 7
Fig. 8
Now, let’s consider another problem. Let us compute the mass M
of a straight thin cylindrical rod with the length l and the cross-sectional
area C=const (see Fig. 9), with the density of material ρ which is variable with respect to its length and is a known continuous function of the
length: ρ=F(x).
In order to compute the mass of the rod we’ll use a method which
is identical to the method we applied for calculating the curvilinear
trapezoid area. Let’s partition the segment [0,l] by points of intersection
x1, … , xn-1 into n small parts and select on each partial interval an arbitrary point ξ i. The mass of the rod segment at each partial interval (xi1,xi) where the density ρ is changed little, can be approximately expressed by the product f(ξ i)∆xi, ∆xi=xi – xi-1, and then the mass of the
whole rod will be approximately expressed by the sum similar to (17),
M ≈ Mn =
n
∑ F(ξi )C∆x i ,
i =1
or
23
n
M ≈ M n = ∑ f (ξi )∆x i ,
(17′)
i =1
where f(x)=CF(x). Let’s continue, as above, to increase unrestrictedly
increasing the number of partitions of the rod [0,l] into segments in the
way, that λn→0. Since for each partition there exists a corresponding
sum Mn (its own approximated value of the mass M), again we obtain
the sequence of sums Mn of the form (17′) – (17). If the limit of the
sequence of sums Mn exists in the above meaning as the number of
partitions is increased unrestrictedly (λn→0), then the limit is taken as
the precise value of the mass M of the rod and we have the expression,
similar to (18),
M = lim
λ n →0
n
∑ f (ξ i ) ∆x i .
(18′)
i =1
As we see, the problems of different physical nature bring us to the
same from (18) for calculating the limit of sums. In mathematical
analysis the limit of the above sums acquired the name of a definite
integral. The process of summation which results in a definite integral is
in general, i. e. apart from the physical meaning of a summed function,
plotted as follows (Fig. 10). Let a continuous function f(x) to be given
on a segment [a,b]. We’ll partition the segment [a,b] by points of intersection x1, … , xn-1 into n small parts, at each partial interval we’ll arbitrarily select
Fig. 9
Fig. 10
a point ξ i∈[xi-1, xi] and compose the sum of products
Sn =
n
∑ f (ξi ) ∆x i ,
i =1
∆ x i = x i − x i−1 ,
(19)
which is called the integral sum. Now, we’ll be unrestrictedly increasing the number of partitions of the segment [a,b] into parts in the way
that λn→0.
If there exists a limit of the obtained sequence of integral sums, i. e.
there exists
24
n
∑ f ( ξi ) ∆ x i ,
λ →0
lim
n
i =1
then it is called the definite integral of the function f(x) on the segment
b
[a,b] and it is denoted by ∫ f ( x )dx . Thus, by definition
a
b
λn
a
n
∑ f (ξ i )∆x i .
→0
∫ f (x )dx = lim
(20)
i =1
It can be proved that if the function f(x) is continuous on the segment [a,b], then the definite integral (20) — is one and the same finite
number at any selection of points xi, ξ i.
From definition (20) it is easy to deduce the basic properties of
definite integral.
1) Integral of the sum is equal to the sum of integrals. Indeed,
b
n
∑ [f (ξ i ) + ϕ(ξi )]∆x i =
→0
∫ [f (x ) + ϕ( x)]dx = lim
λn
a
i =1
n
n


= lim  ∑ f (ξi )∆ x i + ∑ ϕ( ξi )∆x i  = lim ∑ f (ξ i )∆ x i + lim ∑ ϕ(ξ i )∆ x i
λn → 0
λn → 0
 i =1
i =1
 λ n → 0 i =1
i =1
n
n
i. e.
b
b
b
a
a
a
∫ [f (x ) + ϕ( x)]dx = ∫ f (x )dx + ∫ ϕ(x )dx.
In proof of this, we used the fact (see page 6), that the limit of the
sum of sequences (in this case of the sequences of integral sums ) is
equal to the sum of their limits.
2) The constant factor C can be placed after the integral sign. Indeed,
b
a
n
n
n
∑ Cf (ξ i )∆x i = λlim→0 C∑ f (ξ i )∆x i = C λlim→0 ∑ f (ξi )∆x i
λ →0
∫ Cf (x )dx = lim
n
i =1
i=1
n
n
i =1
i. e.
b
b
a
a
∫ Cf ( x)dx = C ∫ f ( x)dx.
25
In proof of this, we used the fact (see page 6), that the constant
factor can be placed after the sign of limit of sequence (in this case —
n
the sequence CS n = C∑ f (ξi )∆ x i ).
i =1
3) When the direction of integration is changed, the integral will reverse the sign. Indeed,
a
n
b
i =1
a
n
∑ f (ξi )(x i−1 − x i ) = − λlim→0 ∑ f (ξi )(x i − x i−1 ) = −∫ f ( x)dx.
λ →0
∫ f (x )dx = lim
i =1
n
b
n
4) Integral with the same upper and lower limit is equal to 0, i. e.
q
∫q f (x )dx = 0,
as in this case in the sum of (20) ∆xi=0.
5) Path of integration can be divided into parts, i. e. (Fig. 11)
b
∫
=
a
c
∫
+
a
b
∫.
c
Indeed, the limit (20) is not depended on the selection of point xi (on the
method of partitioning the integration interval into segments). Therefore, partitioning of the segment [a,b] can be performed in the way, that
point C will remain fixed, while the integral sum (20) will be divided into
two independent ones (see Fig. 11)
n
m
n
i =1
i =1
i = m +1
∑ f (ξi )∆x i = ∑ f (ξ i ) ∆x i + ∑ f (ξ i ) ∆x i ,
∆x i = x i − x i −1 .
And as the limit of the sum is equal to the sum of limits, so
n
m
n
∑ f (ξ i ) ∆x i = λlim→0 ∑ f (ξ i )∆x i + λlim→0 ∑ f (ξi )∆x i .
→0
lim
λn
i =1
n
i =1
n
i =m+1
Hence, basing on the definition (20) of a definite integral, we obtain
the required equation
b
c
b
a
a
c
∫ f (x )dx = ∫ f ( x)dx + ∫ f ( x)dx.
3.2. The mean-value theorem
26
Fig. 11
Fig. 12
The latter property of the definite integral explains its geometrical
meaning with alternating f(x). Indeed, we already know that on intervals, where f(x)≥0, the value of the definite integral is equal to the area
of a curvilinear trapezoid lying above the axis x (Fig. 12, areas S1 and
S3).
On intervals, where f(x)≤0, the value of the definite integral is apparently equal to the area of the curvilinear trapezoid, lying under the
axis x with the negative reversed. Indeed, in order to compute the area
S2 in Fig. 12, in formulas (17) – (18) f(ξ i) should be substituted for
–
f(ξ i) as the area is a nonnegative quantity, i. e.
n
∑ [− f (ξi )]∆x i
λ →0
S2 = lim
n
=
i =1
a2
∫ [− f (x )] dx
a1
a2
= − ∫ f (x )dx,
a1
and, hence,
a2
∫ f (x )dx = −S2 ,
a1
In the general case, when we divide the path of integration into
parts [0,a], [a1,a2], … , [an-1,b], on each of which f(x) is constant in
sign, we obtain
b
a1
a2
a
a
a1
∫ f ( x)dx =
∫ f ( x )dx +
∫ f (x )dx + L +
b
∫ f ( x)dx,
a n−1
from which it follows that the value of the definite integral on the whole
path of integration is numerically equal to the algebraic sum of the areas of curvilinear trapezoids (considering their sign), lying over and under the axis x.
Let’s denote this area by S, i. e.
b
∫ f (x )dx = S.
a
27
Apparently, there exists a rectangle with the base (b–a) and some
height h, such that its area (Fig. 13, rectangle aA1B1b)
(b – a)h = S
(if S<0, then the height h<0, i. e. the rectangle lies under the axis x).
Comparing the last formulas we get that
h=
1 b
f ( x )dx.
b - a ∫a
Fig. 13
Fig. 14
Ending by the geometrical appearance, we can come to a conclusion
(see Fig. 13) that when function f(x) is continuous on [a,b], there should
exist at last one point ξ∈[a,b], at which f(ξ)=h (later on this fact will be
substantiated in detail; there are three such points in Fig. 13 ξ 1, ξ 2, ξ 3).
Assuming for the last equation that h=f(ξ), we obtain
b
∫ f (x )dx = (b − a )f (ξ) .
(21)
a
The exact location of point ξ is usually unknown. It is only known
that ξ∈[a,b]. We have to emphasize that the formula (21) is true for
functions that are continuous on a closed interval [a,b]. It indicates that
the value of the definite integral of a continuous function is equal to the
product of length of the integration interval and the integrand value at
some “midpoint” of this integral. The formula (21) is referred to as the
mean value theorem.
3.3. Indefinite integral. Newton – Leibnitz formula
28
Now, let’s discuss an integral with the variable upper limit of a continuous function
x
∫ f (t )dt.
a
Here, the variable upper limit and the integration variable are denoted by different letters because their nature is different. The upper
limit x is variable in the meaning that it can take on different values.
The integration variable t is variable in other meaning: at each value of
the upper limit x it is running through all the values lying between the
lower a and the upper x limits of integration (see definition (20) of the
integral). The integral with the variable upper limit is a function of its
own upper limit (see Fig. 14). Let’s denote this function by F(x), i. e.
x
F( x ) = ∫ f ( t)dt .
(22)
a
Let’s calculate the derivative of function F(x). We have
F( x + ∆x ) =
x +∆ x
∫ f ( t)dt,
F( x + ∆ x ) − F( x ) =
a
x
x +∆x
a
x
∆F = ∫ f ( t)dt +
∫ f (t )dt
x +∆x
x
a
a
∫ f ( t)dt − ∫ f ( t)dt,
x
x +∆ x
a
x
− ∫ f (t )dt =
∫ f (t )dt
and according to the mean value theorem
∆F = ∆xf (ξ),
where ξ∈[x,x+∆x]. Hence,
∆F
= f ( ξ) ,
∆x
lim
∆x → 0
∆F
= lim f (ξ).
∆x ∆x → 0
As ξ∈[x, x+∆x], so when ∆x→0, the point ξ→x, that is why
lim f (ξ) = lim f (ξ) . Be virtue of the continuity of function f(x), see
∆x →0
ξ→ x
definition (11)
lim f ( ξ ) = f ( x ),
ξ→ x
and we obtain
lim
∆x →0
or
∆F
= f (x ) ,
∆x
F′(x ) = f (x ),
29
or
'
x

 ∫ f (t )dt  = f (x ) .


a

As we see, the derivative of an integral with the variable upper limit of
a continuous function is equal to its integrand. Function f(x), the derivative of which is equal to f(x) (F′(x)=f(x)), is referred to as the primary
of function f(x). Therefore, the integral with the variable upper limit is
the primary of its integrand. It is easy to see that, if F(x) – is the primary of f(x), so is F(x)+C. Indeed, [F(x)+C]′=F′(x)+C′=F′(x)=f(x). It is
easy to establish that all primaries of function f(x) are covered by the
formula F(x)+C, where F(x) — is a certain primary of function f(x).
Indeed, let F1(x), F2(x), be the primaries of function f(x), i. e.
F1′(x)=f(x),
F2′(x)=f(x)
on
(a,b).
Then
F2′(x)–F1′(x)=0,
[F2(x)–F1(x)]′=0 on (a,b). As the rate of change of difference F2–F1 is
equal to zero at all x∈(a,b), so F2–F1=C=const, F2=F1+C on (a,b) as
was to be shown.
The set of all primaries of function f(x) is refereed to as its indefinite integral and is denoted by
∫ f (x )dx.
As all primaries of function f(x) are defined by the formula F(x)+C,
where F(x) is a certain primary, so
∫ f (x )dx = F(x ) + C.
Let a certain primary F(x) of continuous function f(x) be determined.
All primaries differ in constant, and this is why the integral with the
variable upper limit of function f(x), which is the primary of this function, can differ from function F(x) only in constant C, i. e.
x
∫ f (x )dx = F(x ) + C.
a
When x=a, we obtain
0 = F(a) + C,
C = −F(a)
and hence
x
∫ f (x )dx = F(x ) − F(a ).
a
30
Assuming that x=b, we obtain
b
b
∫ f (x )dx = F(b ) − F(a ) = F(x ) a .
(23)
a
The formula (23) is referred to as Newton – Leibnitz formula. It
shows that to calculate the definite integral it is necessary to find any
primary of the integral and calculate the difference of values at lower
and upper limits of the integration.
As all primaries are contained within the indefinite integral (and
differ only in constant) so the formula (23) can be rearranged as follows
b
b
a
a
∫ f (x )dx = (∫ f ( x)dx )
,
(23′)
i. e. to calculate the definite integral it is necessary to find the indefinite
integral and the difference of its values at upper and lower integration
limits.
2
Example. Calculate the integral J = ∫ x 2 dx.
1
3
It is obvious that (x /3) is the primary of function x2, as
(x /3)=3x2⋅x2/3. This is why
3
2
∫ x dx =
x3
+C.
3
According to the formula (23) or (23)′
J =
2
∫1 x dx
2
=
x3
3
2
1
=
8−1
7
= .
3
3
The corresponding curvilinear trapezoid the area of which is defined by the above calculated integral, is shown in Fig. 15.
31
Fig. 15
Fig.16
Example. Calculate the area S of a curvilinear trapezoid formed by
the cosinusoid y=cos x and the axis x on the line segment x∈[0,π/2]
(see Fig. 16). The sought area S is defined by the integral
S=
π/2
∫ cos xdx.
0
As sin x is the primary of function cos x (because (sin x)′=cos x) or
∫ cos xdx = sin x + C,
so according to Newton – Leibnitz formula
S = sin x
π/ 2
= 1 − 0 = 1.
0
It should be noted in conclusion that for comparatively simple functions it is possible to find an analytical expression for their indefinite
integral, i. e. it is possible to “take” the integral. But integrals of the majority of functions that are of practical interest belong to the “non takeable” kind. The examples of such integrals are:
∫
sin x
dx,
x
∫ sin x
2
dx,
∫ cos x
2
dx.
For the majority of practical problems dealing with calculating integrals (with constant or variable limits), we should apply approximated
(numerical) integration methods.
Charter 4. Differential equations and approximated
representations of functions
4.1. Differential equations of n-th order
Many physical problems involve relationships containing the derivative of a sought function. Let’s study the following example.
Let a ball with mass m move (oscillate) under the force of a spring
which is attached at point A (Fig. 17). It is necessary to determine the
law of the ball motion, i. e. the relationship x(t).
32
From the course in physics it is known that the force F of the spring
which is applied to the ball is proportional to the deflection of the ball
from its neutral position x=0 (when the spring is not under tension or
compressed) and is directed to the side which is inverse to deflection, i.
e. F= –kx where k=const — is the proportionality factor, characterizing
the elasticity properties of the spring.
As it follows from Newton’s second law
ma=F,
where a is acceleration, and as a=x′′ according to the physical meaning
of the second derivative we obtain the relation
mx″ = –kx,
or
mx″ + kx = 0,
or
x '' + a 2 x = 0 , a =
k
.
m
(24)
We obtained the relation containing the second derivative of the
sought function as well as the sought function itself. In the problem at
issue the applied force F is a function of the position of the ball. In other
problems a material body with mass m can move straight under the applied force F, the law of change for which is specified with respect to
time, i. e. the function F(t) is given. In this way, for example, moves a
plane or a rocket (with mass m) flying horizontally under the effect of
traction force F of the engine (Fig. 18) that changes with respect to
time according to the given F(t) law. Newton’s second law ma=F in
this case will result in relation
mx″ = F(t).
(25)
Fig. 17
Fig. 18
33
where the newly sought function is found in the form of the second derivative, but the function itself which is sought for is not found.
Any relation which contains the derivative of a sought function and
may contain, the sought function itself and its argument x, is called a
differential equation. The order of the senior derivative in the equation, defines the order of equation. Thus, the equation
f(x, y, y′, y″, ... , y(n)) = 0,
(26)
where x — is the argument, y=y(x) — is the sought function, f — the
known (given) function of its arguments x, y, y′, y′′, … , y(n), is a differential equation of n-th order, written in its general form. Any
function y(x) satisfying the equation (26), i. e. transforming it into identity with respect to the variable x, will be the solution of a differential
equation (26). Differential equations (24), (25) which we obtained in
the above examples, apparently, are differential equations of the second
order with the sought function x(t). The second equation is easily integrated (the process of finding the solution of a differential equation is
briefly called its integration) as the force F is given. Indeed, integrating
the equation x″=F/m once, we obtain
x′ =
1
F(t )dt + C1 ,
m∫
where C1 — is an arbitrary constant. Denoting
1
F( t)dt = Φ( t),
m∫
the obtained equation will be rearranged as
x ′ = Φ(t ) + C 1 .
(27)
Integrating this equation of the first order we find the sought function
x(t):
x = ∫ Φ (t )dt + C1 t + C 2 ,
where C2 — is also the arbitrary constant, or denoting by,
x = Ψ (t ) + C 1 t + C 2 .
In particular, when the traction F of the engine is changing accoding to
the law of linearity
F = a + bt,
where constants a, b are known, we have
34
Φ( t ) =
1
m
Ψ( t) = ∫ Φ(t )dt =
∫ (a + bt) dt =
1
m
1
bt 2 
 at +
,
m
2 

bt 2 
1  t2
t3 




at
+
dt
=
a
+
b
∫
2 
m  2
6 

and, hence,
x( t ) =
1  t2
t3 
 a + b  + C1 t + C 2 .
m 2
6
(28)
Substituting directly the expression (28) into the equation (25) we can
establish that it identically (i. e. for any t and C1, C2) satisfies the equation and is, therefore, its solution.
A notice should be taken of the fact, that the solution (28) of the
differential equation (25) contains arbitrary constants, i. e. the differential equation (25) has not a single but multiple solutions which can be
obtained by fixing this way or other arbitrary constants C1, C2. Moreover, we should pay attention to the fact that the solution (28) of a differential equation of the second order (25) is dependent on two arbitrary constants C1, C2, the solution of a differential equation of the first
order (27) is dependent on one arbitrary constant (C2).
At the same time, examples we can give, when differential equation
doesn’t have any solution at all:
y′2 + y2 = –5
(the sum of squares of real numbers cannot be equal to a negative
number), or when the differential equation has the only solution, not
dependent on arbitrary constants: for example, the equation
y′2 + y2 = 0.
has the only solution y(x)≡0, as the sum of squares can be equal to
zero, only if each addend is equal to zero(when y(x)≡0, apparently,
y′(x)≡0).
Further it will be proved that under certain restrictions imposed on
function f, which determines the structure of the differential equation
(26), the equation (26) has the infinite set of solutions dependent on arbitrary constants, so that the solution of the equation of n-th order is
dependent on n arbitrary constants C1, … , Cn. This particular case,
being of greatest practical interest, will be studied further.
35
4.2. The Cauchy initial value problem.
The solution of the Cauchy problem by Euler method
In order to determine arbitrary constants C1, … , Cn, n conditions
should be imposed on the solution of a differential equation. Usually
they are the value of the sought function and its derivatives of (n–1)-th
order at a certain characteristic point x0 to determine the constants C1,
… , Cn, i. e. the following types of conditions are used
y(x 0 ) = y 0 , y′(x 0 ) = y′0 , K , y ( n −1) (x 0 ) = y (0n −1) ,
(29)
where y0, y0′, … , y(0n −1 ) — are given numbers. If x is the time, then
the initial conditions are usually given at point x0=0, i. e. at the time
point taken as initial. The conditions (29) irrespective of physical meaning of argument x, are called Cauchy’s initial conditions, and the
problem of finding a solution for the equation (26), satisfying Cauchy’s
conditions (29) — the Cauchy problem.
A solution of a differential equation of n-th order, containing n arbitrary constants:
y = ϕ(x, C1, ... , Cn),
(30)
and such that under certain values of constants C1, … , Cn there can be
found from it a solution satisfying Cauchy’s conditions with any given
initial data y0, y0′, … , y(0n −1 ) , is called the general solution of this equation of n-th order.
As any solution y=ψ(x) of a differential equation of n-th order satisfies certain Cauchy’s conditions
ψ(x 0 ) = y 0 , ψ′(x 0 ) = y′0 , K , ψ ( n −1) (x 0 ) = y (0n −1) ,
so the general solution of equation of n-th order is often referred to as
its solution containing arbitrary constants C1, … , Cn (see (30)) and
such that under certain values of constants C1, … , Cn any solution of
the equation can be obtained from it.
Any solution of a differential equation of n-th order that doesn’t
contain arbitrary constants is called the partial solution of equation.
When the general solution (30) is available any partial solution can be
derived from it provided that arbitrary constants C1, … , Cn are properly fixed.
36
Let’s explain the foregoing on examples. As our first example we’ll
examine the solution (28) of the equation (25) and we’ll show that it is
the general solution of the equation (25). Indeed, it contains two arbitrary constants C1, C2 and we only have to prove that any given initial
Cauchy’s conditions can be satisfied with their help
x(0) = x 0 , x ′(0) = x ′0 ,
(31)
where x0, x0′ — are some given numbers.
If we express the solution (28) as
x = Ψ(t) + C1t + C2, Ψ( t) =
1  t2
t3 
 a
+ b 
m 2
6
and satisfy the conditions (31), we obtain
Ψ(0) + C 2 = x 0 ,
 '
Ψ (0) + C 1 = x ′0 .
Hence
C 1 = x ′0 − Ψ′(0), C 2 = x 0 − Ψ (0),
and as
Ψ(0) = 0, Ψ ′(0) =
1
t2
 at + b
m
2

 = 0,
 t =0
then
C 1 = x ′0 , C 2 = x 0 .
( 32)
Therefore
x =
1  t2
t3 
 a + b  + x ′0 t + x 0 .
m 2
6
(33)
Thus, we have established that if we assign the constants C1, C2 in the
form of (32), we well satisfy any given Cauchy’s conditions (31) and it
means that the solution (28) — is the general solution. Setting, for example, in the solution (28) C1=0, C2=0 we find the partial solution of the
equation (25)
x =
1  t2
t3 
 a
+ b .
m 2
6
37
As another example, we’ll try to find the general solution and then
the solution of the Cauchy problem for the equation of oscillations (24)
that was obtained above. From the equation (24) expressed as
x″ = –a2x,
(34)
it follows that the sought function x(t), differentiated twice, returns to its
original form acquiring only the factor (–a2). But this property is possessed by trigonometric functions
x1 = cosat, x2 = sinat,
(35)
the second derivatives of which are respectively equal to
x1′ = − a 2 cos at = − a 2 x 1 , x ′2 = − a 2 sin at = − a 2 x 2 ,
i. e. the functions (35) are partial solutions of the equation (34) and,
therefore of the equation (24), as
x1′ + a 2 x 1 = 0, x ′2 + a 2 x 2 = 0.
(36)
The equation of oscillations (24) belongs to the class of linear equations: and the sought function and its derivative enter in the equation
linearly. That is why the linear combination of solutions x1, x2, i. e. the
function
x(t) = C1x1(t) + C1x2(t),
(37)
where C1, C2 — are arbitrary constants, will also be the solution of the
equation (24). Indeed, substituting the expression (37) into the left side
of the equation (24) we obtain
[C 1 x1 (t ) + C 2 x 2 (t )]′′ + a 2 [C1 x 1 ( t) + C 2 x 2 (t )] =
= C1 [x ′1 ( t) + a 2 x 1 (t )] + C 2 [x ′2 ( t) + a 2 x 2 ( t)] = 0
by virtue of the equations (36).
The expression (37) which we’ll rewrite as follows
x1 = C1cosat + C1sinat,
(38)
contains two arbitrary constants and is, as it was established before, the
solution of the equation (24). Let us find whether it is the general solution. With this purpose we’ll set the Cauchy problem for the equation
(24):
x″ + a2x = 0, x(0) = x0, x′(0) = v0
(39)
and we’ll try to satisfy this equation by duly selecting constants C1, C2.
(In the initial Cauchy’s conditions (39) x0, v0 — are arbitrary given
numbers, defining respectively the initial position x0 and initial velocity
38
v0 of the ball). Substituting the solution (38) into Cauchy’s conditions
(39), we obtain
C1 cos 0 + C 2 sin 0 = x 0 
− aC1 sin 0 + aC 2 cos 0 = v 0 
from which
C1 = x0, C2 = v0/a,
and, hence, the solution of the Cauchy problem (39) at arbitrary initial
data has the form of
x = x 0 cos at +
v0
sin at ,
a
(40)
this form being obtained from the solution (38) under due selection of
constants C1, C2.
Thus, we have established that the expression (38) is the general
solution of the equation of oscillations (24) as well as found the solution
of the Cauchy problem (39) which is often set with respect to an oscillation equation.
In the discussed example, the exact solution of the Cauchy problem
(39) was found in the form of an analytical expression (40). For the
majority of problems of practical interest it is different, i. e. the exact
solution of a differential equation in the majority of cases can not be
obtained in the form of any analytical expression. Hence, approximation
methods of finding solutions of differential equations are of special interest. Approximation methods are subdivided into analytical and numerical. The former give the possibility to find the solution of the considered equation in the form of an analytical expression, approximately
satisfying the given differential equation. The latter allows to develop an
algorithm (a sequence) of calculations with the help of which the approximated numerical values of the sought solution are determined.
With the emergence of quick operating computer equipment numerical
methods turned to be the most powerful means of solving differential
equations. We’ll study Euler method solving the Cauchy problem for a
differential equation of n-th order, resolved for to the derivative, as the
most simple example of a numerical method.
So, let it be required to find the solution y(x) of the equation
y′ = f(x,y),
(41)
39
where f(x,y) — is a known continuous function of variables x, y, satisfying the initial Cauchy’s condition
y(x0) = y0,
(42)
where x0, y0 — are given numbers.
In order to understand the essence of the method, let’s refer to a
geometrical illustration (Fig. 19). Because of the initial conditions (42)
the sought integral curve, i. e. the curve reflecting the solution y=y(x) of
the problem (41) – (42), passes through the point A0(x0,y0).
Fig. 19
Fig. 20
Let’s plot points x1, x2, ... , xn with spacing ∆x=h=const on the axis
x starting from point x0 to the right and try to find the approximate value
of the solution y(x) at these points, i. e. try to calculate approximately
the quantities
y(x1), y(x2), … , y(xn).
Taking into account the geometrical meaning of the derivative, it
can be concluded that any solution y(x) of the equation (14) at any of
its points (x,y) has a tangential the angular coefficient of which k=tgα is
defined by the equation
k=tgα=f(x, y).
Having this in mind let’s draw a line segment through the point
A0(x0,y0) which angular coefficient
k0=f(x0, y0),
(i. e. we’ll draw a tangential segment to an unknown integral curve at
point A0) and assume that the ordinate y1 of the point A1 of its intersection with line x=x1 is the approximate value of the quantity y(x1), i. e.
y1≈y(x1).
Further, let’s draw a line segment through the point A(x1,y1) which angular coefficient corresponds to this point
40
k1=f(x1,y1),
and accept that the ordinate y2 of its point A of intersection with line
x=x2 to be is the approximate value of the quantity y(x2), i. e.
y2≈y(x2).
Operating in this manner further on, we’ll plot a certain broken line
A0 A1 A2 … Am … An, which will generally speaking coincide with the
exact solution y(x) of the problem (41) – (42) the more the less will be
the step value h, because the smaller is the step h the more each section of the broken line differs from the corresponding segment of a true
integral curve.
In order to arrange the computing process we’ll obtain formulas defining the ordinates ym of points Am(xm, ym). It can be seen from Fig.
20 that
ym = ym–1 + htgα m–1 = ym–1 + hf(xm–1,ym–1),
(43)
and if we denote f(xm-1, ym-1)=fm-1, and assign m the values m=1, 2, …
, n, in consecutive order we’ll obtain calculation formulas:
y1 = y 0 + hf 0 ,
y = y + hf ,
2
1
1

KKKKKK
y n = y n −1 + hf n −1 ,
(44)
or reduced
ym = ym–1 + hfm–1, m = 1, 2, ... , n.
(45)
Euler method allows us to find out the form of the solution of the
Cauchy problem (41) – (42) quite easily. We do not dwell on substantiation of the method (on convergence of the sequence ym of approximated values of the solution to the exact solution y(x) at the step h→0,
on method error at selected h, i. e. on evaluation of difference |y(xm)–
ym|, and other).
4.3. Approximate representations of a function.
Taylor’s series. Fourier’s series
In practice there often arises a problem of to calculate a function
which determination doesn’t allow to calculate them directly with sufficient exactness. To such functions refer, for example, sin x, n 1 + x , at
41
any natural n and many others. Indeed, trigonometric sine is determined
by relation (Fig. 21)
Fig. 21
and it is clear that by measuring the segments BC, AB and the angle x
it is not possible to determine however exactly the quantity sin x. Same
applies to the calculation of root y= n 1 + x , which determination brings
us to an expression yn=1+x not fit for calculations (it is necessary to
find the number y which being raised to the n-th power would give the
preset number 1+x). That it is necessity to represent approximately
such (and other) functions that are difficult to calculate in the form of
an easily calculated expression, for example, in the form of a polynomial of degree.
Let us find out what are the possibilities for obtaining such representation. Apparently, any function f(x) determined on a certain interval
(a,b), can be represented on this interval as sum
f (x ) = A 0 + A 1 (x − x 0 ) + L + A n−1 (x − x 0 ) n−1 + R n ( x),
(46)
i. e. as a polynomial to degrees (x–x0), x0∈(a,b) and a certain added
Rn(x), formally equal to Rn(x)=f(x)–Ôn(x), where
Φ n (x ) = A 0 + A1 (x − x 0 ) + L + A n−1 (x − x 0 ) n −1 .
(47)
The index of the polynomial Ôn(x) corresponds to the number of addends entering it; the addend Rn(x) is referred to as the error term.
Coefficients of polynomial Rn(x) are related to function values f(x),
Rn(x) and their derivatives at point x0 by apparent relations
f (x 0 ) = A 0 + R n (x 0 ), f ′(x 0 ) = A 1 + R ′n ( x 0 ), f ′ ( x 0 ) = 2!A 2 + R n′ (x 0 ) , 

( n− 1)
(n − 1)
f ′′ ( x 0 ) = 3!A 3 + R ′n′′(x 0 ), K , f
(x 0 ) = (n − 1)!A n −1 + R n (x 0 ). 
(48)
Let us require that the remainder Rn(x) together with its derivatives
Rn′(x), Rn′′, … , R (nn −1) (x ) would reduce to zero at point x0, i. e. so that
R n ( x 0 ) = 0, R ′n ( x 0 ) = 0, K , R (nn −1) (x 0 ) = 0 .
42
(49)
Then coefficients of polynomial Ôn(x) would be expressed only through
the values of the given function f(x) and its derivatives f′(x), … , f(n1)
(x) at point x0:
A 0 = f (x 0 ), A1 = f ′( x 0 ), A 2 =
A3 =
1
f ′′(x 0 ),
2!
1
1
f ′′′(x 0 ), K , A n −1 =
f ( n−1) (x 0 ).
3!
(n − 1)!
(50)
Equalities (49) will satisfy if the error term Rn(x) will have the structure
of
R n ( x) =
r n ( x)
(x − x 0 ) n ,
n!
(51)
where the factor 1/n! has been introduced purely from esthetical consideration (see formulas (50)); the function rn(x) is subject to definition.
Later, in the course of “Higher mathematics” it will be shown that
rn(x) = f(n)(ξ),
(52)
where ξ is a certain “midpoint” of interval (x0, x), i. e. ξ=x0+θ(x–x0),
0<θ<1. The exact location of point ξ is usually unknown. It is known
only that ξ∈(x0,x). The formula (51) for the error term can be written
now as
R n (x ) =
f ( n ) ( ξ)
(x − x 0 )n .
n!
(53)
The expression (53) is referred to as error term in the Lagrange
form. Substituting the equalities (50) and (53) into the relation (46), obtain
f ′′(x 0 )
(x − x 0 )2 + K +
2!
f ( n ) ( ξ)
+
(x − x 0 ) n .
n!
f (x ) = f (x 0 ) + f ′(x 0 )(x − x 0 ) +
+
f ( n −1 ) (x 0 )
( x − x 0 ) n −1
(n − 1)!
(54)
The relation (54) makes sense if f(x) has the derivatives of up to
the n-th order inclusive at point x0 and its vicinity, what is expected further. Since the function which is continuous on a closed interval is
bounded (properties of functions bounded on a closed interval will be
43
studied in detail later), the derivative f(n)(ξ) is bounded in the certain
vicinity of point x0, i. e. there exists number M>0 such that
|f(n)(ξ)| ≤ M
for any ξ in the closed vicinity x0. But then the error term (53) will be
arbitrary small (Rn(x)<ε) in the sufficiently small vicinity of point x0 (|x–
x0|<δ) and, therefore, in this δ–neighborhood the approximate formula
f (x ) ≈ f (x 0 ) + f ′( x 0 )(x − x 0 ) +
f ′′(x 0 )
(x − x 0 )2 + K +
2!
f (n −1) (x 0 )
+
( x − x 0 ) n −1 ,
(n − 1)!
(55)
the error of which is determined by the quantity of the discarded term
Rn(x) will give the values of function f(x) with the error |Rn(x)|<ε. It is
essential, of course, that there would exist a point x0 on the function
f(x) where function itself and its derivatives of any order can be easily
calculated (for the function sin x, n 1 + x such is the point x0=0). The
formula (54) is referred to as Taylor’s formula, and the polynomial
(55) — Taylor’s polynomial.
We’ll be interested in the case when: 1) function f(x) has continuous derivatives of any order at point x0 and in its certain vicinity (not
necessarily small); 2) at any x from this vicinity
f n ( ξ)
( x − x 0 ) n = 0.
(56)
n→ ∞
n → ∞ n!
Such functions exist and the function sin x, n 1 + x belongs to them.
lim R n ( x) = lim
The condition (56) is of extreme importance, as it implies that with sufficiently large number of terms n in Taylor’s polynomial, the error Rn of
the approximate formula (56) can be made arbitrarily small at any x
from the considered vicinity.
It follows from the first condition that Taylor’s formula (54) has
sense for any n. Basing on this fact we assume in the formula (54) that
n→∞. Then, on the basis of the second condition (formula (56)) we
conclude that
f (x ) = lim Tn ( x),
(57)
n→∞
where
44
Tn (x ) = f (x 0 ) + f ′(x 0 )(x − x 0 ) +
+
f ′′(x 0 )
(x − x 0 )2 + K +
2!
f ( n −1) (x 0 )
(x − x 0 ) n −1 .
(n − 1)!
(58)
As we see, under the above two conditions function f(x) is the limit
of the sequence of Taylor’s polynomials Tn(x). Formally lim Tn has the
n→∞
form of a sum with in finite number of addends (such sums are usually
called infinite series or simply series) and we can rewrite the equality
(57) as follows
f (x ) = f (x 0 ) + f ′(x 0 )(x − x 0 ) +
f ′′(x 0 )
(x − x 0 )2 + K +
2!
f ( n −1 ) ( x 0 )
+
( x − x 0 ) n −1 + K
( n − 1)!
(59)
The series (59) is referred to as Taylor series. The so called partial
sums can be composed from it according to the following rule
S1 (x ) = f (x 0 ), S2 = f (x 0 ) + f ′(x 0 )( x − x 0 ), K , S n ( x) = f ( x 0 ) +
+ f ′(x 0 )(x − x 0 ) + K + K +
f ( n −1 ) (x 0 )
( x − x 0 ) n −1
(n − 1)!
(60)
The partial sums (60) of the series (59) are apparently Taylor’s polynomials of different orders:
S1(x)=T1(x), S2(x)=T2(x), ... , Sn(x)=Tn(x).
(60′)
If we substitute Tn(x) for Sn(x) in the equality (57), it can be rewritten
as follows
f (x ) = lim S n ( x),
(61)
n→∞
shows that function f(x) is the limit of the sequence of partial sums of
the series (59) and the equality (59) should be understood exactly in this
meaning. The above considerations are taken as a basis when summing
any infinite series. For example, we are given a numerical series
a1 + a2 + ... +a n + ...,
(62)
where a1, a2, … , an, … — are given numbers.
Let us arrange the sequence of partial sums of the series
S1 = a1, S2 = a1 + a2, ... , Sn = a1 + a2 + ... +a n, ...
(63)
45
If the numerical sequence Sn is converging at n→∞ and has the
limit S(Sn→S) then the series (62) is called convergent, the limit S is
called the sum of the series and is expressed as
S = a1 + a2 + ... +a n + ... .
If the sequence of partial sums Sn is diverging (doesn’t have a finite
limit at n→∞) then the series (62) is called divergent.
Let us see now a functional series
u1(x) + u2(x) + ... + un(x) + ... ,
(64)
composed of known functions, defined on a certain interval (a, b). If the
sequence of partial sums of the series (64)
S1(x)=u1(x), S2(x)=u1(x) + u2(x), ..., S1(x)=u1(x)+u2(x)+...+un(x), ...
(65)
is converging on interval (a,b) and has the limit of the function S(x) then
the series (64) is called converging on (a,b), the function S(x) is called
its sum and it will be expressed as
S(x) = u1(x) + u2(x) + ... + un(x) + ... .
(66)
It is often said about the notation (66) that function S(x) is expanded
into functions un(x).
If the sequence of partial sums (65) is diverging on (a,b) (doesn’t
have a finite limit at least at one point of interval (a,b)) then the series is
called diverging on (a,b).
In practice, functional series composed of power functions (the
studied Taylor series) are often used. For periodic functions expansion
in trigonometric functions is often used:
f (x ) =
a0
+ a 1 cos x + b1 sin x + a 2 cos 2 x + b 2 sin 2 x + L +
2
+ a n cos nx + b n sin nx + L,
(67)
or compactly
f (x ) =
a0
+
2
∞
∑ (a n cos nx + b n sin nx).
n =1
(68)
where f(x) — is a periodic function with the period 2π (functions
cosnx, sinnx have the same period). If the series (67) – (68) is converging and allows term by term integration similar to finite sums, i. e. the
integral of the sum of series f(x) is equal to the sum of integrals of the
46
terms of series (it is not always hene for infinite sums – series), then
coefficients of the series (67) – (68) are determined by formulas
a0 =
1 2π
1 2π
f
(
x
)
dx
,
a
=
f ( x) cos nxdx,
n
π ∫0
π ∫0
(69)
1 2π
b n = ∫ f ( x) sin nxdx, n = 1, 2, 3, K .
π 0
as it is easy to check, at any m=1, 2, … , n=1, 2, … :
2π
∫ cos nxdx = 0,
0
2π
2π
2π
0
0
∫ sin nxdx = 0,
∫ cos nx sin mxdx = 0
0, ïðè n ≠ m,
ïðè n = m,
∫0 cos nx cos mxdx = π,
(70)
2π
0, ïðè n ≠ m,
∫ sin nx sin mxdx = π, ïðè n = m.
0
The series (67) – (68) with coefficients (69) is referred to as Fourier
series of the function f(x), coefficients (69) – Fourier coefficients.
Example. Represent functions f(x)=sin x in Taylor series in the vicinity of point x0=0.
In the vicinity x0=0 The Taylor series (59) will be rewritten as follows:
f (x ) = f (0) + f ′(0)x + L +
f ( n −1) (0) n −1
x +L =
(n − 1)!
∞
f ( n ) ( 0) n
x
n!
n =0
∑
(71)
(as, according to the definition f(0)(x)=f(x), 0′=1) and will be referred to
as Maclaurin’s series.
We have
f ( x ) = sin x , f ′( x) = cos x, f ′ ( x ) = − sin x, f ′′( x ) = − cos x, f
(IV )
( x ) = sin x
and further repeated. Hence, it is seen that
π

f ( n ) (x ) = sin  x + n  , n = 0, 1, 2, 3, K
2

(72)
and, therefore,
f
(n )
(0) = sin n
at n = 2k ,
 0,
π
= 
k
2
(−1) , at n = 2k + 1.
(73)
47
As a result, the series (71) for f(x)=sin x takes on the following
form
sin x =
∞
∑ (−1)k
k =0
x 2 k+1
x3 x5
x 2 k +1
= x−
+
− L + (−1) k
± L.
( 2k + 1)!
3! 5!
(2k + 1)!
(74)
The function sin x satisfies the conditions of representation in Taylor’s series at any x, and for that reason the series (74) is converging at
any x, i. e. x∈(+∞,–∞) (on the whole numerical axis).
48
Contents
Chapter
1.
Limits
…………………………………………...
1.1. The limit of numerical sequence …………………….
3
1.2. The limit of functional sequence ……………….……
1.3. Continuity of function ………………………………..
9
11
Chapter 2. Derivatives
……………………………………...
2.1. The notion of derivative ……………………………...
2.2. Partial derivative of the function ……………….…….
2.3. Derivatives of higher orders ………………………….
13
Chapter 3. Integrals
………………………………………...
3.1. Definite integrals ……………………………….…….
3.2. The mean-value theorem
……………………………..
3.3. Indefinite integral. Newton – Leibnitz formula ………
18
Chapter 4. Differential equations and approximate
representations of function
……………………
4.1. Differential equations of n-th order ……….………….
4.2. The Cauchy problem. Solution of the Cauchy problem
by the Euler method
.....................................................
4.3. Approximate representations of a function. Taylor’s
series. Fourier’s series
………………………………..
3
13
16
17
18
23
25
28
28
32
37
49