The mechanical meaning of the second order derivative. Equations of the normal and tangent to the graph of a function

Instruction Card No. 20

Takyryby/Subject: « The second derivative and its physical meaning».

Maқsaty / Purpose:

Be able to find the equation of the tangent, as well as the tangent of the angle of inclination of the tangent to the OX axis. Be able to find the rate of change of a function, as well as acceleration.

Create a condition for the formation of skills to compare, classify the facts and concepts studied.

Education of a responsible attitude to educational work, will and perseverance to achieve final results when finding the tangent equation, as well as when finding the rate of change of function and acceleration.

Theoretical material:

(Geometric meaning of the derivative)

The equation for the tangent to the function graph is:

Example 1: Let's find the equation of the tangent to the graph of the function at the point with obscissa 2.

Answer: y = 4x-7

The slope k of the tangent to the graph of the function at the point with the abscissa x o is equal to f / (x o) (k = f / (x o)). The angle of inclination of the tangent to the graph of the function at a given point is

arctg k \u003d arctg f / (x o), i.e. k= f / (x o)= tg

Example 2: At what angle is the sinusoid intersects the x-axis at the origin?

The angle at which the graph of this function intersects the abscissa axis is equal to the angle of inclination a of the tangent drawn to the graph of the function f (x) at this point. Let's find the derivative: Considering the geometric meaning of the derivative, we have: and a = 60°. Answer: =60 0 .

If a function has a derivative at every point in its domain, then its derivative is a function of . The function, in turn, can have a derivative, which is called second order derivative functions (or second derivative) and are denoted by the symbol .

Example 3: Find the second derivative of the function: f(x)=x 3 -4x 2 +2x-7.

At the beginning, we find the first derivative of this function f "(x) \u003d (x 3 -4x 2 + 2x-7) '= 3x 2 -8x + 2,

Then, we find the second derivative of the obtained first derivative

f""x)=(3x 2 -8x+2)''=6x-8. Answer: f""x) = 6x-8.

(Mechanical meaning of the second derivative)

If the point moves in a straight line and the law of its motion is given, then the acceleration of the point is equal to the second derivative of the path with respect to time:

The speed of a material body is equal to the first derivative of the path, that is:

The acceleration of a material body is equal to the first derivative of the speed, that is:

Example 4: The body moves in a straight line according to the law s (t) \u003d 3 + 2t + t 2 (m). Determine its speed and acceleration at time t = 3 s. (The path is measured in meters, the time in seconds).
Solution
v (t) = s΄ (t) =(3+2t+t 2)'= 2 + 2t
a (t) = v΄ (t) =(2+2t)’= 2 (m/s 2)
v(3) = 2 + 2∙3 = 8 (m/s). Answer: 8 m/s; 2 m/s 2 .

Practical part:

1 option	Option 2	3 option	4 option	5 option
Find the tangent of the angle of inclination to the x-axis of the tangent passing through the given point M graph of the function f.
f(x)=x 2 , M(-3;9)	f(x)=x 3 , M(-1;-1)
Write the equation of the tangent to the graph of the function f at the point with the abscissa x 0.
f (x) \u003d x 3 -1, x 0 \u003d 2	f (x) \u003d x 2 +1, x 0 \u003d 1	f (x) \u003d 2x-x 2, x 0 \u003d -1	f(x)=3sinx, x 0 =	f(x)= x 0 = -1
Find the slope of the tangent to the function f at the point with the abscissa x 0.
Find the second derivative of a function:
			f(x)= 2cosx-x 2	f(x)= -2sinx+x 3
The body moves in a straight line according to the law x (t). Determine its speed and acceleration at the moment time t. (Displacement is measured in meters, time in seconds).
x(t)=t 2 -3t, t=4	x(t)=t 3 +2t, t=1	x(t)=2t 3 -t 2 , t=3	x(t)=t 3 -2t 2 +1,t=2	x (t) \u003d t 4 -0.5t 2 \u003d 2, t \u003d 0.5

Control questions:

What do you think is the physical meaning of the derivative - is it instantaneous speed or average speed?

What is the connection between a tangent drawn to the graph of a function through any point and the concept of a derivative?

What is the definition of a tangent to the graph of a function at the point M (x 0; f (x 0))?

What is the mechanical meaning of the second derivative?

Derivative. Consider some function y= f (x) at two points x 0 and x 0 + : f(x 0) and f (x 0 + ). Here, denoted by some small change in the argument, called argument increment; respectively, the difference between the two values ​​of the function: f(x 0 + ) - f (x 0) is called function increment. derivative functions y= f (x) at the point x 0 is called the limit:

If this limit exists, then the function f (x) is called differentiable at the point x 0 . Derivative of a function f (x) is denoted as follows:

The geometric meaning of the derivative. Consider the graph of the function y= f (x):

It can be seen from Fig. 1 that for any two points A and B of the graph of the function:

Where is the angle of inclination of the secant AB.

Thus, the difference ratio is equal to the slope of the secant. If we fix point A and move point B towards it, then it decreases indefinitely and approaches 0, and the secant AB approaches the tangent AC. Therefore, the limit of the difference ratio is equal to the slope of the tangent at point A. It follows from this: the derivative of a function at a point is the slope of the tangent to the graph of that function at that point. This is what it consists geometric meaning derivative.

Tangent equation. Let us derive the equation of the tangent to the graph of the function at the point A ( x 0 , f (x 0)). In the general case, the equation of a straight line with a slope f ’(x 0) has the form:

y = f ’(x 0) · x + b .

To find b, we use the fact that the tangent passes through the point A:

f (x 0) = f ’(x 0) · x 0 +b,

from here b = f (x 0) – f ’(x 0) · x 0 , and substituting this expression for b, we will get tangent equation:

y =f (x 0) + f ’(x 0) · ( x-x 0) .

The mechanical meaning of the derivative. Consider the simplest case: the movement of a material point along the coordinate axis, and the law of motion is given: coordinate x moving point is a known function x (t) time t. During the time interval from t 0 to t 0 + the point moves a distance: x (t 0 + ) -x (t 0) = , and its average speed is equal to: va = / . At 0, the value of the average speed tends to a certain value, which is called instantaneous speed v(t 0) material point at time t 0 . But by the definition of a derivative, we have:

from here v(t 0)= x'(t 0) , i.e. Velocity is the derivative of the coordinate with respect to time. This is what it consists mechanical sense derivative . Likewise, acceleration is the derivative of speed with respect to time: a = v'(t).

Task examples

Task 1. Write the equation of the common tangent to the graphs of the functions and .

A straight line is a common tangent to the graphs of functions and if it touches both one and the other graphs, but not necessarily at the same point.

- the equation of the tangent to the graph of the function y=x2 at the point with the abscissa x0

- the equation of the tangent to the graph of the function y=x3 at the point with the abscissa x1

Straight lines coincide if their slopes and free terms are equal. From here

The solution of the system will be

The general tangent equations have the form:

16. Rules of differentiation. Derivatives of complex, inverse and implicit functions.
Differentiation rules
When differentiating, a constant can be taken out as a derivative:

The rule for differentiating the sum of functions:

Function difference differentiation rule:

The rule of differentiation of the product of functions (Leibniz's rule):

The rule for differentiating quotient functions:

Rule for differentiating a function to the power of another function:

Compound function differentiation rule:

Logarithm rule when differentiating a function:

Derivative of a complex function

A "two-layer" complex function is written as where u = g(x) is the inner function, which, in turn, is an argument for the outer function f. If f and g are differentiable functions, then the complex function is also differentiable with respect to x and its derivative is equal to This formula shows that the derivative of a complex function is equal to the product of the derivative of the outer function and the derivative of the inner function. It is important, however, that the derivative of the inner function is calculated at the point x, and the derivative of the outer function is calculated at the point u = g(x)! This formula is easily generalized to the case when a complex function consists of several "layers" nested hierarchically into each other. Consider a few examples illustrating the rule for the derivative of a complex function. This rule is widely used in many other problems in the "Differentiation" section.

Example 1

Find the derivative of the function . Solution. Since , then, according to the rule of the derivative of a complex function, we obtain

A function is complex if it can be represented as a function of a function y = f[φ(x)], where y = f(u), au=φ(x), where u is an intermediate argument. Any complex function can be represented as elementary functions (simple), which are its intermediate arguments.

Examples:

Simple functions: Complex functions:

y \u003d x 2 y \u003d (x + 1) 2; u \u003d (x + 1); y \u003d u 2;

y = sinx; y \u003d sin2x; u \u003d 2x; y=sinu;

y \u003d e x y \u003d e 2x; u \u003d 2x; y \u003d e u;

y \u003d lnx y \u003d ln (x + 2); u \u003d x + 2; y=lnu.

The general rule for differentiating a complex function is given by the above theorem without proof.

If the function u \u003d φ (x) has a derivative u "x \u003d φ" (x) at the point x, and the function y \u003d f (u) has a derivative y "u \u003d f " (u) at the corresponding point u, then the derivative of the complex function y \u003d f [φ (x)] at the point x is found by the formula: y "x \u003d f " (u) u "(x).

A less precise but shorter formulation of this theorem is often used. : the derivative of a complex function is equal to the product of the derivative with respect to the intermediate variable and the derivative of the intermediate variable with respect to the independent variable.

Example: y=sin2x 2 ; u \u003d 2x 2; y=sinu;

y "x \u003d (sinu)" u (2x 2) "x \u003d cosu 4x \u003d 4x cos2x 2.

3. Derivative of the second order. Mechanical meaning of the second derivative.

The derivative of the function y \u003d f (x) is called the first order derivative or simply the first derivative of the function. This derivative is a function of x and can be differentiated a second time. The derivative of the derivative is called the second order derivative or the second derivative. It is denoted: y "xx - (y two strokes on x); f "(x) – ( eff two strokes on x); d 2 y / dx 2 - (de two y on de x twice); d 2 f / dx 2 - (de two ef on de x twice).

Based on the definition of the second derivative, we can write:

y "xx \u003d (y" x) "x; f" (x) \u003d "x d 2 y / dx 2 \u003d d / dx (dy / dx).

The second derivative, in turn, is a function of x and can be differentiated to obtain a third-order derivative, and so on.

Example: y \u003d 2x 3 + x 2; y "xx \u003d [(2x 3 + x 2)" x] "x \u003d (6x 2 + 2x)" x \u003d 12x + 2;

The mechanical meaning of the second derivative is explained on the basis of instantaneous acceleration, which characterizes variable motion.

If S=f(t) is the equation of motion, then=S" t ; A cf. =;

A inst. =
A cf =
=" t ; A inst. = " t = (S" t)" t = S" tt .

Thus, the second derivative of the path with respect to time is equal to the instantaneous acceleration of the variable motion. This is the physical (mechanical) meaning of the 2nd derivative.

Example: Let the rectilinear motion of a material point occur according to the law S=t 3 /3. The acceleration of a material point will be defined as the second derivative of S "tt: A\u003d S "tt \u003d (t 3 / 3)" \u003d 2t.

4. Function differential.

Closely related to the concept of a derivative is the concept of a differential of a function, which has important practical applications.

Function f( X) has a derivative
= f " (X);

According to the theorem (we do not consider the theorem) on the connection of an infinitesimal quantity α(∆х)(
α(∆х)=0) with derivative: = f " (х)+ α (∆х), whence ∆f = f " (х) ∆х+α(∆х) ∆х.

It follows from the last equality that the increment of the function consists of a sum, each term of which is an infinitesimal value as ∆х→ 0.

Let us determine the order of smallness of each infinitely small value of this sum with respect to the infinitely small ∆x:

Therefore, infinitesimal f (х) ∆х and ∆x have the same order of magnitude.

Therefore, the infinitesimal value α(∆х)∆х has a higher order of smallness in relation to the infinitesimal value ∆х. This means that in the expressions for ∆f the second term α(∆х)∆х tends to 0 faster as ∆х→0 than the first term f " (x)∆x.

This is the first term f " (x)∆x is called the differential of the function at the point x. It is denoted dy (de y) or df (de ef). So dy=df= f " (x)∆x or dy= f " (x)dx, because the differential dx of the argument is equal to its increment ∆x (if in the formula df= f " (x)dx accept that f(x)=x, then we getdf=dx=x"x ∆x, butx"x =1, i.e. dx=∆x). So, the differential of a function is equal to the product of this function and the differential of the argument.

The analytical meaning of the differential lies in the fact that the differential of a function is the main part of the increment of the function ∆f, linear with respect to the argument ∆x. The function differential differs from the function increment by an infinitesimal value α(∆х)∆х higher order of smallness than ∆х. Indeed ∆f=f " (х)∆х+α(∆х)∆х or ∆f=df+α(∆х)∆х; whence df= ∆f- α(∆х)∆х.

Example: y \u003d 2x 3 + x 2; dy \u003d? dy \u003d y "dx \u003d (2x 3 + x 2)" x dx \u003d (6x 2 + 2x) dx.

Neglecting the infinitely small value α(∆x)∆x of a higher order smallness than ∆X, we get df≈∆f≈ f " (x)dx i.e. the differential of a function can be used to approximate the increment of a function, since the differential is usually easier to calculate. The differential can also be applied to the approximate calculation of the value of a function. Let us know the function y= f(x) and its derivative at the point x. It is necessary to find the value of the function f(x+∆x) at some close point (x+∆x). To do this, we use the approximate equality ∆у ≈dy or ∆у ≈f " (x) ∆x. Considering that ∆y=f(x+∆x)-f(x), we get f(x+∆x)-f (x) ≈f " (x) dx , whence f(х+∆х) = f(х)+f " (x) dx. The resulting formula solves the problem.

Derivative(functions at a point) - the basic concept of differential calculus, characterizing the rate of change of a function (at a given point). It is defined as the limit of the ratio of the increment of a function to the increment of its argument as the increment of the argument tends to zero, if such a limit exists. A function that has a finite derivative (at some point) is called differentiable (at a given point).

Derivative. Consider some function y = f (x ) at two points x 0 and x 0 + : f (x 0) and f (x 0 + ). Here, denoted by some small change in the argument, called argument increment; respectively, the difference between the two values ​​of the function: f (x 0 + )  f (x 0 ) is called function increment.derivative functions y = f (x ) at the point x 0 called the limit:

If this limit exists, then the function f (x ) is called differentiable at the point x 0 . Derivative of a function f (x ) is denoted as follows:

The geometric meaning of the derivative. Consider the graph of the function y = f (x ):

It can be seen from Fig. 1 that for any two points A and B of the graph of the function:

where is the angle of inclination of the secant AB.

Thus, the difference ratio is equal to the slope of the secant. If we fix point A and move point B towards it, then it decreases indefinitely and approaches 0, and the secant AB approaches the tangent AC. Therefore, the limit of the difference ratio is equal to the slope of the tangent at point A. It follows from this: the derivative of a function at a point is the slope of the tangent to the graph of that function at that point. This is what it consists geometric meaning derivative.

Tangent equation. Let us derive the equation of the tangent to the graph of the function at the point A ( x 0 , f (x 0 )). In the general case, the equation of a straight line with a slope f ’(x 0 ) looks like:

y = f ’(x 0 ) · x + b .

To find b, we use the fact that the tangent passes through point A:

f (x 0 ) = f ’(x 0 ) · x 0 +b ,

from here b = f (x 0 ) – f ’(x 0 ) · x 0 , and substituting this expression for b, we will get tangent equation:

y =f (x 0 ) + f ’(x 0 ) · ( x-x 0 ) .

The mechanical meaning of the derivative. Consider the simplest case: the motion of a material point along the coordinate axis, and the law of motion is given: coordinate x moving point is a known function x (t) time t. During the time interval from t 0 to t 0 + the point moves a distance: x (t 0 + )  x (t 0) = , and its average speed is equal to: v a =  . At 0, the value of the average speed tends to a certain value, which is called instant speed v ( t 0 ) material point at time t 0 . But by the definition of a derivative, we have:

from here v (t 0 ) = x' (t 0 ) , i.e. speed is the derivative of the coordinate By time. This is what it consists mechanical sense derivative . Likewise, acceleration is the derivative of speed with respect to time: a = v' (t).

8. Table of derivatives and differentiation rules

We talked about what a derivative is in the article “The geometric meaning of the derivative”. If a function is given by a graph, its derivative at each point is equal to the tangent of the slope of the tangent to the graph of the function. And if the function is given by a formula, the table of derivatives and the rules of differentiation will help you, that is, the rules for finding the derivative.

Derivative(functions at a point) - the basic concept of differential calculus, characterizing the rate of change of a function (at a given point). It is defined as the limit of the ratio of the increment of a function to the increment of its argument as the increment of the argument tends to zero, if such a limit exists. A function that has a finite derivative (at some point) is called differentiable (at a given point).

Derivative. Consider some function y = f (x ) at two points x 0 and x 0 + : f (x 0) and f (x 0 + ). Here, denoted by some small change in the argument, called argument increment; respectively, the difference between the two values ​​of the function: f (x 0 + )  f (x 0 ) is called function increment.derivative functions y = f (x ) at the point x 0 called the limit:

If this limit exists, then the function f (x ) is called differentiable at the point x 0 . Derivative of a function f (x ) is denoted as follows:

The geometric meaning of the derivative. Consider the graph of the function y = f (x ):

It can be seen from Fig. 1 that for any two points A and B of the graph of the function:

where is the angle of inclination of the secant AB.

Thus, the difference ratio is equal to the slope of the secant. If we fix point A and move point B towards it, then it decreases indefinitely and approaches 0, and the secant AB approaches the tangent AC. Therefore, the limit of the difference ratio is equal to the slope of the tangent at point A. It follows from this: the derivative of a function at a point is the slope of the tangent to the graph of that function at that point. This is what it consists geometric meaning derivative.

Tangent equation. Let us derive the equation of the tangent to the graph of the function at the point A ( x 0 , f (x 0 )). In the general case, the equation of a straight line with a slope f ’(x 0 ) looks like:

y = f ’(x 0 ) · x + b .

To find b, we use the fact that the tangent passes through point A:

f (x 0 ) = f ’(x 0 ) · x 0 +b ,

from here b = f (x 0 ) – f ’(x 0 ) · x 0 , and substituting this expression for b, we will get tangent equation:

y =f (x 0 ) + f ’(x 0 ) · ( x-x 0 ) .

The mechanical meaning of the derivative. Consider the simplest case: the movement of a material point along the coordinate axis, and the law of motion is given: coordinate x moving point is a known function x (t) time t. During the time interval from t 0 to t 0 + the point moves a distance: x (t 0 + ) x (t 0) = , and its average speed is equal to: v a =  . At 0, the value of the average speed tends to a certain value, which is called instant speed v ( t 0 ) material point at time t 0 . But by the definition of a derivative, we have:

from here v (t 0 ) = x' (t 0 ) , i.e. speed is the derivative of the coordinate By time. This is what it consists mechanical sense derivative . Likewise, acceleration is the derivative of speed with respect to time: a = v' (t).

8. Table of derivatives and differentiation rules

We talked about what a derivative is in the article “The geometric meaning of the derivative”. If a function is given by a graph, its derivative at each point is equal to the tangent of the slope of the tangent to the graph of the function. And if the function is given by a formula, the table of derivatives and the rules of differentiation will help you, that is, the rules for finding the derivative.

§ 2. Definition of a derivative.

Let the function y= f(x) defined on the interval ( a;b). Consider the value of the argument

(a;b) . Let's increment the argument ∆ x 0 so that the condition ( x 0 +∆ x)

a;b). Let us denote the corresponding values ​​of the function through y 0 and y 1:

y 0 = f(x 0 ), y 1 = f(x 0 +∆ x). When moving from x 0 To x 0 +∆ x the function will be incremented

∆y= y 1 -y 0 = f(x 0 +∆ x) -f(x 0 ). If, in striving ∆ x to zero there is a limit to the ratio of the increment of the function ∆y to the argument increment that called it ∆ x,

those. there is a limit

=

,

then this limit is called the derivative of the function y= f(x) at the point x 0 . So the derivative of the function y= f(x) at the point x=x 0 there is a limit to the ratio of the increment of the function to the increment of the argument when the increment of the argument tends to zero. Derivative of a function y= f(x) at the point x denoted by symbols (x) or (x). The designations are also used , , ,. The last three notations emphasize the fact that the derivative is taken with respect to the variable x.

If the function y= f(x) has a derivative at every point of some interval, then on this interval the derivative ( x) is an argument function x.

§ 3. Mechanical and geometric meaning of the derivative.

Equations of the normal and tangent to the function graph.

As shown in § 1, the instantaneous velocity of a point is

v = .

But this means that the speed v is the derivative of the distance traveled S by time t ,

v =. Thus, if the function y= f(x) describes the law of rectilinear motion of a material point, where y is the path traveled by a material point from the moment of the beginning of the movement to the moment of time x, then the derivative ( x) determines the instantaneous speed of a point at a time x. This is the mechanical meaning of the derivative.

In § 1 we also found the slope of the tangent to the graph of the function y= f(x) k= tgα= . This relation means that the slope of the tangent is equal to the derivative ( x). More strictly speaking, the derivative ( x) functions y= f(x) , calculated with the value of the argument equal to x, is equal to the slope of the tangent to the graph of this function at a point whose abscissa is equal to x. This is the geometric meaning of the derivative.

Let at x=x 0 function y= f(x) takes on the value y 0 =f(x 0 ) , and the graph of this function has a tangent at the point with coordinates ( x 0 ;y 0). Then the slope of the tangent

k = ( x 0). Using the equation of a straight line, known from the course of analytical geometry, passing through a given point in a given direction ( y-y 0 =k(x-x 0)), we write the equation of the tangent:

The line passing through the point of contact perpendicular to the tangent is called the normal to the curve. Since the normal is perpendicular to the tangent, its slope k norms is related to the slope of the tangent k the relation known from analytic geometry: k norms = ─ , i.e. for a normal passing through a point with coordinates ( x 0 ;y 0),k norm = ─ . Therefore, the equation for this normal is:

(provided that

).

§ 4. Examples of calculating the derivative.

To calculate the derivative of a function y= f(x) at the point x, necessary:

Argument x increment ∆ x;

Find the corresponding increment of the function ∆ y=f(x+∆x) -f(x);

Compose a relation ;

Find the limit of this ratio for ∆ x→0.

Example 4.1. Find the derivative of a function y=C=const.

Argument x give an increment ∆ x.

Whatever x, ∆y=0: ∆y=f(x+∆x) ─f(x)=С─С=0;

From here =0 and =0, i.e. =0.

Example 4.2. Find the derivative of a function y=x.

∆ y=f(x+∆x) ─f(x)= x+∆x– x=∆ x;

1, =1, i.e. =1.

Example 4.3. Find the derivative of a function y=x 2.

∆ y= (x+∆ x)2–x 2= 2 x∙∆ x+ (∆ x)2;

= 2 x+ ∆ x, = 2 x, i.e. =2 x.

Example 4.4. Find the derivative of the function y=sin x.

∆ y=sin( x+∆x) -sin x= 2sin cos( x+);

=

;

=



= cos x, i.e. = cos x.

Example 4.5. Find the derivative of a function y=

.

=

, i.e. = .

MECHANICAL MEANING OF THE DERIVATIVE

It is known from physics that the law of uniform motion has the form s = v t, Where s- path traveled up to the point in time t, v is the speed of uniform motion.

However, since most of the movements occurring in nature are uneven, then in the general case, the speed, and, consequently, the distance s will depend on time t, i.e. will be a function of time.

So, let the material point move in a straight line in one direction according to the law s=s(t).

Note a moment in time t 0 . By this point, the point has passed the path s=s(t 0 ). Let's determine the speed v material point at time t 0 .

To do this, consider some other moment in time t 0 + Δ t. It corresponds to the distance traveled s =s(t 0 + Δ t). Then for the time interval Δ t the point has traveled the path Δs =s(t 0 + Δ t)–s(t).

Let's consider the relationship. It is called the average speed in the time interval Δ t. The average speed cannot accurately characterize the speed of movement of a point at the moment t 0 (because the movement is uneven). In order to more accurately express this true speed using the average speed, you need to take a smaller time interval Δ t.

So, the speed of movement at a given time t 0 (instantaneous speed) is the limit of the average speed in the interval from t 0 to t 0 +Δ t when Δ t→0:

,

those. speed of uneven movement is the derivative of the distance traveled with respect to time.

GEOMETRIC MEANING OF THE DERIVATIVE

Let us first introduce the definition of a tangent to a curve at a given point.

Let we have a curve and a fixed point on it M 0(see figure). Consider another point M this curve and draw a secant M 0 M. If point M starts to move along the curve, and the point M 0 remains stationary, the secant changes its position. If, with unlimited approximation of the point M curve to point M 0 on any side, the secant tends to take the position of a certain straight line M 0 T, then the straight line M 0 T is called the tangent to the curve at the given point M 0.

That., tangent to the curve at a given point M 0 called the limit position of the secant M 0 M when the point M tends along the curve to a point M 0.

Consider now the continuous function y=f(x) and the curve corresponding to this function. For some value X 0 function takes a value y0=f(x0). These values x 0 and y 0 on the curve corresponds to a point M 0 (x 0; y 0). Let's give an argument x0 increment Δ X. The new value of the argument corresponds to the incremented value of the function y 0 +Δ y=f(x 0 –Δ x). We get a point M(x 0+Δ x; y 0+Δ y). Let's draw a secant M 0 M and denote by φ the angle formed by the secant with the positive direction of the axis Ox. Let's make a relation and note that .

If now Δ x→0, then, due to the continuity of the function Δ at→0, and therefore the point M, moving along the curve, indefinitely approaches the point M 0. Then the secant M 0 M will tend to take the position of a tangent to the curve at the point M 0, and the angle φ→α at Δ x→0, where α denotes the angle between the tangent and the positive direction of the axis Ox. Since the function tg φ continuously depends on φ at φ≠π/2, then at φ→α tg φ → tg α and, therefore, the slope of the tangent will be:

those. f"(x)= tgα .

Thus, geometrically y "(x 0) represents the slope of the tangent to the graph of this function at the point x0, i.e. for a given value of the argument x, the derivative is equal to the tangent of the angle formed by the tangent to the graph of the function f(x) at the corresponding point M 0 (x; y) with positive axis direction Ox.

Example. Find the slope of the tangent to the curve y = x 2 at point M(-1; 1).

We have already seen that ( x 2)" = 2X. But the slope of the tangent to the curve is tg α = y"| x=-1 = - 2.

Geometric, mechanical, economic meaning of the derivative

Definition of a derivative.

Lecture №7-8

Bibliography

1 Ukhobotov, V. I. Mathematics: Textbook.- Chelyabinsk: Chelyab. state un-t, 2006.- 251 p.

2 Ermakov, V.I. Collection of problems in higher mathematics. Tutorial. -M.: INFRA-M, 2006. - 575 p.

3 Ermakov, V.I. General course of higher mathematics. Textbook. -M.: INFRA-M, 2003. - 656 p.

Theme "Derivative"

Target: explain the concept of a derivative, trace the relationship between the continuity and differentiability of a function, show the applicability of using a derivative with examples.

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This limit in economics is called the marginal cost of production.

Definition of a derivative. The geometric and mechanical meaning of the derivative, the equation of a function tangent to the graph.

Need a short answer (no extra water)

Dead_white_snow

Derivative is the basic concept of differential calculus, which characterizes the rate of change of a function.
Geometric?
Tangent to function at point... .
Function increase condition: f "(x) > 0.
Decreasing function condition: f "(x)< 0.
Inflection point (necessary condition): f " " (x0) = 0.
Convex up: f " " (x) Convex down: f " " (x) >0
Normal Equation: y=f(x0)-(1/f `(x0))(x-x0)
Mechanical?
Velocity is the derivative with respect to distance, acceleration is the derivative with respect to velocity, and the second derivative with respect to distance...
The equation of the tangent to the graph of the function f at the point x0
y=f(x0)+f `(x0)(x-x0)

User deleted

If there is a limit on the ratio delta y to delta x of the increment of the function delta y to the increment of the argument delta x that caused it, when delta x tends to zero, then this limit is called the derivative of the function y = f (x) at a given point x and is denoted by y "or f "(x)
The speed v of rectilinear motion is the derivative of the path s with respect to time t: v = ds/dt. This is the mechanical meaning of the derivative.
The slope of the tangent to the curve y \u003d f (x) at the point with the abscissa x zero is the derivative of f "(x zero). This is the geometric meaning of the derivative.
The tangent curve at the point M zero is called the straight line M zero T, the slope of which is equal to the limit of the slope of the secant M zero M one when delta x tends to zero.
tg phi = lim tg alpha as delta x approaches zero = lim (delta x/delta y) as delta x approaches zero
From the geometric meaning of the derivative, the tangent equation will take the form:
y - y null = f "(x null) (x - x null)