Protecting variable. Open Library - Open educational library

medical and biological physics

Lecture №1

Derivative and differential function.

Private derivatives.

1. The concept of the derivative, its mechanical and geometric meaning.

but ) The increment of argument and function.

Let the function y \u003d f (x) be given, where the value of the argument from the function of determining the function. If you select two values \u200b\u200bof the argument x o and x from a certain interval of the function definition area, the difference between the two values \u200b\u200bof the argument is called the increment of the argument: x - x o \u003d Δh.

The value of the X argument can be determined by x 0 and its increment: x \u003d x o + Δh.

The difference between the two values \u200b\u200bof the function is called the increment of the function: ΔY \u003d ΔF \u003d F (x o + Δh) - F (x o).

The increment of arguments of the function can be represented graphically (Fig. 1). The increment of the argument and the increment of the function can be both positive and negative. As follows from Fig. 1, the geometrically increment of the argument ΔХ is depicted by the increment of the abscissa, and the increment of the function Δu is the increment of ordinate. The calculation of the increment function should be carried out in the following order:

we give the argument the increment ΔХ and get the value - X + ΔX;

2) we find the value of the function for the value of the argument (x + Δh) - f (x + Δh);

3) We find the increment of the function ΔF \u003d F (x + Δh) - F (x).

Example:Determine the increment of the function y \u003d x 2 if the argument has changed from x o \u003d 1 to x \u003d 3. For the point x about the value of the function f (x o) \u003d x² about; For a point (x o + Δh), the value of the function f (x o + Δh) \u003d (x o + δh) 2 \u003d x² about + 2x o Δh + Δh 2, from where Δf \u003d f (x o + ΔХ) -f (x o) \u003d (x o + Δh) 2 -х² O \u003d x² about + 2x Δh + Δh 2-x² O \u003d 2x о Δх + ΔХ 2; Δf \u003d 2x o Δh + Δh 2; Δh \u003d 3-1 \u003d 2; ΔF \u003d 2 · 1 · 2 + 4 \u003d 8.

b)Tasks leading to the concept of derivative. Determining the derivative, its physical meaning.

The concept of the increment of argument and the function is necessary for the introduction of the concept of a derivative, which historically originated on the need to determine the speed of certain processes.

Consider how it is possible to determine the rate of rectilinear movement. Let the body move straightly according to the law: Δѕ \u003d  · Δt. For evaluator movement:  \u003d Δѕ / ΔT.

For variable motion, the value Δѕ / ΔTodetes value of the CP. , i.e.  cf. \u003d Δѕ / Δt. But average speed It does not allow to reflect the features of the body movement and give an idea of \u200b\u200bthe true speed at time t. With a decrease in the period of time, i.e. When Δt → 0, the average speed is rated to its limit - Instant Speed:

 MGN. \u003d.
 Wed \u003d.
Δѕ / ΔT.

In the same way, the instantaneous chemical reaction rate is determined:

 MGN. \u003d.
 Wed \u003d.
ΔХ / Δt,

where x is the amount of substance formed during a chemical reaction during T. Such problems in determining the speed of various processes led to the introduction in mathematics the concept of a derivative function.

Let the continuous function f (x), determined on the interval] A, in [Ei, the increment Δf \u003d f (x + Δh) -f (x).
it is a function ΔХ and expresses the average speed of change of function.

Limit of the relationship when ΔХ → 0, provided that this limit exists, is called a derived function :

y "x \u003d

.

The derivative is indicated:
- (Sharpery of the Barcode of X); f " (x) - (EF Barcode of X) ; y "- (Shark Barcode); DY / DX – (Degrea for DE X); - (play with a point).

Based on the definition of the derivative, it can be said that the instantaneous speed of the straight movement is derived from the time of time:

 MGN. \u003d S "T \u003d F " (t).

Thus, it can be concluded that the derivative of the argument X is an instantaneous rate of change in the function F (x):

u "X \u003d F " (x) \u003d  MGN.

This is the physical meaning of the derivative. The process of finding a derivative is called differentiation, therefore the expression "indity function" is equivalent to the expression "find a derivative function".

in)Geometric meaning derivative.

P
the operating function y \u003d f (x) has a simple geometric meaning associated with the concept of a lines curve at some point. At the same time, tangent, i.e. The direct line is analytically expressed in the form of y \u003d kh \u003d tg · x, where – the angle of inclination of tangential (straight) to the x axis will present a continuous curve as a function y \u003d f (x), take the point of M 1 close to it on the curve and give the securing point. Its angular coefficient to sec \u003d Tg β \u003d . If you bring the point M 1 to m, then the increment of the argument ΔХ it will strive for zero, and the sequer at β \u003d α will take the position of tangent. Figure 2 follows: TGα \u003d
tGβ \u003d.
\u003d y "x. But Tgαins the angular coefficient tangent to the graph of the function:

k \u003d TGα \u003d
\u003d y "x \u003d f " (x). So, the angular coefficient of tangential to the graph of the function at this point is equal to its derivative at the point of touch. This is the geometric meaning of the derivative.

d)The general rule of finding a derivative.

Based on the derivative determination, the differentiation process of the function can be represented as follows:

f (x + Δh) \u003d f (x) + Δf;

find the increment of the function: ΔF \u003d F (x + Δh) - F (x);

the ratio of the increment of the function to the increment of the argument is:

;

Example:f (x) \u003d x 2; F. " (x) \u003d?.

However, as can be seen even from this simple example, the use of the specified sequence when taking derivatives is a time-consuming process and complex. Therefore, for various functions are entered general formulas differentiation, which are presented in the form of a table "Basic formulas differentiation of functions".

Not always in life, we are interested in accurate values \u200b\u200bof any values. Sometimes it is interesting to know the change in this value, for example, the average speed of the bus, the ratio of the magnitude of the movement to the time interval, etc. To compare the values \u200b\u200bof the function at some point with the values \u200b\u200bof the same function at other points, it is convenient to use such concepts as the "increment of the function" and the "argument increment".

The concepts of "increment of the function" and "the increment of the argument"

Suppose X is some arbitrary point that lies in any neighborhood of the point x0. The increment of the argument at the point x0 is the difference X-X0. The increment is referred to as follows: ΔХ.

ΔХ \u003d x-x0.

Sometimes this magnitude is also called the increment of an independent variable at the point x0. From the formula follows: x \u003d x0 + Δh. In such cases, it is said that the initial value of an independent variable x0, obtained increment to ΔХ.

If we change the argument, the value of the function will also change.

f (x) - f (x0) \u003d f (x0 + Δh) - F (x0).

Increment of the function F at point x0, The corresponding increment is Δh called the difference F (x0 + Δh) - F (x0). The increment of the function is indicated as follows ΔF. Thus, we obtain, by definition:

Δf \u003d f (x0 + Δx) - F (x0).

Sometimes, Δf is also called the increment of the dependent variable and use ΔU to designate if the function was, for example, y \u003d f (x).

Geometric meaning of increment

Look at the next drawing.

As you can see, increment shows the change in the ordinate and the abscissa of the point. And the ratio of the increment of the function to increment the argument determines the angle of inclination of the sequential passing through the initial and final position of the point.

Consider examples of the increment of the function and argument

Example 1. Find the increment of the argument Δh and the increment of the function ΔF at the point x0, if f (x) \u003d x 2, x0 \u003d 2 a) x \u003d 1.9 b) x \u003d 2.1

We use the formulas shown above:

a) Δh \u003d x-x0 \u003d 1.9 - 2 \u003d -0.1;

ΔF \u003d f (1.9) - f (2) \u003d 1.9 2 - 2 2 \u003d -0.39;

b) Δx \u003d x - x0 \u003d 2.1-2 \u003d 0.1;

Δf \u003d f (2.1) - f (2) \u003d 2.1 2 - 2 2 \u003d 0.41.

Example 2. Calculate the increment Δf for the function f (x) \u003d 1 / x at the point x0, if the argument increment is Δx.

Again, we use the formulas obtained above.

Δf \u003d f (x0 + Δx) - f (x0) \u003d 1 / (x0-δx) - 1 / x0 \u003d (x0 - (x0 + Δx)) / (x0 * (x0 + Δx)) \u003d - Δx / ((x0 * (x0 + Δx)).

Let X be an arbitrary point that flies in some surroundings of the fixed point x 0. The difference X - x 0 is taken to call the increment by an independent variable (or incrementing the argument) at point x 0 and denotes Δx. In this way,

Δx \u003d x -x 0,

from where it follows that

Protect function -the difference between the two values \u200b\u200bof the function.

Let a function specify w. = f (x)defined with an argument value equal h. 0. Let's give an argument increment d h., ᴛ.ᴇ. Consider the value of the argument͵ equal x. 0 + D. h.. Suppose that this value of the argument is also included in the definition area of \u200b\u200bthis function. Then the difference D. y. = f (X. 0 + D. x) – f (x 0) It is customary to be called the increment of the function. Protecting function f.(x.) At point x. - function is usually indicated Δ X F from the new variable Δ x. defined as

Δ X F(Δ x.) = f.(x. + Δ x.) − f.(x.).

Find the increment of the argument and the increment of the function at point x 0, if

Example 2. Find the increment of the function f (x) \u003d x 2, if x \u003d 1, Δh \u003d 0.1

Solution: F (x) \u003d x 2, f (x + Δh) \u003d (x + Δh) 2

Find the increment of the function Δf \u003d f (x + Δx) - f (x) \u003d (x + δx) 2 - x 2 \u003d x 2 + 2x * Δx + Δx 2 - x 2 \u003d 2x * Δx + Δx 2 /

We substitute the values \u200b\u200bx \u003d 1 and Δh \u003d 0.1, we obtain Δf \u003d 2 * 1 * 0.1 + (0,1) 2 \u003d 0.2 + 0.01 \u003d 0.21

Find the increment of the argument and increment the function at points x 0

2.f (x) \u003d 2x 3. x 0 \u003d 3 x \u003d 2.4

3. f (x) \u003d 2x 2 +2 x 0 \u003d 1 x \u003d 0.8

4. F (x) \u003d 3x + 4 x 0 \u003d 4 x \u003d 3.8

Definition: Derivative Functions At the point, it is customary to call the limit (if it exists and finite) the ratio of the increment of the function to the increment of the argument provided that the latter tends to zero.

The following designations of the derivative are most common:

In this way,

Finding a derivative called call differentiation . Introduced definition of differentiable function: Function F that has a derivative at each point of some gap, is called differentiable at a given interval.

Suppose that in some neighborhood despillas, the function of the performance function is called such a number that the function in the surrounding area U.(x. 0) can be represented as

f.(x. 0 + h.) = f.(x. 0) + AH. + o.(h.)

if there is.

Definition of the derivative function at the point.

Let the function f (x) Defined on the interval (a; b)and - points of this gap.

Definition. Derived function f (x) At the point it is customary to call the limit of the relationship of the function of the function to increments the argument at. Denotes.

When the last limit takes a concrete final value, then there is talking about existence finite derivative at point. In case the limit is infinite, they say that derivative infinite at this point. In case, if the limit does not exist, then derivative function at this point does not exist.

Function f (x) Called differentiable at the point when it has a finite derivative in it.

In case the function f (x) differentiable at every point of some interval (a; b)The function is called differentiable at this interval. Τᴀᴋᴎᴍ ᴏϭᴩᴀᴈᴏᴍ, any point x. From the gap (a; b) You can put in accordance with the value of the derivative function at this point, that is, we have the ability to determine the new function called the derived function f (x) At the interval (a; b).

The operation of finding a derivative is customary to be called differentiation.

Definition 1.

If for each pair of $ (x, y) $ of the values \u200b\u200bof two independent variables from a certain area is put in accordance with a certain value of $ z $, then it is said that $ z $ is the function of two variables $ (x, y) $. Designation: $ z \u003d f (x, y) $.

With respect to the function $ z \u003d f (x, y) $ we consider the concept of general (full) and private increments of the function.

Let the function $ z \u003d f (x, y) $ two independent variables $ (x, y) $.

Note 1.

Since the variables $ (x, y) $ are independent, then one of them can be changed, and the other to maintain a constant value.

Let us give a variable $ x $ increment $ \\ Delta X $, while saving the value of the variable $ y $ unchanged.

Then the function $ z \u003d f (x, y) $ will receive an increment that will be referred to as a private increment of the function $ z \u003d f (x, y) $ for a variable $ x $. Designation:

Similarly, we give a variable $ y $ increment $ \\ delta y $, while saving the value of the $ x $ variable is unchanged.

Then the function $ z \u003d f (x, y) $ will receive an increment that will be called a private increment of the function $ z \u003d f (x, y) $ along the $ y $ variable. Designation:

If $ x $ argument is an increment of $ \\ Delta X $, and the $ y $ argument is the increment of $ \\ delta y $, then the complete increment of the specified function is $ z \u003d f (x, y) $. Designation:

Thus, we have:

$ \\ Delta _ (x) z \u003d f (x + \\ deelta x, y) -f (x, y) $ - the private increment of the function $ z \u003d f (x, y) $ for $ x $;

$ \\ Delta _ (y) z \u003d f (x, y + \\ deelta y) -f (x, y) $ - the private increment of the function $ z \u003d f (x, y) $ for $ y $;

$ \\ Delta z \u003d f (x + \\ deelta x, y + \\ deelta y) -f (x, y) $ is the complete increment of the function $ z \u003d f (x, y) $.

Example 1.

Decision:

$ \\ Delta _ (x) z \u003d x + \\ deelta x + y $ - the private increment of the function $ z \u003d f (x, y) $ to $ x $;

$ \\ Delta _ (y) z \u003d x + y + \\ Delta y $ is the private increment of the function $ z \u003d f (x, y) $ for $ y $.

$ \\ Delta z \u003d x + \\ Delta X + Y + \\ Delta y $ - the complete increment of the function $ z \u003d f (x, y) $.

Example 2.

Calculate the private and complete increment of the function $ z \u003d xy $ at point $ (1; 2) $ with $ \\ deelta x \u003d 0.1; \\, \\, \\ deelta y \u003d 0.1 $.

Decision:

By definition of private increment, we find:

$ \\ Delta _ (x) z \u003d (x + \\ deelta x) \\ cdot y $ - Private increment of the function $ z \u003d f (x, y) $ for $ x $

$ \\ Delta _ (y) z \u003d x \\ cdot (y + \\ deelta y) $ - the private increment of the function $ z \u003d f (x, y) $ for $ y $;

By definition of complete increment, we find:

$ \\ Delta Z \u003d (x + \\ Delta X) \\ CDOT (Y + \\ Delta Y) $ - the complete increment of the function $ z \u003d f (x, y) $.

Hence,

\\ [\\ Delta _ (x) z \u003d (1 + 0.1) \\ Cdot 2 \u003d 2.2 \\] \\ [\\ Delta _ (y) z \u003d 1 \\ Cdot (2 + 0.1) \u003d 2.1 \\] \\ [\\ Delta Z \u003d (1 + 0.1) \\ Cdot (2 + 0.1) \u003d 1.1 \\ Cdot 2,1 \u003d 2.31. \\]

Note 2.

The complete increment of the specified function $ z \u003d f (x, y) $ is not equal to the sum of its private increments of $ \\ delta _ (x) z $ and $ \\ delta _ (y) z $. Mathematical recording: $ \\ delta z \\ ne \\ delta _ (x) z + \\ deelta _ (y) z $.

Example 3.

Check approval comments for function

Decision:

$ \\ Delta _ (x) z \u003d x + \\ deelta x + y $; $ \\ Delta _ (y) z \u003d x + y + \\ delta y $; $ \\ Delta z \u003d x + \\ deelta x + y + \\ delta y $ (obtained in example 1)

We will find the amount of private increments of the specified function $ z \u003d f (x, y) $

\\ [\\ Delta _ (x) z + \\ deelta _ (y) z \u003d x + \\ deelta x + y + (x + y + \\ deelta y) \u003d 2 \\ Cdot (x + y) + \\ Delta X + \\ Delta y. \\]

\\ [\\ Delta _ (x) Z + \\ Delta _ (y) z \\ ne \\ delta z. \\]

Definition 2.

If for each three $ (x, y, z) $ of the values \u200b\u200bof three independent variables from a certain area put in accordance with a certain value of $ W $, then it is said that $ W $ is a function of three variables $ (x, y, z) $ in This area.

Designation: $ w \u003d f (x, y, z) $.

Definition 3.

If for each totality of $ (x, y, z, ..., t) $, the values \u200b\u200bof independent variables from a certain region are put in accordance with a certain value of $ W $, then it is said that $ W $ is the function of variables $ (x, y, z, ..., t) $ in this area.

Designation: $ w \u003d f (x, y, z, ..., t) $.

For a function of three and more variables, it is similar to how the function of two variables are determined by private increments for each of the variables:

$ \\ Delta _ (z) w \u003d f (x, y, z + \\ deelta z) -f (x, y, z) $ - the private increment of the function $ w \u003d f (x, y, z, ..., t ) $ for $ z $;

$ \\ Delta _ (t) w \u003d f (x, y, z, ..., t + \\ deelta t) -f (x, y, z, ..., t) $ - Private increment of the function $ w \u003d F (x, y, z, ..., t) $ for $ t $.

Example 4.

Write private and complete increment of the function

Decision:

By definition of private increment, we find:

$ \\ Delta _ (x) w \u003d ((x + \\ deelta x) + y) \\ CDOT Z $ - Private increment of the function $ w \u003d f (x, y, z) $ for $ x $

$ \\ Delta _ (y) w \u003d (x + (y + \\ deelta y)) \\ Cdot z $ - the private increment of the function $ w \u003d f (x, y, z) $ for $ y $;

$ \\ Delta _ (z) w \u003d (x + y) \\ Cdot (Z + \\ Delta z) $ - the private increment of the function $ w \u003d f (x, y, z) $ for $ z $;

By definition of complete increment, we find:

$ \\ Delta W \u003d ((X + \\ Delta X) + (Y + \\ Delta Y)) \\ CDOT (Z + \\ Delta Z) $ - the complete increment of the function $ w \u003d f (x, y, z) $.

Example 5.

Calculate the private and complete increment of the function $ w \u003d xyz $ at point $ (1; 2; 1) $ with $ \\ deelta x \u003d 0.1; \\, \\, \\, \\, \\, \\ delta z \u003d 0.1 $.

Decision:

By definition of private increment, we find:

$ \\ Delta _ (x) w \u003d (x + \\ deelta x) \\ cdot y \\ cdot z $ - Private increment of the function $ w \u003d f (x, y, z) $ for $ x $

$ \\ Deelta _ (y) w \u003d x \\ cdot (y + \\ deelta y) \\ Cdot z $ - the private increment of the function $ w \u003d f (x, y, z) $ for $ y $;

$ \\ Delta _ (z) w \u003d x \\ cdot y \\ cdot (z + \\ delta z) $ - the private increment of the function $ w \u003d f (x, y, z) $ for $ z $;

By definition of complete increment, we find:

$ \\ Delta W \u003d (X + \\ Delta X) \\ CDOT (Y + \\ Delta Y) \\ CDOT (Z + \\ Delta Z) $ - the complete increment of the function $ w \u003d f (x, y, z) $.

Hence,

\\ [\\ Delta _ (x) w \u003d (1 + 0.1) \\ cdot 2 \\ cdot 1 \u003d 2.2 \\] \\ [\\ delta _ (y) w \u003d 1 \\ Cdot (2 + 0,1) \\ \\ CDOT (2 + 0.1) \\ Cdot (1 + 0.1) \u003d 1.1 \\ CDot 2.1 \\ Cdot 1.1 \u003d 2.541. \\]

From a geometric point of view, the complete increment of the function $ z \u003d f (x, y) $ (by definition $ \\ deelta z \u003d f (x + \\ deelta x, y + \\ deelta y) -f (x, y) $) is equal to increment of the application of the graph The functions $ z \u003d f (x, y) $ in the transition from the point $ m (x, y) $ to the point $ M_ (1) (X + \\ Delta X, Y + \\ Delta y) $ (Fig. 1).

Picture 1.