On which we examined the simplest derivatives, and also became acquainted with the rules of differentiation and some technical techniques for finding derivatives. Thus, if you are not very good with derivatives of functions or some points in this article are not entirely clear, then first read the above lesson. Please get in a serious mood - the material is not simple, but I will still try to present it simply and clearly.

In practice, you have to deal with the derivative of a complex function very often, I would even say, almost always, when you are given tasks to find derivatives.

We look at the table at the rule (No. 5) for differentiating a complex function:

Let's figure it out. First of all, let's pay attention to the entry. Here we have two functions - and , and the function, figuratively speaking, is nested within the function . A function of this type (when one function is nested within another) is called a complex function.

I will call the function external function, and the function – internal (or nested) function.

! These definitions are not theoretical and should not appear in the final design of assignments. I use informal expressions “external function”, “internal” function only to make it easier for you to understand the material.

To clarify the situation, consider:

Example 1

Find the derivative of a function

Under the sine we have not just the letter “X”, but an entire expression, so finding the derivative right away from the table will not work. We also notice that it is impossible to apply the first four rules here, there seems to be a difference, but the fact is that the sine cannot be “torn into pieces”:

In this example, it is already intuitively clear from my explanations that a function is a complex function, and the polynomial is an internal function (embedding), and an external function.

First step what you need to do when finding the derivative of a complex function is to understand which function is internal and which is external.

In the case of simple examples, it seems clear that a polynomial is embedded under the sine. But what if everything is not obvious? How to accurately determine which function is external and which is internal? To do this, I suggest using the following technique, which can be done mentally or in a draft.

Let's imagine that we need to calculate the value of the expression at on a calculator (instead of one there can be any number).

What will we calculate first? First of all you will need to perform the following action: , therefore the polynomial will be an internal function:

Secondly will need to be found, so sine – will be an external function:

After we SOLD OUT with internal and external functions, it’s time to apply the rule of differentiation of complex functions .

Let's start deciding. From the lesson How to find the derivative? we remember that the design of a solution to any derivative always begins like this - we enclose the expression in brackets and put a stroke at the top right:

At first we find the derivative of the external function (sine), look at the table of derivatives of elementary functions and notice that . All table formulas are also applicable if “x” is replaced with a complex expression, in this case:

Please note that the inner function hasn't changed, we don't touch it.

Well, it's quite obvious that

The result of applying the formula in its final form it looks like this:

The constant factor is usually placed at the beginning of the expression:

If there is any misunderstanding, write the solution down on paper and read the explanations again.

Example 2

Find the derivative of a function

Example 3

Find the derivative of a function

As always, we write down:

Let's figure out where we have an external function and where we have an internal one. To do this, we try (mentally or in a draft) to calculate the value of the expression at . What should you do first? First of all, you need to calculate what the base is equal to: therefore, the polynomial is the internal function:

And only then is the exponentiation performed, therefore, the power function is an external function:

According to the formula , first you need to find the derivative of the external function, in this case, the degree. We look for the required formula in the table: . We repeat again: any tabular formula is valid not only for “X”, but also for a complex expression. Thus, the result of applying the rule for differentiating a complex function next:

I emphasize again that when we take the derivative of the external function, our internal function does not change:

Now all that remains is to find a very simple derivative of the internal function and tweak the result a little:

Example 4

Find the derivative of a function

This is an example for you to solve on your own (answer at the end of the lesson).

To consolidate your understanding of the derivative of a complex function, I will give an example without comments, try to figure it out on your own, reason where the external and where the internal function is, why the tasks are solved this way?

Example 5

a) Find the derivative of the function

b) Find the derivative of the function

Example 6

Find the derivative of a function

Here we have a root, and in order to differentiate the root, it must be represented as a power. Thus, first we bring the function into the form appropriate for differentiation:

Analyzing the function, we come to the conclusion that the sum of the three terms is an internal function, and raising to a power is an external function. We apply the rule of differentiation of complex functions :

We again represent the degree as a radical (root), and for the derivative of the internal function we apply a simple rule for differentiating the sum:

Ready. You can also reduce the expression to a common denominator in brackets and write everything down as one fraction. It’s beautiful, of course, but when you get cumbersome long derivatives, it’s better not to do this (it’s easy to get confused, make an unnecessary mistake, and it will be inconvenient for the teacher to check).

Example 7

Find the derivative of a function

This is an example for you to solve on your own (answer at the end of the lesson).

It is interesting to note that sometimes instead of the rule for differentiating a complex function, you can use the rule for differentiating a quotient , but such a solution will look like an unusual perversion. Here is a typical example:

Example 8

Find the derivative of a function

Here you can use the rule of differentiation of the quotient , but it is much more profitable to find the derivative through the rule of differentiation of a complex function:

We prepare the function for differentiation - we move the minus out of the derivative sign, and raise the cosine into the numerator:

Cosine is an internal function, exponentiation is an external function.
Let's use our rule :

We find the derivative of the internal function and reset the cosine back down:

Ready. In the example considered, it is important not to get confused in the signs. By the way, try to solve it using the rule , the answers must match.

Example 9

Find the derivative of a function

This is an example for you to solve on your own (answer at the end of the lesson).

So far we have looked at cases where we had only one nesting in a complex function. In practical tasks, you can often find derivatives, where, like nesting dolls, one inside the other, 3 or even 4-5 functions are nested at once.

Example 10

Find the derivative of a function

Let's understand the attachments of this function. Let's try to calculate the expression using the experimental value. How would we count on a calculator?

First you need to find , which means the arcsine is the deepest embedding:

This arcsine of one should then be squared:

And finally, we raise seven to a power:

That is, in this example we have three different functions and two embeddings, while the innermost function is the arcsine, and the outermost function is the exponential function.

Let's start deciding

According to the rule First you need to take the derivative of the outer function. We look at the table of derivatives and find the derivative of the exponential function: The only difference is that instead of “x” we have a complex expression, which does not negate the validity of this formula. So, the result of applying the rule for differentiating a complex function next.

Solving physical problems or examples in mathematics is completely impossible without knowledge of the derivative and methods for calculating it. The derivative is one of the most important concepts in mathematical analysis. We decided to devote today’s article to this fundamental topic. What is a derivative, what is its physical and geometric meaning, how to calculate the derivative of a function? All these questions can be combined into one: how to understand the derivative?

Geometric and physical meaning of derivative

Let there be a function f(x) , specified in a certain interval (a, b) . Points x and x0 belong to this interval. When x changes, the function itself changes. Changing the argument - the difference in its values x-x0 . This difference is written as delta x and is called argument increment. A change or increment of a function is the difference between the values ​​of a function at two points. Definition of derivative:

The derivative of a function at a point is the limit of the ratio of the increment of the function at a given point to the increment of the argument when the latter tends to zero.

Otherwise it can be written like this:

What's the point of finding such a limit? And here's what it is:

the derivative of a function at a point is equal to the tangent of the angle between the OX axis and the tangent to the graph of the function at a given point.

Physical meaning of the derivative: the derivative of the path with respect to time is equal to the speed of rectilinear motion.

Indeed, since school days everyone knows that speed is a particular path x=f(t) and time t . Average speed over a certain period of time:

To find out the speed of movement at a moment in time t0 you need to calculate the limit:

Rule one: set a constant

The constant can be taken out of the derivative sign. Moreover, this must be done. When solving examples in mathematics, take it as a rule - If you can simplify an expression, be sure to simplify it .

Example. Let's calculate the derivative:

Rule two: derivative of the sum of functions

The derivative of the sum of two functions is equal to the sum of the derivatives of these functions. The same is true for the derivative of the difference of functions.

We will not give a proof of this theorem, but rather consider a practical example.

Find the derivative of the function:

Rule three: derivative of the product of functions

The derivative of the product of two differentiable functions is calculated by the formula:

Example: find the derivative of a function:

Solution:

It is important to talk about calculating derivatives of complex functions here. The derivative of a complex function is equal to the product of the derivative of this function with respect to the intermediate argument and the derivative of the intermediate argument with respect to the independent variable.

In the above example we come across the expression:

In this case, the intermediate argument is 8x to the fifth power. In order to calculate the derivative of such an expression, we first calculate the derivative of the external function with respect to the intermediate argument, and then multiply by the derivative of the intermediate argument itself with respect to the independent variable.

Rule four: derivative of the quotient of two functions

Formula for determining the derivative of the quotient of two functions:

We tried to talk about derivatives for dummies from scratch. This topic is not as simple as it seems, so be warned: there are often pitfalls in the examples, so be careful when calculating derivatives.

With any questions on this and other topics, you can contact the student service. In a short time, we will help you solve the most difficult test and understand the tasks, even if you have never done derivative calculations before.

The content of the article

MATHEMATICAL ANALYSIS, a branch of mathematics that provides methods for the quantitative study of various processes of change; deals with the study of the rate of change (differential calculus) and the determination of the lengths of curves, areas and volumes of figures bounded by curved contours and surfaces (integral calculus). It is typical for problems of mathematical analysis that their solution is associated with the concept of a limit.

The beginning of mathematical analysis was laid in 1665 by I. Newton and (around 1675) independently by G. Leibniz, although important preparatory work was carried out by I. Kepler (1571–1630), F. Cavalieri (1598–1647), P. Fermat (1601– 1665), J. Wallis (1616–1703) and I. Barrow (1630–1677).

To make the presentation more vivid, we will resort to the language of graphics. Therefore, it may be useful for the reader to look into the article ANALYTICAL GEOMETRY before starting to read this article.

DIFFERENTIAL CALCULUS

Tangents.

In Fig. 1 shows a fragment of the curve y = 2x – x 2, enclosed between x= –1 and x= 3. Sufficiently small segments of this curve look straight. In other words, if R is an arbitrary point of this curve, then there is a certain straight line passing through this point and which is an approximation of the curve in a small neighborhood of the point R, and the smaller the neighborhood, the better the approximation. Such a line is called tangent to the curve at the point R. The main task of differential calculus is to construct a general method that allows one to find the direction of a tangent at any point on a curve at which a tangent exists. It is not difficult to imagine a curve with a sharp break (Fig. 2). If R is the top of such a break, then we can construct an approximating straight line P.T. 1 – to the right of the point R and another approximating straight line RT 2 – to the left of the point R. But there is no single straight line passing through a point R, which approached the curve equally well in the vicinity of the point P both on the right and on the left, therefore the tangent at the point P does not exist.

In Fig. 1 tangent FROM drawn through the origin ABOUT= (0,0). The slope of this line is 2, i.e. when the abscissa changes by 1, the ordinate increases by 2. If x And y– coordinates of an arbitrary point on FROM, then, moving away from ABOUT to a distance X units to the right, we are moving away from ABOUT on 2 y units up. Hence, y/x= 2, or y = 2x. This is the tangent equation FROM to the curve y = 2x – x 2 at point ABOUT.

It is now necessary to explain why, out of the set of lines passing through the point ABOUT, the straight line is chosen FROM. How does a straight line with a slope of 2 differ from other straight lines? There is one simple answer, and it is difficult to resist the temptation to give it using the analogy of a tangent to a circle: the tangent FROM has only one common point with the curve, while any other non-vertical line passing through the point ABOUT, intersects the curve twice. This can be verified as follows.

Since the expression y = 2x – x 2 can be obtained by subtraction X 2 of y = 2x(equations of straight line FROM), then the values y there is less knowledge for the graph y for a straight line at all points except the point x= 0. Therefore, the graph is everywhere except the point ABOUT, located below FROM, and this line and the graph have only one common point. Moreover, if y = mx- equation of some other line passing through a point ABOUT, then there will definitely be two points of intersection. Really, mx = 2x – x 2 not only when x= 0, but also at x = 2 – m. And only when m= 2 both intersection points coincide. In Fig. 3 shows the case when m is less than 2, so to the right of ABOUT a second intersection point appears.

What FROM– the only non-vertical straight line passing through a point ABOUT and having only one common point with the graph, not its most important property. Indeed, if we turn to other graphs, it will soon become clear that the tangent property we noted is not satisfied in the general case. For example, from Fig. 4 it is clear that near the point (1,1) the graph of the curve y = x 3 is well approximated by a straight line RT which, however, has more than one common point with it. However, we would like to consider RT tangent to this graph at point R. Therefore, it is necessary to find some other way to highlight the tangent than the one that served us so well in the first example.

Let us assume that through the point ABOUT and an arbitrary point Q = (h,k) on the curve graph y = 2x – x 2 (Fig. 5) a straight line (called a secant) is drawn. Substituting the values ​​into the equation of the curve x = h And y = k, we get that k = 2h – h 2, therefore, the angular coefficient of the secant is equal to

At very small h meaning m close to 2. Moreover, choosing h close enough to 0 we can do m arbitrarily close to 2. We can say that m"tends to the limit" equal to 2 when h tends to zero, or whatever the limit m equals 2 at h tending to zero. Symbolically it is written like this:

Then the tangent to the graph at the point ABOUT is defined as a straight line passing through a point ABOUT, with a slope equal to this limit. This definition of a tangent is applicable in the general case.

Let's show the advantages of this approach with one more example: let's find the slope of the tangent to the graph of the curve y = 2x – x 2 at any point P = (x,y), not limited to the simplest case when P = (0,0).

Let Q = (x + h, y + k) – the second point on the graph, located at a distance h to the right of R(Fig. 6). We need to find the slope k/h secant PQ. Dot Q is at a distance

above the axis X.

Opening the brackets, we find:

Subtracting from this equation y = 2x – x 2, find the vertical distance from the point R to the point Q:

Therefore, the slope m secant PQ equals

Now that h tends to zero, m tends to 2 – 2 x; We will take the last value as the angular coefficient of the tangent P.T.. (The same result will occur if h takes negative values, which corresponds to the selection of a point Q on the left of P.) Note that when x= 0 the result obtained coincides with the previous one.

Expression 2 – 2 x called the derivative of 2 x – x 2. In the old days, the derivative was also called "differential ratio" and "differential coefficient". If by expression 2 x – x 2 designate f(x), i.e.

then the derivative can be denoted

In order to find out the slope of the tangent to the graph of the function y = f(x) at some point, it is necessary to substitute in fў ( x) value corresponding to this point X. Thus, the slope fў (0) = 2 at X = 0, fў (0) = 0 at X= 1 and fў (2) = –2 at X = 2.

The derivative is also denoted atў , dy/dx, D x y And Du.

The fact that the curve y = 2x – x 2 near a given point is practically indistinguishable from its tangent at this point, allows us to speak of the angular coefficient of the tangent as the “angular coefficient of the curve” at the point of tangency. Thus, we can say that the slope of the curve we are considering has a slope of 2 at the point (0,0). We can also say that when x= 0 rate of change y relatively x is equal to 2. At point (2,0) the slope of the tangent (and the curve) is –2. (The minus sign means that as we increase x variable y decreases.) At the point (1,1) the tangent is horizontal. We say it's a curve y = 2x – x 2 has a stationary value at this point.

Highs and lows.

We have just shown that the curve f(x) = 2x – x 2 is stationary at point (1,1). Because fў ( x) = 2 – 2x = 2(1 – x), it is clear that when x, less than 1, fў ( x) is positive, and therefore y increases; at x, large 1, fў ( x) is negative, and therefore y decreases. Thus, in the vicinity of the point (1,1), indicated in Fig. 6 letter M, meaning at grows to a point M, stationary at point M and decreases after the point M. This point is called “maximum” because the value at at this point exceeds any of its values ​​in a sufficiently small neighborhood. Similarly, the “minimum” is defined as the point in the vicinity of which all values y exceed the value at at this very point. It may also happen that although the derivative of f(x) at a certain point and vanishes; its sign in the vicinity of this point does not change. Such a point, which is neither a maximum nor a minimum, is called an inflection point.

As an example, let's find the stationary point of the curve

The derivative of this function is equal to

and goes to zero at x = 0, X= 1 and X= –1; those. at points (0,0), (1, –2/15) and (–1, 2/15). If X a little less than –1, then fў ( x) is negative; If X a little more than –1, then fў ( x) is positive. Therefore, the point (–1, 2/15) is the maximum. Similarly, it can be shown that the point (1, –2/15) is a minimum. But the derivative fў ( x) is negative both before the point (0,0) and after it. Therefore, (0,0) is the inflection point.

The study of the shape of the curve, as well as the fact that the curve intersects the axis X at f(x) = 0 (i.e. when X= 0 or ) allow us to present its graph approximately as shown in Fig. 7.

In general, if we exclude unusual cases (curves containing straight segments or an infinite number of bends), there are four options for the relative position of the curve and the tangent in the vicinity of the tangent point R. (Cm. rice. 8, on which the tangent has a positive slope.)

1) On both sides of the point R the curve lies above the tangent (Fig. 8, A). In this case they say that the curve at the point R convex down or concave.

2) On both sides of the point R the curve is located below the tangent (Fig. 8, b). In this case, the curve is said to be convex upward or simply convex.

3) and 4) The curve is located above the tangent on one side of the point R and below - on the other. In this case R– inflection point.

Comparing values fў ( x) on both sides of R with its value at the point R, one can determine which of these four cases one has to deal with in a particular problem.

Applications.

All of the above has important applications in various fields. For example, if a body is thrown vertically upward with an initial speed of 200 feet per second, then the height s, on which they will be located through t seconds compared to the starting point will be

Proceeding in the same way as in the examples we considered, we find

this quantity goes to zero at c. Derivative fў ( x) is positive up to the value c and negative after this time. Hence, s increases to , then becomes stationary, and then decreases. This is a general description of the movement of a body thrown upward. From it we know when the body reaches its highest point. Next, substituting t= 25/4 V f(t), we get 625 feet, the maximum lift height. In this problem fў ( t) has a physical meaning. This derivative shows the speed at which the body is moving at an instant t.

Let us now consider an application of another type (Fig. 9). From a sheet of cardboard with an area of ​​75 cm2, you need to make a box with a square bottom. What should be the dimensions of this box in order for it to have maximum volume? If X– side of the base of the box and h is its height, then the volume of the box is V = x 2 h, and the surface area is 75 = x 2 + 4xh. Transforming the equation, we get:

Derivative of V turns out to be equal

and goes to zero at X= 5. Then

And V= 125/2. Graph of a function V = (75x – x 3)/4 is shown in Fig. 10 (negative values X omitted as having no physical meaning in this problem).

Derivatives.

An important task of differential calculus is the creation of methods that allow you to quickly and conveniently find derivatives. For example, it is easy to calculate that

(The derivative of a constant is, of course, zero.) It is not difficult to derive a general rule:

Where n– any whole number or fraction. For example,

(This example shows how useful fractional exponents are.)

Here are some of the most important formulas:

There are also the following rules: 1) if each of the two functions g(x) And f(x) has derivatives, then the derivative of their sum is equal to the sum of the derivatives of these functions, and the derivative of the difference is equal to the difference of the derivatives, i.e.

2) the derivative of the product of two functions is calculated by the formula:

3) the derivative of the ratio of two functions has the form

4) the derivative of a function multiplied by a constant is equal to the constant multiplied by the derivative of this function, i.e.

It often happens that the values ​​of a function have to be calculated step by step. For example, to calculate sin x 2, we need to first find u = x 2, and then calculate the sine of the number u. We find the derivative of such complex functions using the so-called “chain rule”:

In our example f(u) = sin u, fў ( u) = cos u, hence,

These and other similar rules allow you to immediately write down derivatives of many functions.

Linear approximations.

The fact that, knowing the derivative, we can in many cases replace the graph of a function near a certain point with its tangent at this point is of great importance, since it is easier to work with straight lines.

This idea finds direct application in calculating approximate values ​​of functions. For example, it is quite difficult to calculate the value when x= 1.033. But you can use the fact that the number 1.033 is close to 1 and that . Up close x= 1 we can replace the graph with a tangent curve without making any serious mistakes. The angular coefficient of such a tangent is equal to the value of the derivative ( x 1/3)ў = (1/3) x–2/3 at x = 1, i.e. 1/3. Since point (1,1) lies on the curve and the angular coefficient of the tangent to the curve at this point is equal to 1/3, the tangent equation has the form

On this straight line X = 1,033

Received value y should be very close to the true value y; and, indeed, it is only 0.00012 more than the true one. In mathematical analysis, methods have been developed that make it possible to increase the accuracy of this kind of linear approximations. These methods ensure the reliability of our approximate calculations.

The procedure just described suggests one useful notation. Let P– point corresponding to the function graph f variable X, and let the function f(x) is differentiable. Let's replace the graph of the curve near the point R tangent to it drawn at this point. If X change by value h, then the ordinate of the tangent will change by the amount h H f ў ( x). If h is very small, then the latter value serves as a good approximation to the true change in the ordinate y graphic arts. If instead h we will write a symbol dx(this is not a product!), but a change in ordinate y let's denote dy, then we get dy = f ў ( x)dx, or dy/dx = f ў ( x) (cm. rice. eleven). Therefore, instead of Dy or f ў ( x) the symbol is often used to denote a derivative dy/dx. The convenience of this notation depends mainly on the explicit appearance of the chain rule (differentiation of a complex function); in the new notation this formula looks like this:

where it is implied that at depends on u, A u in turn depends on X.

Magnitude dy called differential at; in reality it depends on two variables, namely: from X and increments dx. When the increment dx very small size dy is close to the corresponding change in value y. But assume that the increment dx little, no need.

Derivative of a function y = f(x) we designated f ў ( x) or dy/dx. It is often possible to take the derivative of the derivative. The result is called the second derivative of f (x) and is denoted f ўў ( x) or d 2 y/dx 2. For example, if f(x) = x 3 – 3x 2, then f ў ( x) = 3x 2 – 6x And f ўў ( x) = 6x– 6. Similar notation is used for higher order derivatives. However, to avoid a large number of strokes (equal to the order of the derivative), the fourth derivative (for example) can be written as f (4) (x), and the derivative n-th order as f (n) (x).

It can be shown that the curve at a point is convex downward if the second derivative is positive, and convex upward if the second derivative is negative.

If a function has a second derivative, then the change in value y, corresponding to the increment dx variable X, can be approximately calculated using the formula

This approximation is usually better than that given by the differential fў ( x)dx. It corresponds to replacing part of the curve not with a straight line, but with a parabola.

If the function f(x) there are derivatives of higher orders, then

The remainder term has the form

Where x- some number between x And x + dx. The above result is called Taylor's formula with remainder term. If f(x) has derivatives of all orders, then usually Rn® 0 at n ® Ґ .

INTEGRAL CALCULUS

Squares.

When studying the areas of curvilinear plane figures, new aspects of mathematical analysis are revealed. The ancient Greeks tried to solve problems of this kind, for whom determining, for example, the area of ​​a circle was one of the most difficult tasks. Archimedes achieved great success in solving this problem, who also managed to find the area of ​​a parabolic segment (Fig. 12). Using very complex reasoning, Archimedes proved that the area of ​​a parabolic segment is 2/3 of the area of ​​the circumscribed rectangle and, therefore, in this case is equal to (2/3)(16) = 32/3. As we will see later, this result can be easily obtained by methods of mathematical analysis.

The predecessors of Newton and Leibniz, mainly Kepler and Cavalieri, solved problems of calculating the areas of curvilinear figures using a method that can hardly be called logically sound, but which turned out to be extremely fruitful. When Wallis in 1655 combined the methods of Kepler and Cavalieri with the methods of Descartes (analytic geometry) and took advantage of the newly emerging algebra, the stage was fully prepared for the appearance of Newton.

Wallis divided the figure, the area of ​​which needed to be calculated, into very narrow strips, each of which he approximately considered a rectangle. Then he added up the areas of the approximating rectangles and in the simplest cases obtained the value to which the sum of the areas of the rectangles tended when the number of strips tended to infinity. In Fig. Figure 13 shows rectangles corresponding to some division into strips of the area under the curve y = x 2 .

Main theorem.

The great discovery of Newton and Leibniz made it possible to eliminate the laborious process of going to the limit of the sum of areas. This was done thanks to a new look at the concept of area. The point is that we must imagine the area under the curve as generated by an ordinate moving from left to right and ask at what rate the area swept by the ordinates changes. We will get the key to answering this question if we consider two special cases in which the area is known in advance.

Let's start with the area under the graph of a linear function y = 1 + x, since in this case the area can be calculated using elementary geometry.

Let A(x) – part of the plane enclosed between the straight line y = 1 + x and a segment OQ(Fig. 14). When driving QP right area A(x) increases. At what speed? It is not difficult to answer this question, since we know that the area of ​​a trapezoid is equal to the product of its height and half the sum of its bases. Hence,

Rate of area change A(x) is determined by its derivative

We see that Aў ( x) coincides with the ordinate at points R. Is this a coincidence? Let's try to check on the parabola shown in Fig. 15. Area A (x) under the parabola at = X 2 in the range from 0 to X equal to A(x) = (1 / 3)(x)(x 2) = x 3/3. The rate of change of this area is determined by the expression

which exactly coincides with the ordinate at moving point R.

If we assume that this rule holds in the general case such that

is the rate of change of the area under the graph of the function y = f(x), then this can be used for calculations and other areas. In fact, the ratio Aў ( x) = f(x) expresses a fundamental theorem that could be formulated as follows: the derivative, or rate of change of area as a function of X, equal to the function value f (x) at point X.

For example, to find the area under the graph of a function y = x 3 from 0 to X(Fig. 16), let's put

A possible answer reads:

since the derivative of X 4 /4 is really equal X 3. Besides, A(x) is equal to zero at X= 0, as it should be if A(x) is indeed an area.

Mathematical analysis proves that there is no other answer other than the above expression for A(x), does not exist. Let us show that this statement is plausible using the following heuristic (non-rigorous) reasoning. Suppose there is some second solution IN(x). If A(x) And IN(x) “start” simultaneously from zero value at X= 0 and change at the same rate all the time, then their values ​​​​cannot be X cannot become different. They must coincide everywhere; therefore, there is a unique solution.

How can you justify the relationship? Aў ( x) = f(x) in general? This question can only be answered by studying the rate of change of area as a function of X in general. Let m– the smallest value of the function f (x) in the range from X before ( x + h), A M– the largest value of this function in the same interval. Then the increase in area when moving from X To ( x + h) must be enclosed between the areas of two rectangles (Fig. 17). The bases of both rectangles are equal h. The smaller rectangle has a height m and area mh, larger, respectively, M And Mh. On the graph of area versus X(Fig. 18) it is clear that when the abscissa changes to h, the ordinate value (i.e. area) increases by the amount between mh And Mh. The secant slope on this graph is between m And M. what happens when h tends to zero? If the graph of a function y = f(x) is continuous (i.e. does not contain discontinuities), then M, And m tend to f(x). Therefore, the slope Aў ( x) graph of area as a function of X equals f(x). This is precisely the conclusion that needed to be reached.

Leibniz proposed for the area under a curve y = f(x) from 0 to A designation

In a rigorous approach, this so-called definite integral should be defined as the limit of certain sums in the manner of Wallis. Considering the result obtained above, it is clear that this integral is calculated provided that we can find such a function A(x), which vanishes when X= 0 and has a derivative Aў ( x), equal to f (x). Finding such a function is usually called integration, although it would be more appropriate to call this operation anti-differentiation, meaning that it is in some sense the inverse of differentiation. In the case of a polynomial, integration is simple. For example, if

which is easy to verify by differentiating A(x).

To calculate the area A 1 under the curve y = 1 + x + x 2 /2, enclosed between ordinates 0 and 1, we simply write

and, substituting X= 1, we get A 1 = 1 + 1/2 + 1/6 = 5/3. Square A(x) from 0 to 2 is equal to A 2 = 2 + 4/2 + 8/6 = 16/3. As can be seen from Fig. 19, the area enclosed between ordinates 1 and 2 is equal to A 2 – A 1 = 11/3. It is usually written as a definite integral

Volumes.

Similar reasoning makes it surprisingly easy to calculate the volumes of bodies of revolution. Let's demonstrate this with the example of calculating the volume of a sphere, another classical problem that the ancient Greeks, using the methods known to them, managed to solve with great difficulty.

Let's rotate part of the plane contained inside a quarter circle of radius r, at an angle of 360° around the axis X. As a result, we get a hemisphere (Fig. 20), the volume of which we denote V(x). We need to determine the rate at which it increases V(x) with increasing x. Moving from X To X + h, it is easy to verify that the increment in volume is less than the volume p(r 2 – x 2)h circular cylinder with radius and height h, and more than volume p[r 2 – (x + h) 2 ]h cylinder radius and height h. Therefore, on the graph of the function V(x) the angular coefficient of the secant is between p(r 2 – x 2) and p[r 2 – (x + h) 2 ]. When h tends to zero, the slope tends to

At x = r we get

for the volume of the hemisphere, and therefore 4 p r 3/3 for the volume of the entire ball.

A similar method allows one to find the lengths of curves and the areas of curved surfaces. For example, if a(x) – arc length PR in Fig. 21, then our task is to calculate aў( x). At the heuristic level, we will use a technique that allows us not to resort to the usual passage to the limit, which is necessary for a rigorous proof of the result. Let us assume that the rate of change of the function A(x) at point R the same as it would be if the curve were replaced by its tangent P.T. at the point P. But from Fig. 21 is directly visible when stepping h to the right or left of the point X along RT meaning A(x) changes to

Therefore, the rate of change of the function a(x) is

To find the function itself a(x), you just need to integrate the expression on the right side of the equality. It turns out that integration is quite difficult for most functions. Therefore, the development of methods of integral calculus constitutes a large part of mathematical analysis.

Antiderivatives.

Every function whose derivative is equal to the given function f(x), is called antiderivative (or primitive) for f(x). For example, X 3 /3 – antiderivative for the function X 2 since ( x 3 /3)ў = x 2. Of course X 3 /3 is not the only antiderivative of the function X 2 because x 3 /3 + C is also a derivative for X 2 for any constant WITH. However, in what follows we agree to omit such additive constants. In general

Where n is a positive integer, since ( x n + 1/(n+ 1))ў = x n. Relation (1) is satisfied in an even more general sense if n replace with any rational number k, except –1.

An arbitrary antiderivative function for a given function f(x) is usually called the indefinite integral of f(x) and denote it in the form

For example, since (sin x)ў = cos x, the formula is valid

In many cases where there is a formula for the indefinite integral of a given function, it can be found in numerous widely published tables of indefinite integrals. Integrals from elementary functions are tabular (they include powers, logarithms, exponential functions, trigonometric functions, inverse trigonometric functions, as well as their finite combinations obtained using the operations of addition, subtraction, multiplication and division). Using table integrals you can calculate integrals of more complex functions. There are many ways to calculate indefinite integrals; The most common of these is the variable substitution or substitution method. It consists in the fact that if we want to replace in the indefinite integral (2) x to some differentiable function x = g(u), then for the integral to remain unchanged, it is necessary x replaced by gў ( u)du. In other words, the equality

(substitution 2 x = u, from where 2 dx = du).

Let us present another integration method - the method of integration by parts. It is based on the already known formula

By integrating the left and right sides, and taking into account that

This formula is called the integration by parts formula.

Example 2. You need to find . Since cos x= (sin x)ў , we can write that

From (5), assuming u = x And v= sin x, we get

And since (–cos x)ў = sin x we find that

It should be emphasized that we have limited ourselves to only a very brief introduction to a very vast subject in which numerous ingenious techniques have been accumulated.

Functions of two variables.

Due to the curve y = f(x) we considered two problems.

1) Find the angular coefficient of the tangent to the curve at a given point. This problem is solved by calculating the value of the derivative fў ( x) at the specified point.

2) Find the area under the curve above the axis segment X, bounded by vertical lines X = A And X = b. This problem is solved by calculating a definite integral.

Each of these problems has an analogue in the case of a surface z = f(x,y).

1) Find the tangent plane to the surface at a given point.

2) Find the volume under the surface above the part of the plane xy, bounded by a curve WITH, and from the side – perpendicular to the plane xy passing through the points of the boundary curve WITH (cm. rice. 22).

The following examples show how these problems are solved.

Example 4. Find the tangent plane to the surface

at point (0,0,2).

A plane is defined if two intersecting lines lying in it are given. One of these straight lines ( l 1) we get in the plane xz (at= 0), second ( l 2) – in the plane yz (x = 0) (cm. rice. 23).

First of all, if at= 0, then z = f(x,0) = 2 – 2x – 3x 2. Derivative with respect to X, denoted fў x(x,0) = –2 – 6x, at X= 0 has a value of –2. Straight l 1 given by the equations z = 2 – 2x, at= 0 – tangent to WITH 1, lines of intersection of the surface with the plane at= 0. Similarly, if X= 0, then f(0,y) = 2 – y – y 2 , and the derivative with respect to at looks like

Because fў y(0,0) = –1, curve WITH 2 – line of intersection of the surface with the plane yz– has a tangent l 2 given by the equations z = 2 – y, X= 0. The desired tangent plane contains both lines l 1 and l 2 and is written by the equation

This is the equation of a plane. In addition, we receive direct l 1 and l 2, assuming, accordingly, at= 0 and X = 0.

The fact that equation (7) really defines a tangent plane can be verified at a heuristic level by noting that this equation contains first-order terms included in equation (6), and that second-order terms can be represented in the form -. Since this expression is negative for all values X And at, except X = at= 0, surface (6) lies below plane (7) everywhere, except for the point R= (0,0,0). We can say that surface (6) is convex upward at the point R.

Example 5. Find the tangent plane to the surface z = f(x,y) = x 2 – y 2 at origin 0.

On surface at= 0 we have: z = f(x,0) = x 2 and fў x(x,0) = 2x. On WITH 1, intersection lines, z = x 2. At the point O the slope is equal to fў x(0,0) = 0. On the plane X= 0 we have: z = f(0,y) = –y 2 and fў y(0,y) = –2y. On WITH 2, intersection lines, z = –y 2. At the point O curve slope WITH 2 is equal fў y(0,0) = 0. Since the tangents to WITH 1 and WITH 2 are axes X And at, the tangent plane containing them is the plane z = 0.

However, in the neighborhood of the origin, our surface is not on the same side of the tangent plane. Indeed, a curve WITH 1 everywhere, except point 0, lies above the tangent plane, and the curve WITH 2 – respectively below it. Surface intersects tangent plane z= 0 in straight lines at = X And at = –X. Such a surface is said to have a saddle point at the origin (Fig. 24).

Partial derivatives.

In previous examples we used derivatives of f (x,y) By X and by at. Let us now consider such derivatives in a more general sense. If we have a function of two variables, for example, F(x,y) = x 2 – xy, then we can determine at each point its two “partial derivatives”, one by differentiating the function with respect to X and fixing at, the other – differentiating by at and fixing X. The first of these derivatives is denoted as fў x(x,y) or ¶ f/¶ x; second - how f f ў y. If both mixed derivatives (by X And at, By at And X) are continuous, then ¶ 2 f/¶ x¶ y= ¶ 2 f/¶ y¶ x; in our example ¶ 2 f/¶ x¶ y= ¶ 2 f/¶ y¶ x = –1.

Partial derivative fў x(x,y) indicates the rate of change of the function f at point ( x,y) in the direction of increasing X, A fў y(x,y) – rate of change of function f in the direction of increasing at. Rate of change of function f at point ( X,at) in the direction of a straight line making an angle q with positive axis direction X, is called the derivative of the function f towards; its value is a combination of two partial derivatives of the function f in the tangent plane is almost equal (at small dx And dy) true change z on the surface, but calculating the differential is usually easier.

The formula from the variable change method that we have already considered, known as the derivative of a complex function or the chain rule, in the one-dimensional case when at depends on X, A X depends on t, has the form:

For functions of two variables, a similar formula has the form:

The concepts and notations of partial differentiation are easy to generalize to higher dimensions. In particular, if the surface is specified implicitly by the equation f(x,y,z) = 0, the equation of the tangent plane to the surface can be given a more symmetrical form: the equation of the tangent plane at the point ( x(x 2 /4)], then integrated over X from 0 to 1. The final result is 3/4.

Formula (10) can also be interpreted as a so-called double integral, i.e. as the limit of the sum of the volumes of elementary “cells”. Each such cell has a base D x D y and a height equal to the height of the surface above some point of the rectangular base ( cm. rice. 26). It can be shown that both points of view on formula (10) are equivalent. Double integrals are used to find centers of gravity and numerous moments encountered in mechanics.

A more rigorous justification of the mathematical apparatus.

So far we have presented the concepts and methods of mathematical analysis on an intuitive level and have not hesitated to resort to geometric figures. It remains for us to briefly consider the more rigorous methods that emerged in the 19th and 20th centuries.

At the beginning of the 19th century, when the era of storm and pressure in the “creation of mathematical analysis” ended, questions of its justification came to the fore. In the works of Abel, Cauchy and a number of other outstanding mathematicians, the concepts of “limit”, “continuous function”, “convergent series” were precisely defined. This was necessary in order to introduce logical order into the basis of mathematical analysis in order to make it a reliable research tool. The need for a thorough justification became even more obvious after the discovery in 1872 by Weierstrass of functions that were everywhere continuous but nowhere differentiable (the graph of such functions has a kink at each point). This result had a stunning effect on mathematicians, since it clearly contradicted their geometric intuition. An even more striking example of the unreliability of geometric intuition was the continuous curve constructed by D. Peano, which completely fills a certain square, i.e. passing through all its points. These and other discoveries gave rise to the program of “arithmetization” of mathematics, i.e. making it more reliable by grounding all mathematical concepts using the concept of number. The almost puritanical abstinence from clarity in works on the foundations of mathematics had its historical justification.

According to modern canons of logical rigor, it is unacceptable to talk about the area under the curve y = f(x) and above the axis segment X, even f- a continuous function, without first defining the exact meaning of the term “area” and without establishing that the area thus defined actually exists. This problem was successfully solved in 1854 by B. Riemann, who gave a precise definition of the concept of a definite integral. Since then, the idea of ​​summation behind the concept of a definite integral has been the subject of many in-depth studies and generalizations. As a result, today it is possible to give meaning to the definite integral, even if the integrand is discontinuous everywhere. New concepts of integration, to the creation of which A. Lebesgue (1875–1941) and other mathematicians made a great contribution, increased the power and beauty of modern mathematical analysis.

It would hardly be appropriate to go into detail about all these and other concepts. We will limit ourselves only to giving strict definitions of the limit and the definite integral.

In conclusion, let us say that mathematical analysis, being an extremely valuable tool in the hands of a scientist and engineer, still attracts the attention of mathematicians today as a source of fruitful ideas. At the same time, modern development seems to indicate that mathematical analysis is increasingly being absorbed by those dominant in the 20th century. branches of mathematics such as abstract algebra and topology.

Mathematical analysis.

Workshop.

For university students in the specialty:

"State and municipal administration"

T.Z. Pavlova

Kolpashevo 2008

Chapter 1: Introduction to Analysis

1.1 Functions. General properties

1.2 Limit theory

1.3 Continuity of function

2.1 Definition of derivative

2.4 Function research

2.4.1 Full function study design

2.4.2 Function study examples

2.4.3. The largest and smallest value of a function on a segment

2.5 L'Hopital's rule

3.1 Indefinite integral

3.1.1 Definitions and properties

3.1.2 Table of integrals

3.1.3 Basic integration methods

3.2 Definite integral

3.2.2 Methods for calculating the definite integral

Chapter 4. Functions of several variables

4.1 Basic concepts

4.2 Limits and continuity of functions of several variables

4.3.3 Total differential and its application to approximate calculations

Chapter 5. Classical optimization methods

6.1 Utility function.

6.2 Lines of indifference

6.3 Budget set

Home test assignments

1.1 Functions. General properties

A numerical function is defined on the set D of real numbers if each value of the variable is associated with some well-defined real value of the variable y, where D is the domain of definition of the function.

Analytical representation of a function:

explicitly: ;

implicitly: ;

in parametric form:

different formulas in the area of ​​definition:

Properties.

Even function: . For example, the function is even, because .

Odd function: . For example, the function is odd, because .

Periodic function: , where T is the period of the function, . For example, trigonometric functions.

Monotonic function. If for any of the domain of definition the function is increasing, then it is decreasing. For example, - increasing, and - decreasing.

Limited function. If there is a number M such that . For example, functions and , because .

Example 1. Find the domain of definition of the functions.

+ 2 – 3 +

1.2 Limit theory

Definition 1. The limit of a function at is a number b if for any ( is an arbitrarily small positive number) one can find a value of the argument starting from which the inequality holds.

Designation: .

Definition 2. The limit of a function at is a number b if for any ( is an arbitrarily small positive number) there is a positive number such that for all values ​​of x satisfying the inequality the inequality is satisfied.

Designation: .

Definition 3. A function is said to be infinitesimal for or if or.

Properties.

1. The algebraic sum of a finite number of infinitesimal quantities is an infinitesimal quantity.

2. The product of an infinitesimal quantity and a bounded function (a constant, another infinitesimal quantity) is an infinitesimal quantity.

3. The quotient of dividing an infinitesimal quantity by a function whose limit is non-zero is an infinitesimal quantity.

Definition 4. A function is said to be infinitely large if .

Properties.

1. The product of an infinitely large quantity and a function whose limit is different from zero is an infinitely large quantity.

2. The sum of an infinitely large quantity and a limited function is an infinitely large quantity.

3. The quotient of dividing an infinitely large quantity by a function that has a limit is an infinitely large quantity.

Theorem.(The relationship between an infinitesimal quantity and an infinitely large quantity.) If a function is infinitesimal at (), then the function is an infinitely large quantity at (). And, conversely, if the function is infinitely large at (), then the function is an infinitesimal value at ().

Limit theorems.

1. A function cannot have more than one limit.

2. The limit of the algebraic sum of several functions is equal to the algebraic sum of the limits of these functions:

3. The limit of the product of several functions is equal to the product of the limits of these functions:

4. The limit of the degree is equal to the degree of the limit:

5. The limit of the quotient is equal to the quotient of the limits if the limit of the divisor exists:

.

6. The first wonderful limit.

Consequences:

7. Second remarkable limit:

Consequences:

Equivalent infinitesimal quantities at:

Calculation of limits.

When calculating limits, the basic theorems about limits, properties of continuous functions and rules arising from these theorems and properties are used.

Rule 1. To find the limit at a point of a function that is continuous at this point, you need to substitute its limit value into the function under the limit sign instead of the argument x.

Example 2. Find

Rule 2. If, when finding the limit of a fraction, the limit of the denominator is equal to zero, and the limit of the numerator is different from zero, then the limit of such a function is equal to .

Example 3. Find

Rule 3. If, when finding the limit of a fraction, the limit of the denominator is equal to , and the limit of the numerator is different from zero, then the limit of such a function is equal to zero.

Example 4. Find

Often, substituting the limit value of an argument results in undefined expressions of the form

.

Finding the limit of a function in these cases is called uncertainty discovery. To reveal the uncertainty, it is necessary to transform this expression before moving to the limit. Various techniques are used to reveal uncertainties.

Rule 4. The uncertainty of the type is revealed by transforming the sublimit function so that in the numerator and denominator one can isolate a factor whose limit is equal to zero, and, reducing the fraction by it, find the limit of the quotient. To do this, the numerator and denominator are either factored or multiplied by the expressions conjugate to the numerator and denominator.

Rule 5. If the sublimit expression contains trigonometric functions, then the first remarkable limit is used to resolve the uncertainty of the form.

.

Rule 6. To reveal the uncertainty of the form at , the numerator and denominator of the sublimit fraction must be divided by the highest power of the argument and then the limit of the quotient must be found.

Possible results:

1) the required limit is equal to the ratio of the coefficients of the highest powers of the argument of the numerator and denominator, if these powers are the same;

2) the limit is equal to infinity if the degree of the numerator argument is higher than the degree of the denominator argument;

3) the limit is equal to zero if the degree of the numerator argument is lower than the degree of the denominator argument.

A)

because

The powers are equal, which means that the limit is equal to the ratio of the coefficients of the higher powers, i.e. .

b)

The degree of the numerator and denominator is 1, which means the limit is

V)

The degree of the numerator is 1, the denominator is , which means the limit is 0.

Rule 7. To reveal the uncertainty of the form, the numerator and denominator of the sublimit fraction must be multiplied by the conjugate expression.

Example 10.

Rule 8. To reveal the uncertainty of the species, the second remarkable limit and its consequences are used.

It can be proven that

Example 11.

Example 12.

Example 13.

Rule 9. When revealing uncertainties whose sublimit function contains b.m.v., it is necessary to replace the limits of these b.m.v. to the limits of b.m. equivalent to them.

Example 14.

Example 15.

Rule 10. L'Hopital's rule (see 2.6).

1.3 Continuity of function

A function is continuous at a point if the limit of the function, as the argument tends to a, exists and is equal to the value of the function at this point.

Equivalent conditions:

1. ;

3.

Classification of break points:

1st kind rupture

Removable – one-sided limits exist and are equal;

Irreducible (jump) – one-sided limits are not equal;

discontinuity of the second kind: the limit of a function at a point does not exist.

Example 16. Establish the nature of the discontinuity of a function at a point or prove the continuity of a function at this point.

at the function is not defined, therefore, it is not continuous at this point. Because and correspondingly, , then is a point of removable discontinuity of the first kind.

b)

Compared to assignment (a), the function is further defined at the point so that , which means that this function is continuous at this point.

When the function is not defined;

.

Because one of the one-sided limits is infinite, then this is a discontinuity point of the second kind.

Chapter 2. Differential calculus

2.1 Definition of derivative

Definition of derivative

The derivative or of a given function is the limit of the ratio of the increment of the function to the corresponding increment of the argument, when the increment of the argument tends to zero:

Or .

The mechanical meaning of a derivative is the rate of change of a function. The geometric meaning of the derivative is the tangent of the angle of inclination of the tangent to the graph of the function:

2.2 Basic rules of differentiation

Name	Function	Derivative
Multiplying by a constant factor
Algebraic sum of two functions
Product of two functions
Quotient of two functions
Complex function

Derivatives of basic elementary functions

No.	Function name	Function and its derivative
1	constant
2	power function special cases
3	exponential function special case
4	logarithmic function special case
5	trigonometric functions
6	reverse trigonometric

b)

2.3 Higher order derivatives

Second order derivative of a function

Second order derivative of the function:

Example 18.

a) Find the second-order derivative of the function.

Solution. Let us first find the first order derivative .

From the first-order derivative, let us take the derivative again.

Example 19. Find the third-order derivative of the function.

2.4 Function research

2.4.1 Full function study plan:

Full function study plan:

1. Elementary research:

Find the domain of definition and range of values;

Find out general properties: evenness (oddness), periodicity;

Find the points of intersection with the coordinate axes;

Determine areas of constant sign.

2. Asymptotes:

Find vertical asymptotes if ;

Find oblique asymptotes: .

If any number, then – horizontal asymptotes.

3. Research using:

Find the critical points, those. points at which or does not exist;

Determine the intervals of increase, those. intervals on which the function decreases – ;

Determine extrema: points through which the sign changes from “+” to “–” are points of maximum, from “–” to “+” are points of minimum.

4. Research using:

Find points at which or does not exist;

Find areas of convexity, i.e. intervals on which and concavities – ;

Find inflection points, i.e. points when passing through which the sign changes.

1. Individual elements of the study are plotted on the graph gradually, as they are found.

2. If difficulties arise with constructing a graph of a function, then the values ​​of the function are found at some additional points.

3. The purpose of the study is to describe the nature of the behavior of the function. Therefore, not an exact graph is built, but an approximation of it, on which the found elements are clearly marked (extrema, inflection points, asymptotes, etc.).

4. It is not necessary to strictly adhere to the given plan; It is important not to miss the characteristic elements of the function's behavior.

2.4.2 Examples of function research:

1)

2) Odd function:

.

3) Asymptotes.

– vertical asymptotes, because

Oblique asymptote.

5)

– inflection point.

2) Odd function:

3) Asymptotes: There are no vertical asymptotes.

Oblique:

– oblique asymptotes

4) – the function increases.

– inflection point.

Schematic graph of this function:

2) General function

3) Asymptotes

– there are no inclined asymptotes

– horizontal asymptote at

– inflection point

Schematic graph of this function:

2) Asymptotes.

– vertical asymptote, because

– there are no inclined asymptotes

, – horizontal asymptote

Schematic graph of this function:

2) Asymptotes

– vertical asymptote at , because

– there are no inclined asymptotes

, – horizontal asymptote

3) – the function decreases on each of the intervals.

Schematic graph of this function:

To find the largest and smallest values ​​of a function on a segment, you can use the following diagram:

1. Find the derivative of the function.

2. Find the critical points of the function at which or does not exist.

3. Find the value of the function at critical points belonging to a given segment and at its ends and select the largest and smallest from them.

Example. Find the smallest and largest value of the function on a given segment.

25. in between

2) – critical points

26. in the interval.

The derivative does not exist for , but 1 does not belong to this interval. The function decreases on the interval, which means that there is no greatest value, but the smallest value is .

2.5 L'Hopital's rule

Theorem. The limit of the ratio of two infinitesimal or infinitely large functions is equal to the limit of the ratio of their derivatives (finite or infinite), if the latter exists in the indicated sense.

Those. when disclosing uncertainties of the type or you can use the formula:

.

27.

Chapter 3. Integral calculus

3.1 Indefinite integral

3.1.1 Definitions and properties

Definition 1. A function is called antiderivative for if .

Definition 2. An indefinite integral of a function f(x) is the set of all antiderivatives for this function.

Designation: , where c is an arbitrary constant.

Properties of the indefinite integral

1. Derivative of the indefinite integral:

2. Differential of the indefinite integral:

3. Indefinite integral of the differential:

4. Indefinite integral of the sum (difference) of two functions:

5. Extending the constant factor beyond the sign of the indefinite integral:

3.1.2 Table of integrals

.1.3 Basic integration methods

1. Using the properties of the indefinite integral.

Example 29.

2. Submitting the differential sign.

Example 30.

3. Variable replacement method:

a) replacement in the integral

Where - a function that is easier to integrate than the original one; - function inverse to function; - antiderivative of function.

Example 31.

b) replacement in the integral of the form:

Example 32.

Example 33.

4. Method of integration by parts:

Example 34.

Example 35.

Let us take separately the integral

Let's return to our integral:

3.2 Definite integral

3.2.1 The concept of a definite integral and its properties

Definition. Let a continuous function be given on a certain interval. Let's build a graph of it.

A figure bounded above by a curve, on the left and right by straight lines and below by a segment of the abscissa axis between points a and b is called a curvilinear trapezoid.

S – area – curvilinear trapezoid.

Divide the interval with dots and get:

Cumulative sum:

Definition. A definite integral is the limit of an integral sum.

Properties of the definite integral:

1. The constant factor can be taken out of the integral sign:

2. The integral of the algebraic sum of two functions is equal to the algebraic sum of the integrals of these functions:

3. If the integration segment is divided into parts, then the integral on the entire segment is equal to the sum of the integrals for each of the resulting parts, i.e. for any a, b, c:

4. If on the segment , then

5. The limits of integration can be swapped, and the sign of the integral changes:

6.

7. The integral at the point is equal to 0:

8.

9. (“about the mean”) Let y = f(x) be a function integrable on . Then , where , f(c) – average value of f(x) on:

10. Newton-Leibniz formula

,

where F(x) is the antiderivative of f(x).

3.2.2 Methods for calculating the definite integral.

1. Direct integration

Example 35.

A)

b)

V)

d)

2. Change of variables under the definite integral sign .

Example 36.

2. Integration by parts in a definite integral .

Example 37.

A)

b)

d)

3.2.3 Applications of the definite integral

Characteristic	Function type	Formula
	in Cartesian coordinates
curvilinear sector area	in polar coordinates
area of ​​a curved trapezoid	in parametric form
arc length	in Cartesian coordinates
arc length	in polar coordinates
arc length	in parametric form
body volume rotation	in Cartesian coordinates
volume of a body with a given transverse cross section

Example 38. Calculate the area of ​​a figure bounded by lines: And .

Solution: Let's find the intersection points of the graphs of these functions. To do this, we equate the functions and solve the equation

So, the points of intersection and .

Find the area of ​​the figure using the formula

.

In our case

Answer: Area is (square units).

4.1 Basic concepts

Definition. If each pair of mutually independent numbers from a certain set is assigned, according to some rule, one or more values ​​of the variable z, then the variable z is called a function of two variables.

Definition. The domain of definition of a function z is the set of pairs for which the function z exists.

The domain of definition of a function of two variables is a certain set of points on the Oxy coordinate plane. The z coordinate is called an applicate, and then the function itself is depicted as a surface in the space E 3 . For example:

Example 39. Find the domain of the function.

A)

The expression on the right side makes sense only when . This means that the domain of definition of this function is the set of all points lying inside and on the boundary of a circle of radius R with a center at the origin.

The domain of definition of this function is all points of the plane, except points of straight lines, i.e. coordinate axes.

Definition. Function level lines are a family of curves on the coordinate plane, described by equations of the form.

Example 40. Find function level lines .

Solution. The level lines of a given function are a family of curves on the plane, described by the equation

The last equation describes a family of circles with a center at point O 1 (1, 1) of radius . The surface of revolution (paraboloid) described by this function becomes “steeper” as it moves away from the axis, which is given by the equations x = 1, y = 1. (Fig. 4)

4.2 Limits and continuity of functions of several variables.

1. Limits.

Definition. A number A is called the limit of a function as a point tends to a point if for every arbitrarily small number there is a number such that for any point the condition is true, and the condition is also true . Write down: .

Example 41. Find limits:

those. the limit depends on , which means it does not exist.

2. Continuity.

Definition. Let the point belong to the domain of definition of the function. Then a function is called continuous at a point if

(1)

and the point tends to the point in an arbitrary manner.

If at any point condition (1) is not satisfied, then this point is called the break point of the function. This may be in the following cases:

1) The function is not defined at point .

2) There is no limit.

3) This limit exists, but it is not equal to .

Example 42. Determine whether a given function is continuous at the point if .

Got that This means that this function is continuous at the point.

the limit depends on k, i.e. it does not exist at this point, which means the function has a discontinuity at this point.

4.3 Derivatives and differentials of functions of several variables

4.3.1 First order partial derivatives

The partial derivative of a function with respect to the argument x is the ordinary derivative of a function of one variable x for a fixed value of the variable y and is denoted:

The partial derivative of a function with respect to the argument y is the ordinary derivative of a function of one variable y for a fixed value of the variable x and is denoted:

Example 43. Find partial derivatives of functions.

4.3.2 Second order partial derivatives

Second order partial derivatives are partial derivatives of first order partial derivatives. For a function of two variables of the form, four types of second-order partial derivatives are possible:

Second-order partial derivatives, in which differentiation is carried out with respect to different variables, are called mixed derivatives. The second order mixed derivatives of a twice differentiable function are equal.

Example 44. Find second-order partial derivatives.

4.3.3 Total differential and its application to approximate calculations.

Definition. The first order differential of a function of two variables is found by the formula

.

Example 45. Find the complete differential for the function.

Solution. Let's find the partial derivatives:

.

For small increments of arguments x and y, the function receives an increment approximately equal to dz, i.e. .

Formula for finding the approximate value of a function at a point if its exact value at a point is known:

Example 46. Find .

Solution. Let ,

Then we use the formula

Answer. .

Example 47. Calculate approximately .

Solution. Let's consider the function. We have

Example 48. Calculate approximately .

Solution. Consider the function . We get:

Answer. .

4.3.4 Differentiation of an implicit function

Definition. A function is called implicit if it is given by an equation that is not solvable with respect to z.

The partial derivatives of such a function are found by the formulas:

Example 49: Find the partial derivatives of the function z given by the equation .

Solution.

Definition. A function is called implicit if it is given by an equation that is not solvable with respect to y.

The derivative of such a function is found by the formula:

.

Example 50. Find derivatives of these functions.

5.1 Local extremum of a function of several variables

Definition 1. A function has a maximum at point if

Definition 2. A function has a minimum at point if for all points sufficiently close to the point and different from it.

A necessary condition for an extremum. If a function reaches an extremum at a point, then the partial derivatives of the function vanish or do not exist at this point.

The points at which partial derivatives vanish or do not exist are called critical.

A sufficient sign of an extremum. Let the function be defined in some neighborhood of the critical point and have continuous second-order partial derivatives at this point

1) has a local maximum at the point if and ;

2) has a local minimum at the point if and ;

3) does not have a local extremum at the point if ;

Scheme of research on the extremum of a function of two variables.

1. Find the partial derivatives of the functions: and.

2. Solve the system of equations and find the critical points of the function.

3. Find second-order partial derivatives, calculate their values ​​at critical points and, using a sufficient condition, draw a conclusion about the presence of extrema.

4. Find the extrema of the function.

Example 51. Find extrema of a function .

1) Let's find the partial derivatives.

2) Let's solve the system of equations

4) Let us find the second order partial derivatives and their values ​​at critical points: . At the point we get:

This means that there is no extremum at the point. At the point we get:

This means that there is a minimum at the point.

5.2 Global extremum (the largest and smallest value of the function)

The largest and smallest values ​​of a function of several variables, continuous on some closed set, are achieved either at extremum points or at the boundary of the set.

Scheme for finding the largest and smallest values.

1) Find critical points lying inside the region, calculate the value of the function at these points.

2) Investigate the function at the boundary of the region; if the border consists of several different lines, then the study must be carried out for each section separately.

3) Compare the obtained function values ​​and select the largest and smallest.

Example 52. Find the largest and smallest values ​​of a function in a rectangle.

Solution. 1) Let’s find the critical points of the function, for this we’ll find the partial derivatives: , and solve the system of equations:

We have obtained a critical point A. The resulting point lies inside the given region,

The boundary of the region is made up of four segments: i. Let's find the largest and smallest value of the function on each segment.

4) Let us compare the results obtained and find that at the points .

Chapter 6. Model of consumer choice

We will assume that there are n different goods. Then we will denote a certain set of goods by an n-dimensional vector , where is the quantity of the i-th product. The set of all sets of goods X is called a space.

The choice of an individual consumer is characterized by a preference relationship: it is believed that the consumer can say about any two sets which is more desirable, or he does not see the difference between them. The preference relation is transitive: if a set is preferable to a set, and a set is preferable to a set, then the set is preferable to a set. We will assume that consumer behavior is completely described by the axiom of the individual consumer: each individual consumer makes decisions about consumption, purchases, etc., based on his system of preferences.

6.1 Utility function

A function is defined on the set of consumer sets X , the value of which on the consumer set is equal to the individual’s consumer assessment for this set. The function is called the consumer utility function or consumer preference function. Those. Each consumer has his own utility function. But the entire set of consumers can be divided into certain classes of consumers (by age, property status, etc.) and each class can be assigned a certain, perhaps averaged, utility function.

Thus, the function is a consumer assessment or the level of satisfaction of an individual’s needs when purchasing a given set. If a set is preferable to a set for a given individual, then .

Properties of the utility function.

1.

The first partial derivatives of the utility function are called marginal utilities of products. From this property it follows that an increase in the consumption of one product while the consumption of other products remains unchanged leads to an increase in consumer evaluation. Vector is the gradient of the function, it shows the direction of greatest growth of the function. For a function, its gradient is a vector of marginal utilities of products.

2.

Those. The marginal utility of any good decreases as consumption increases.

3.

Those. The marginal utility of each product increases as the quantity of the other product increases.

Some types of utility functions.

1) Neoclassical: .

2) Quadratic: , where the matrix is ​​negative definite and For .

3) Logarithmic function: .

6.2 Lines of indifference

In applied problems and models of consumer choice, a special case of a set of two goods is often used, i.e. when the utility function depends on two variables. The line of indifference is a line connecting consumer sets that have the same level of satisfaction of the individual's needs. In essence, indifference lines are function level lines. Equations of indifference lines: .

Basic properties of indifference lines.

1. Lines of indifference corresponding to different levels of need satisfaction do not touch or intersect.

2. Lines of indifference decrease.

3. Indifference lines are convex downward.

Property 2 implies an important approximate equality.

This ratio shows how much an individual should increase (decrease) the consumption of the second product when decreasing (increasing) the consumption of the first product by one unit without changing the level of satisfaction of his needs. The ratio is called the rate of replacement of the first product by the second, and the value is called the marginal rate of replacement of the first product by the second.

Example 53. If the marginal utility of the first good is 6, and the second is 2, then if the consumption of the first good is reduced by one unit, the consumption of the second good must be increased by 3 units at the same level of satisfaction of needs.

6.3 Budget set

Let – vector of prices for a set of n products; I is the individual’s income, which he is willing to spend on purchasing a set of products. The set of sets of goods costing no more than I at given prices is called the budget set B. Moreover, the set of sets costing I is called the boundary G of the budget set B. Thus. the set B is bounded by the boundary G and natural restrictions.

The budget set is described by a system of inequalities:

For the case of a set of two goods, the budget set B (Fig. 1) is a triangle in the coordinate system, limited by the coordinate axes and the straight line.

6.4 Theory of consumer demand

In consumption theory, it is believed that the consumer always strives to maximize his utility and the only limitation for him is the limited income I that he can spend on purchasing a set of goods. In general, the problem of consumer choice (the problem of rational consumer behavior in the market) is formulated as follows: find the consumer set , which maximizes its utility function under a given budget constraint. Mathematical model of this problem:

In the case of a set of two products:

Geometrically, the solution to this problem is the point of tangency between the boundary of the budget set G and the indifference line.

The solution to this problem comes down to solving the system of equations:

(1)

The solution to this system is the solution to the consumer choice problem.

The solution to the consumer choice problem is called the demand point. This point of demand depends on prices and income I. I.e. the demand point is a function of demand. In turn, the demand function is a set of n functions, each of which depends on an argument:

These functions are called demand functions for the corresponding goods.

Example 54. For a set of two goods on the market, known prices for them and income I, find the demand functions if the utility function has the form .

Solution. Let's differentiate the utility function:

.

Let us substitute the resulting expressions into (1) and obtain a system of equations:

In this case, the expense for each product will be half of the consumer’s income, and the quantity of the product purchased is equal to the amount spent on it divided by the price of the product.

Example 55. Let the utility function for the first good, second,

price of the first product, price of the second. Income . How much of a good should a consumer purchase to maximize utility?

Solution. Let's find the derivatives of the utility functions, substitute them into system (1) and solve it:

This set of goods is optimal for the consumer from the point of view of maximizing utility.

The test must be completed in accordance with the option selected by the last digit of the grade book number in a separate notebook. Each problem must contain a condition, a detailed solution and a conclusion.

1. Introduction to mathematical analysis

Task 1. Find the domain of definition of the function.

5.

Task 2. Find the limits of the functions.

.

Task 3. Find the discontinuity points of the function and determine their type.

1. 2. 3.

Chapter 2. Differential calculus of a function of one variable

Task 4. Find derivatives of these functions.

1. a); b) c) y = ;

d) y = x 6 + + + 5; e) y = x tan x + ln sin x + e 3x ;

e) y = 2 x - arcsin x.

2. a) ; b) y = ; c) y = ; d) y = x 2 –+ 3; e) y = e cos; e) y = .

3. a) y = lnx; b) y =; c) y = ln;

4. a) y = ; b) y = (e 5 x – 1) 6 ; c) y = ; d) y = ; e) y = x 8 ++ + 5; e) y = 3 x - arcsin x.

5. a) y = 2x 3 - + e x ; b) y = ; c) y = ;

d) y = ; e) y = 2 cos; e) y = .

6. a) y = lnx; b) y =; c) y = ln;

d) y = ; e) y = x 7 + + 1; e) y = 2.

7. a) ; b) y = ; c)y = ; d)y = x 2 + xsinx + ; e) y = e cos; e) y = .

8. a) y = ; b) y = (3 x – 4) 6 ; c) y = sintg;

d) y = 3x 4 – – 9+ 9; e) y = ;

e)y = x 2 + arcsin x - x.

9. a); b) ; c) y = ; d) y = 5 sin 3 x ; e) y = x 3 – – 6+ 3; e) y = 4x 4 + ln.

10. a) b) y = ; c) y = (3 x – 4) 6 ; d) y = ; e)y = x 2 - x; e) y = e sin 3 x + 2.

Task 5. Explore the function and build its graph.

1. a) b) c) .

2. a) b) V) .

3. a) b) V) .

4. b) V)

5. a) b) V) .

6. a) b) V) .

7. a) b) c) .

8. a) b) c) .

9. a) b) c) .

10. a) b) V) .

Task 6. Find the largest and smallest value of the function on a given segment.

1. .

3. .

6. .

8. .

9. .

10. .

Chapter 3. Integral calculus

Problem 7. Find indefinite integrals.

1. a) b);

2. a) ;b) c) d) .

4. G)

5. a) ; b); V) ; G).

6. a) ; b); V); G)

7. a) ; b) ; V) ; G)

8. a) ; b); V) ; G) .

9. a) ; b) c); G).

10. a) b) V) ; G) .

Problem 8. Calculate definite integrals.

1.

2.

3.

4.

5.

6.

7. .

8.

9.

10.

Problem 9. Find improper integrals or prove that they diverge.

1. .

2. .

3. .

4. .

5. .

6. .

7. .

8. .

9. .

10. .

Problem 10. Find the area of ​​the region bounded by the curves

1. .2. .

5. 6.

7. , .8..

10. , .

Chapter 4. Differential calculus of functions of several variables.

Task 11. Find the domain of definition of the function (show in the drawing).

Problem 12. Investigate the continuity of the function at

Problem 13. Find the derivative of an implicitly given function.

Problem 14. Calculate approximately

1. a) ;b) ; V)

2. a) ; b) ; V) .

3. a) ; b) ; V) .

4. a) ; b) ; V) .

5. a); b) ; V) .

6. a); b) ; V) .

7. a); b) ; V) .

8. a) ;b) ; V)

9. a) ; b) ; V) .

10. a) ;b) ; V)

Problem 15. Investigate the function for extrema.

7. .

8. .

9. .

10. .

Problem 16. Find the largest and smallest value of the function in a given closed region.

1. in a rectangle

2.

3. in a rectangle

4. in the area limited by a parabola

And the x-axis.

5. squared

6. in a triangle limited by the coordinate axes and the straight line

7. in a triangle limited by the coordinate axes and the straight line

8. in a triangle bounded by the coordinate axes and the straight line

9. in the area limited by a parabola

And the x-axis.

10. in the area limited by a parabola

And the x-axis.

Main

1. M.S. Krass, B.P. Chuprynov. Fundamentals of mathematics and its application in economic education: Textbook. – 4th ed., Spanish. – M.: Delo, 2003.

2. M.S. Krass, B.P. Chuprynov. Mathematics for economic specialties: Textbook. – 4th ed., Spanish. – M.: Delo, 2003.

3. M.S. Krass, B.P. Chuprynov. Mathematics for economic bachelor's degree. Textbook. – 4th ed., Spanish. – M.: Delo, 2005.

4. Higher mathematics for economists. Textbook for universities / N.Sh. Kremer, B.A. Putko, I.M. Trishin, M.N. Friedman; Ed. prof. N.Sh. Kremer, - 2nd ed., revised. and additional – M: UNITY, 2003.

5. Kremer N.Sh., Putko B.A., Trishin I.M., Fridman M.N.. Higher mathematics for economic specialties. Textbook and Workshop (parts I and II) / Ed. prof. N.Sh. Kremer, - 2nd ed., revised. and additional – M: Higher Education, 2007. – 893 p. – (Fundamentals of Sciences)

6. Danko P.E., Popov A.G., Kozhevnikova T.Ya. Higher mathematics in exercises and problems. M. Higher School. 1999.

Additional

1. I.I. Bavrin, V.L. Sailors. Higher mathematics. "Humanitarian Publishing Center Vlados", 2002.

2. I.A. Zaitsev. Higher mathematics. "Higher School", 1998.

3. A.S. Solodovnikov, V.A. Babaytsev, A.V. Brailov, I.G. Shandra. Mathematics in economics / in two parts/. M. Finance and Statistics. 1999.