Mathematics for Foreign Students: Limits and Derivatives

The Ministry of Science and Higher Education of the Russian Federation Kazan National Research Technological University











MATHEMATICS FOR FOREIGN STUDENTS: LIMITS AND DERIVATIVES


        Tutorial














Kazan
KNRTU Press

2022

         UDC 51(075)


Published by the decision of the Editorial Review Board of the Kazan National Research Technological University

Reviewers:
PhD in Physics and Mathematics, Associate Professor Z. Yakupov PhD in Physics and Mathematics, Associate Professor D. Shevchenko

Authors:
D. Bikmukhametova, N. Gazizova, S. Enikeeva,
A. Mindubaeva, N. Nikonova, D. Giliazova







         Bikmukhametova D.
         Mathematics for Foreign Students: Limits and Derivatives : tutorial / D. Bikmukhametova, N. Gazizova, S. Enikeeva [et. al.]; The Ministry of Education and Science of the Russian Federation, Kazan National Research Technological University. - Kazan : KNRTU Press, 2022. - 116 p.

         ISBN 978-5-7882-3155-6

      The tutorial contains theoretical information on and encompasses the applied problems of how to introduce a calculus or differential calculus.
      The tutorial is intended for students studying Mathematics and Calculus.
      Prepared at the Department of Advanced Mathematics.

                                               UDC 51(075)

ISBN 978-5-7882-3155-6    © D. Bikmukhametova, N. Gazizova, S. Enikeeva,
                              A. Mindubaeva, N. Nikonova, D. Giliazova, 2022 © Kazan National Research Technological
University, 2022


2

                INTRODUCTION





       This Tutorial is intended for international 1st and 2nd-year full-time, internal, and part-time students studying Mathematics and Calculus. It fully covers the material in Advanced Mathematics for students majoring in engineering.
       Studying this Tutorial allows forming the students’ general cultural and professional competencies:
       - independent work skills;
       - self-organization and self-education skills;
       -        skills to generalize and analyze information, set goals and select ways to achieve goals;
       -        readiness to apply fundamental mathematical, natural science, and general engineering knowledge in general professional activities.
       To prepare the present Tutorial we considered, studied and analyzed the works of Russian and foreign authors. The Tutorial consists of the theoretical part, examples of solving typical problems, tests, and answer keys. The theoretical part includes all necessary information on how to prepare for tests, colloquia and final examination. Texts are illustrated with a large number of examples and figures.
       In addition to the basic formulae and definitions, the authors offer a detailed analysis of the tests. The Tutorial provides a set of tasks that can be used by both lecturers for organising their classes and classroom tests and students for their self-study and self-preparation for tests, colloquia and examinations.


3

                NUMERICAL SEQUENCES





      A numerical sequence is an infinite set of numbers enumerated by a positive integer index in ascending order of values of the index.
      In other words, a sequence is a function f (n) of a discrete variable n, whose domain consists of the set of all natural numbers.
      The elements of a sequence are called the terms. The term f (n) (that is, the n-th term) is called the general term or variable of the sequence.
      The general term is represented by a lower case letter with the subscript n: bₙ , xₙ , etc.
      The general term put into braces denotes a sequence:[an|, [bn}, [xn|, etc.
      Examples:
      1)        The general term bₙ = n² determines the - sequence of -number: 1, 4, 9, 16, ..., n²,--:
      2)        The general term b1 = b, bn+1 = bn • q determines an infinite geometric progression with the common ratio q.
      3)        The general term yₙ = 2ⁿ determines the sequence of number: 2 2² 23
2,2 ,2 ,•••;> ² •>••••
       Let Sn be the sum of the first n elements of a sequence [bn} :
n
Sn =Z bk
k=1
Then the set S1,S2,...,Sₙ,... is also a sequence [Sn} which is called the sequence of the partial sums of the sequence [bn |.
      A sequence [ xn} is said to be an upper-bounded sequence if there exists a finite number M such that |xₙ| < M for all natural numbers n. The number M is said to be an upper bound of [ xn} .
      A sequence [ xn} is called a lower-bounded sequence if there exists a finite number m such that \xₙ | > m for each natural number n. The number m is called a lower bound of [xₙ}.

4

      A sequence is called bounded if there exist two finite numbers, m and M, such that m < |xn| < M for all terms of the sequence. Otherwise, the sequence is unbounded.
      Examples:
      1)       The sequence 2, 4, 6, 8, ..., is lower-bounded , since aₙ = 2n > 0 for all natural numbers n.
      2)       The sequence Xₙ = -n² (-1,-4,-9,-16,...) is upper-bounded (x= -n² < 0) for all natural numbers n.

    ~            1   1      (1111
    3)    The sequence b =— — — _ _
ⁿ 3ⁿ  13,9,27,81

is bounded since

0 < b = — < 1 for all natural numbers n.
     ⁿ  3 з
      4)        The sequence |(-1)ⁿ • ₙ} (-1,2, -3,4, • • •) is unbounded since it has no finite bounds.
      A sequence {xₙ | is called a monotone increasing sequence if

xn< xn+1

for each natural number n.

      A sequence {xₙ | is called a monotone decreasing sequence if

x<x n+1 n

for each natural number n.
     Examples of monotone increasing sequences:

        x=2n+1:
        x  =2(n+1)+1=2n+3>2n+1=x

for all natural numbers n.
     Examples of monotone decreasing sequences:

1

1

yₙ

72: Уп = ~ n      n²

1
⁽ⁿ+1)2,


    n = 1,2,3,...


Examples of non-monotone sequences:
                    n


        bn = — n


1,

1  11

2,  3,4,  5

1

5

                LIMITS





Limits of numerical sequences


       Intuitive Definition of the Limit:
       The limit of a sequence {Xₙ} is a number a such that the terms Xn remain arbitrarily close to a when n is sufficiently large.
       This statement is written symbolically in any of the following form: lim xₙ = a, n ^W

lim xₙ = a,      xₙ ^ a,     as n ^w .
       Formal Definition of the Limit:
       Number a is called the limit of a sequence {Xₙ} if for any arbitrary small number 8 > 0 there exists a number N such that

        |Xn - a\ < ⁸

for each n> N.
       Geometrically, the inequality |xₙ - a| < 8 can be interpreted as the open interval (a-8,a+8).

       In Calculus, the interval (a-8,a+8) is usually named “delta neighborhood” (or “delta vicinity”) of the point a.
       In terms of 8 -neighborhood, the limit of a sequence can be defined by the following wording:
       Number a is the limit of a sequence {Xₙ} if any arbitrary small delta neighborhood of the point a contains all terms of the sequence, starting from a suitable term.
       If a sequence has a limit a such that a is a finite number, it is said that the sequence converges to the number a , and the sequence is called convergent. Otherwise, the sequence is called divergent.

      Examples:
      1)       The sequence | Xₙ = 1 + (-1)ⁿ | is divergent since it has no limit as n ^w

6

      2)         The sequence {Хп = 1 + 2 +... + ft} is divergent since it approaches infinity as number П ^ X.
      3)       The sequence Jx = sin — \ is divergent since it has no limit as

                        I п     2 I


п ^X

     4)     The sequence {x =(-5)п} is divergent since it has no limit as
п ^X

      5)       Prove that the sequence 2, —, ..., П converges to the num
    2        п -1

ber 1.
      To prove this statement rigorously, we have to show that for any arbitrary small number 6 > 0 there exists a number N such that the condition n > N implies the inequality x -1| < 6 or



п
п -1

Indeed,

п-1

1=

п-(п -1)  1

п-1


    ^ — < 6, ^ п -1 п - 1


  .
п-1
11

    —, п > —+ 1.

66


    Setting N > — + 2

            6

we

obtain that the inequality

X - 1| = А

—

п > N

Implies


п >¹ + 2, 6

and hence,


—— 1
п - 1

6,


no matter how small positive value of 5 is chosen. Therefore,

lim хп = 1.
п ^X


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Infinitesimal sequences and infinite large sequences


A sequence {an} is called infinitesimal if it converges to zero: lim a„ = 0.
                                 n ^W
       The formal definition is the following.
       A sequence {an } is called an infinitesimal sequence if for any arbitrary small positive number S there exists a number N such that the inequality n > N implies |an| < S.
       A sequence {Xn} is called infinite large (or divergent) if Xn approaches infinity as n ^ W
       The formal definition is the following:
       A sequence {xn} is called an infinite large sequence if for any arbitrary large number M > 0 there exists a number N such that |xn| >M for each n> N

       Notations:

                             lim x =w, n,
                              n ^W

or xn ^ w as n ^ W

      Examples:

      1) The sequence

— ^ 0 as n ^ W .
n


{an }={¹I i n j

is an infinitesimal sequence since

2) The sequence [aₙ }

’ —;= > is an infinitesimal sequence as n ^ W. i Vn J

8

I

    3) The sequence [an J = ’ Sin — r


is an infinitesimal sequence as

n ^»

      4) The sequence [an J = J--In +1J


ris an infinitesimal sequence as

n ^»

5) The sequence [ₐ

is an infinitesimal sequence as П ^Ж .



      6)  The sequence [p J = J —— Г is an infinite large sequence since n y In+11

— >ᵥ as n ^» .
n +1
      7)       The sequence [Д J = [Vn + 1 + n- -1 J is an infinite large sequence since n+ +1 + n- -1 ^ ж as n ^ж .



    Properties of infinitesimal sequences


       Property 1. If [an J    is an infinitesimal sequence and [bn J is a
bounded sequence, then [anbn J is an infinitesimal sequence.
       Property 2. If [an J and [yn J are infinitesimal sequences, then [anynJ is also an infinitesimal sequence.
       Corollary. The sum of any finite number of infinitesimal variables is an infinitesimal variable.
       Property 3. If [an J   is an infinitesimal sequence then J — I is an
Ian J

infinite large sequence and vice versa.


9

Properties of limits of sequences



      Property 1. lim (k • xₙ ) = k • lim xₙ, ke R. n ^да V             n     n^да

      Property 2. If there exist finite limits of sequences {xₙ} and |yₙ},

then

              lim (xn ± yn ) = lim Xn ± lim yₙ. n ^да         n ^да   n ^да
Property 3. If there exist finite limits of sequences {Xₙ} and |yₙ |,

then

lim(xₙ • yₙ) = lim xₙ •lim yₙ.
                        n^да V        n^да   n^да

      Property 4. If there exist finite limits of sequences {xₙ | and {yₙ} and


        lim yₙ Ф 0, then

n ^да

lim
n ^да

lim xₙ
n ^да
       . lim yₙ n ^да

      Property 5. If there exist finite limits of sequences | Xn | and

{},

then

lim
n ^да

                          (m limxₙ) n ^да



m = 1,2,3,

Theorems of sequences

     Theorem 1.       Each monotone increasing upper-bounded sequence has a finite limit.
     Theorem 2.       Each monotone decreasing lower-bounded sequence has a finite limit.
     Theorem 3.       A monotone increasing sequence is divergent if it has no upper bound.
     Theorem 4.       A monotone decreasing sequence is divergent if it has no lower bound.


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