Reference_book_on_Higher_Mathemanics_Part_I_F

CONTENTS

ARITHMETIC .....................................................................................................

5

Vocabulary .............................................................................................................

5

Reading of formulas ...............................................................................................

6

Tests for divisibility by 2, 3 and 5 ..........................................................................

7

GCF(a, b) and LCM(a, b) ........................................................................................

7

Common fraction ....................................................................................................

7

Addition, subtraction, multiplication and division of common fractions ...............

7

Decimal ...................................................................................................................

8

Ratio, proportion and percent .................................................................................

8

ALGEBRA .............................................................................................................

9

Vocabulary ..............................................................................................................

9

Fundamental concepts .............................................................................................

10

The real number system ..........................................................................................

10

The interval notations ..............................................................................................

10

Exponents, radicals and their properties .................................................................

11

Polynomials .............................................................................................................

11

Special product formulas.........................................................................................

12

Rational fraction (algebraic fraction) ......................................................................

12

Logarithms and their properties ..............................................................................

12

Trigonometric functions of an acute angle..............................................................

13

Trigonometric circle ..............................................................................................

14

Trigonometric formulas...........................................................................................

14

Equations (linear, quadratic, exponential, logarithmic and trigonometric) ............

15

COMPLEX NUMBERS........................................................................................

17

Vocabulary ..............................................................................................................

17

The equality relation and the arithmetical operations .............................................

16

Imaginary unit .........................................................................................................

17

Operations on complex numbers.............................................................................

18

FUNCTIONS..........................................................................................................

20

Vocabulary ..............................................................................................................

20

Definitions ...............................................................................................................

20

Table for finding the domain of a composite function ...........................................

21

Graphs of functions in the Cartesian System of Coordinates ................................

22

Power functions .......................................................................................................

22

Parabolas .................................................................................................................

22

Hyperbolas ..............................................................................................................

23

Exponential function ..............................................................................................

23

Logarithm function .................................................................................................

23

Trigonometric functions ..........................................................................................

24

Inverse trigonometric functions...............................................................................

25

Hyperbolic functions ...............................................................................................

25

Curves of the second order .....................................................................................

26

Ellipse ......................................................................................................................

26

Hyperbola ................................................................................................................

26

Parabola ...................................................................................................................

26

Witch of Agnesi.......................................................................................................

26

Curve of Gauss ........................................................................................................

26

Loops .......................................................................................................................

27

Curves given by parametric equations ...................................................................

28

Curves in the polar system of coordinates .............................................................

28

LINIEAR ALGEBRA ...........................................................................................

31

Vocabulary .............................................................................................................

31

Matrices, determinants, systems of linear equations...............................................

32

Methods of solution of linear systems.....................................................................

34

Matrix method ........................................................................................................

34

Cramer’s rule ..........................................................................................................

34

Gauss method .........................................................................................................

35

VECTORS .............................................................................................................

37

Vocabulary .............................................................................................................

37

Theorems, definitions and formulas for calculation ...............................................

37

ANALYTIC GEOMETRY ...................................................................................

41

Vocabulary .............................................................................................................

41

A plane in a space ...................................................................................................

42

A straight line in a space ........................................................................................

42

A plane and a straight line in a space .....................................................................

45

A straight line in a plane..........................................................................................

44

Second order curves ...............................................................................................

45

A general equation of a plane .................................................................................

46

Particular cases of a plane in a space ......................................................................

47

Quadric surfaces .....................................................................................................

48

Elliptic, hyperbolic and parabolic cylynders...........................................................

48

Elliptic cone ............................................................................................................

48

Hyperbolic paraboloid ............................................................................................

48

Elliptic paraboloid ..................................................................................................

49

Two-sheeted and one-sheeted hyperboloids ..........................................................

49

Ellipsoid ..................................................................................................................

49

LIMITS...................................................................................................................

50

Vocabulary ..............................................................................................................

50

Operations on limits ................................................................................................

50

Some recommends for calculation of limits............................................................

52

DIFFERENTIAL CALCULUS............................................................................

53

Vocabulary ..............................................................................................................

53

Table of derivatives .................................................................................................

54

Using of a derivative ...............................................................................................

55

Functions of several variables ................................................................................

56

RUSSIAN – ENGLISH VOCABULARY ...........................................................

58

дробь в смешанное число

decimal – десятичная дробь

decrease

– уменьшать

denominator – знаменатель

difference

– разность

divide (by) – делить

dividend – делимое

divisibility

– делимость

division – деление

divisor – делитель

evaluate –

вычислять

example – пример

factor – множитель

fraction

– дробь

greater than or equal to – больше или равно

Greatest Common Factor

– наибольший общий делитель

how much

– сколько

improper fraction– неправильная дробь

increase

– увеличивать

is divisible by – делится на

known

– известный

Least Common Multiple

– наименьшее общее кратное

less than or equal to – меньше или равно

long division– деление в столбик

minuend

– уменьшаемое

quotient – частное (от деления)

remainder

– остаток

repeating decimal

– периодическая дробь

subtract – вычитать

subtraction

– вычитание

subtrahend

– вычитаемое

sum

– сумма

summand

– слагаемое

terminating decimal – конечная десятичная дробь

the least common denominator– наименьший общий знаменатель

three times

– трижды

to (find) – для того, чтобы (найти)

twice

– дважды

unknown

– неизвестный

Arithmetic Operations and Properties

The formula

Read

a + b = c

• a plus b equals c

• b added to a is c

• the sum of a and b is c

a – b = c

• a minus b equals (is) c

• the

difference of a

and b is c

a x b = c, a · b = c;

• a multiplied by

b equals c

• a by b is c

• a times b is c

• the product of a and b is c

a : b = a / b =

a

= c

• a divided by b is c

b

• a quotient of a and b is c

a n

a to the nth power

Properties of addition and multiplication

Name

Formula

Commutative property

• a + b = b + a

• a · b = b · a

Associative property

• (a + b) + c = a + (b + c)

• a(bc) = (ab)c

Distributive property

• a(b ± c) = ab ± ac

Identity property

• There is a unique number, namely 0, such that for

any number a: a + 0 = 0 + a = a

• There is a unique whole number, namely 1, such

that for every number a:

a ·1 = 1 ·a = a

If a ≠ 0, then

Property of zero

• 0 : a = 0 ;

• 0 ·a = a ·0 = 0

Divisor

a is a divisor of b if a is a factor of b

Multiple of a

b is a multiple of a if b is divisible by a

Test for divisibility by 2

A number is divisible by 2 if and only if its ones

digit is 0, 2, 4, 6, or 8

Test for divisibility by 3

A number is divisible by 3 if and only if the sum of

its digits is divisible by 3.

Test for divisibility by 5

A number is divisible by 5 if and only if its ones

digit is 0 or 5.

245

Two hundred forty five.

2436

Two thousand four hundred thirty six, or twenty

four thirty six.

The Greatest Common Factor

The greatest common factor of two nonzero whole

(GCF(a, b))

numbers a and b is the largest whole number that is

a factor of both these numbers.

The Least Common Multiple

The least common multiple of two nonzero whole

numbers a and b is the smallest nonzero

whole

LCM(a, b)

number that is the multiple of each

of

these

numbers.

•

a

, or a / b, represents a of b equivalent parts

Common fraction

b

• a is called the numerator

• b is called the denominator.

Proper fraction

A

fraction

whose

numerator

is

less

than

a

denominator.

Improper fraction

A fraction

whose

numerator

is greater

than

or

equal to a denominator.

Mixed numbers

A number containing the whole an fractional parts.

2/3

Two of thirds or

two over

three

(the

proper

fraction).

3

Four and three of the fifths, or four and three over

five (the mixed number).

4

5

Addition and subtraction of

a

±

c

=

a ± c

fractions with common

b

b

b

denominators

Addition and subtraction of

1)find the

least common denominator

of

these

fractions ;

fractions with different

2) subtract them as before.

denominators

Multiplication of fractions

a

c

=

a c

b

d

b d

Multiplication of a mixed

4

2

7 = 4 7

2 7

= 28

14

number by a whole number

15

15

15

Multiplication of a mixed

2

2

1

3

=

32

11

=

32 11

=

44

= 2

14

15

number by a mixed number

15

8

15

8

15 8

15

a

:

c

=

a

d

Division of fractions

b

d

b

c

Division of a mixed number by

1

19

:1

5

=

40

:

11

=

40 6

2

=

80

= 1

3

a mixed number

21

6

21

6

7 21 11

77

77

Decimal is a common fraction denominator of

Decimal

which is 10 n , where n = 1,2,... . For example:

123

=

123

= 0.0123

10000

104

0.0123

read

“zero point, zero, one, two, three”.

1.875

read “one point, eight, seven, five”.

a

has a terminating decimal representation if and

Terminating decimals

b

only if b contains only 2s and (or) 5s in its prime

factorization:

3

3

3 5 2 2

60

6

=

=

=

=

= 0.06

50

2 5 5

(2 5) (5 2) (5 2)

1000

100

zero point, zero, six.

a

has a repeating decimal representation that does

Repeating decimals

b

not terminate if and only if b has a prime factor

other than 2 or 5 in its prime factorization:

34

.

= 0.343434... = 0.(34) = 0.34

99

a : b =

a

is the ordered pair of numbers, with b ≠ 0.

A ratio of two numbers a and b

b

a

=

c

is a

statement

that two

given

ratios are

A proportion

b

d

equal.

is one hundredth of this number : a

1

=

a

1 percent (1% ) of a number a

100

100

Name

Formula

Let A and B be sets then:

x is an element of the set A ( x belongs to A)

x A

an empty set

A = Ǿ

a union of A and B

A B

an intersection of A and B

A ∩ B

A is a subset of B

A B

to follow

if and only if

any, for any

to exist

such that

:

The Real Number System

Natural numbers (positive integers)

N = {1,2,3,..., n,...}

Integers (whole numbers)

Z = {...,−3,−2,−1,0,1,2,3....}

Rational numbers –

Q =

m

, where m, n Z , n ≠ 0

{all terminating or repeating decimals}

n

all nonterminating

Irrational numbers

nonrepeating decimals

Real numbers

R = {all rational and irrational numbers}

Absolute value of the real number a

a

if a ≥ 0

a

=

− a

if a < 0

The interval notations:

(a, b)={x : a < x < b} - an open interval

all real numbers between a and b, not

including a and not including b.

[a, b]= {x : a ≤ x ≤ b} a closed interval

all real numbers between a and b, including a

and including b.

(a, b]= {x : a < x ≤ b}

all real numbers between a and b, not

including a and including b

[a, b) = {x : a ≤ x < b}

all real numbers between a and b, including a

and not including b.

(− ∞, b) = {x : x < b}

all real numbers less than b.

(− ∞, b]= {x : x ≤ b}

all real numbers less than or equal to b.

(a, ∞)

= {x : x > a}

all real numbers greater than a.

[a, ∞)

= {x : x ≥ a}

all real numbers greater than or equal to a.

(− ∞, ∞) = R

all real numbers