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Let lim f (x) = L1 and lim g(x) = L2 , then

x→x0

Sum rule and difference rule

lim ( f (x) ± g(x)) = L1 ± L2

x→x0

Constant multiple rule

lim (kf (x)) = kL1 for any number k

x→x0

Product rule

lim ( f (x) g(x)) = L1 L2

x→x0

Quotient rule

(x)

lim

if L ≠ 0

x→x0 g(x)

Infinitesimal function

Function limit of which equals 0 when

x → x0 ,

Infinitely large function

Function limit of which equals ∞ , when

x → x0

lim α (x) = 0

= ∞

x→x0

then

lim

x→ x0 α (x)

lim

f (x) = ∞

= 0

x→x0

then

lim

x→ x0 f (x)

lim

α (x)

= 1 α (x) ~ β (x), if x → x .

Equivalent

infinitesimal

functions

x→x0 β (x)

(infinite functions) for x → x0 :

Ratio

infinitesimal

functions

∞

(infinite functions),

∞

difference of infinite functions,

[∞ - ∞],

powers with bases tending to 1 and

[1∞ ]

infinitely large exponents

are called inderteminacies

lim

sin u

= lim

= 1

The first remarkable limit

u →0

u →0 sin u

Equivalent infinitesimal functions

sin u ~

tgu ~

arcsin u ~ arctg u ~ u if

u → 0

1 x

lim 1 +

= e , where е≈ 2,718 –

The second remarkable limit

x→∞

irrational.

kx+m

lim 1 +

= e ak / b

x→∞

bx + c

Polynomial

equivalent

its

Pn (x) = an x n

+ an−1 x n−1 + L + a1 x + a0 ~

leading monomial as x → ∞

x n if

x → ∞ .

Some recommends for calculating limits

Type of indeterminacy

Recommends

P (x)

Devide the numarator and the

lim

dinominator by (x − x

)

15.

(x)

x→x0

f (x)

, where f (x), g(x)-

To carry irrationality from one part of the

lim

fraction to another knowing that

x→x0 g(x)

16.

• (

a −

b ) ( a +

b )= a − b

are irrational

) (3

• (3

m 3

+ 3

± 3

a 2

b 2

= a ± b

f (x)

(

)

(

)

To transform

f (x)

in such a way to use

17.

lim

where

x , g

g(x)

x→x0 g(x)

contain trigonometric or inverse

(12)

trigonometric functions

f (x)

∞

Use (14)

lim

x→∞ g(x)

∞

lim( f (x))g (x )

= [1∞ ]

Transform ( f (x))g (x ) in such a way to

use (13)

x→∞

•Any monotone and bounded sequence has a limit as n → +∞

•A sum of any finite number of infinitesimal functions is an infinitesimal

•The product of a bounded function and an infinitesimal is an infinitesimal

•A product of any finite number of infinitesimal functions is an infinitesimal

DIFFERENTIAL CALCULUS

Vocabulary

acceleration – ускорение

angular velocity – угловая скорость concavity – вогнутый

convexity – выпуклый

critical (stationary) point – критическая (стационарная) точка decrease – убывать

decreasing – убывающий derivative - производная

derivatives of y with respect to x – производная y по x differentiable - дифференцируемый

differential - дифференциал differentiate - дифференцировать

directional derivative – производная по направлению exstremum – экстремум

force of current – сила тока

function of several variables – функция многих переменных gradient – градиент

inclined asymptote – наклонная асимптота increase – возрастать

increasing – возрастающий

increment of a function – приращение функции increment of an argument – приращение аргумента inflection point – точка перегиба

kinetic energy – кинетическая энергия low of motion – закон движения maximum – максимум

minimum – минимум normal – нормаль

partial derivative – частная производная rectilinear motion – прямолинейное движение second derivative – производная второго порядка tangent line – касательная

tangent plane – касательная плоскость total differential – полный дифференциал velocity – скорость

vertical asymptote – вертикальная асимптота

Table of Derivatives

C′ = 0

x′ = 1

(u ± v)′ = u′ ± v′

(uv)′

= u′v + uv′

(Cu )′

= Cu′

′

u′v − uv′

Cv′

C ′

= −

(logau )′ =

u′

ulna

u′

(lnu )′ =

(sin u )′ = cosu u′

(cosu )′ = −sin u u′

Y(tan u )′ = u′

cos2 u

′= − u′(cot u )

		2
		sin u
		sin u

′

= au

a−1

)

u′

(

)′ =

u′

)′ =

u′

n n un−1

(au )′

= auu′ln a

(

u )′

III

= e

u′

(uv )′

= uvv′ln u + vuv−1u′

(arcsin u )′ =

u′

1 − u2

(arccosu )′

= −

u′

1 − u

(arctan u )′

u′

1 + u2

u′

(arc cot u )′

= −

1 + u

(sinh u )′ = cosh u u′

(cosh u )′ = sinh u u′

YII*

)′

u′

(

= cosh2 u

tanh u

where C, e, a,α , n – are constants, u = u(x), v = v(x) – are functions.

№

Name

Formula, equation

Increment of an argument

x = x − x0

Increment of a function y = f (x):

y = f (x) − f (x0 )

Definition of derivative

of a

function

y′ = lim

y(x) at a point x0

x→0

Using of a derivative

Equation of a tangent line at a point

M 0 (x0 , y0 ) belonging to the graph of a

y = f ′(x0 ) (x − x0 ) + y0

function y = f (x)

y = −

(x − x

0 ) + y0

Equation of a normal at x = x0

j ′(x

)

L’Hospitale’s Rule

lim

f (x)

∞

= lim

f ′(x)

x→x0 g(x)

∞

x→x0 g ′(x)

x0 D(y ) is called critical point

f ′(x0 ) = 0 , or does not exist

y = f (x) increases if

f ′(x) > 0

y = f (x) decreases if

f ′(x) < 0

Critical point x0 is the point of maximum

1). f ′(x) changes its sign from

of a function y = f (x) if

“+”to “– “ at x0

2). f ′′(x0 ) < 0

Critical point x0 is the point of minimum

1). f ′(x) changes its sign from “– “

of a function y = f (x) if

to “+” at x0

2). f ′′(x0 ) > 0

x0 is the inflection point of a function

f ′′(x) changes its sign at a point

y = f (x) if

Inflection points can be only points where the function exists and its second

derivative is zero or does not exist.

Inflection points split a domain of a function into intervals of concavity or of

convexity.

Arc of the curve y = f (x) is concave if

f ′′(x) > 0

Arc of the curve y = f (x) is convex if

f ′′(x) < 0

17A straight line Г is called an asymptote of a curve L if the distance between

the moving point of the curve line L and the line tends to zero as the distance between this point and the origin increases infinitely

Equation of vertical asymptote of the

x = x0 , where x0 is such that

curve y = f (x)

lim f (x) = ± ∞

x→ x0

y = kx + b ,

where k = lim

f (x)

19 Equations of inclined asymptote of the

x→±∞

curve y = f (x)

and

b = lim ( f (x) − kx)

x→±∞

1. Find D(y ), E(y), asymptotes,

Investigation of functions and

roots, intervals of constant sign.

constructing their graphs

Define if the function is a)

periodic or not, b) even or odd.

2. Find critical points, intervals of

increasing and decreasing of the

function, extremums.

3. Find inflection points, intervals

of concavity and convexity.

Differential of a function y = y(x)

dy = y′dx

Differentiation of a function giving in

yt′

x = x (t )

xt′

parametric form:

y = y (t )

Functions of Several Variables

Let a function u = f (x, y, z ) be given

∂u

= f

′

y =C ;

∂u

= f

′

x=C

;

∂x

z =C

∂y

z =C

Partial derivatives of the first order

∂u

= f

′

x=C

∂z

y =C

A total differential of u = f (x, y, z ) at

du =

∂ u(P0 )

dx +

∂ u(P0 )

dy +

a point P0 (x0 ; y0 ; z0 )

∂ x

∂ y

∂ u(P0 )

∂ z

Gradient of

u = f (x, y, z ) at a point

∂ u(P

)

∂ u(P

)

∂ u(P

)

gradu(P )

P (x

;

; y

; z

)

∂

Directional derivative in the direction

of the vector

= (a x ; a y ; a z )

∂ u(P0 )

= (gradu

= (

α; cos β ; cos γ

)

a 0

cos

, where

∂a

a x

(

)

∂ (

)

∂

cosα

cosα +

cos β +

a x2 + a 2y + a z2

∂x

∂y

a y

∂u(Po )

cos β =

cos γ

∂z

a x2 + a 2y + a z2

cos γ =

a z

a x2 + a 2y + a z2

Let a function z = f

(x, y )be given

27	Partial derivatives of the second		∂ 2 z
	order of a function z = f (x, y,)	•	∂ 2 z			=
		•	∂ x		2	=
			∂ x		2
		•	∂ 2 z			=
		•	∂ y		2	=
			∂ y		2
		•		∂ 2 z
		•	∂ x∂ y
			∂ x∂ y
		•		∂ 2 z
		•		∂ y∂ x
				∂ y∂ x

∂ z ∂ ∂ x

∂ x

∂ z ∂ ∂ y

∂ y

		∂ z
	∂
	∂
=		∂ x
=		∂ y
		∂ y
		∂ z
		∂
		∂
=		∂ y
=			∂ x
			∂ x

Equation

of a

tangent plane to a

z − z

= f ′(P)(x − x ) + f ′ (P )(y − y

)

y 0

surface

z = f (x, y )

point

P0 (x0 ; y0 ; z0 )

x − x0

y − y0

z − z0

Equation of a normal to a surface

f ′ (P )

− 1

z = f (x, y ) at a point P0 (x0 , y0 , z0 )

Exstremum of Function of Two

Variables z = f (x, y )

Stationary point P (x

∂z

= 0

, y

∂x

∂z

= 0

∂y

∂ 2 z(P )

A B

Let

A ,

= B

, and

= C , and

= AC − B 2

∂x

∂x∂y

∂y 2

B C

> 0 then extremum exists, and if

a) A > 0 (B > 0)

it is minimum,

Sufficient Conditions of Exstremum

b) A < 0 (B < 0)

it is maximum,

at a point P0 (x0 , y0 )

< 0 extremum doesn’t exist,

= 0 we can

say nothing.

RUSSIAN - ENGLISH VOCABULARY

абсолютная величина – absolute value

алгебраическое дополнение элементаaij − cofactor of an element aij

арккосинуc – anticosine, arccosine арккотангенс – anticotangent, arccotangent арксинус – antisine, arcsine

арктангенс – antitangent, arctangent асимптота – asymptote

астроида – Astroid

большая ось – major axis большая полуось – semimajor axis

больше или равно – greater than or equal to

вектор – vector

x) − common logarithm.

векторное произведение – vector product (cross product)

векторное уравнение – vector equation

вершина параболы – vertex of the parabola

взаимно-однозначная функция – one-to-one function

возрастающий – to be increasing

вычислять – evaluate

вычитаемое – subtrahend вычитание – subtraction

вычитание матриц –subtraction of matrices вычитать – subtract

геометрическая интерпритация – geometrical representation

гипербола – hyperbola

гиперболические функции – hyperbolic Functions гиперболический косинус – hyperbolic cosine гиперболический котангенс – hyperbolic cotangent гиперболический параболоид – hyperbolic paraboloid гиперболический синус – hyperbolic sin гиперболический тангенс – hyperbolic tangent гиперболический цилиндр – hyperbolic cylinder

главная диагональ – principal (main) diagonal главный аргумент – principal argument график функции – graph of a function

дважды – twice

двуполостный гиперболоид – two-sheeted hyperboloid действительная ось – real axis

действительная часть – real part действительные числа – real number

деление – division

деление в столбик – long division

делимое – dividend делимость – divisibility

делитель – divisor

делить – divide (by) делиться на – be divisible by

десятичная дробь – decimal десятичный логарифм (log10 диагональная матрица – diagonal matrix директриса – directrix

дискриминант – discriminant

для того, чтобы (найти) – to (find) дописать, добавить – supplement

дробь – fraction

единичная матрица – unit-matrix

единичный вектор – ort of a vector (unit vector)

единственное решение – unique solution

заглавная буква – capital letter закрытый интервал – closed interval

знаменатель – denominator

известный – known

интервал – interval

иррациональные числа – irrational numbers

канонические уравнения – canonical equations кардиоида – сardioid

квадратичная функция – quadratic function квадратная матрица – square matrix коллинеарный – collinear

компланарный – coplanar комплексное число – complex number

конечная десятичная дробь – terminating decimal

конечная точка – terminal (end) point

корень или ноль функции – root or a zero of a function

корень энной степени – nth root of корень, радикал – radical косинусоида – сosine curve котангенсоида – сotangent curve

коэффициент при старшей степени – leading coefficient кривая Гаусса – сurve of Gauss

кривые второго порядка – second order curves кубическая парабола – cube parabola

левая тройка – left-handed triple

лемниската Бернулли – lemniscate of Bernoulli

линейная функция – linear function

лист Декарта – Folium of Descartes

логарифмическая функция – logarithm function

локон Аньези – witch of Agnesi

максимум – maximum

матрица (матрицы) – matrix (matrices)

матрица порядка n – matrix of order n

меньше или равно – less than or equal to минимум – minimum

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