MATHEMATICAL ANALYSIS
branch of mathematics giving methods quantitative research different 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). For problems of mathematical analysis, it is characteristic that their solution is associated with the concept of a limit. Mathematical analysis was initiated in 1665 by I. Newton and (about 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 lively, we will resort to the language of graphs. Therefore, the reader may find it useful to look at the article
ANALYTIC GEOMETRY ,
before reading this article.
DIFFERENTIAL CALCULUS
Tangents. In fig. 1 shows a fragment of the curve y = 2x - x2, enclosed between x = -1 and x = 3. Sufficiently small segments of this curve look straight. In other words, if P is an arbitrary point of this curve, then there is some straight line passing through this point and being an approximation of the curve in a small neighborhood of the point P, and the smaller the neighborhood, the better the approximation. Such a straight line is called the tangent line to the curve at the point P. The main problem of differential calculus is to construct general method that allows you to find the direction of the tangent at any point on the curve where the tangent exists. It is not difficult to imagine a curve with a sharp break (Fig. 2). If P is the vertex of such a kink, then it is possible to construct an approximating line PT1 - to the right of point P and another approximating line PT2 - to the left of point P. But there is no single straight line passing through point P, which equally well approached the curve in the vicinity of point P both on the right and on the left, hence the tangent at the point P does not exist.
In fig. 1 tangent OT is drawn through the origin of coordinates O = (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 are the coordinates of an arbitrary point on OT, then, moving away from O at a distance of x units to the right, we move away from O by 2y units up. Therefore, y / x = 2, or y = 2x. This is the equation of the tangent OT to the curve y = 2x - x2 at the point O. It is now necessary to explain why the line OT was chosen from the set of lines passing through the point O. What is the difference between a straight line with a slope of 2 and other straight lines? There is one simple answer, and we find it difficult to resist the temptation to bring it using the analogy with the tangent to the circle: the tangent OT has only one common point with the curve, while any other non-vertical line passing through the point O intersects the curve twice. This can be verified as follows. Since the expression y = 2x - x2 can be obtained by subtracting x2 from y = 2x (equations of the straight line OT), the values of y for the graph turn out to be less than the knowledge of y for the line at all points, except for the point x = 0. Therefore, the graph is everywhere, except for the point Oh, located below OT, and this line and the graph have only one point in common. In addition, if y = mx is the equation of some other straight line passing through the point O, then there will certainly be two intersection points. Indeed, mx = 2x - x2 not only for x = 0, but also for x = 2 - m. And only for m = 2, both points of intersection coincide. In fig. 3 shows the case when m is less than 2, so the second intersection point appears to the right of O.
The fact that OT is the only non-vertical straight line passing through point O and having only one common point with the graph is not its most important property. Indeed, if we turn to other graphs, it will soon become clear that the property of the tangent that we have noted in the general case is not fulfilled. For example, from Fig. 4 it can be seen that near the point (1,1) the graph of the curve y = x3 is well approximated by the straight line PT, which, however, has more than one common point with it. Nevertheless, we would like to consider PT as tangent to this graph at point P. Therefore, we need to find some other way to highlight the tangent than the one that served us so well in the first example.
Suppose that a straight line (called a secant) is drawn through the point O and an arbitrary point Q = (h, k) on the graph of the curve y = 2x - x2 (Fig. 5). Substituting the values x = h and y = k into the equation of the curve, we obtain that k = 2h - h2, therefore, the slope of the secant is
For very small h, m is close to 2. Moreover, by choosing h sufficiently close to 0, we can make m arbitrarily close to 2. We can say that m "tends to the limit" of 2 as h tends to zero, or that the limit of m is 2 as h tends to zero. This is symbolically written as follows:
Then the tangent to the graph at point O is defined as a straight line passing through point O, with a slope equal to this limit. This definition of a tangent is generally applicable. Let us show the advantages of this approach using one more example: find the slope of the tangent to the graph of the curve y = 2x - x2 at an arbitrary point P = (x, y), not limiting ourselves to the simplest case when P = (0,0). Let Q = (x + h, y + k) be the second point on the graph, located at a distance h to the right of P (Fig. 6). It is required to find the slope k / h of the secant PQ. Point Q is at a distance
over the x-axis. Expanding the brackets, we find:
Subtracting y = 2x - x2 from this equation, we find the vertical distance from point P to point Q:
Therefore, the slope m of the secant PQ is
Now, as h tends to zero, m tends to 2 - 2x; we will take the last value as the slope of the tangent PT. (The same result will be obtained if h takes negative values, which corresponds to the choice of point Q to the left of P.) Note that for x = 0 the result obtained coincides with the previous one. The expression 2 - 2x is called the derivative of 2x - x2. In the old days, the derivative was also called "differential ratio" and "differential coefficient". If the expression 2x - x2 denotes 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 the value x corresponding to this point in f "(x). Thus, the slope f" (0) = 2 at x = 0, f "(0) = 0 at x = 1 and f" (2) = -2 at x = 2. The derivative is also denoted by y ", dy / dx, Dхy and Dу. The fact that the curve y = 2x - x2 near a given point is practically indistinguishable from its tangent at this point, allows us to speak of the slope of the tangent as a "slope of the curve" at the point of tangency. Thus, we can assert that the slope of the curve we are considering has at the point (0 , 0) slope 2. We can also say that at x = 0 the rate of change of y relative to x is 2. At point (2,0), the slope of the tangent (and the curve) is -2. (The minus sign means that with increasing x the variable y decreases.) At the point (1,1) the tangent is horizontal. We say that the curve y = 2x - x2 has a stationary value at this point e.
Highs and Lows. We have just shown that the curve f (x) = 2x - x2 is stationary at the point (1,1). Since f "(x) = 2 - 2x = 2 (1 - x), it is clear that for x less than 1, f" (x) is positive, and therefore y increases; for x greater than 1, f "(x) is negative, and therefore y decreases. Thus, in the vicinity of the point (1,1), denoted in Fig. 6 by the letter M, the value of y increases to the point M, is stationary at the point M and decreases after point M. Such a point is called a "maximum" because the value of y at this point exceeds any of its values in a sufficiently small vicinity of it. Similarly, a "minimum" is defined as a point in the vicinity of which all values of y exceed the value of y at this very point . It may also happen that although the derivative of f (x) vanishes at some point, its sign does not change in the vicinity of this point. Such a point, which is neither a maximum nor a minimum, is called an inflection point. curve point
The derivative of this function is
And vanishes at x = 0, x = 1 and x = -1; those. at points (0,0), (1, -2/15) and (-1, 2/15). If x is slightly less than -1, then f "(x) is negative; if x is slightly greater than -1, then f" (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 f "(x) is negative both before the point (0,0) and after it. Therefore, (0,0) is an inflection point. The conducted study of the shape of the curve, as well as the fact that the curve intersects the x-axis at f (x) = 0 (i.e. when x = 0 or
In general, if we exclude unusual cases (curves containing straight line 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 P. (See Fig. 8, where the tangent has a positive slope.) 1 ) On both sides of the point P, the curve lies above the tangent (Fig. 8, a). In this case, the curve at the point P is said to be convex downward or concave.
2) On both sides of the point P, 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 P and below - on the other. In this case, P is the inflection point. By comparing the values of f "(x) on both sides of P with its value at point P, one can determine which of these four cases has to be dealt with in a particular problem.
Applications. All of the above finds important applications in various fields. For example, if the body is thrown vertically upward with an initial speed of 200 feet per second, then the height s at which they will be in t seconds compared to the starting point will be
Acting in the same way as in the examples we have considered, we find
this quantity vanishes at
, then becomes stationary, and then decreases. This is general description movements of the body thrown up. From it we know when the body reaches its highest point. Next, substituting t = 25/4 into f (t), we get 625 feet, the maximum lift. In this problem, f "(t) has a physical meaning. This derivative shows the speed with which the body moves at time t. Now consider another type of application (Fig. 9). It is required to make a box with a square bottom from a piece of cardboard with an area of 75 cm2. How big should this box be to have the maximum volume? If x is the side of the base of the box and h is its height, then the volume of the box is V = x2h, and the surface area is 75 = x2 + 4xh. Transforming the equation, we get:> " >
, then becomes stationary, and then decreases. This is a general description of the movement of a body thrown upwards. From it we know when the body reaches its highest point. Next, substituting t = 25/4 into f (t), we get 625 feet, the maximum lift. In this problem, f "(t) has a physical meaning. This derivative shows the speed with which the body moves at time t. Now consider another type of application (Fig. 9). It is required to make a box with a square bottom from a piece of cardboard with an area of 75 cm2. How large should this box be to have the maximum volume? If x is the side of the base of the box and h is its height, then the volume of the box is V = x2h, and the surface area is 75 = x2 + 4xh. Transforming the equation, we get: ">
with. The derivative f "(x) is positive up to the value
s and is negative after this time. Therefore, s increases to
, then becomes stationary, and then decreases. This is a general description of the movement of a body thrown upwards. From it we know when the body reaches its highest point. Next, substituting t = 25/4 into f (t), we get 625 feet, the maximum lift. In this problem, f "(t) has a physical meaning. This derivative shows the speed with which the body moves at time t. Now consider another type of application (Fig. 9). It is required to make a box with a square bottom from a piece of cardboard with an area of 75 cm2. How large should this box be to have the maximum volume? If x is the side of the base of the box and h is its height, then the volume of the box is V = x2h, and the surface area is 75 = x2 + 4xh. Transforming the equation, we get:
where
The derivative of V turns out to be
and vanishes at x = 5. Then
And V = 125/2. The graph of the function V = (75x - x3) / 4 is shown in Fig. 10 (negative x values are omitted as they have 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 the constant is, of course, zero.) It is not difficult to deduce the general rule:
where n is any integer or fraction. For example,
(This example shows how useful fractional exponents are.) Here are some important formulas:
There are also the following rules: 1) if each of 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 the two functions has the form
4) the derivative of a function multiplied by a constant is equal to a constant multiplied by the derivative of this function, i.e.
It often happens that the values of a function have to be calculated in stages. For example, to calculate sin x2, we need to first find u = x2, and then calculate the sine of u. We find the derivative of such complex functions using the so-called "chain rule":
In our example, f (u) = sin u, f "(u) = cos u, therefore,
where
These and other similar rules allow one to immediately write out the derivatives of many functions.
Linear approximations. The fact that, knowing the derivative, we can in many cases replace the graph of a function near some point of its tangent at this point is of great importance, since straight lines are easier to work with. This idea finds direct application in the calculation of approximate values of functions. For example, it is quite difficult to compute the value
Max-width = "": = "" height: = "" auto = "" width: = "">
tangent without making any serious mistake. The slope of such a tangent is equal to the value of the derivative (x1 / 3) "= (1/3) x -2/3 at x = 1, i.e. 1/3. Since the point (1,1) lies on the curve and the angular the coefficient of the tangent to the curve at this point is 1/3, the equation of the tangent has the form> ">
tangent without making any serious mistake. The slope of such a tangent is equal to the value of the derivative (x1 / 3) "= (1/3) x -2/3 at x = 1, i.e. 1/3. Since the point (1,1) lies on the curve and the angular the coefficient of the tangent to the curve at this point is 1/3, the equation of the tangent is ">
at x = 1.033. But you can take advantage of the fact that the number 1.033 is close to 1 and that
... Near x = 1 we can replace the graph with the curve
tangent without making any serious mistake. The slope of such a tangent is equal to the value of the derivative (x1 / 3) "= (1/3) x -2/3 at x = 1, i.e. 1/3. Since the point (1,1) lies on the curve and the angular the coefficient of the tangent to the curve at this point is 1/3, the equation of the tangent is
or
On this straight line at x = 1.033
The y-value obtained should be very close to the true y-value; and, indeed, it is only 0.00012 more than the true one. In mathematical analysis, methods have been developed to improve the accuracy of this kind of linear approximation. These methods ensure the reliability of our approximate calculations. The procedure just described suggests a useful notation. Let P be the point corresponding to the variable x on the graph of the function f, and let the function f (x) be differentiable. Replace the graph of the curve near the point P tangent to it, drawn at this point. If x is changed by h, then the ordinate of the tangent will change by h * f "(x). If h is very small, then the latter value is a good approximation to the true change in the y ordinate of the graph. If instead of h we write the symbol dx (this is not a product !), and we denote the change in the ordinate y by dy, then we get dy = f "(x) dx, or dy / dx = f" (x) (see Fig. 11). Therefore, instead of Dy or f "(x) to denote the derivative often uses the dy / dx symbol. 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 y is meant to depend on u, and u, in turn, depends on x. The quantity dy is called the differential y; it actually depends on two variables, namely x and the increment dx. When the dx increment is very small, the dy value is close to the corresponding change in y. But there is no need to assume that the dx increment is small. We denote the derivative of the function y = f (x) by f "(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 by f" (x) or d 2y / dx2. For example, if f (x) = x3 - 3x2, then f "(x) = 3x2 - 6x and f" (x) = 6x - 6. Similar notation is used for higher order derivatives. However, to avoid a large number strokes (equal to the order of the derivative), the fourth derivative (for example) can be written as f (4) (x), and the derivative of the nth 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 the function has a second derivative, then the change in the value y corresponding to the increment dx of the variable x can be approximately calculated by the formula
This approximation, as a rule, is better than the one that gives the differential f "(x) dx. It corresponds to replacing a part of the curve with a parabola instead of a straight line. If the function f (x) has derivatives of higher orders, then
The remainder is
where x is some number between x and x + dx. The above result is called the remainder Taylor formula. If f (x) has derivatives of all orders, then usually Rn (r) 0 for n (r) Ґ.
INTEGRAL CALCULUS
Squares. When studying the areas of curvilinear plane figures, new aspects of mathematical analysis are revealed. Even the ancient Greeks tried to solve such problems, for whom the determination, for example, of 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). With the help of very complex reasoning, Archimedes proved that the area of a parabolic segment is 2/3 of the area of the described 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 the problem of calculating the areas of curvilinear figures using a method that can hardly be called logical, 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 used the newly born algebra, the stage for Newton's appearance was completely prepared. Wallis divided the figure, the area of which was required to be calculated, into very narrow strips, each of which was 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 tends when the number of stripes tends to infinity. In fig. 13 shows rectangles corresponding to some striping of the area under the curve y = x2.
The main theorem. The great discovery of Newton and Leibniz made it possible to exclude the laborious process of going over to the limit of the sum of areas. This was done thanks to a new look at the concept of square. The point is that we have to imagine the area under the curve as generated by the ordinate moving from left to right and ask how fast the area swept out by the ordinates changes. We will get the key to the answer to 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 the linear function y = 1 + x, since in this case the area can be calculated using elementary geometry. Let A (x) be the part of the plane enclosed between the straight line y = 1 + x and the segment OQ (Fig. 14). As QP moves to the right, the area A (x) increases. How fast? 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,
The rate of change of the area A (x) is determined by its derivative
We see that A "(x) coincides with the ordinate at point P. Is this accidental? Let's try to check on the parabola shown in Fig. 15. The area A (x) under the parabola y = x2 in the interval from 0 to x is equal to A ( x) = (1/3) (x) (x2) = x3 / 3. The rate of change of this area is determined by the expression
Which exactly coincides with the ordinate of the moving point P. If we assume that this rule is satisfied in the general case so 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 relation A "(x) = f (x) expresses a fundamental theorem that could be formulated as follows: the derivative, or the rate of change of the area as a function of x, is equal to the value of the function f (x) at the point x. For example , to find the area under the graph of the function y = x3 from 0 to x (Fig. 16), we put
A possible answer reads:
since the derivative of x4 / 4 is indeed equal to x3. In addition, A (x) is 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). Let us show that this statement is plausible using the following heuristic (non-rigorous) reasoning. Suppose there is some second solution B (x). If A (x) and B (x) "start" simultaneously from zero value at x = 0 and change at the same rate all the time, then their values cannot become different at any x. They must be the same everywhere; therefore, there is only one solution. How can one justify the relation A "(x) = f (x) in the general case? This question can be answered only by studying the rate of change of the area as a function of x in the general case. Let m be the smallest value of the function f (x) in the interval from x to (x + h), and M - highest value this function in the same interval. Then the increment of the area during the transition from x to (x + h) must be enclosed between the areas of two rectangles (Fig. 17). The bases of both rectangles are equal to h. The smaller rectangle has a height m and an area mh, the larger one M and Mh, respectively. The plot of the area versus x (Fig. 18) shows that when the abscissa changes by h, the value of the ordinate (ie the area) increases by the value between mh and Mh. The slope of the secant in this graph is between m and M. What happens when h goes to zero? If the graph of the function y = f (x) is continuous (i.e., does not contain discontinuities), then both M and m tend to f (x). Therefore, the slope A "(x) of the plot of area as a function of x is equal to f (x). This is exactly the conclusion that was required to come.
Leibniz proposed for the area under the curve y = f (x) from 0 to a the notation
In a rigorous approach, this so-called definite integral must be defined as the limit of certain sums in the manner of Wallis. Taking into account the above result, it is clear that this integral is calculated provided that we can find a function A (x) that vanishes at x = 0 and has a derivative A "(x) equal to f (x). Finding such functions are usually called integration, although it would be more appropriate to call this operation antidifferentiation, meaning that it is in some sense inverse to differentiation. In the case of a polynomial, integration is simple. For example, if
That
which is easy to verify by differentiating A (x). To calculate the area of A1 under the curve y = 1 + x + x2 / 2, enclosed between ordinates 0 and 1, we simply write
And, substituting x = 1, we get A1 = 1 + 1/2 + 1/6 = 5/3. The area A (x) from 0 to 2 is A2 = 2 + 4/2 + 8/6 = 16/3. As can be seen from Fig. 19, the area between ordinates 1 and 2 is A2 - A1 = 11/3. It is usually written as a definite integral
Volumes. Similar reasoning makes it surprisingly simple to calculate the volumes of bodies of revolution. Let us demonstrate this by the example of calculating the volume of a ball, another classical problem that the ancient Greeks, using the methods known to them, managed to solve with great difficulty. Let us rotate a part of the plane, enclosed within a quarter of a circle of radius r, by an angle of 360 ° around the x-axis. As a result, we will receive a hemisphere (Fig. 20), the volume of which will be denoted by V (x). It is required to determine the rate at which V (x) increases with increasing x. Passing from x to x + h, it is easy to verify that the volume increment is less than the volume p (r2 - x2) h of a circular cylinder of radius
Max-width = "": = "" height: = "" auto = "" width: = "">
and height h. Therefore, on the graph of the function V (x), the slope of the secant is between p (r2 - x2) and p []. As h tends to zero, the slope tends to ">
and height h, and is greater than the volume p [] h of a cylinder of radius
and height h. Therefore, on the graph of the function V (x), the slope of the secant is between p (r2 - x2) and p []. As h tends to zero, the slope tends to
Hence,
For x = r we get
For the volume of the hemisphere, and therefore 4pr3 / 3 for the volume of the entire ball. A similar method allows you to find the lengths of curves and areas of curved surfaces. For example, if a (x) is the length of the PR arc in Fig. 21, then our task is to calculate a "(x). Let us use at the heuristic level a technique that allows us not to resort to the usual passage to the limit, which is necessary in a rigorous proof of the result. Suppose that the rate of change of the function a (x) at the point P is the same , what it would be if the curve was replaced by its tangent PT at the point P. But from Fig. 21 it is directly seen that with a step h to the right or left of the point x along PT the value of a (x) changes to
Therefore, the rate of change of the function a (x) is
To find the function a (x) itself, it is only necessary to integrate the expression on the right-hand side of the equality. It turns out that for most functions, the integration is quite difficult. Therefore, the development of methods of integral calculus is a large part of mathematical analysis.
Antiderivatives. Each function whose derivative is equal to a given function f (x) is called the antiderivative (or primitive) for f (x). For example, x3 / 3 is the antiderivative for the function x2, since (x3 / 3) "= x2. Of course, x3 / 3 is not the only antiderivative of the function x2, since x3 / 3 + C is also the derivative for x2 for any constant C . However, in what follows we agree to omit such additive constants.
where n is a positive integer, since (xn + 1 / (n + 1)) "= xn. Relation (1) is fulfilled in an even more general sense if n is replaced by any rational number k, except -1. An arbitrary antiderivative function for a given function f (x) it is customary to call the indefinite integral of f (x) and denote it as
For example, since (sin x) "= cos x, the following formula is valid
From formula (1) it follows that
for n not equal to -1. Since (lnx) "= x-1, then
.
In many cases, when there is a formula for an indefinite integral of a given function, it can be found in numerous widely published tables of indefinite integrals. Integrals of elementary functions are tabular (these include degrees, logarithms, exponential function, trigonometric functions, inverse trigonometric functions, as well as their finite combinations obtained using the operations of addition, subtraction, multiplication and division). Using tabular 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 x in the indefinite integral (2) by some differentiable function x = g (u), then, so that the integral does not change, x must be replaced by g "(u) du. In other words, it is true equality
Example 1.
(substitution 2x = u, whence 2dx = du). Here is another integration method - the method of integration by parts. It is based on the already known formula
It can be written like this:
By integrating the left and right sides, and taking into account that
get
This formula is called the formula for integration by parts.
Example 2. It is required to find
... Since cos x = (sin x) ", we can write that
From (5), setting u = x and v = sin x, we obtain
And since (-cos x) "= sin x we find that
and
Example 3.
It should be emphasized that we have limited ourselves to only very brief introduction into a very extensive subject in which numerous witty devices have been accumulated.
Functions of two variables. In connection with the curve y = f (x), we considered two problems. 1) Find the slope of the tangent to the curve at a given point. This problem is solved by calculating the value of the derivative f "(x) at the specified point. 2) Find the area under the curve above the segment of the x axis, bounded by the vertical lines x = a and x = b. This problem is solved by calculating the definite integral
The following examples show how these tasks are accomplished.
Example 4. Find the tangent plane to a surface
At the point (0,0,2). A plane is defined if two intersecting lines lying in it are given. We obtain one of these straight lines (l1) in the xz plane (y = 0), the second (l2) - in the yz plane (x = 0) (see Fig. 23).
First of all, if y = 0, then z = f (x, 0) = 2 - 2x - 3x2. The derivative with respect to x, denoted f "x (x, 0) = -2 - 6x, at x = 0 has a value of -2. The straight line l1, given by the equations z = 2 - 2x, y = 0 is the tangent to C1, the line of intersection of the surface with the plane y = 0. Similarly, if x = 0, then f (0, y) = 2 - y - y2, and the derivative with respect to y has the form
Since f "y (0,0) = -1, the curve C2 - the line of intersection of the surface with the plane yz - has a tangent l2, given by the equations z = 2 - y, x = 0. The desired tangent plane contains both straight lines l1 and l2 and is written by the equation
This is the equation of the plane. In addition, we obtain the straight lines l1 and l2, setting, respectively, y = 0 and x = 0. The fact that equation (7) does indeed define the tangent plane can be verified at the heuristic level if we note that this equation contains the terms of the first order included in equation (6), and that the terms of the second order can be represented in the form - []. Since this expression is negative for all values of x and y, except for x = y = 0, the surface (6) lies everywhere below the plane (7), except for the point P = (0,0,0). We can say that surface (6) is convex upward at point P.
Example 5. Find the tangent plane to the surface z = f (x, y) = x2 - y2 at the origin of coordinates 0. On the plane y = 0 we have: z = f (x, 0) = x2 and f "x (x, 0) = 2x . On C1, the intersection line, z = x2. At point O, the slope is f "x (0,0) = 0. On the x = 0 plane, we have: z = f (0, y) = -y2 and f" y (0, y) = -2y. On C2, the intersection line, z = -y2. At point O, the slope of the curve C2 is f "y (0,0) = 0. Since the tangents to C1 and C2 are the x-axes and y, the tangent plane containing them is the plane z = 0. However, in the vicinity of the origin, our surface is not on one side of the tangent plane. Indeed, the curve C1 everywhere, with the exception of point 0, lies above the tangent plane, and the curve C2, respectively, below it. The surface intersects the tangent plane z = 0 along the straight lines y = x and y = -x. Such a surface is said to have a saddle point at the origin (fig. 24).
Partial derivatives. In the previous examples, we used the x and y derivatives of f (x, y). Let us now consider such derivatives more generally. If we have a function of two variables, for example, F (x, y) = x2 - xy, then we can determine at each point two of its "partial derivatives", one by differentiating the function with respect to x and fixing y, the other by differentiating with respect to y and fixing x. The first of these derivatives is denoted as f "x (x, y) or df / dx; the second as f" y (x, y) or df / dy. If f (x, y) = x2 - xy, then df / dx = 2x - y and df / dy = -x. Note that partial derivatives of any function are, generally speaking, new functions. In practice, these functions, in turn, are differentiable. Partial derivatives of f "x with respect to x and y are usually denoted by and or d2f / dx2 and d2f / dxdy; similar designations are used for partial derivatives of f" y. If both mixed derivatives (with respect to x and y, and with respect to y and x) are continuous, then d2f / dxdy = d2f / dydx; in our example d2f / dxdy = d2f / dydx = -1. The partial derivative f "x (x, y) indicates the rate of change of the function f at the point (x, y) in the direction of increasing x, and f" y (x, y) is the rate of change of the function f in the direction of increasing y. The rate of change of the function f at the point (x, y) in the direction of the straight line making an angle q with the positive direction of the x-axis is called the derivative of the function f in the direction; its value is a combination of two partial derivatives of the function f - with respect to x and with respect to y, and is equal to
As we have already seen in particular cases, the tangent plane to the surface z = f (x, y) at the point (x0, y0) has the equation
If we denote x - x0 by dx, and y - y0 by dy, then the equation of the tangent plane means that the change in dz = z - z0 in the tangent plane, when x changes by dx and y - by dy, is equal to dz = f "x (x0, y0) dx + f "y (x0, y0) dy. This quantity is called the differential of the function f. If f has continuous partial derivatives, then the change in dz in the tangent plane is almost equal (for small dx and dy) to the true change in z on the surface, but calculating the differential is usually easier. The formula we have already considered from the method of variable change, known as the derivative of a complex function or the chain rule, in the one-dimensional case, when y depends on x, and x depends on t, has the form:
For functions of two variables, a similar formula is:
It is easy to generalize the concepts and notation of partial differentiation to higher dimensions. In particular, if the surface is implicitly given by the equation f (x, y, z) = 0, the equation of the tangent plane to the surface can be given a more symmetric form: the equation of the tangent plane at the point (x0, y0, z0) has the form
If a surface f (x, y, z) = 0 is given and we want to know what is happening on the surface, then usually any two of the three variables can be considered independent, and the third variable can be considered as dependent on them. Sometimes to denote partial derivatives in this case, the symbol (dz / dx) y is used to emphasize that differentiation is made with respect to x, and y is considered an independent variable. We have:
this formula emphasizes that we cannot give an independent meaning to the symbols dx, dy, dz or regard dz / dx as the ratio of dz to dx. Let us now turn to the example of the second problem, i.e. calculating volumes.
Example 6. Find the volume of a body enclosed between a surface
and above the unit square, see fig. 25.
Let V (x) be a volume bounded by a surface and five planes, namely z = 0, y = 0, y = 1, x = 0 and a PQRS plane perpendicular to the x axis and intersecting this axis at a distance x from the origin. It is easy to see that the derivative V "(x) is equal to A (x), the cross-sectional area PQRS. Thus,
But A (x) is the area under the curve
Hence,
Where integration is carried out over y, and x is considered as a constant. Substituting (9) into (8), we write V in the form of the iterated integral
In formula (10), it is assumed that the internal integration is performed first. The result of this integration, the expression [[(5/6) - (x2 / 4)]], is then integrated over x from 0 to 1. The final result is 3/4. Formula (10) can also be interpreted as the so-called double integral, i.e. as the limit of the sum of the volumes of elementary "cells". Each such cell has a base DxDy and a height equal to the height of the surface above some point of the rectangular base (see Fig. 26). It can be shown that both points of view on formula (10) are equivalent. Double integrals are used to find the centers of gravity and numerous moments found in mechanics.
More rigorous substantiation of the mathematical apparatus. So far, we have presented the concepts and methods of calculus on an intuitive level and have not hesitated to resort to geometric shapes. It remains for us to briefly review the more rigorous methods that appeared in the 19th and 20th centuries. At the beginning of the 19th century, when the era of assault and onslaught in the "creation of mathematical analysis" ended, the 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", and "converging series" were precisely defined. This was necessary in order to bring a logical order to 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 everywhere continuous, but nowhere differentiable functions (the graph of such functions has a break at each point). This result made an overwhelming impression 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, completely filling 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 justifying all mathematical concepts using the concept of a number. The almost puritanical abstinence from clarity in works on foundations of mathematics had its historical justification. According to modern canons of logical rigor, it is inadmissible to speak of the area under the curve y = f (x) and over the segment of the x-axis, even if f is a continuous function, without first defining the exact meaning of the term "area" and not establishing that the area thus determined 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 notion of a definite integral has been the subject of many deep studies and generalizations. As a result, today it is possible to give meaning to a 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 the details of all these and other concepts. We will only restrict ourselves to giving rigorous definitions of the limit and definite integral. 1) The number L is called the limit of the function f (x) as x tends to a if for any arbitrarily small number e there is a corresponding positive number d such that
Compiled by Yu.V. Obrubov
Kaluga - 2012
Introduction to mathematical analysis.
Real numbers. Variables and constants.
One of the basic concepts of mathematics is number.
Positive numbers 1,2,3, ..., which are obtained when counting, are called natural.
Numbers ... -3, -2, -1,0,1,2,3, ... are called integers. Numbers that can be represented as a finite ratio of two integers ( 
) are called rational.
These include whole and fractional, positive and negative numbers. Numbers that represent infinite non-periodic fractions are called irrational.
Examples of irrational numbers are 
,
... In the set of irrational numbers, one distinguishes transcendental
numbers. These are numbers that are the result of non-algebraic actions. The most famous of these are the number 
and neper number 
... Rational and irrational numbers are called valid
... Real numbers are represented by dots on the numerical axis. Each point on the numerical axis corresponds to one single real number and, conversely, each real number corresponds to a single point on the numerical axis. Thus, a one-to-one correspondence is established between the real numbers and the points of the number line. This makes it possible to interchangeably use the terms “number a” and “point a”.
In the process of studying various physical, economic, social processes, one often has to deal with quantities that represent the numerical values of the parameters of the phenomena under study. In this case, some of them change, while others retain their values.
Variable
called a quantity that takes on different numerical values. A quantity whose numerical value does not change in a given problem or experiment is called constant.
Variables are usually denoted in Latin letters 
and constant 
.
Variable 
is considered given if the set of values that it can take is known. This set is called the range of the variable.
There are different kinds of sets of values for a numeric variable.
Interval
is the set of values x enclosed between the numbers a and b, while the numbers a and b do not belong to the set under consideration. The interval is denoted by: (a, b); a By segment
is the set of values of x, enclosed between the numbers a and b, while the numbers a and b belong to the set under consideration. The segment is denoted by a≤x≤b. The set of all real numbers is an open interval. Denoted: (- ∞, + ∞), -∞<х <+∞, R. Near point x
0
is called an arbitrary interval (a, b) containing the point x 0, all points of this interval satisfy the inequality a ε
- the neighborhood of point a
is called an interval centered at point a satisfying the inequality a – ε Function is one of the basic concepts of calculus. Let X and Y be arbitrary sets of real numbers. If, according to some rule or law, each number x X is associated with a unique well-defined real number yY, then they say that given function
with a domain of definition of X and a set of values of Y. Denote by y = f (x). The variable x is called argument
functions. In the definition of a function, two points are essential: an indication of the scope of definition and the establishment of a law of correspondence. The scope of
or area of existence
function is called the set of argument values for which the function exists, that is, it makes sense. The scope of change
function is called the set of values of y, which it takes for admissible values of x. Methods for setting a function.
An analytical way of defining a function. With this method of defining a function, the law of correspondence is written in the form of a formula (analytical expression), indicating by means of which mathematical transformations, according to the known value of the argument x, the corresponding value of y can be found. A function can be specified by one analytical expression on its entire domain of definition, or it can be a collection of several analytical expressions. For example: y = sin (x 2 + 1) 2. Tabular way of setting the function As a result of direct observation or experimental study of any phenomenon or process, the values of the argument x and the corresponding values of y are written out in a certain order. This table defines the function of y versus x. An example of a tabular way of setting a function are tables of trigonometric functions, tables of logarithms, dates and exchange rates, temperature and humidity, etc. 3. Graphical way of setting the function. The graphical way of setting the function consists in displaying points (x, y) on the coordinate plane by means of technical devices. The graphical method of defining a function in mathematical analysis is not used, but a graphical illustration of analytically set functions is always resorted to. MATHEMATICAL ANALYSIS part of mathematics, in a cut functions and their generalizations are studied by the method limits. The concept of a limit is closely related to the concept of an infinitesimal quantity; therefore, we can also say that M. and. studies functions and their generalizations by the method of infinitesimal. The name "M. a." - an abbreviated modification of the old name of this part of mathematics - "Analysis of the infinitesimal"; the latter reveals the content more fully, but it is also abbreviated (the title "Analysis by means of the infinitesimal" would describe the subject more accurately). In classical M. and. the objects of study (analysis) are primarily functions. "First of all" because M.'s development and. led to the possibility of studying it by methods of more complex formations than, - functionals, operators, etc. In nature and technology, movements and processes are encountered everywhere, which are described by functions; the laws of natural phenomena are also usually described by functions. Hence the objective importance of M. and. as a means of learning functions. M. a. in the broadest sense of this term covers a very large part of mathematics. It includes differential, integral calculus, functions of complex variable theory, theory ordinary differential equations, theory partial differential equations, theory integral equations, calculus of variations, functional analysis and some other mathematical. discipline. Modern numbers theory and probability theory apply and develop M.'s methods and. Yet M.'s term and. often used to name only the foundations of mathematical analysis, combining theory real number, theory of limits, theory rows, differential and integral calculus and their direct applications, such as the theory of maxima and minima, theory implicit functions, Fourier series, Fourier integrals. Function. In M. and. proceed from the definition of the function according to Lobachevsky and Dirichlet. If each number x from some set F of numbers, by virtue of the K.-L. the law is listed y, then this defines the function from one variable NS. The function from variables, where x =(x 1 , ..., x n) -
point in n-dimensional space; also consider the functions from points x =(x 1 , NS 2 ,
...) of a certain infinite-dimensional space, which, however, are more often called functionals. Elementary functions. Fundamental value in M. and. play elementary functions. In practice, they mainly operate with elementary functions, they approximate functions of a more complex nature. Elementary functions can be considered not only for real, but also for complex x; then the ideas about these functions become, in a certain sense, complete. In this regard, an important branch of M. and., Called. the theory of functions of a complex variable, or the theory analytical functions. A real number. The concept of a function is essentially based on the concept of a real (rational and irrational) number. It was finally formed only at the end of the 19th century. In particular, a logically flawless connection has been established between numbers and points of geometrical. straight line, which led to the formal substantiation of the ideas of R. Descartes (R. Descartes, middle 17th century), who introduced into mathematics rectangular coordinate systems and the representation of functions in them by graphs. Limit. In M. and. the method of learning functions is. Distinguish between a sequence limit and a function limit. These concepts were finally formed only in the 19th century, although ancient Greek people had an idea of them. scientists. Suffice it to say that Archimedes (3rd century BC) was able to calculate a segment of a parabola using a process that we would call the passage to the limit (cf. Exhaustion method). Continuous functions. An important function studied in M. and. Is formed by continuous functions. One of the possible definitions of this concept: function y = f(x). from one variable NS, set on the interval ( a, b),
called continuous at point NS, if The function is continuous on the interval ( a, b),
if it is continuous at all its points; then it is a continuous curve in the everyday sense of the word. Derivative and. Among continuous functions, one should single out functions that have derivative. Function derivative at point x, the rate of its change at this point, i.e., the limit If you leave the coordinate of a point moving along the ordinate in time NS, then f "(x). is the instantaneous velocity of the point at the moment of time NS. By the sign of the derivative f "(x) .
judge the nature of the change in f (x): if f "(z)> 0 ( f "(x) <0
).
on the interval ( s, d), then the function / increases (decreases) on this interval. If the function / at the point x attains a local extremum (maximum or minimum) and has a derivative at this point, then the latter is equal to zero at this point f "(x 0) = 0. Equality (1) can be replaced by the equivalent equality where is infinitesimal when, i.e., if the function f has a derivative at the point NS, then its increment at this point is decomposed into two terms. Of these, the first is from (proportional), the second tends to zero faster than The quantity (2) is called. differential functions corresponding to the increment At small values can be considered approximately equal to dy: The above reasoning about the differential is characteristic of M. and. They extend to functions of several variables and to functionals. For example, if the function from variables has continuous partial derivatives at the point x =(x 1 , ..., x n), then its increment
corresponding to the increments of the independent variables can be written in the form where if that is if all Here, the first term on the right-hand side of (3) is the differential dz function f. It depends linearly on and the second term tends to zero at faster than Let it be given (see Art. Calculus of variations) extended to function classes x (t) ,
having a continuous derivative on the interval and satisfying the boundary conditions x ( t 0)= x 0, x ( t 1)= x l, where x 0, x 1 - data numbers; let, further, be the class of the function h (t) ,
having a continuous derivative on and such that h ( t 0)= h(t 1) = 0. Obviously if In the calculus of variations, it is proved that under certain conditions on L, the increment of the functional J (x). Can be written in the form where and, thus, the second term on the right-hand side of (4) tends to zero faster than || h ||, and the first term depends linearly on The first term in (4) is called. variation of the functional and is denoted by dJ ( x, h). Integral. Along with the derivative, it is of fundamental importance in M. and. There are indefinite and definite integrals. The indefinite integral is closely related to the antiderivative function. Function F (x). the antiderivative of the function f on the interval ( a, b) if on this interval F "(x) = f(x). The definite integral (Riemann) of the function / on the segment [ a, b] there is a limit If the function f is positive and continuous on the segment [ a, b],
then the integral of it on this segment is equal to the area of the figure bounded by the curve y = f(x), axis Oh and direct x = a, x = b. The class of Riemann integrable functions contains all continuous on [ a, b] functions and some discontinuous functions. But they are all necessarily limited. For unbounded functions that do not grow very quickly, as well as for certain functions given on infinite intervals, the so-called. improper integrals, requiring a double transition to the limit for their definition. The concept of the Riemann integral for a function of one variable extends to functions of several variables (see Sec. Multiple integral). On the other hand, M.'s needs and. led to a generalization of the integral in a completely different direction, I mean Lebesgue integral or more general Lebesgue-Stieltjes integral. Essential in the definition of these integrals is the introduction for certain sets called measurable, the concept of their measure and, on this basis, the concept of a measurable function. For measurable functions and, the Lebesgue - Stieltjes integral is introduced. At the same time, a wide range of different measures and the corresponding classes of measurable sets and functions are considered. This makes it possible to adapt one or another integral to a specific specific problem. Formula of Newton - Leibniz. There is a connection between the derivative and the integral, expressed by the Newton - Leibniz formula (theorem) Here f (x) is continuous on [ a, b] function, a F (x) -
its antiderivative. Formula and Taylor. Along with the derivative and the integral, the most important concept (research tool) in M. and. are Taylor n Taylor series. If the function f (x) , a called its Taylor polynomial (degree n). in degrees x-x 0: (Taylor's formula); in this case, the approximation error tends to zero at faster than Thus, the function f (x) in the vicinity of the point x 0 can be approximated with any degree of accuracy by a very simple function (polynomial) that requires only arithmetic for its calculation. operations - addition, subtraction and multiplication. Particularly important are the so-called. functions analytic in a certain neighborhood x 0, having an infinite number of derivatives, such that for them in this neighborhood at they can be represented in the form of an infinite Taylor power series: Taylor expansions under certain conditions are also possible for functions of several variables, as well as functionals and operators. Historical reference. Until the 17th century. M. a. represented a set of solutions to disparate particular problems; For example, in integral calculus, these are tasks for calculating the areas of figures, volumes of bodies with curved boundaries, work of variable force, etc. article Infinitesimal calculus),
M. a. as a single and systematic. the whole was formed in the works of I. Newton, G. Leibniz, L. Euler, J. Lagrange and other scientists of the 17th -18th centuries, and his - the theory of limits - was developed by O. Komi (A. Cauchy) in the beginning. 19th century A deep analysis of M.'s initial concepts and. was associated with the development in the 19th and 20th centuries. set theory, measure theory, theory of functions of a real variable and led to various generalizations. Lit.: La Vallée - Pussen S.-J. e e, A course in the analysis of the infinitesimal, trans. from French, t. 1-2, M., 1933; Ilyin V.A., Poznyak E.G., Fundamentals of Mathematical Analysis, 3rd ed., Part 1, Moscow, 1971; 2nd ed., P. 2, M., 1980; And l and V. A. N., Sadovnichy V. A., Seyidov B. X., Mathematical analysis, M., 1979; Kudryavtsev L.D., Mathematical analysis, 2nd ed., T. 1-2, M., 1973; Nikol'skiy S.M., Course of mathematical analysis, 2nd ed., V. 1-2, M., 1975; Uitteker E.T., Vatson J. N., The course of modern analysis, trans. from English, parts 1-2, 2nd ed., M., 1962-63; Fichtengol'ts GM, Course of differential and integral calculus, 7th ed., T. 1-2, Moscow, 1970; 5th ed., Vol. 3, M., 1970. S. M. Nikolsky. Encyclopedia of Mathematics. - M .: Soviet encyclopedia... I. M. Vinogradov. 1977-1985. MATHEMATICAL ANALYSIS, a set of sections of mathematics devoted to the study of functions by methods of differential calculus and integral calculus ... Modern encyclopedia A set of sections of mathematics devoted to the study of functions by methods of differential and integral calculus. The term is rather pedagogical than scientific: courses in mathematical analysis are taught in universities and technical schools ... Big Encyclopedic Dictionary English. mathematical analysis; German mathematische Analysis. A branch of mathematics devoted to the study of functions by methods of differential and integral calculus. Antinazi. Encyclopedia of Sociology, 2009 ... Encyclopedia of Sociology Nus., Number of synonyms: 2 matan (2) calculus (2) ASIS synonym dictionary. V.N. Trishin. 2013 ... Synonym dictionary MATHEMATICAL ANALYSIS- MATHEMATICAL ANALYSIS. A set of sections of mathematics devoted to the study of mathematical functions by methods of differential and integral calculus. Using M.'s methods and. is an effective means of solving the most important ... ... New Dictionary of Methodological Terms and Concepts (Theory and Practice of Language Teaching) mathematical analysis- - EN mathematical analysis The branch of mathematics most explicitly concerned with the limit process or the concept of convergence; includes the theories of differentiation, ... ... Technical translator's guide Mathematical analysis- MATHEMATICAL ANALYSIS, a set of sections of mathematics devoted to the study of functions by methods of differential calculus and integral calculus. ... Illustrated Encyclopedic Dictionary A pile of terrible formulas, manuals on higher mathematics, which you open and then close, painful searches for a solution to a seemingly quite simple problem .... This situation is not uncommon, especially when a mathematics textbook was last opened in the distant 11th grade. Meanwhile, in universities, the curricula of many specialties provide for the study of everyone's favorite higher mathematics. And in this situation, you often feel like a complete teapot in front of a heap of terrible mathematical gibberish. Moreover, a similar situation can arise when studying any subject, especially from the cycle of natural sciences. What to do? For a full-time student, everything is much easier, unless, of course, the subject is too neglected. You can consult a teacher, classmates, and just cheat from a neighbor on the desk. Even a full kettle in higher mathematics will survive the session in such a scenario. And if a person studies at the correspondence department of a university, and higher mathematics, to put it mildly, is unlikely to be required in the future? Besides, there is absolutely no time for classes. This is how it is, in most cases it is, but no one canceled the execution of tests and passing the exam (most often, written). With tests in higher mathematics, everything is easier, you are a teapot, or not a teapot - test work in mathematics can be ordered... For example, mine. And for the rest of the subjects you can also order. Not here anymore. But the completion and delivery of the test papers for review will not yet lead to the coveted record in the grade book. It often happens that a work of art, made to order, needs to be protected, and explain why this formula follows from these letters. In addition, exams are coming, and there you will have to solve determinants, limits and derivatives YOURSELF. Unless, of course, the teacher does not accept valuable gifts, or there is no hired well-wisher outside the classroom. Let me give you some very important advice. On tests, exams in exact and natural sciences, IT IS VERY IMPORTANT TO UNDERSTAND SOMETHING. Remember, AT LEAST SOMETHING. The complete absence of thought processes just infuriates the teacher, I know of cases when correspondence students were wrapped 5-6 times. I remember that one young man passed the test 4 times, and after each retake he turned to me for a free warranty consultation. In the end, I noticed that in the answer he wrote the letter "pe" instead of the letter "pi", which was followed by severe sanctions from the reviewer. The student DIDN'T EVEN WANT TO INTELLIGENCE the assignment he casually rewrote You can be a complete teapot in higher mathematics, but it is highly desirable to know that the derivative of a constant is equal to zero. Because if you answer some nonsense to an elementary question, then there is a high probability that your studies at the university will end for you. Teachers are much more supportive of the student who AT LEAST TRIES to understand the subject, to the one who, albeit mistakenly, tries to solve, explain or prove something. And this statement is true for all disciplines. Therefore, the position “I don’t know anything, I don’t understand anything” should be resolutely dismissed. The second important tip is to ATTEND THE LECTURES, even if they are few. I already mentioned this on the main page of the site. Mathematics for correspondence students... There is no point in repeating why this is VERY important, read there. So, what to do if a test, an exam in higher mathematics is just around the corner, and things are deplorable - the state of a full, or, more precisely, an empty teapot? One option is to hire a tutor. The largest database of tutors can be found (mainly in Moscow) or (mainly in St. Petersburg). Using a search engine, it is quite possible to find a tutor in your city, or to look at local advertising newspapers. The price for a tutor's services can vary from 400 rubles or more per hour, depending on the qualifications of the teacher. It should be noted that cheap does not mean bad, especially if you have good mathematical background. At the same time, for 2-3K rubles you will receive a LITTLE one. Nobody takes such money in vain, and nobody pays such money in vain ;-). The only important point is to try to choose a tutor with specialized pedagogical education. Indeed, we do not go to the dentist for legal help. Recently, the online tutoring service has been gaining popularity. It is very convenient when you urgently need to solve one or two problems, understand a topic or prepare for an exam. An undoubted advantage is the prices, which are several times lower than those of an offline tutor + savings in travel time, which is especially important for residents of megalopolises. In the course of higher mathematics, it is very difficult to master some things without a tutor, you need a “live” explanation. Nevertheless, it is quite possible to figure out many types of problems on your own, and the purpose of this section of the site is to teach you how to solve typical examples and problems that are almost always found on exams. Moreover, for a number of tasks there are “hard” algorithms, where there is “nowhere to get away” from the correct solution. And, to the best of my knowledge, I will try to help you, especially since there is a pedagogical education and work experience in the specialty. Let's start raking mathematical gibberish. It's okay, even if you are a teapot, higher mathematics is really simple and really affordable. And you need to start by repeating the school mathematics course. Repetition is the mother of torment. Before you start studying my teaching materials, and indeed you start studying any materials on higher mathematics, I STRONGLY RECOMMEND that you read the following. SAVE A MICROCALCULATOR. From programs - Excel (great choice!). I have uploaded the manual for dummies to the library. There is? Already good. – Permutation of terms - the sum does not change: . It is simply impossible to rearrange the "X" and "four". At the same time, we recall the iconic letter "X", which in mathematics denotes an unknown or variable quantity.
– Permutation of factors - the product does not change: . – Remembering the rules for opening brackets: In general, it is enough to remember that TWO MINUSES GIVE A PLUS, a THREE MINUSES - GIVES A MINUS... And, try not to get confused in solving problems in higher mathematics (a very common and annoying mistake). – We recall the reduction of similar terms You should have a good understanding of the following action: – Remember what a degree is: , , The degree is just a simple multiplication. –Remember that fractions can be reduced: (reduced by 2), (reduced by five), (reduced by). – Remembering actions with fractions: ADVICE: all INTERMEDIATE calculations in higher mathematics are best done in ORDINARY RIGHT AND WRONG FRACTIONS, even if you get terrible fractions like. This fraction should NOT be represented in the form, and, moreover, it should NOT be divided by the numerator by the denominator on the calculator, getting 4.334552102…. The EXCEPTION of the rule is the final answer of the task, that's when it is better to write down or. – The equation... It has a left side and a right side. For example: You can transfer any term to another part by changing its sign: In the history of mathematics, one can conditionally distinguish two main periods: elementary and modern mathematics. The borderline from which it is customary to count the era of new (sometimes they say - higher) mathematics was the 17th century - the century of the emergence of mathematical analysis. By the end of the 17th century. I. Newton, G. Leibniz and their predecessors created the apparatus of a new differential calculus and integral calculus, which forms the basis of mathematical analysis and even, perhaps, the mathematical basis of all modern natural science. Mathematical analysis is a vast area of mathematics with a characteristic object of study (variable), a kind of research method (analysis by means of infinitesimal or by means of limit transitions), a defined system of basic concepts (function, limit, derivative, differential, integral, series) and constantly improving and a developing apparatus based on differential and integral calculus. Let's try to give an idea of what kind of mathematical revolution took place in the 17th century, what characterizes the transition from elementary mathematics to what is now the subject of research in mathematical analysis associated with the birth of mathematical analysis, and what explains its fundamental role in the entire modern system of theoretical and applied knowledge ... Imagine that before you is a beautifully executed color photograph of a stormy ocean wave running ashore: a mighty stooped back, a steep but slightly sunken chest, already tilted forward and ready to fall with a head torn by the wind with a gray mane. You stopped the moment, you managed to catch the wave, and now you can carefully study it in all its details without haste. The wave can be measured, and using the means of elementary mathematics, you will draw many important conclusions about this wave, and therefore all of its ocean sisters. But by stopping the wave, you have deprived it of movement and life. Its inception, development, run, the force with which it hits the shore - all this turned out to be out of your field of vision, because you do not yet have any language or mathematical apparatus suitable for describing and studying not static, but developing, dynamic processes, variables and their relationships. "Mathematical analysis is no less comprehensive than nature itself: it defines all tangible relationships, measures times, spaces, forces, temperatures." J. Fourier Movement, variables and their interconnections surround us everywhere. Various types of motion and their patterns constitute the main object of study of specific sciences: physics, geology, biology, sociology, etc. Therefore, an exact language and appropriate mathematical methods for describing and studying variable quantities turned out to be necessary in all areas of knowledge to approximately the same extent as and arithmetic are necessary when describing quantitative relationships. So, mathematical analysis is the basis of the language and mathematical methods for describing variables and their relationships. Nowadays, without mathematical analysis, it is impossible not only to calculate space trajectories, the operation of nuclear reactors, the running of the ocean wave and the patterns of cyclone development, but also to economically manage production, resource allocation, organization of technological processes, predict the course of chemical reactions or changes in the number of various species interconnected in nature. animals and plants, because these are all dynamic processes. Elementary mathematics was mainly the mathematics of constants; it studied mainly the relationships between the elements of geometric figures, the arithmetic properties of numbers, and algebraic equations. To some extent, her attitude to reality can be compared with a careful, even thorough and complete study of each fixed frame of a film that captures a changeable, developing living world in its movement, which, however, is not visible in a separate frame and which can be observed only by looking tape as a whole. But just as cinema is unthinkable without photography, so modern mathematics is impossible without that part of it, which we conventionally call elementary, without the ideas and achievements of many outstanding scientists, sometimes separated by tens of centuries. Mathematics is unified, and its "higher" part is connected with the "elementary" in about the same way as the next floor of a building under construction is connected with the previous one, and the width of the horizons that mathematics opens up to us in the world around us depends on which floor of this building we managed rise. Born in the 17th century. mathematical analysis opened up opportunities for scientific description, quantitative and qualitative study of variables and motion in the broadest sense of the word. What are the prerequisites for the emergence of mathematical analysis? By the end of the 17th century. the following situation has developed. Firstly, within the framework of mathematics itself, over the years, some important classes of similar problems have accumulated (for example, the problem of measuring the areas and volumes of non-standard figures, the problem of drawing tangents to curves) and methods for their solution have appeared in various special cases. Secondly, it turned out that these problems are closely related to the problems of describing arbitrary (not necessarily uniform) mechanical motion, and in particular with calculating its instantaneous characteristics (speed, acceleration at any time), as well as with finding the value of the distance traveled for movement occurring at a given variable speed. The solution of these problems was necessary for the development of physics, astronomy, and technology. Finally, thirdly, by the middle of the 17th century. the works of R. Descartes and P. Fermat laid the foundations of the analytical method of coordinates (the so-called analytical geometry), which made it possible to formulate geometrical and physical problems of various origins in a common (analytical) language of numbers and numerical dependencies, or, as we now say, numerical functions. NIKOLAY NIKOLAEVICH LUZIN
NN Luzin - Soviet mathematician, founder of the Soviet school of function theory, academician (1929). Luzin was born in Tomsk, studied at the Tomsk gymnasium. The formalism of the gymnasium course in mathematics alienated the talented young man, and only a capable tutor was able to reveal to him the beauty and greatness of mathematical science. In 1901, Luzin entered the Mathematics Department of the Physics and Mathematics Faculty of Moscow University. From the first years of study, issues related to infinity fell into the circle of his interests. At the end of the XIX century. German scientist G. Cantor created the general theory of infinite sets, which has received numerous applications in the study of discontinuous functions. Luzin began to study this theory, but his studies were interrupted in 1905. A student who took part in revolutionary activities had to leave for France for a while. There he listened to lectures by the most prominent French mathematicians of the time. Upon his return to Russia, Luzin graduated from the university and was left to prepare for a professorship. Soon he again went to Paris, and then to Göttingen, where he became close with many scientists and wrote the first scientific works. The main problem that interested the scientist was the question of whether there can exist sets containing more elements than the set of natural numbers, but less than the set of points of the segment (the problem of the continuum). For any infinite set that could be obtained from segments using the operations of joining and intersecting countable collections of sets, this hypothesis was fulfilled, and in order to solve the problem, it was necessary to find out what other ways of constructing the sets were. At the same time, Luzin studied the question of whether it is possible to represent any periodic function, even if it has infinitely many points of discontinuity, as a sum of a trigonometric series, i.e. the sum of an infinite set of harmonic vibrations. Luzin obtained a number of significant results on these issues and in 1915 defended his dissertation "Integral and Trigonometric Series", for which he was immediately awarded the degree of Doctor of Pure Mathematics, bypassing the intermediate master's degree that existed at that time. In 1917 Luzin became a professor at Moscow University. A talented teacher, he attracted the brightest students and young mathematicians. The Luzin school reached its peak in the first post-revolutionary years. Luzin's students formed a creative team that was jokingly called "Lusitania". Many of them received first-class scientific results while still in college. For example, PS Aleksandrov and M. Ya. Suslin (1894-1919) discovered a new method for constructing sets, which was the beginning of the development of a new direction - descriptive set theory. Research in this area, carried out by Luzin and his students, showed that the usual methods of set theory are not enough to solve many of the problems that have arisen in it. Luzin's scientific predictions were fully confirmed in the 60s. XX century. Many of Luzin's students later became academicians and corresponding members of the USSR Academy of Sciences. Among them is P.S. Aleksandrov. A. N. Kolmogorov. M. A. Lavrent'ev, L. A. Lyusternik, D. E. Menshov, P. S. Novikov. L. G. Shnirelman and others. Contemporary Soviet and foreign mathematicians in their works develop the ideas of N. N. Luzin. The confluence of these circumstances and led to the fact that at the end of the XVII century. two scientists - I. Newton and G. Leibniz - independently of each other managed to create a mathematical apparatus for solving these problems, summarizing and generalizing individual results of their predecessors, including the ancient scientist Archimedes and contemporaries of Newton and Leibniz - B. Cavalieri, B. Pascal , D. Gregory, I. Barrow. This apparatus formed the basis of mathematical analysis - a new branch of mathematics that studies various developing processes, i.e. interconnections of variables, which in mathematics are called functional dependencies or, in other words, functions. By the way, the term "function" itself was required and naturally arose precisely in the 17th century, and by now it has acquired not only general mathematical, but also general scientific meaning. Initial information about the basic concepts and the mathematical apparatus of analysis is given in the articles "Differential calculus" and "Integral calculus". In conclusion, I would like to dwell on only one principle of mathematical abstraction that is common for all mathematics and characteristic of analysis, and in this regard, explain in what form mathematical analysis studies variable quantities and what is the secret of such universality of its methods for studying all kinds of specific developing processes and their interrelationships. ... Let's consider a few illustrative examples and analogies. Sometimes we no longer realize that, for example, a mathematical relationship written not for apples, chairs or elephants, but in an abstract form abstracted from concrete objects, is an outstanding scientific achievement. This is a mathematical law that experience has shown is applicable to various specific objects. This means that by studying the general properties of abstract, abstract numbers in mathematics, we thereby study the quantitative relations of the real world. For example, it is known from a school mathematics course that, therefore, in a specific situation, you could say: “If two six-ton dump trucks are not allocated to me for transporting 12 tons of soil, then you can request three four-ton trucks and the work will be done, and if only one four-ton truck is given, then she will have to make three flights. " So the abstract numbers and numerical patterns that are now familiar to us are associated with their specific manifestations and applications. The laws of change of concrete variables and developing processes of nature are connected in approximately the same way with the abstract, abstract form-function in which they appear and are studied in mathematical analysis. For example, an abstract relationship may reflect the dependence of a cinema's box office on the number of tickets sold, if 20 is 20 kopecks - the price of one ticket. But if we ride a bicycle along the highway, driving 20 km per hour, then the same ratio can be interpreted as the relationship between the time (hours) of our bike ride and the distance covered during this time (kilometers)., You can always argue that, for example, a change by several times leads to a proportional (i.e., by the same number of times) change in the value, and if, then the opposite conclusion is also true. This means, in particular, to double the box office of a movie theater, you will have to attract twice as many viewers, and in order to ride a bicycle at the same speed twice the distance, you will have to ride twice as long. Mathematics studies both the simplest dependence and other, much more complex dependences in a general, abstract form abstracted from a particular interpretation. The properties of a function or methods for studying these properties identified in such a study will be in the nature of general mathematical techniques, conclusions, laws and conclusions applicable to each specific phenomenon in which the function studied in an abstract form occurs, regardless of which area of knowledge this phenomenon belongs to. ... So, mathematical analysis as a branch of mathematics took shape at the end of the 17th century. The subject of study in mathematical analysis (as it is presented from modern positions) are functions, or, in other words, dependencies between variables. With the emergence of mathematical analysis, mathematics became available to study and reflect the developing processes of the real world; variables and motion entered mathematics.Function. Basic definitions and concepts.
See what "MATHEMATICAL ANALYSIS" is in other dictionaries:
In order to successfully solve problems in higher mathematics, it is NECESSARY:
But these are completely different things:
With division, such a trick will not work, and these are two completely different fractions and the permutation of the numerator with the denominator does not do without consequences.
We also remember that the multiplication sign ("dot") is usually not written:,
- here the signs of the terms do not change
- and here they are reversed.
And for multiplication: 
, 
.
and also, a very important rule for reducing fractions to a common denominator: 
If these examples are difficult to understand, refer to school textbooks.
Without this it will be TIGHT.
Let's transfer, for example, all terms to the left side:
Or to the right: 
(1883-1950)
