Total differential and its application to approximate calculations. Applying differential to approximate calculations

23. The concept of differential function. Properties. Application of differential in approx.y calculations.

Concept of differential function

Let the function y=ƒ(x) have a nonzero derivative at the point x.

Then, according to the theorem about the connection between a function, its limit and an infinitesimal function, we can write  у/х=ƒ"(x)+α, where α→0 at ∆х→0, or ∆у=ƒ"(x) ∆х+α ∆х.

Thus, the increment of the function ∆у is the sum of two terms ƒ"(x) ∆x and a ∆x, which are infinitesimal for ∆x→0. In this case, the first term is infinitely small function of the same order as ∆x, since and the second term is an infinitesimal function of more high order, than ∆x:

Therefore, the first term ƒ"(x) ∆x is called the main part of the increment functions ∆у.

Differential function y=ƒ(x) at the point x is called the main part of its increment, equal to the product of the derivative of the function and the increment of the argument, and is denoted dу (or dƒ(x)):

dy=ƒ"(x) ∆x. (1)

The dу differential is also called first order differential. Let's find the differential of the independent variable x, i.e. the differential of the function y=x.

Since y"=x"=1, then, according to formula (1), we have dy=dx=∆x, i.e. the differential of the independent variable equal to increment this variable: dх=∆х.

Therefore, formula (1) can be written as follows:

dy=ƒ"(х)dх, (2)

in other words, the differential of a function is equal to the product of the derivative of this function and the differential of the independent variable.

From formula (2) follows the equality dy/dx=ƒ"(x). Now the notation

the derivative dy/dx can be considered as the ratio of the differentials dy and dx.

Differentialhas the following main properties.

1. d(With)=0.

2. d(u+w-v)= du+dw-dv.

3. d(uv)=du·v+u·dv.

d(Withu)=Withd(u).

4. .

5. y= f(z), , ,

The form of the differential is invariant (unchanging): it is always equal to the product derivative of a function by the differential of the argument, regardless of whether the argument is simple or complex.

Applying differential to approximate calculations

As is already known, the increment ∆у of the function y=ƒ(x) at point x can be represented as ∆у=ƒ"(x) ∆х+α ∆х, where α→0 at ∆х→0, or ∆у= dy+α ∆х. Discarding the infinitesimal α ∆х of a higher order than ∆х, we obtain an approximate equality

∆y≈dy, (3)

Moreover, this equality is more accurate, the smaller ∆х.

This equality allows us to approximately calculate the increment of any differentiable function with great accuracy.

The differential is usually much simpler to find than the increment of a function, so formula (3) is widely used in computing practice.

24. Antiderivative function and indefiniteth integral.

THE CONCEPT OF A PRIMITIVE FUNCTION AND AN INDEMNITE INTEGRAL

Function F (X) is called antiderivative function for this function f (X) (or, in short, antiderivative this function f (X)) on a given interval, if on this interval . Example. The function is an antiderivative of the function on the entire number axis, since for any X. Note that, together with a function, an antiderivative for is any function of the form , where WITH- arbitrary constant number(this follows from the fact that the derivative of a constant is zero). This property also holds in the general case.

Theorem 1. If and are two antiderivatives for the function f (X) in a certain interval, then the difference between them in this interval is equal to a constant number. From this theorem it follows that if any antiderivative is known F (X) of this function f (X), then the entire set of antiderivatives for f (X) is exhausted by functions F (X) + WITH. Expression F (X) + WITH, Where F (X) - antiderivative of function f (X) And WITH- an arbitrary constant, called indefinite integral from function f (X) and is denoted by the symbol, and f (X) is called integrand function ; - integrand , X - integration variable ; ∫ - sign of the indefinite integral . Thus, by definition If . The question arises: for everyone functions f (X) there is an antiderivative, and therefore an indefinite integral? Theorem 2. If the function f (X) continuous on [ a ; b], then on this segment for the function f (X) there is an antiderivative . Below we will talk about antiderivatives only for continuous functions. Therefore, the integrals we consider later in this section exist.

25. Properties of the indefiniteAndintegral. Integrals from basic elementary functions.

Properties of the indefinite integral

In the formulas below f And g- variable functions x, F- antiderivative of function f, a, k, C- constant values.



Integrals of elementary functions

List of integrals from rational functions

(the antiderivative of zero is a constant; within any limits of integration, the integral of zero is equal to zero)

List of integrals of logarithmic functions

List of integrals of exponential functions

List of integrals from irrational functions

("long logarithm")

list of integrals of trigonometric functions , list of integrals of inverse trigonometric functions

26. Substitution methods variable, method of integration by parts in the indefinite integral.

Variable replacement method (substitution method)

The method of integration by substitution involves introducing a new integration variable (that is, substitution). In this case, the given integral is reduced to a new integral, which is tabular or reducible to it. Common methods there is no selection of substitutions. The ability to correctly determine substitution is acquired through practice.

Concept of differential

Let the function y = f(x) is differentiable for some value of the variable x. Therefore, at the point x there is a finite derivative

Then, by definition of the limit of a function, the difference

is an infinitesimal value at . Expressing the increment of the function from equality (1), we obtain

(2)

(the value does not depend on , i.e. remains constant at ).

If , then on the right side of equality (2) the first term is linear with respect to . Therefore, when

it is infinitesimal of the same order of smallness as . The second term is an infinitesimal of a higher order of smallness than the first, since their ratio tends to zero as

Therefore, they say that the first term of formula (2) is the main, relatively linear part of the increment of the function; the smaller , the larger the proportion of the increment this part makes up. Therefore, for small values (and for ) the increment of the function can be approximately replaced by main part, i.e.

This main part the increment of a function is called the differential of the given function at the point x and denote

Hence,

(5)

So, the differential of the function y = f(x) is equal to the product of its derivative and the increment of the independent variable.

Comment. It must be remembered that if x– original argument value,

The incremented value, then the derivative in the differential expression is taken in starting point x; in formula (5) this is evident from the record, in formula (4) it is not.

The differential of a function can be written in another form:

Geometric meaning differential. Function differential y = f(x) is equal to the increment of the ordinate of the tangent drawn to the graph of this function at the point ( x; y), when it changes x by the amount.

Differential properties. Invariance of differential shape

In this and the next paragraphs, we will consider each of the functions to be differentiable for all considered values of its arguments.

The differential has properties similar to those of the derivative:

(WITH - constant) (8)

(9)

(10)

(12)

Formulas (8) – (12) are obtained from the corresponding formulas for the derivative by multiplying both sides of each equality by .

Consider the differential complex function. Let be a complex function:

Differential

this function, using the formula for the derivative of a complex function, can be written in the form

But there is a differential function, so

(13)

Here the differential is written in the same form as in formula (7), although the argument is not an independent variable, but a function. Therefore, expressing the differential of a function as the product of the derivative of this function and the differential of its argument is valid regardless of whether the argument is an independent variable or a function of another variable. This property is called invariance(invariance) of the differential shape.

We emphasize that in formula (13) cannot be replaced by , since

for any function except linear.

Example 2. Write the differential of the function

in two ways, expressing it: through the differential of the intermediate variable and through the differential of the variable x. Check the matching of the resulting expressions.

Solution. Let's put

and the differential will be written in the form

Substituting into this equality

We get

Application of differential in approximate calculations

The approximate equality established in the first paragraph

allows you to use a differential for approximate calculations of function values.

Let us write down the approximate equality in more detail. Because

Example 3. Using the concept of differential, calculate approximately ln 1.01.

Solution. The number ln 1.01 is one of the values of the function y= log x. Formula (15) in in this case will take the form

Hence,

which is a very good approximation: table value ln 1.01 = 0.0100.

Example 4. Using the concept of differential, calculate approximately

Solution. Number
is one of the function values

Since the derivative of this function

then formula (15) will take the form

we get

(tabular value

Using the approximate value of a number, you need to be able to judge the degree of its accuracy. For this purpose, its absolute and relative error.

The absolute error of the approximate number is absolute value the difference between the exact number and its approximate value:

The relative error of an approximate number is the ratio of the absolute error of this number to the absolute value of the corresponding exact number:

Multiplying by 4/3, we find

Taking the table value of the root

for the exact number, we estimate using formulas (16) and (17) the absolute and relative errors of the approximate value:

By analogy with the linearization of a function of one variable, when approximately calculating the values of a function of several variables that is differentiable at a certain point, one can replace its increment with a differential. Thus, you can find the approximate value of a function of several (for example, two) variables using the formula:

Example.

Calculate approximate value.

Consider the function and choose x 0 = 1, y 0 = 2. Then Δ x = 1.02 – 1 = 0.02; Δ y = 1.97 – 2 = -0.03. We'll find ,

Therefore, given that f ( 1, 2) = 3, we get:

Differentiation of complex functions.

Let the function arguments z = f (x, y) are, in turn, functions of variables u And v: x = x (u, v), y = y (u, v). Then the function f there is also a function from u And v. Let's find out how to find its partial derivatives with respect to the arguments u And v, without making a direct substitution

z = f (x(u, v), y(u, v)). In this case, we will assume that all the functions under consideration have partial derivatives with respect to all their arguments.

Let's set the argument u increment Δu, without changing the argument v. Then

If you set the increment only to the argument v, we get: . (2.8)

Let us divide both sides of equality (2.7) by Δ u, and equalities (2.8) – on Δ v and move to the limit, respectively, at Δ u→ 0 and Δ v→ 0. Let us take into account that due to the continuity of functions X And at. Hence,

Let's consider some special cases.

Let x = x(t), y = y(t). Then the function f(x,y) is actually a function of one variable t, and it is possible, using formulas (2.9) and replacing the partial derivatives in them X And at By u And v to ordinary derivatives with respect to t(of course, provided that the functions are differentiable x(t) And y(t)) , get an expression for :

(2.10)

Let us now assume that as t acts as a variable X, that is X And at related by the relation y = y(x). In this case, as in the previous case, the function f is a function of one variable X. Using formula (2.10) with t = x and taking into account that , we get that

. (2.11)

Let us pay attention to the fact that this formula contains two derivatives of the function f by argument X: on the left is the so-called total derivative, in contrast to the private one on the right.

Examples.

1. Let z = xy, Where x = u² + v, y = uv². Let's find and. To do this, we first calculate the partial derivatives of three specified functions for each of its arguments:

Then from formula (2.9) we obtain:

(In the final result we substitute expressions for X And at as functions u And v).

2. Find the total derivative of the function z = sin( x+y²), where y= cos x.

Invariance of the shape of the differential.

Using formulas (2.5) and (2.9), we express full differential functions z = f (x, y), Where x = x(u,v), y = y(u,v), through differentials of variables u And v:

(2.12)

Therefore, the differential form is preserved for arguments u And v same as for functions of these arguments X And at, that is, is invariant(unchangeable).

Implicit functions, conditions for their existence. Differentiation of implicit functions. Partial derivatives and differentials of higher orders, their properties.

Definition 3.1. Function at from X, defined by the equation

F(x,y)= 0 , (3.1)

called implicit function.

Of course, not every equation of the form (3.1) determines at as a unique (and, moreover, continuous) function of X. For example, the equation of the ellipse

sets at as a two-valued function of X: For

Conditions for the existence of a unique and continuous implicit function are determined by the following theorem:

Theorem 3.1(no proof). Let be:

1) function F(x,y) defined and continuous in a certain rectangle centered at the point ( x 0, y 0);

2) F (x 0 , y 0) = 0 ;

3) at constant xF(x,y) monotonically increases (or decreases) with increasing at.

a) in some neighborhood of the point ( x 0, y 0) equation (3.1) defines at as a single-valued function of X: y = f(x);

b) at x = x 0 this function takes the value y 0 : f (x 0) = y 0;

c) function f(x) continuous.

Let us find, if the specified conditions are met, the derivative of the function y = f(x) By X.

Theorem 3.2. Let the function at from X is given implicitly by equation (3.1), where the function F(x,y) satisfies the conditions of Theorem 3.1. Let, in addition, - continuous functions in some area D, containing a point (x,y), whose coordinates satisfy equation (3.1), and at this point . Then the function at from X has a derivative

Example.

Calculate approximate value
.

Consider the function
and choose X 0 = 1, at 0 = 2. Then Δ x = 1.02 – 1 = 0.02; Δ y = 1.97 – 2 = -0.03. We'll find
,

Therefore, given that f ( 1, 2) = 3, we get:

Differentiation of complex functions.

Let the function arguments z = f (x, y) u And v: x = x (u, v), y = y (u, v). Then the function f there is also a function from u And v. Let's find out how to find its partial derivatives with respect to the arguments u And v, without making a direct substitution

z = f (x(u, v), y(u, v)). In this case, we will assume that all the functions under consideration have partial derivatives with respect to all their arguments.

Let's set the argument u increment Δ u, without changing the argument v. Then

If you set the increment only to the argument v, we get: .

(2.8) u Let us divide both sides of equality (2.7) by Δ v, and equalities (2.8) – on Δ u→ and move to the limit, respectively, at Δ v→ 0 and Δ X And at. Hence,

Let's consider some special cases.

Let x = x(t), y = y(t). Then the function f (x, y) is actually a function of one variable t, and it is possible, using formulas (2.9) and replacing the partial derivatives in them X And at By u And v to ordinary derivatives with respect to t(of course, provided that the functions are differentiable x(t) And y(t) ) , get an expression for :

(2.10)

. (2.11)

Examples.

Then from formula (2.9) we obtain:

(In the final result we substitute expressions for X And at as functions u And v).

Let's find the complete derivative of the function z = sin( x + y²), where y = cos x.

Invariance of the shape of the differential.

Using formulas (2.5) and (2.9), we express the total differential of the function z = f (x, y) , Where x = x(u, v), y = y(u, v), through differentials of variables u And v:

(2.12)

Therefore, the differential form is preserved for arguments u And v same as for functions of these arguments X And at, that is, is invariant(unchangeable).

Implicit functions, conditions for their existence. Differentiation of implicit functions. Partial derivatives and differentials of higher orders, their properties.

Definition 3.1. Function at from X, defined by the equation

F(x,y)= 0 , (3.1)

called implicit function.

Of course, not every equation of the form (3.1) determines at as a unique (and, moreover, continuous) function of X. For example, the equation of the ellipse

sets at as a two-valued function of X:
For

The conditions for the existence of a unique and continuous implicit function are determined by the following theorem:

Theorem 3.1 (no proof). Let be:

a) in some neighborhood of the point ( X 0 , y 0 ) equation (3.1) defines at as a single-valued function of X: y = f(x) ;

b) at x = x 0 this function takes the value at 0 : f (x 0 ) = y 0 ;

c) function f (x) continuous.

Let us find, if the specified conditions are met, the derivative of the function y = f (x) By X.

Theorem 3.2. Let the function at from X is given implicitly by equation (3.1), where the function F (x, y) satisfies the conditions of Theorem 3.1. Let, in addition,
- continuous functions in some area D, containing a point (x,y), whose coordinates satisfy equation (3.1), and at this point
. Then the function at from X has a derivative

(3.2)

Example. We'll find , If
. We'll find
,
.

Then from formula (3.2) we obtain:
.

Derivatives and differentials of higher orders.

Partial derivative functions z = f (x, y) are, in turn, functions of variables X And at. Therefore, one can find their partial derivatives with respect to these variables. Let's designate them like this:

Thus, four partial derivatives of the 2nd order are obtained. Each of them can be differentiated again according to X and by at and get eight partial derivatives of the 3rd order, etc. Let us define derivatives of higher orders as follows:

Definition 3.2.Partial derivativen -th order a function of several variables is called the first derivative of the derivative ( n– 1)th order.

Partial derivatives have important property: the result of differentiation does not depend on the order of differentiation (for example,
). Let's prove this statement.

Theorem 3.3. If the function z = f (x, y) and its partial derivatives
defined and continuous at a point M(x,y) and in some of its vicinity, then at this point

(3.3)

Consequence. This property is true for derivatives of any order and for functions of any number of variables.