Bounded function on the set x. Basic properties of functions
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TwitterPlease note: in all definitions, a number set X appears, which is part of the scope of the function: X with D (f). In practice, most often there are cases when X is a numerical interval (segment, interval, ray, etc.).
Definition 1.
A function y \u003d f (x) is called increasing on a set X with D (f) if for any two points x 1 and x 2 of the set X such that x 1< х 2 , выполняется неравенство f(х 1 < f(х 2).
Definition 2.
A function y \u003d f (x) is called decreasing on a set X with D (f) if for any monotonicity of two points x 1 and x 2 of the set X such that x 1< х 2 , функции выполняется неравенство f(x 1) > f (x 2).
In practice, it is more convenient to use the following formulations: the function increases if a larger value of the argument corresponds to a larger value of the function; the function decreases if the larger value of the argument corresponds to the smaller value of the function.
In the 7th and 8th grades, we used the following geometric interpretation of the concepts of increasing or decreasing a function: moving along the graph of an increasing function from left to right, we seem to climb a hill (Fig. 55); moving along the graph of the decreasing function from left to right, as if descending a hill (Fig. 56).
Usually, the terms "increasing function", "decreasing function" combine the general name of a monotonic function, and the study of a function for increasing or decreasing is called a study of a function for monotonicity.
Note one more circumstance: if a function increases (or decreases) in its natural domain of definition, then it is usually said that the function is increasing (or decreasing) - without specifying the numerical set X.
Example 1.
Examine the function for monotony:
and) y \u003d x 3 + 2; b) y \u003d 5 - 2x.
Decision:
a) Take arbitrary values \u200b\u200bof the argument x 1 and x 2 and let x 1<х 2 . Тогда, по свойствам числовых неравенств (мы с вами изучали их в курсе алгебры 8-го класса), будем иметь:
The last inequality means that f (x 1)< f(х 2). Итак, из х 1 < х 2 следует f{х 1) < f(х 2), а это означает, что заданная функция возрастает (на всей числовой прямой).
So from x 1< х 2 следует f(х 1) > f (x 2), which means that the given function decreases (on the whole number line).
Definition 3.
A function y - f (x) is called lower bounded on the set X with D (f) if all values \u200b\u200bof the function on the set X are greater than some number (in other words, if there is a number m such that for any value x є X the inequality f ( x)\u003e m).
Definition 4.
A function y \u003d f (x) is called upper bounded on a set X with D (f) if all values \u200b\u200bof the function are less than a certain number (in other words, if there is a number M such that for any value x є X the inequality f (x)< М).
If the set X is not indicated, then it is understood that we are talking about the boundedness of the function from below or from above in the entire domain of definition.
If a function is bounded both from below and from above, then it is called bounded.
The boundedness of a function is easy to read from its graph: if the function is bounded from below, then its graph is entirely located above some horizontal line y \u003d m (Fig. 57); if the function is bounded from above, then its graph is entirely located below some horizontal line y \u003d M (Fig. 58).
Example 2. Investigate for boundedness of function
Decision.On the one hand, the inequality is quite obvious (by the definition of the square root, this means that the function is bounded from below. On the other hand, we have, and therefore
This means that the function is bounded from above. Now look at the graph of the given function (Fig. 52 from the previous paragraph). The limited function of both the top and the bottom is easy to read from the graph.
Definition 5.
The number m is called the smallest value of the function y \u003d f (x) on the set X С D (f) if:
1) in X there is a point x 0 such that f (x 0) \u003d m;
2) for all x from X the inequality m\u003e f (x 0) holds.
Definition 6.
The number M is called the largest value of the function y \u003d f (x) on the set X C D (f) if:
1) in X there is a point x 0 such that f (x 0) \u003d M;
2) for all x from X the inequality
We denoted the smallest value of the function in both the 7th and 8th grades by the symbol y, and the largest by the symbol y.
If the set X is not indicated, then it is understood that we are talking about finding the smallest or largest value of the function in the entire domain of definition.
The following useful statements are fairly obvious:
1) If the function has Y, then it is bounded from below.
2) If the function has Y, then it is bounded from above.
3) If the function is not bounded from below, then Y does not exist.
4) If the function is not bounded above, then Y does not exist.
Example 3.
Find the smallest and largest values \u200b\u200bof a function
Decision.
It is quite obvious, especially if we resort to using the function graph (Fig. 52), that \u003d 0 (the function reaches this value at the points x \u003d -3 and x \u003d 3), and \u003d 3 (the function reaches this value at the point x \u003d 0.
In the 7th and 8th grades, we mentioned two more properties of functions. The first was called the convexity property of a function. It is considered that the function is convex downward on the interval X, if, having connected any two points of its graph (with abscissas from X) by a straight line segment, we find that the corresponding part of the graph lies below the drawn segment (Fig. 59). continuity A function is convex upward on the interval X, if, by connecting any two points of its graph (with abscissas from X) by a straight line segment, we will find that the corresponding part of the graph lies above the drawn segment (Fig. 60).
The second property - the continuity of the function on the interval X - means that the graph of the function on the interval X is solid, i.e. has no punctures and jumps.
Comment.
In fact, in mathematics, everything is, as they say, "exactly the opposite": the graph of a function is depicted as a solid line (without punctures and jumps) only when the continuity of the function is proved. But the formal definition of the continuity of a function, rather complicated and subtle, is still beyond our strength. The same can be said about the convexity of a function. In discussing these two properties of functions, we will continue to rely on visual intuitive representations.
Now let's review our knowledge. Remembering the functions that we studied in the 7th and 8th grades, we will clarify how their graphs look and list the properties of the function, adhering to a certain order, for example, this: scope; monotone; limitation; ,; continuity; range of values; convex.
Subsequently, new properties of functions will appear, and the list of properties will change accordingly.
1. Constant function y \u003d C
The graph of the function y \u003d C is shown in Fig. 61 - straight line parallel to the x axis. This is such an uninteresting function that it makes no sense to list its properties.
The graph of the function y \u003d kx + m is a straight line (Fig. 62, 63).
Properties of the function y \u003d kx + m:
1) 
2) increases if k\u003e 0 (Fig. 62), decreases if k< 0 (рис. 63);
4) there is neither the highest nor the lowest values;
5) the function is continuous;
6) 
7) it makes no sense to talk about convexity.
The graph of the function y \u003d kx 2 is a parabola with a vertex at the origin and with branches directed upwards if k\u003e 0 (Fig. 64), and down if k< 0 (рис. 65). Прямая х = 0 (ось у) является осью параболы.
Properties of the function y - kx 2:
For the case k\u003e 0 (Fig. 64):
1) D (f) \u003d (-oo, + oo);
4) \u003d does not exist;
5) continuous;
6) E (f) \u003d the function decreases, and in the interval, decreases on the ray;
7) convex upward.
The graph of the function y \u003d f (x) is plotted point by point; the more points of the form (x; f (x)) we take, the more accurate representation of the graph we will get. If you take a lot of these points, then the idea of \u200b\u200bthe graph will be more complete. It is in this case that intuition tells us that the graph should be depicted as a solid line (in this case, as a parabola). And then, reading the graph, we draw conclusions about the continuity of the function, about its convexity down or up, about the range of values \u200b\u200bof the function. You should understand that of the seven properties listed above, only properties 1), 2), 3), 4) are “legitimate” in the sense that we are able to substantiate them by referring to precise definitions. We have only visual and intuitive ideas about the rest of the properties. By the way, there is nothing wrong with that. From the history of the development of mathematics, it is known that mankind has often and for a long time used various properties of certain objects, without knowing the exact definitions. Then, when such definitions could be formulated, everything fell into place. 
The graph of the function is a hyperbola, the coordinate axes serve as the asymptotes of the hyperbola (Fig. 66, 67).
1) D (f) \u003d (-00.0) 1U (0, + oo);
2) if k\u003e 0, then the function decreases on an open beam (-oo, 0) and on an open beam (0, + oo) (Fig. 66); if to< 0, то функция возрастает на (-оо, 0) и на (0, +оо) (рис. 67);
3) is not limited either from below or from above;
4) there is neither the smallest nor the largest values;
5) the function is continuous on an open beam (-oo, 0) and on an open beam (0, + oo);
6) E (f) \u003d (-oo, 0) U (0, + oo);
7) if k\u003e 0, then the function is convex upward for x< 0, т.е. на открытом луче (-оо, 0), и выпукла вниз при х > 0, i.e. on an open beam (0, + oo) (Fig. 66). If to< 0, то функция выпукла вверх при х > O and convex down at x< О (рис. 67).
The graph of the function is a branch of the parabola (Fig. 68). Function properties:
1) D (f) \u003d, increases on the ray (set A), then it will be bounded on it both above and below.
Indeed, to show its boundedness from above, one must consider the predicate
and show that there is (exists) such M such that for all x taken on the interval [–2; 1], it will be true
It is not difficult to find such an M. We can assume M \u003d 7, the existential quantifier implies finding at least one value of M. The presence of such M and confirms the fact that the function on the interval [–2; 1] is bounded from above.
To prove its boundedness from below, it is necessary to consider the predicate
The value of M that ensures the truth of this predicate is, for example, M \u003d –100.
It can be proved that the function will also be limited in modulus: for all x from the interval [–2; 1], the values \u200b\u200bof the function coincide with the values, therefore, as M, we can take, for example, the previous value M \u003d 7.
Let us show that the same function, but on the interval, will be unlimited, that is
To show that such x exist, consider the statement
Looking for the required values \u200b\u200bof x among the positive values \u200b\u200bof the argument, we get
This means that no matter what positive Mm we take, the values \u200b\u200bof x that ensure that the inequality
are obtained from the ratio.
Considering the function on the entire real axis, one can show that it is unlimited in absolute value.
Indeed, from the inequality
That is, no matter how large the positive M, or will ensure the fulfillment of the inequality.
EXTREME FUNCTION.
The function has at the point from local maximum (minimum) if there exists a neighborhood of this point such that for x¹ from from this neighborhood the inequality
especially that the extremum point can only be the inner point of the interval and f (x) must be defined there. Possible cases of the absence of an extremum are shown in Fig. 8.8.
If the function increases (decreases) over some interval and decreases (increases) over some interval, then the point from is the point of the local maximum (minimum).
The absence of the maximum of the function f (x) at the point from can be formulated as follows:
_______________________
f (x) has a maximum at point c
This means that if the point c is not a local maximum point, then whatever the neighborhood that includes the point c as an internal one, there will be at least one value x not equal to c, at which. Thus, if there is no maximum at point c, then at this point there may be no extremum at all, or it may be a minimum point (Fig. 8.9).
The concept of an extremum gives a comparative estimate of the value of a function at any point in relation to those nearby. A similar comparison of the values \u200b\u200bof functions can be carried out for all points of a certain interval.
The LARGE (LOWEST) value of a function on a set will mean its value at a point from this set such that - for. The largest value of the function is achieved at the interior point of the segment, and the smallest – at its left end.
To determine the largest (smallest) value of a function given on a segment, it is necessary to choose the largest (smallest) number among all the values \u200b\u200bof its maxima (minima), as well as the values \u200b\u200btaken at the ends of the interval. It will be the largest (smallest) value of the function. This rule will be further refined.
The problem of finding the largest and smallest values \u200b\u200bof a function in an open interval is not always easy to solve. For example, the function
in the interval (Fig. 8.11) they do not.
Let's make sure, for example, that this function doesn't matter most. Indeed, given the monotonicity of the function, it can be argued that no matter how close we set the values \u200b\u200bof x to the left of unity, there will be other x in which the values \u200b\u200bof the function will be greater than its values \u200b\u200bat the fixed points, but still less than one.
Function concept. Limited features.
Definition of the function: If each number x from the set of numbers D is associated with a single number y, then they say that a function f is given on the set D and write y \u003d f (x), where x is called the independent variable or argument of this function, and the set D is the scope of this function.
Limited and unlimited functions.The function is called limitedif there is such a positive number Mwhat | f( x) | M for all values x.If no such number exists, then the function - unlimited.
EXAMPLES
Functions are even, odd, monotone.
Even and odd functions.If for any xfrom the scope of the function definition takes place: f(- x) = f (x), then the function is called even; if it takes place: f(- x) = - f (x), then the function is called odd... Even function graph symmetric about Y-axis (Fig. 5), a graph of an odd function symmetrical about origin (fig. 6).
Monotonic function.If for any two values \u200b\u200bof the argument x 1 and x 2 from condition x 2 > x 1 follows f( x 2 ) > f( x 1), then the function f( x) called increasing; if for any x 1 and x 2 from condition x 2 > x 1 follows f( x 2 ) < f( x 1 ), then the function f( x) is called decreasing... A function that only increases or only decreases is called monotonous.
3. Numerical sequences. Definition and examples.
We will say that the variable x there is ordered variable, if the area of \u200b\u200bits change is known, and about each of its two values, we can say which of them is the previous and which is the next. A special case of an ordered variable is a variable whose values \u200b\u200bform numerical sequence x 1, x 2,…, x n,… For such values \u200b\u200bat i< j, i, j Î N , value x i is considered to be preceding, and x j - subsequent regardless of which of these values \u200b\u200bis greater. Thus, a numerical sequence is a variable, the consecutive values \u200b\u200bof which can be renumbered. The numerical sequence will be denoted by. The individual numbers of the sequence are called her elements.
For example, the following values \u200b\u200bform a numerical sequence:
3., where a, d Are constant numbers.
Limit of a numerical sequence.
Number a called limit sequences x = {x n) if for an arbitrary predetermined arbitrarily small positive number ε there is a natural number Nthat for all n\u003e N the inequality | x n - a |< ε.
If the number a there is a sequence limit x = {x n), then they say that x n strives for a, and write.
To formulate this definition in geometric terms, we introduce the following concept. Near point x 0 is called an arbitrary interval ( a, b) containing this point inside itself. The neighborhood of the point is often considered x 0, for which x 0 is the middle, then x 0 called center neighborhood, and the quantity ( b–a)/2 – radiusneighborhood.
So, let's find out what the concept of the limit of a numerical sequence means geometrically. For this, we write the last inequality from the definition in the form This inequality means that all elements of the sequence with numbers n\u003e N must lie in the interval (a - ε; a + ε).
Therefore, the constant number a is the limit of the numerical sequence ( x n) if for any small neighborhood centered at the point a radius ε (ε are neighborhoods of the point a) there is an element of the sequence with number Nthat all subsequent elements with numbers n\u003e N will be inside this neighborhood.
Examples.
1. Let the variable x sequentially takes values
Let us prove that the limit of this numerical sequence is 1. Take an arbitrary positive number ε. We need to find such a natural number Nthat for all n\u003e N inequality holds | x n - 1| < ε. Действительно, т.к.
then for the relation | x n - a |< ε достаточно, чтобы или . Поэтому, взяв в качестве N any natural number satisfying the inequality, we get what we need. So if we take, for example, then, putting N \u003d6, for everyone n\u003e 6 will have.
2. Using the definition of the limit of a numerical sequence, prove that.
Take arbitrary ε\u003e 0. Consider Then, if or, i.e. ... Therefore, we choose any natural number satisfying the inequality.
Examples.
3. Consider. When x → 1 the numerator of the fraction tends to 1, and the denominator tends to 0. But since, i.e. is an infinitesimal function for x →1, then
Theorem 4. Given three functions f (x), u (x) and v (x)satisfying the inequalities u (x) ≤f (x) ≤ v (x)... If functions u (x) and v (x) have the same limit at x → a (or x → ∞), then the function f (x) tends to the same limit, i.e. if
Theorem 5. If at x → a (or x → ∞) function y \u003d f (x) takes non-negative values y≥0 and at the same time tends to the limit b, then this limit cannot be negative: b≥0.
Evidence... We carry out the proof by contradiction. Let's pretend that b<0 then | y - b | ≥ | b | and, therefore, the modulus of the difference does not tend to zero as x → a... But then y does not tend to the limit b at x → a, which contradicts the condition of the theorem.
Theorem 6. If two functions f (x)and g (x) for all values \u200b\u200bof the argument x satisfy the inequality f (x) ≥ g (x) and have limits, then the inequality b≥c.
Evidence. By the hypothesis of the theorem f (x) -g (x) ≥0, therefore, by Theorem 5, or.
6. Disclosure of uncertainties (0/0), ∞ -∞
I. Uncertainty.
When factoring the numerator into factors, we used the rule of dividing a polynomial by a polynomial "angle". Since the number x\u003d 1 is the root of the polynomial x 3 – 6x 2 + 11x- 6, then when dividing we get
7. Limit of sequence ... The concept of the natural logarithm.
SECOND WONDERFUL LIMIT
Examples:
Logarithm base e (e - a transcendental number, approximately equal to 2.718281828 ...) is called natural logarithm... Natural logarithm of a number x denoted by ln x... Natural logarithms are widely used in mathematics, physics, and engineering calculations.
Logarithms are widely used in
the base, called natural. Natural logarithms are indicated by the symbol
Function limit concept.
The concept of continuity of a function is directly related to the concept of the limit of a function.
A number A is called the limit of a function f at a point a, which is a limit for a set E, if, for any neighborhood V (A) of a point A, there exists a punctured neighborhood of a such that its image under the mapping f is a subset of a given neighborhood V (A) of A.
The limit of a function f at a point a, which is a limit for a set E, is denoted as follows: or, if we can omit the mention of the set E.
Since each neighborhood can be associated with its own correct (symmetric) neighborhood, the definition of the limit can be formulated in the language -δ in the form that is accepted in mathematical analysis:
The limit of the function at the point f at the point a, which is the limit for the set E, is directly related to the limit of the sequence.
We will consider all possible sequences of points of the set E that have the point a as their limit, and the corresponding sequences of values \u200b\u200bof the function at the points of the sequence. If the limit of the function f at point a exists, then this limit will be the limit of each sequence.
The converse is also true: if all sequences converge to the same value, then the function has a limit equal to the given value.
FIRST WONDERFUL LIMIT
Function undefined when x\u003d 0 because both the numerator and denominator of the fraction vanish. The function graph is shown in the figure.
However, one can find the limit of this function at x→0.
Let us give a proof of the written formula. Consider a circle of radius 1 and assume that the angle α, expressed in radians, lies within 0< α < π/2. (Так как четная функция и ее значения не изменяются при изменении знака α, то достаточно рассмотреть случай, когда α > 0.) It can be seen from the figure that
S ΔOAC .
Since the indicated areas are respectively equal
S Δ OAC=0,5∙OC∙OA∙ sin α= 0.5sinα, S sects. OAC \u003d0,5∙OC 2 ∙ α \u003d 0.5α, S Δ OBC=0,5∙OC∙BC \u003d0.5tgα.
Consequently,
sin α< α < tg α.
We divide all terms of the inequality by sin α\u003e 0:.
But. Therefore, based on Theorem 4 on the limits, we conclude that the derived formula is called the first remarkable limit.
Thus, the first remarkable limit serves to uncover uncertainty. Note that the resulting formula should not be confused with the limits Examples.
11.The Limit and its associated limits.
SECOND WONDERFUL LIMIT
The second remarkable limit serves to disclose the uncertainty 1 ∞ and looks as follows
Let's pay attention to the fact that in the formula for the second remarkable limit in the exponent there should be an expression inverse to the one that is added to one at the base (since in this case you can introduce a change of variables and reduce the sought limit to the second remarkable limit)
Examples.
1. Function f (x)=(x-1) 2 is infinitesimal for x→ 1, since (see fig.).
2. Function f (x) \u003d tg x - infinitely small at x→0.
3. f (x) \u003d ln (1+ x) Is infinitely small for x→0.
4. f (x) = 1/x- infinitely small at x→∞.
Let us establish the following important relation:
Theorem. If the function y \u003d f (x) representable at x → aas the sum of a constant number b and infinitesimal value α (x): f (x) \u003d b + α (x) then.
Conversely, if, then f (x) \u003d b + α (x)where a (x) - infinitely small at x → a.
Evidence.
1. Let us prove the first part of the statement. From equality f (x) \u003d b + α (x) should | f (x) - b | \u003d | α |... But since a (x) Is infinitesimal, then for arbitrary ε there is δ - a neighborhood of the point a, with all x from which, the values a (x) satisfy the relation | α (x) |< ε. Then | f (x) - b |< ε. And this means that.
2. If, then for any ε >0 for all x from some δ is a neighborhood of the point a will be | f (x) - b |< ε. But if we denote f (x) - b \u003d αthen | α (x) |< ε, which means that a - infinitely small.
Consider the basic properties of infinitesimal functions.
Theorem 1. The algebraic sum of two, three and, in general, any finite number of infinitesimal is an infinitesimal function.
Evidence... Let us give a proof for two terms. Let be f (x) \u003d α (x) + β (x)where and. We need to prove that for an arbitrary arbitrarily small ε > 0 is found δ> 0 such that for xsatisfying the inequality | x - a |<δ , performed | f (x) |< ε.
So, we fix an arbitrary number ε > 0. Since by the hypothesis of the theorem α (x) Is an infinitely small function, then there is δ 1 > 0, which for | x - a |< δ 1 we have | α (x) |< ε / 2. Similarly, since β (x) Is infinitely small, then there is δ 2 > 0, which for | x - a |< δ 2 we have | β (x) |< ε / 2.
Let's take δ \u003d min (δ 1 , δ 2 } . Then in the vicinity of the point a radius δ each of the inequalities | α (x) |< ε / 2 and | β (x) |< ε / 2. Therefore, in this neighborhood there will be
| f (x) | \u003d | α (x) + β (x)| ≤ | α (x) | + | β (x) |< ε /2 + ε /2= ε,
those. | f (x) |< ε, as required.
Theorem 2. Product of an infinitesimal function a (x) for limited function f (x) at x → a (or at x → ∞) is an infinitesimal function.
Evidence... Since the function f (x) is limited, then there is a number M such that for all values x from some neighborhood of the point a | f (x) | ≤M. Moreover, since a (x) Is an infinitely small function at x → a, then for arbitrary ε > 0 there is a neighborhood of the point a, in which the inequality | α (x) |< ε / M... Then in the smaller of these neighborhoods we have | αf |< ε / M\u003d ε. And this means that af - infinitely small. For the occasion x → ∞ the proof is similar.
It follows from the proved theorem:
Corollary 1. If u, then
Corollary 2. If and c \u003dconst, then.
Theorem 3. The ratio of an infinitesimal function α (x) per function f (x), the limit of which is nonzero, is an infinitesimal function.
Evidence... Let be . Then 1 / f (x) there is a limited function. Therefore, the fraction is the product of an infinitesimal function and a bounded function, i.e. function is infinitesimal.
Examples.
1. It is clear that for x → + ∞function y \u003d x 2 +1 is infinitely large. But then, according to the theorem formulated above, the function is infinitely small for x → + ∞, i.e. ...
The converse theorem can also be proved.
Theorem 2. If the function f (x) - infinitely small at x → a (or x → ∞) and does not vanish, then y \u003d1/ f (x) is an infinitely large function.
Prove the theorem yourself.
Examples.
3., since the functions and are infinitesimal for x → + ∞, then, how the sum of infinitesimal functions is an infinitesimal function. The function is the sum of a constant number and an infinitesimal function. Therefore, by Theorem 1, for infinitesimal functions, we obtain the required equality.
Thus, the simplest properties of infinitely small and infinitely large functions can be written using the following conditional relations: A≠ 0
13. Infinitesimal functions of the same order, equivalent to infinitesimal ones.
Infinitesimal functions and are called infinitesimal of the same order of smallness, if, denote. And, finally, if it does not exist, then infinitesimal functions are incomparable.
EXAMPLE 2. Comparison of infinitesimal functions
Equivalent infinitesimal functions.
If, then infinitesimal functions are called equivalent, denote ~.
Locally equivalent functions:
If if
Some equivalences(at):
One-sided limits.
So far, we have considered the definition of the limit of a function when x → a in an arbitrary way, i.e. the limit of the function did not depend on how the x towards a, left or right of a... However, quite often you can find functions that do not have a limit under this condition, but they do have a limit if x → astaying on one side of and, left or right (see fig.). Therefore, the concept of one-sided limits is introduced.
If f (x)tends to the limit b at x tending to a certain number a so that xtakes only values \u200b\u200bless than athen write and call b by the limit of the function f (x) at the point a on the left.
So the number b called the limit of the function y \u003d f (x) at x → aon the left, if whatever the positive number ε, there is such a number δ (less a
Similarly, if x → a and takes on large values athen write and call b the limit of the function at the point and on right. Those. number b called limit of the function y \u003d f (x) as x → a on the right, if whatever the positive number ε, there is such a number δ (greater and) such that the inequality holds for all.
Note that if the limits on the left and right at the point a for function f (x) do not coincide, then the function has no (two-sided) limit at the point and.
Examples.
1. Consider the function y \u003d f (x)defined on the segment as follows
Find the limits of the function f (x) at x →3. Obviously, but
In other words, for any arbitrarily small number of epsilons, there is such a number of deltas, depending on epsilon, that from the fact that for any x satisfying the inequality it follows that the difference between the values \u200b\u200bof the function at these points will be arbitrarily small.
Continuity criterion for a function at a point:
Function will be continuous at point A if and only if it is continuous at point A both to the right and to the left, that is, so that at point A there are two one-sided limits, they are equal to each other and equal to the value of the function at point A.
Definition 2: The function is continuous on a set if it is continuous at all points of this set.
Derivative of a function at a point
Let the given be defined in the neighborhood. Consider
If this limit exists, then it is called derivative of the function f at the point.
Function derivative - the limit of the ratio of the function increment to the argument increment, with the argument increment.
The operation of calculating or finding a derivative at a point is called differentiation .
Differentiation rules.
Derivative function f (x) at the point x \u003d x 0 is the ratio of the increment of the function at this point to the increment of the argument, as the latter tends to zero. Finding the derivative is called differentiation... The derivative of the function is calculated according to the general rule of differentiation: f (x) \u003d u, g (x) \u003d v- functions differentiable at the point x. Basic rules for differentiation 1) (the derivative of the sum is equal to the sum of the derivatives) 2) (this, in particular, implies that the derivative of the product of a function and a constant is equal to the product of the derivative of this function by a constant) 3) Derivative of the quotient: if g 0 4) Derivative of a complex function: 5) If the function is given parametrically:, then
Examples.
1. y = x a is a power function with an arbitrary exponent.
Implicitly specified function
If the function is given by the equation y \u003d ƒ (x), resolved with respect to y, then the function is given explicitly (explicit function).
Under implicit assignment functions understand the definition of a function in the form of an equation F (x; y) \u003d 0, not resolved with respect to y.
Any explicitly given function y \u003d ƒ (x) can be written as implicitly given by the equation ƒ (x) -y \u003d 0, but not vice versa.
It is not always easy, and sometimes impossible, to solve the equation for y (for example, y + 2x + cozy-1 \u003d 0 or 2 y -x + y \u003d 0).
If the implicit function is given by the equation F (x; y) \u003d 0, then to find the derivative of y with respect to x, there is no need to solve the equation for y: it is enough to differentiate this equation with respect to x, considering y as a function of x, and then solve the equation obtained for y ".
The derivative of an implicit function is expressed in terms of the argument x and the function y.
Example:
Find the derivative of the function y given by the equation x 3 + y 3 -3xy \u003d 0.
Solution: Function y is set implicitly. We differentiate the equality x 3 + y 3 -3xy \u003d 0 with respect to x. From the obtained relation
3x 2 + 3y 2 y "-3 (1 y + x y") \u003d 0
it follows that y 2 y "-xy" \u003d y-x 2, that is, y "\u003d (y-x 2) / (y 2 -x).
Higher order derivatives
It is clear that the derivative
function y \u003d f (x)there is also a function from x:
y "\u003d f" (x)
If the function f "(x) is differentiable, then its derivative is denoted by the symbol y "" \u003d f "" (x) xtwice.
Derivative of the second derivative, i.e. function y "" \u003d f "" (x)is called the third derivative of the function y \u003d f (x)or derivative of the function f (x) of the third order and denoted by symbols
Generally n-th derivative or derivative n-th order function y \u003d f (x)denoted by symbols
F-la Leibniz:
Suppose that the functions and are differentiable together with their derivatives up to the nth order inclusive. Applying the rule of differentiation of the product of two functions, we obtain
Let's compare these expressions with the powers of the binomial:
The correspondence rule is striking: to obtain a formula for the derivative of the 1st, 2nd or 3rd orders of the product of functions and, you need to replace the degrees and in the expression for (where n \u003d 1,2,3) derivatives of the corresponding orders. In addition, the zero degrees of the quantities and should be replaced by derivatives of the zero order, meaning the functions and:
Generalizing this rule to the case of a derivative of arbitrary order n, we get leibniz formula,
where are binomial coefficients:
Rolle's theorem.
This theorem makes it possible to find critical points, and then, with the help of sufficient conditions, investigate the function for extrema.
Let 1) the fth f (x) be defined and continuous on some closed interval; 2) there is a finite derivative, at least in the open interval (a; b); 3) at the ends of the interval the f-i takes equal values \u200b\u200bf (a) \u003d f (b). Then between the points a and b there is a point c such that the derivative at this point will be \u003d 0.
By the theorem on the property of f-th, continuous on a segment, f-th f (x) takes on this segment its max and min values.
f (x 1) \u003d M - max, f (x 2) \u003d m - min; x 1; x 2 Î
1) Let M \u003d m, i.e. m £ f (x) £ M
Þ f-i f (x) will take constant values \u200b\u200bon the interval from a to b, and Þ its derivative will be equal to zero. f '(x) \u003d 0
2) Let M\u003e m
Because by the conditions of the theorem, f (a) \u003d f (b) Þ its smallest or largest value of φ will take not at the ends of the segment, but Þ will take M or m at the interior point of this segment. Then by Fermat's theorem f '(c) \u003d 0.
Lagrange's theorem.
Finite increment formula or lagrangian mean value theorem asserts that if the function f is continuous on the segment [ a;b] and differentiable in the interval ( a;b), then there is a point such that
Cauchy's theorem.
If functions f (x) and g (x) are continuous on an interval and differentiable on an interval (a, b) and g ¢ (x) ¹ 0 on an interval (a, b), then there is at least one point e, a< e < b, такая, что
Those. the ratio of the increments of functions on a given segment is equal to the ratio of derivatives at the point e. Examples of problem solving course of lectures Calculating the volume of a body from the known areas of its parallel sections Integral calculus
Examples of course work Electrical engineering
At first glance, it is very convenient to use Lagrange's theorem to prove this theorem. Write down the finite difference formula for each function and then divide them by each other. However, this view is erroneous, since the point e for each of the functions is generally different. Of course, in some special cases this point of the interval may turn out to be the same for both functions, but this is a very rare coincidence, and not a rule, therefore it cannot be used to prove the theorem.
Evidence. Consider the helper function
When x → x 0, the quantity c also tends to x 0; we pass in the previous equality to the limit:
Because then.
therefore
(the limit of the ratio of two infinitesimal is equal to the limit of the ratio of their derivatives, if the latter exists)
L'Hôpital's rule, at ∞ / ∞.
1) Domain of function and domain of function values.
Function scope is the set of all valid valid argument values x (variable x) for which the function y \u003d f (x) defined. The range of values \u200b\u200bof a function is the set of all real values ythat the function accepts.
In elementary mathematics, functions are studied only on the set of real numbers.
2) Function zeros.
Function zero is an argument value at which the function value is equal to zero.
3) Intervals of constancy of function.
The intervals of constant sign of a function are such sets of argument values \u200b\u200bon which the function values \u200b\u200bare only positive or only negative.
4) Monotonicity of function.
An increasing function (in a certain interval) is a function for which a larger value of the argument from this interval corresponds to a larger value of the function.
Decreasing function (in a certain interval) - a function in which the larger value of the argument from this interval corresponds to the smaller value of the function.
5) Parity (odd) function.
An even function is a function whose domain of definition is symmetric about the origin and for any x from the domain, the equality f (-x) \u003d f (x)... The graph of an even function is symmetric about the ordinate axis.
An odd function is a function whose domain of definition is symmetric about the origin and for any x from the domain of definition, the equality f (-x) \u003d - f (x). The graph of an odd function is symmetric about the origin.
6) Limited and unlimited functions.
A function is called bounded if there exists a positive number M such that | f (x) | ≤ M for all values \u200b\u200bof x. If there is no such number, then the function is unlimited.
7) Periodicity of function.
A function f (x) is periodic if there exists a nonzero number T such that for any x from the domain of the function the following holds: f (x + T) \u003d f (x). This smallest number is called the period of the function. All trigonometric functions are periodic. (Trigonometric formulas).
19. Basic elementary functions, their properties and graphics. Application of functions in economics.
Basic elementary functions. Their properties and graphs
1. Linear function.
Linear function called a function of the form, where x is a variable, a and b are real numbers.
Number and called the slope of a straight line, it is equal to the tangent of the angle of inclination of this straight line to the positive direction of the abscissa axis. The graph of a linear function is a straight line. It is defined by two points.
Linear function properties
1. Domain of definition - the set of all real numbers: D (y) \u003d R
2. The set of values \u200b\u200bis the set of all real numbers: E (y) \u003d R
3. The function takes on a zero value for or.
4. The function increases (decreases) over the entire domain of definition.
5. The linear function is continuous on the whole domain of definition, differentiable and.
2. Quadratic function.
A function of the form, where x is a variable, the coefficients a, b, c are real numbers, is called quadratic.
Odds a, b, c determine the location of the graph on the coordinate plane
The coefficient a determines the direction of the branches. The graph of a quadratic function is a parabola. The coordinates of the vertex of the parabola are found by the formulas:
Function properties:
2. The set of values \u200b\u200bof one of the intervals: or.
3. The function takes zero values \u200b\u200bwhen 
, where the discriminant is calculated by the formula :.
4. The function is continuous on the entire domain and the derivative of the function is
The theorem on the limit of a monotone function. A proof of the theorem is given using two methods. Also, definitions of strictly increasing, non-decreasing, strictly decreasing and non-increasing functions are given. Definition of a monotone function.
ContentThe function is not limited from above
1.1. Let the number b be finite:.
1.1.2. Let the function be not bounded above.
.
at.
Let us denote. Then exists for any, so that
at.
This means that the left limit at b is equal (see "Definitions of one-sided infinite limits of a function at an end point").
b early plus infinity
Function bounded from above
1. Let the function not decrease on the interval.
1.2.1. Let the function be bounded from above by the number M: for.
Let us prove that in this case there is a limit.
Since the function is bounded from above, there is a finite upper bound
.
According to the definition of the exact top face, the following conditions are met:
;
for any positive there is an argument for which
.
Since the function does not decrease, then at. Then at. Or
at.
So, we found that for anyone there is a number, so
at.
"Definitions of one-sided limits at infinity").
The function is not limited from above
1. Let the function not decrease on the interval.
1.2. Let the number b be equal to plus infinity:.
1.2.2. Let the function be not bounded above.
Let us prove that in this case there is a limit.
Since the function is not bounded from above, for any number M there exists an argument for which
.
Since the function does not decrease, then at. Then at.
So, for anyone there is a number, so
at.
This means that the limit at is equal to (see "Definitions of one-sided infinite limits at infinity").
Function does not increase
Now consider the case where the function does not increase. You can, as above, consider each option separately. But we'll cover them right away. For this we use. Let us prove that in this case there is a limit.
Consider the finite lower bound of the set of values \u200b\u200bof the function:
.
Here B can be either a finite number or an infinitely distant point. According to the definition of the exact bottom edge, the following conditions are met:
;
for any neighborhood of the point B, there is an argument for which
.
By the condition of the theorem,. Therefore .
Since the function does not increase, then at. Since, then
at.
Or
at.
Further, we note that the inequality defines the left punctured neighborhood of the point b.
So, we found that for any neighborhood of the point, there exists a punctured left neighborhood of the point b such that
at.
This means that the limit on the left at b is:
(see the universal definition of the limit of a function according to Cauchy).
Limit at point a
Now we will show that there is a limit at point a and find its value.
Let's consider a function. By the hypothesis of the theorem, the function is monotone for. Replace x with - x (or substitute and then replace t with x). Then the function is monotonic for. Multiplying inequalities by -1 and changing their order, we come to the conclusion that the function is monotonic for.
In a similar way, it is easy to show that if it does not decrease, then it does not increase. Then, according to what was proved above, there is a limit
.
If it does not increase, then it does not decrease. In this case, there is a limit
.
Now it remains to show that if there is a limit of the function at, then there is a limit of the function at, and these limits are equal:
.
Let's introduce the notation:
(1)
.
Let us express f in terms of g:
.
Let's take an arbitrary positive number. Let there be an epsilon neighborhood of point A. Epsilon neighborhood is defined for both finite and infinite values \u200b\u200bof A (see “Point neighborhood”). Since there is a limit (1), then, according to the definition of the limit, for any there exists such that
at.
Let a be a finite number. Let us express the left punctured neighborhood of the point -a using the inequalities:
at.
Replace x with -x and note that:
at.
The last two inequalities define the punctured right-hand neighborhood of the point a. Then
at.
Let a be an infinite number,. We repeat the reasoning.
at;
at;
at;
at.
So, we found that for anyone there is such that
at.
It means that
.
The theorem is proved.
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