The definition of a derivative is its physical meaning. Lesson topic "the geometric meaning of the derivative"

Derivative of the function.

1. Definition of the derivative, its geometric meaning.

2. Derivative of a complex function.

3. Derivative of the inverse function.

4. Higher order derivatives.

5. Parametrically defined functions and implicitly.

6. Differentiation of functions specified parametrically and implicitly.

Introduction.

The source of differential calculus was two questions raised by the demands of science and technology in the 17th century.

1) The question of calculating the speed for an arbitrarily given law of motion.

2) The question of finding (by means of calculations) the tangent to an arbitrary curve.

The problem of drawing a tangent to some curves was solved by the ancient Greek scientist Archimedes (287-212 BC), using the method of drawing.

But only in the 17th and 18th centuries, in connection with the progress of natural science and technology, these questions received their proper development.

One of the important questions in the study of any physical phenomenon is usually the question of the speed, rapidity of the occurring phenomenon.

The speed at which an airplane or car moves is always the most important indicator of its performance. The rapidity of population growth in a given state is one of the main characteristics of its social development.

The initial idea of ​​speed is clear to everyone. However, this general idea is not enough to solve most practical problems. It is necessary to have such a quantitative definition of this quantity, which we call speed. The need for such a precise quantitative definition has historically been one of the main drivers for the creation of mathematical analysis. An entire section of mathematical analysis is devoted to solving this basic problem and the conclusions from this solution. We turn to the study of this section.

Definition of a derivative, its geometric meaning.

Let there be given a function defined in some interval (a, c) and continuous in it.

1. Let's give an argument NS increment, then the function will receive

increment:

2. Let's compose the relation .

3. Passing to the limit at and, assuming that the limit

exists, we get a value that is called

derivative of the function with respect to the argument NS.

Definition. The derivative of a function at a point is the limit of the ratio of the increment of the function to the increment of the argument when → 0.

The value of the derivative obviously depends on the point NS, in which it is found, therefore the derivative of the function is, in turn, some function of NS... It is indicated.

By definition, we have

or (3)

Example. Find the derivative of the function.

1. ;

Consider the graph of some function y = f (x).

Mark on it some point A with coordinates (x, f (x)) and not far from it point B with coordinates (x + h, f (x + h). Draw a straight line (AB) through these points. Consider the expression ... The difference f (x + h) -f (x) is equal to the distance BL, and the distance AL is equal to h. The BL / AL ratio is the tangent ε of the angle - the angle of inclination of the straight line (AB). Now let's imagine that the value of h is very, very small. Then the line (AB) almost coincides with the tangent at the point x to the graph of the function y = f (x).

So, let's give definitions.

The derivative of the function y = f (x) at the point x is the limit of the ratio as h tends to zero. They write:

The geometric meaning of the derivative is the tangent of the angle of inclination of the tangent.

The derivative also has a physical meaning. In the elementary grades, the definition of speed was given as distance divided by time. However, in real life, the speed of, for example, a car is not constant throughout the journey. Let the path be some function of time - S (t). Let us fix the moment of time t. In a short period of time from t to t + h, the car will cover the path S (t + h) -S (t). For a small period of time, the speed will not change much and therefore, you can use the definition of speed, known from elementary school ... And when h tends to zero, this will be the derivative.

The derivative of the function f (x) at the point x0 is the limit (if it exists) of the ratio of the increment of the function at the point x0 to the increment of the argument Δx, if the increment of the argument tends to zero and is denoted by f '(x0). The action of finding the derivative of a function is called differentiation.
The derivative of a function has the following physical meaning: the derivative of a function at a given point is the rate of change of a function at a given point.

The geometric meaning of the derivative... The derivative at the point x0 is equal to the slope of the tangent to the graph of the function y = f (x) at this point.

The physical meaning of the derivative. If a point moves along the x-axis and its coordinate changes according to the law x (t), then the instantaneous velocity of the point is:

Differential concept, its properties. Differentiation rules. Examples.

Definition. The differential of a function at some point x is the main, linear part of the increment of the function. The differential of the function y = f (x) is equal to the product of its derivative and the increment of the independent variable x (argument).

It is written like this:

or

Or


Differential properties
The differential has properties similar to those of the derivative:





TO basic rules of differentiation include:
1) taking a constant factor out of the sign of the derivative
2) derivative of the sum, derivative of the difference
3) the derivative of the product of functions
4) the derivative of the quotient of two functions (derivative of a fraction)

Examples.
Let us prove the formula: By the definition of the derivative, we have:

An arbitrary factor can be moved outside the sign of the passage to the limit (this is known from the properties of the limit), therefore

For example: Find the derivative of a function
Solution: Let us use the rule of taking the factor out of the sign of the derivative :

Quite often, you first have to simplify the form of the differentiated function in order to use the table of derivatives and the rules for finding derivatives. The following examples clearly confirm this.

Differentiation formulas. Differential application in approximate calculations. Examples.

The use of the differential in approximate calculations allows you to use the differential for approximate calculations of the values ​​of a function.
Examples of.
Using the differential, calculate approximately
To calculate this value, we apply the formula from the theory
Let us introduce into consideration the function and represent the given value in the form
then Calculate

Substituting everything into the formula, we finally get
Answer:

16. L'Hôpital's rule for disclosing uncertainties of the form 0/0 Or ∞ / ∞. Examples.
The limit of the ratio of two infinitesimal or two infinitely large quantities is equal to the limit of the ratio of their derivatives.

1)

17. Increase and decrease of function. Extremum function. Algorithm for studying a function for monotonicity and extremum. Examples.

Function is increasing on the interval if the inequality holds for any two points of this interval related by the relation. That is, a larger value of the argument corresponds to a larger value of the function, and its graph goes “from bottom to top”. The demo function grows on the interval

Similarly, the function decreases on the interval, if for any two points of the given interval, such that, the inequality is true. That is, a larger value of the argument corresponds to a smaller value of the function, and its graph goes “from top to bottom”. Ours decreases at intervals decreases at intervals .

Extremes A point is called the maximum point of the function y = f (x) if the inequality holds for all x from its neighborhood. The value of the function at the maximum point is called maximum function and denote.
A point is called a minimum point of the function y = f (x) if the inequality holds for all x from its neighborhood. The value of the function at the minimum point is called minimum function and denote.
The neighborhood of a point is understood as the interval , where is a sufficiently small positive number.
The minimum and maximum points are called extremum points, and the values ​​of the function corresponding to the extremum points are called extrema of the function.

To investigate the function on monotony, use the following scheme:
- Find the domain of the function;
- Find the derivative of the function and the domain of the derivative;
- Find the zeros of the derivative, i.e. the value of the argument for which the derivative is zero;
- On the numerical rays, mark the common part of the region of definition of the function and the region of definition of its derivative, and on it - the zeros of the derivative;
- Determine the signs of the derivative at each of the intervals obtained;
- By the signs of the derivative, determine at which intervals the function increases and at which it decreases;
- Write down the appropriate spaces separated by semicolons.

Algorithm for studying a continuous function y = f (x) for monotonicity and extrema:
1) Find the derivative f ′ (x).
2) Find stationary (f ′ (x) = 0) and critical (f ′ (x) does not exist) points of the function y = f (x).
3) Mark the stationary and critical points on the number line and determine the signs of the derivative on the resulting intervals.
4) Draw conclusions about the monotonicity of the function and its extremum points.

18. Convexity of a function. Inflection points. Algorithm for studying a function for convexity (concavity) Examples.

convex down on the interval X, if its graph is located not lower than the tangent to it at any point of the interval X.

The function to be differentiated is called convex up on the interval X, if its graph is located not higher than the tangent to it at any point of the interval X.

The point of the formula is called inflection point function y = f (x), if at a given point there is a tangent to the graph of the function (it can be parallel to the Oy axis) and there is such a neighborhood of the point formula, within which the graph of the function has different directions of convexity to the left and to the right of the point M.

Finding the intervals for convexity:

If the function y = f (x) has a finite second derivative on the interval X and if the inequality (), then the graph of the function has a bulge directed down (up) on X.
This theorem allows us to find the intervals of concavity and convexity of a function; it is only necessary to solve the inequalities and, respectively, on the domain of the original function.

Example: Find out the intervals at which the graph of the function Find out the intervals at which the graph of the function has an upward bulge and a downward bulge. has an upward bulge and a downward bulge.
Solution: The domain of this function is the entire set of real numbers.
Let's find the second derivative.

The domain of definition of the second derivative coincides with the domain of definition of the original function, therefore, in order to find out the intervals of concavity and convexity, it is sufficient to solve and, respectively. Therefore, the function is convex down on the interval formula and convex up on the interval formula.

19) Asymptotes of a function. Examples.

The straight line is called vertical asymptote function graph if at least one of the limit values ​​is either equal to or.

Comment. A straight line cannot be a vertical asymptote if the function is continuous at a point. Therefore, the vertical asymptotes should be sought at the points of discontinuity of the function.

The straight line is called horizontal asymptote function graph if at least one of the limit values ​​or is equal.

Comment. The function graph can have only the right horizontal asymptote or only the left one.

The straight line is called oblique asymptote function graph if

EXAMPLE:

Exercise. Find the asymptotes of the graph of a function

Solution. Function scope:

a) vertical asymptotes: straight line - vertical asymptote, since

b) horizontal asymptotes: we find the limit of the function at infinity:

that is, there are no horizontal asymptotes.

c) oblique asymptotes:

Thus, the oblique asymptote is:.

Answer. The vertical asymptote is straight.

The oblique asymptote is straight.

20) The general scheme of the study of the function and the construction of the graph. Example.

a.
Find ODZ and discontinuity points of the function.

b. Find the points of intersection of the graph of the function with the coordinate axes.

2. Conduct a study of the function using the first derivative, that is, find the extremum points of the function and the intervals of increasing and decreasing.

3. Investigate the function using the second-order derivative, that is, find the inflection points of the graph of the function and the intervals of its convexity and concavity.

4. Find the asymptotes of the graph of the function: a) vertical, b) oblique.

5. On the basis of the study, build a graph of the function.

Note that before plotting the graph, it is useful to determine whether the given function is odd or even.

Recall that a function is called even if the value of the function does not change when the argument sign changes: f (-x) = f (x) and a function is called odd if f (-x) = -f (x).

In this case, it is enough to investigate the function and build its graph for positive values ​​of the argument belonging to the ODZ. For negative values ​​of the argument, the graph is completed on the basis that for an even function it is symmetric about the axis Oy, and for odd relative to the origin.

Examples. Explore functions and plot their graphs.

Function scope D (y) = (–∞; + ∞). There are no break points.

Axis intersection Ox: x = 0,y = 0.

The function is odd, therefore, it can only be investigated in the interval)