Математика. Математический анализ 1

ФЕДЕРАЛЬНОЕ ГОСУДАРСТВЕННОЕ ОБРАЗОВАТЕЛЬНОЕ  
БЮДЖЕТНОЕ УЧРЕЖДЕНИЕ ВЫСШЕГО ОБРАЗОВАНИЯ  
«ФИНАНСОВЫЙ УНИВЕРСИТЕТ ПРИ ПРАВИТЕЛЬСТВЕ  
РОССИЙСКОЙ ФЕДЕРАЦИИ»  
(ФИНАНСОВЫЙ УНИВЕРСИТЕТ)

Департамент математики

В.Б. Гисин

МАТЕМАТИКА

МАТЕМАТИЧЕСКИЙ АНАЛИЗ 1

Учебное пособие 

для студентов, обучающихся 
по направлению «Экономика»

МОСКВА
2023

ISBN 978-5-00172-481-0

УДК 519.85
ББК 22.16
Г51

Рецензенты:
А.В. Чечкин, доктор физико-математических наук, профессор департамента математики Финансового университета 
при Правительстве Российской Федерации;
О.Е.Орел, кандидат физико-математических наук, 
доцент кафедры высшей математики Московского физикотехнического института (МФТИ).

Г51
Гисин В.Б.
Математика. Математический анализ 1: Учебное пособие / В.Б. Гисин. — М.: Прометей, 2023. —  
148 с.

ISBN 978-5-00172-481-0

Математика. Математический анализ 1: учебное пособие 
для студентов, обучающихся по направлению «Экономика» 
38.03.01 — программа подготовки бакалавра
Пособие предназначено для студентов, изучающих математику на английском языке. Пособие содержит учебный 
материал, относящийся к введению в математический анализ 
и таким его разделам, как дифференциальное и интегральное 
исчисление функций одной переменной.

© В.Б. Гисин, 2023
© Издательство «Прометей», 2023

PREFACE

The mathematics needed for the study of economics 
changes with each passing year. The share of “theoretical 
math” decreases giving place to computer methods. The 
manual is designed to present a thorough, easily understood 
introduction to univariate calculus. The best way to study 
mathematics with this manual is to combine the study 
of calculus with the study of digital mathematics using 
computers.
The book offers a brief review of basic concepts and 
notations for those who are going to learn math in English. 
The theory-and-solved-problem format of each chapter 
provides concise explanations illustrated by examples. No 
mathematical proficiency beyond the high school level is 
assumed at the start.
Formulas and tables are numbered inside each section. 
The figures are numbered end-to-end.

CONTENTS

PRECALCULUS .............................................................................6
1.1. BASIC NOTATIONS ................................................ 6
Signs and Symbols. Functions. Sets. Greek Alphabet
1.2. GRAPHS ON THE CARTESIAN PLANE....................11
Rectangular coordinates. The Graph of an Equation
1.3. FUNCTIONS .........................................................16
Basic Notions. Elementary Functions. Symmetry.
Transformations. Basic Shapes
1.4. COMPLEX NUMBERS ...........................................36
The Set of Complex Numbers. Addition, 
Subtraction, and Multiplication of Complex 
Numbers. Complex Conjugate. A Quotient of 
Complex Numbers. Complex Solutions of a 
Quadratic Equation. Fundamental Theorem of 
Algebra. The Complex Plane. Operations with 
Complex Numbers in Polar Form. Roots of Degree n
1.5. SEQUENCES AND LIMITS .....................................49
Finite and Infinite Sequences. Arithmetic 
Sequences. Geometric Sequences. Properties 
of Sequences. Limit of a Sequence. Selected 
Sequences. Properties of limits. Convergence to 
Infinity. Properties of Infinite Limits

UNIVARIATE CALCULUS  .........................................................62
2.1. LIMITS AND CONTINUITY ....................................62
Limits. Properties of Limits. Continuity. 
Combinations of Continuous Functions. Properties 
of Continuous Functions. Locating Roots of 
Equations. Infinite Limits. Vertical Asymptotes. 
Limits at Infinity (the End Behavior). Oblique 
Asymptotes. Infinite Limits at Infinity. Selected Limits
2.2. THE DERIVATIVE ................................................80
Differentiability at a Point. The Derivative 
Function. Differential. Techniques of Differentiation. 

Higher Derivatives. Implicit Differentiation 
L’Hopital’s Rule. Mean-Value Theorem.  Graph 
Sketching. A General Graphing Procedure
2.3. INTEGRATION .....................................................99
The Indefinite Integral. Properties of the Indefinite 
Integral. The Definite Integral (Riemann Integral)
The Fundamental Theorem of Calculus. Improper 
Integrals. Specificity of the integration techniques 
for evaluating definite integrals
2.4. TAYLOR POLYNOMIAL ....................................... 122
2.5. SERIES ............................................................. 127
The Integral Test. The Comparison Test. 
Alternating Series Test. The Ratio Test. The Root 
Test. Power Series. The Taylor series.
INDEX ........................................................................................140

PRECALCULUS

1.1. BASIC NOTATIONS

Signs and Symbols

N = { , ,...}
1 2
set of natural numbers

Z { ,
,
,...}
0
1
2
set of integer numbers

Q
a
b a b
Z b
{ | ,
,
}
0

set of rational numbers

R
set of real numbers

C
a
bi a b
R i
{
| ,
,
}
 2
1
set of complex numbers

a
b
=
a equals b, a is equal to b, 
identity

a
b
≠
a is not equal to b, a does not 
equal to b, a is other than b

a
b
<
a is less than b 

a
b
≤
a is less than or equal to b 

a
b
>
a is greater than b 

a
b
≥
a is greater than or equal to 
b

a > 0
a is a positive number, a is 
greater than 0

a ≥ 0
a is a non-negative number

a
b
+
addition, a plus b, sum of a 
and b, a, b are the summands 

a
b
−
subtraction, a minus b, 
difference between a and b

a b
⋅
multiplication, a times b, 
product of a and b, a, b are 
the factors

a
b

division, a divided by b, a 
over b, quotient of a and b,
a is the numerator, b is the 
denominator
½, 2/3, 2½
one half, two thirds, two and 
a half
0.01
nought point nought one
5%, 2/5%
five per cent, two fifth per 
cent

a
a
a
i
i

n

n
1
1
...

sum over ai of i equals 1 up 
to n

a
a
a
i
i

n

n
1
1 ...

product over ai of i equals 1 
up to n

∞
infinity

Functions

f
X
Y
:
→
f is a transformation of X into Y

X , D f
f
Y
( )
( )
1
domain of f

Y , R f
f X
( )
(
)
=
codomain of f, range of f

xn
x to the power of n, nth power of x for 
n ≥ 0

x2
x squared

x3
x cubed

x , 
x
n
square root of x, n-th root of x

n!
n factorial

| |
x
absolute value of x

sgn x
sign of x (sgn5
1
=
, sgn3
1,

sgn0
0
=
)

ex , exp x
exponential function of x, e to the 
power of x

log a x
logarithm (log) of x (to) base a

ln x
natural logarithm (log) of x (to base e)

sin x
sine of x

cos x
cosine of x

sec
cos
x
x
=
1
secant of x

csc
sin
x
x
=
1
cosecant of x

tan x , tg x
tangent of x

cot x , ctg x
cotangent of x

arcsin x , sin−1 x
arc sine of x, inverse sine of x

arccos x , cos−1 x
arc cosine of x, inverse cosine of x

arctan x , arctg x
arc tangent of x, inverse tangent of x

Sets

a b c
, , ,...
set with the elements a, b, c, …

a
A
∈
a is an element of A, e.g. 3∈Z  (3 is an integer)

a
A
∉
a is an element of A, e.g. 
2 ∉Q (
2  is 
not rational number)

x
M P x
| ( )
The set of all elements in M satisfying 
condition P, e.g. x
x
Z|
,
2
5
3 4

A
B
⊆
A is a subset of B (any element of A is an 
element of B), A is included in B

A
B
⊂
A is a proper subset of B (is a subset and 
unequal)

A
B
∪
A union B, A or B, includes all occurring 
elements (x is in A
B
∪
 if and only if x is 
in A or x is in B)

A
B
∩
A intersection B, A and B, includes all 
common elements (x is in A
B
∪
 if and 
only if x is in A and x is in B)

A
B
\
A not B include all elements of A that 
are not in B

A
B
×
A cross B, cartesian product of A and B, 
set of all (ordered) pairs from A and B

A
A
A

n

n
...
the set of all ordered n-tuples (
,...,
)
x
xn
1
, 

x
A
i ∈
, i
n
= 1,...,
, e.g. R
R
2
3
,

∅
Empty set

Greek Alphabet

Letters (upper 
case, lower case)
Name
Pronunciation

alpha
′ælfə

beta
′bi:tə

gamma
′gæmə

delta
′deltə

epsilon
′epsəlɔn

xi
ksai

eta
′i:tə

zeta
′zi:tə

iota
ai′əυtə

kappa
′kæpə

lambda
′læmbə

mu
′mju:

nu
′nju:

omicron
′əυməkrυn

pi
pai

rho
rəυ

sigma
′sigmə

tau
taυ

upsilon
′ʌpsəlυn

theta
′θi:tə

phi
fai

psi
psai

chi
kai

omega
əυ′migə