The concept of a function. Limited features.
Function definition: If each number x from the set of numbers D is associated with a single number y, then they say that the function f is given on the set D and write y \u003d f (x), where x is called an independent variable or argument of this function, and the set D is the domain of this function.
Limited and unlimited functions. The function is called limited if there is such a positive number M what | f(x) | M for all values x . If no such number exists, then the function is unlimited.
EXAMPLES.
Functions are even, odd, monotonic.
Even and odd functions. If for any x from the scope of the function definition takes place: f(- x) = f (x), then the function is called even; if it does: f(- x) = - f (x), then the function is called odd. Graph of an even function symmetrical about the Y axis(Fig.5), a graph of an odd function symmetrical about origin(Fig. 6).
monotonic function. If for any two values of the argument x 1 and x 2 of the 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 of the condition x 2 >x 1 follows f(x 2 ) <f(x 1 ), then the function f(x) is called waning. 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 orderly variable , if the area of its change is known, and for each of any two of its values it is possible to say which of them is the previous and which is the next. A special case of an ordered variable is a variable whose values form number sequence x 1 ,x 2 ,…,x n ,… For such values at i< j, i, j Î N , meaning x i considered to be prior, xj– subsequent, regardless of which of these values is greater. Thus, a numerical sequence is a variable whose successive values can be renumbered. The numerical sequence will be denoted by . The individual numbers of a sequence are called its elements.
For example, the numerical sequence is formed by the following quantities:
3. , where a, d are constant numbers.
Limit number sequence.
Number a called limit sequences x = {x n) if for an arbitrary preassigned arbitrarily small positive number ε there exists such natural number N, that for all n>N the inequality |x n - a|< ε.
If number a there is a sequence limit x = {x n), then they say that x n tends to a, and write .
To formulate this definition in geometric terms, we introduce the following notion. Neighborhood of point x 0 is called an arbitrary interval ( a, b) containing this point inside itself. The neighborhood of a point is often considered x0, for which x0 is the middle, then x0 called center neighborhood, and the quantity ( b–a)/2 – radius neighborhood.
So, let's find out what the concept of the limit of a numerical sequence means geometrically. To do this, we write the last inequality from the definition in the form This inequality means that all elements of the sequence with numbers n>N must lie in the interval (a – ε; a + ε).
Therefore, a constant number a is the limit of the numerical sequence ( x n) if for any small neighborhood centered at a point a radius ε (ε are neighborhoods of the point a) there is such an element of the sequence with number N that all subsequent elements with numbers n>N will be within this neighborhood.
Examples.
1. Let the variable x takes values sequentially
Let us prove that the limit of this numerical sequence is equal to 1. Take an arbitrary positive number ε. We need to find such a natural number N, that for all n>N the inequality | x n - 1| < ε. Действительно, т.к.
then to fulfill the relation |x n - a|< ε достаточно, чтобы или . Поэтому, взяв в качестве N any natural number that satisfies the inequality , we get what we need. So if we take, for example, , then, setting N= 6, for everyone n>6 we will have .
2. Using the definition of the limit of a numerical sequence, prove that .
Take an arbitrary ε > 0. Consider Then , if or , i.e. . Therefore, we choose any natural number that satisfies the inequality .
Examples.
3. Consider. At 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. Let three functions be given 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 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=f(x) takes non-negative values y≥0 and tends to the limit b, then this limit cannot be negative: b≥0.
Proof. The proof will be carried out by contradiction. Let's pretend that b<0 , then |y – b|≥|b| and, therefore, the modulus of the difference does not tend to zero at x→a. But then y does not go 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 of the argument x satisfy the inequality f(x)≥ g(x) and have limits , then we have the inequality b≥c.
Proof. According to the theorem f(x)-g(x) ≥0, therefore, by Theorem 5 , or .
6. Disclosure of uncertainties (0/0), ∞ -∞
I. Uncertainty.
When decomposing the numerator into factors, we used the rule for dividing a polynomial by a polynomial by an “angle”. Since the number x=1 is the root of the polynomial x 3 – 6x2 + 11x– 6, then when dividing we get
7. Sequence limit . The concept of the natural logarithm.
SECOND REMARKABLE LIMIT
Examples:
base logarithm e (e- a transcendental number approximately equal to 2.718281828 ...) is called natural logarithm. Natural logarithm of a number x denoted ln x. Natural logarithms are widely used in mathematics, physics and engineering calculations.
Logarithms are widely used
base, called natural. Natural logarithms are denoted by the symbol
The concept of the limit of a function.
The concept of the 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 limiting for a set E, if for any neighborhood V(A) of the point A, there exists a punctured neighborhood of the point a such that its image under the mapping f is a subset of the given neighborhood V(A) of the point A.
The limit of the function f at the point a, which is the limit for the set E, is denoted as follows: or , if it is possible to omit the mention of the set E.
Since each neighborhood can be associated with its own regular (symmetric) neighborhood, the definition of the limit can be formulated in the -δ language in the form that is customary 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 function values at the points of the sequence. If the limit of the function function f at the 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 REMARKABLE LIMIT
The function is not defined when x=0, since the numerator and denominator of the fraction vanish. The graph of the function is shown in the figure.
However, one can find the limit of this function at X→0.
We present the proof of the written formula. Consider a circle of radius 1 and assume that the angle α, expressed in radians, is 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
SΔOAC .
Since the indicated areas are respectively equal
S∆OAC=0,5∙OC∙OA sin α= 0.5sinα, S sect. OAC= 0,5∙OC 2 ∙α=0.5α, S ∆ OBC=0,5∙OC∙BC= 0.5tga.
Consequently,
sinα< α < tg α.
We divide all the terms of the inequality by sin α > 0: .
But . Therefore, on the basis of Theorem 4 on limits, we conclude that the derived formula is called the first remarkable limit.
Thus, the first remarkable limit serves to reveal the uncertainty. Note that the resulting formula should not be confused with the limits Examples.
11.Limit and related limits.
SECOND REMARKABLE LIMIT
The second remarkable limit serves to reveal the uncertainty 1 ∞ and looks like this
Let us pay attention to the fact that in the formula for the second remarkable limit, the exponent must contain an expression that is the opposite of that which is added to the unit in the base (since in this case it is possible to introduce a change of variables and reduce the desired limit to the second remarkable limit)
Examples.
1. Function f(x)=(x-1) 2 is infinitely small for x→1, since (see Fig.).
2. Function f(x)=tg x is infinitely small at x→0.
3. f(x)= log(1+ x) is infinitely small at x→0.
4. f(x) = 1/x is infinitely small at x→∞.
Let's establish the following important relation:
Theorem. If the function y=f(x) representable at x→a as a sum of a constant number b and infinitely small α(x): f(x)=b+ α(x) then .
Conversely, if , then f(x)=b+α(x), where a(x) is infinitely small at x→a.
Proof.
1. Let us prove the first part of the assertion. From equality f(x)=b+α(x) should |f(x) – b|=| α|. But since a(x) is infinitesimal, then for arbitrary ε there is δ, a neighborhood of the point a, for all x from which, 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= α, then |α(x)|< ε, which means that a- infinitely small.
Let us consider the main properties of infinitesimal functions.
Theorem 1. The algebraic sum of two, three, and in general any finite number of infinitesimals is an infinitesimal function.
Proof. Let us give a proof for two terms. Let f(x)=α(x)+β(x), where and . We need to prove that for arbitrary arbitrarily small ε > 0 there δ> 0, such that for x satisfying the inequality |x- a|<δ , performed |f(x)|< ε.
Thus, we fix an arbitrary number ε > 0. Since, according to the hypothesis of the theorem, α(x) is an infinitesimal function, then there exists δ 1 > 0, which at |x – a|< δ 1 we have |α(x)|< ε / 2. Likewise, since β(x) is infinitesimal, then there is such a δ 2 > 0, which at |x – a|< δ 2 we have | β(x)|< ε / 2.
Let's take δ=min(δ1 , δ2 } .Then in a neighborhood of the point a radius δ each of the inequalities will be satisfied |α(x)|< ε / 2 and | β(x)|< ε / 2. Therefore, in this neighborhood there will be
|f(x)|=| α(x)+β(x)| ≤ |α(x)| + | β(x)|< ε /2 + ε /2= ε,
those. |f(x)|< ε, which was to be proved.
Theorem 2. Product of an infinitesimal function a(x) for limited function f(x) at x→a(or when x→∞) is an infinitesimal function.
Proof. 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. In addition, since a(x) is an infinitesimal function for 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= ε. And this means that af- infinitely small. For the case x→∞ the proof is carried out in a similar way.
From the proved theorem it follows:
Consequence 1. If and then
Consequence 2. If and c= const, then .
Theorem 3. Ratio of an infinitesimal function α(x) per function f(x), whose limit is nonzero, is an infinitesimal function.
Proof. Let . Then 1 /f(x) there is a limited function. Therefore, a fraction is a product of an infinitesimal function and a bounded function, i.e. function is infinitesimal.
Examples.
1. It is clear that for x→+∞ function y=x 2 + 1 is infinite. But then, according to the theorem formulated above, the function is infinitesimal at 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= 1/f(x) is an infinite function.
Prove the theorem yourself.
Examples.
3. , since the functions and are infinitesimal for x→+∞, then as the sum of infinitesimal functions is an infinitesimal function. A function is the sum of a constant number and an infinitely small 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. Infinitely small functions of the same order, equivalent infinitely small.
Infinitely small functions and are called infinitesimal of the same order of smallness if , denote . And, finally, if does not exist, then infinitesimal functions and are incomparable.
EXAMPLE 2. Comparison of infinitesimal functions
Equivalent infinitesimal functions.
If , then infinitesimal functions and are called equivalent, denote ~ .
Locally equivalent functions:
When if
Some equivalences(at ):
Unilateral limits.
So far, we have considered the definition of the limit of a function when x→a arbitrarily, i.e. the limit of the function did not depend on how the x towards a, to the left or right of a. However, it is quite common to find functions that have no limit under this condition, but they do have a limit if x→a, staying on one side of a, left or right (see fig.). Therefore, the concept of one-sided limits is introduced.
If a f(x) tends to the limit b at x striving for some number a so x takes only values less than a, then write and call blimit of the function f(x) at point a on the left.
So the number b is called the limit of the function y=f(x) at x→a on the left, if there is any positive number ε, there is a number δ (smaller than a
Similarly, if x→a and takes on large values a, then write and call b function limit at a point a on right. Those. number b called limit of the function y=f(x) at x→a on the right, if there is any positive number ε, there is such a number δ (greater than a) that the inequality holds for all .
Note that if the limits are left and right at a point a for function f(x) do not match, then the function has no (two-sided) limit at the point a.
Examples.
1. Consider the function y=f(x), defined on the segment as follows
Let's find the limits of the function f(x) at x→ 3. Obviously, a
In other words, for any arbitrarily small number of epsilons, there is such a delta, depending on epsilons, that from the fact that for any x satisfying the inequality it follows that the difference in the values of the function at these points will be arbitrarily small.
Criterion for the continuity of a function at a point:
Function will be continuous at point A if and only if it is continuous at point A both on the right and on the left, i.e. in order for two one-sided limits to exist at point A, 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 given be defined in a neighborhood of . Consider
If this limit exists, then it is called the derivative of the function f at the point .
Function derivative- the limit of the ratio of the function increment to the argument increment, when the argument is incremented.
The operation of calculating or finding the derivative at a point is called differentiation .
Differentiation rules.
derivative functions f(x) at the point x=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 a function is calculated according to the general differentiation rule: Let us denote f(x) = u, g(x) = v- functions differentiable at a point X. Basic rules of differentiation 1) (the derivative of the sum is equal to the sum of the derivatives) 2) (hence, in particular, it follows 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 a quotient: if g 0 4) Derivative of a complex function: 5) If the function is set parametrically: , then
Examples.
1. y = x a - power function with an arbitrary index.
Implicit function
If the function is given by the equation y=ƒ(x) resolved with respect to y, then the function is given explicitly (explicit function).
Under implicit assignment functions understand the assignment of a function in the form of an equation F(x;y)=0, not allowed with respect to y.
Any explicitly given function y=ƒ(x) can be written as implicitly given by the equationƒ(x)-y=0, but not vice versa.
It is not always easy, and sometimes impossible, to solve an equation for y (for example, y+2x+cozy-1=0 or 2y-x+y=0).
If the implicit function is given by the equation F(x; y)=0, then to find the derivative of y with respect to x there is no need to solve the equation with respect to y: it suffices to differentiate this equation with respect to x, while considering y as a function of x, and then solve the resulting equation with respect to 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=0.
Solution: The function y is implicitly defined. Differentiate with respect to x the equality x 3 +y 3 -3xy=0. From the resulting ratio
3x 2 + 3y 2 y "-3 (1 y + x y") \u003d 0
it follows that y 2 y "-xy" \u003d y-x 2, i.e. y "= (y-x 2) / (y 2 -x).
Derivatives of higher orders
It is clear that the derivative
functions y=f(x) there is also a function from x:
y"=f" (x)
If the function f"(x) is differentiable, then its derivative is denoted by the symbol y""=f""(x) x twice.
The derivative of the second derivative, i.e. functions y""=f""(x), is called third derivative of the function y=f(x) or derivative of the function f(x) of the third order and is symbolized
Generally n-i derivative or derivative n-th order function y=f(x) denoted by symbols
F-la Leibniz:
Let us assume 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: in order to obtain a formula for the derivative of the 1st, 2nd or 3rd orders from the product of functions and , you need to replace the degrees and in the expression for (where n= 1,2,3) derivatives of the corresponding orders. In addition, zero powers of and should be replaced by derivatives zero order, meaning by them the functions and :
Generalizing this rule to the case of an arbitrary order derivative n, we get Leibniz formula,
where are the binomial coefficients:
Rolle's theorem.
This theorem makes it possible to find critical points and then, with the help of sufficient conditions, to investigate f-th for extrema.
Let 1) the f-th 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 interval f-i takes equal values f(a) = f(b). Then between points a and b there is such a point c that the derivative at this point will be = 0.
According to the theorem on the property of f-ths that are continuous on a segment, the 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 О
1) Let M = m, i.e. m £ f(x) £ M
Þ f-th f(x) will take on the interval from a to b constant values, and Þ its derivative will be equal to zero. f'(x)=0
2) Let M>m
Because by the conditions of the theorem, f(a) = f(b) z is its least or greatest f-th value will take not at the ends of the segment, but Þ will take M or m at an interior point of this segment. Then by Fermat's theorem f'(c)=0.
Lagrange's theorem.
Finite Increment Formula or Lagrange mean value theorem states that if the function f continuous on the segment [ a;b] and differentiable in the interval ( a;b), then there is a point such that
Cauchy's theorem.
If the functions f(x) and g(x) are continuous on the interval and differentiable on the interval (a, b) and g¢(x) ¹ 0 on the interval (a, b), then there is at least one point e, a< e < b, такая, что
Those. the ratio of 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 Calculation of body volume by famous squares his parallel sections Integral calculus
Examples of course work electrical engineering
To prove this theorem, at first glance, it is very convenient to use Lagrange's theorem. Write down the finite difference formula for each function, and then divide them by each other. However, this view is erroneous, because the point e for each of the functions is generally different. Of course, in some special cases this interval point may be the same for both functions, but this is a very rare coincidence, not a rule, and therefore cannot be used to prove the theorem.
Proof. Consider the helper function
When x→x 0, the value of c also tends to x 0; let's pass in the previous equality to the limit:
Because , then .
That's why
(the limit of the ratio of two infinitesimals is equal to the limit of the ratio of their derivatives, if the latter exists)
L'Hopital's rule, at ∞ / ∞.
Note that all definitions include a numeric set X, which is part of the domain of the function: X with D(f). In practice, most often there are cases when X - number gap(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(x2).
In practice, it is more convenient to use the following formulations: the function increases if the larger value of the argument corresponds to the larger value of the function; the function is decreasing 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 functions: moving along the graph of an increasing function from left to right, we sort of climb up the hill (Fig. 55); moving along the graph of a decreasing function from left to right, as if we were going down a hill (Fig. 56).
Usually the terms “increasing function”, “decreasing function” are united by the common name monotonic function, and the study of a function for increasing or decreasing is called the study of a function for monotonicity.
We note one more circumstance: if a function is increasing (or decreasing) 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 monotonicity:
a) y \u003d x 3 + 2; b) y \u003d 5 - 2x.
Solution:
a) Take arbitrary values of 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 is decreasing (on the entire number line).
Definition 3.
The function y - f(x) is called bounded from below on the set X with D (f) if all the values of the function on the set X are greater 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) >m).
Definition 4.
The function y \u003d f (x) is called bounded from above on the set X with D (f) if all values of 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 specified, then it is assumed that the function is bounded 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 easily 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 a function for boundedness
Solution. On the one hand, the inequality is quite obvious (by definition 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 boundedness of the function both from above and from below is read quite easily from the graph.
Definition 5.
The number m is called the smallest value of the function y \u003d f (x) on the set X C D (f), if:
1) in X there is such a point x 0 that f(x 0) = m;
2) for all x from X the inequality m>f(х 0) is fulfilled.
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 such a point x 0 that f(x 0) = M;
2) for all x from X, the inequality
We denoted the smallest value of the function both in the 7th and 8th grades by the symbol y, and the largest value by the symbol y.
If the set X is not specified, then it is assumed that we are talking about finding the smallest or the greatest value functions in the entire domain of definition.
The following useful statements are quite obvious:
1) If a function has Y, then it is bounded from below.
2) If a function has Y, then it is bounded from above.
3) If the function is not bounded below, then Y does not exist.
4) If the function is not bounded from above, then Y does not exist.
Example 3
Find the smallest and largest values of a function
Solution.
It is quite obvious, especially if you resort to the function graph (Fig. 52), that \u003d 0 (the function reaches this value at the points x \u003d -3 and x \u003d 3), a \u003d 3 (the function reaches this value at the point x \u003d 0.
In 7th and 8th grade, we mentioned two more properties of functions. The first was called the convexity property of a function. It is considered that a function is convex downward on the interval X if, by connecting any two points of its graph (with abscissas from X) with 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 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 continuous, 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, which is quite complex and subtle, is beyond our powers yet. The same can be said about the convexity of a function. 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: domain of definition; 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 > 0 (Fig. 62), decreases if k< 0 (рис. 63);
4) there is neither the largest nor the smallest 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 O (Fig. 64), and downwards if k< 0 (рис. 65). Прямая х = 0 (ось у) является осью параболы.
Properties of the function y - kx 2:
For the case k > 0 (Fig. 64):
1) D(f) = (-oo,+oo);
4) = does not exist;
5) continuous;
6) Е(f) = the function decreases, and on the interval , decreases on the ray;
7) convex upwards.
The graph of the function y \u003d f (x) is built point by point; the more points of the form (x; f (x)) we take, the more accurate idea of the graph we get. If we take a lot of these points, then the idea of the graph will be more complete. It is in this case that intuition tells us that the graph should be drawn 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 downwards or upwards, about the range of the function. You must understand that of the listed seven properties, only properties 1), 2), 3), 4) are "legitimate" in the sense that we are able to substantiate them, referring to precise definitions. We have only visual-intuitive representations about the remaining properties. By the way, there's nothing wrong with that. From the history of the development of mathematics, it is known that mankind often and for a long time used various properties of certain objects, not 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 asymptotes of the hyperbola (Fig. 66, 67).
1) D(f) = (-00.0)1U (0.+oo);
2) if k > 0, then the function decreases on the open ray (-oo, 0) and on the open ray (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 the open ray (-oo, 0) and on the open ray (0, +oo);
6) E(f) = (-oo, 0) U (0, + oo);
7) if k > 0, then the function is convex upwards at x< 0, т.е. на открытом луче (-оо, 0), и выпукла вниз при х >0, i.e. on the 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) = , increases on the ray. On this segment $16-x^2≤16$ or $\sqrt(16-x^2)≤4$, but this means boundedness from above.
Answer: our function is limited by two lines $y=0$ and $y=4$.
Highest and lowest value
The smallest value of the function y= f(x) on the set Х⊂D(f) is some number m, such that:
b) For any xϵX, $f(x)≥f(x0)$ holds.
The greatest value of the function y=f(x) on the set Х⊂D(f) is some number m, such that:
a) There is some x0 such that $f(x0)=m$.
b) For any xϵX, $f(x)≤f(x0)$ is satisfied.
The largest and smallest value is usually denoted by y max. and y name. .
The concepts of boundedness and the largest with the smallest value of a function are closely related. The following statements are true:
a) If there is a smallest value for a function, then it is bounded from below.
b) If there is a maximum value for a function, then it is bounded from above.
c) If the function is not bounded from above, then there is no maximum value.
d) If the function is not bounded below, then the smallest value does not exist.
Find the largest and smallest value of the function $y=\sqrt(9-4x^2+16x)$.
Solution: $f(x)=y=\sqrt(9-4x^2+16x)=\sqrt(9-(x-4)^2+16)=\sqrt(25-(x-4)^2 )≤5$.
For $x=4$ $f(4)=5$, for all other values, the function takes smaller values or does not exist, that is, this is the largest value of the function.
By definition: $9-4x^2+16x≥0$. Let's find the roots square trinomial$(2x+1)(2x-9)≥0$. At $x=-0.5$ and $x=4.5$ the function vanishes, at all other points it is greater than zero. Then, by definition, the smallest value of the function is zero.
Answer: y max. =5 and y min. =0.
Guys, we have also studied the concepts of convexity of a function. When solving some problems, we may need this property. This property is also easily determined using graphs.
The function is convex down if any two points of the graph of the original function are connected, and the graph of the function is below the line connecting the points.
The function is convex upward if any two points of the graph of the original function are connected, and the graph of the function is above the line connecting the points.
A function is continuous if the graph of our function has no discontinuities, such as the graph of the function above.
If you want to find the properties of a function, then the sequence of searching for properties is as follows:
a) Domain of definition.
b) Monotony.
c) limitation.
d) The largest and smallest value.
e) Continuity.
f) Range of values.
Find the properties of the function $y=-2x+5$.
Solution.
a) Domain of definition D(y)=(-∞;+∞).
b) Monotony. Let's check for any values x1 and x2 and let x1< x2.
$f(x1)=-2x1+2$.
$f(x2)=-2x2+2$.
Because x1< x2, то f(x1) < f(x2), то есть большему значению аргумента, соответствует меньшее значение функции. Функция убывает.
c) limitation. Obviously, the function is not limited.
d) The largest and smallest value. Since the function is not bounded, there is no maximum or minimum value.
e) Continuity. The graph of our function has no gaps, then the function is continuous.
f) Range of values. E(y)=(-∞;+∞).
Tasks on the properties of a function for independent solution
Find function properties:a) $y=2x+7$,
b) $y=3x^2$,
c) $y=\frac(4)(x)$.
Limit theorem monotonic function. The proof of the theorem is given using two methods. Definitions of strictly increasing, non-decreasing, strictly decreasing and non-increasing functions are also given. Definition of a monotonic function.
Definitions
Definitions of Increasing and Decreasing Functions
Let the function f (x) is defined on some set of real numbers X .
The function is called strictly increasing (strictly decreasing), if for all x′, x′′ ∈ X such that x′< x′′
выполняется неравенство:
f (x′)< f(x′′)
( f (x′) > f(x′′) )
.
The function is called non-decreasing (non-increasing), if for all x′, x′′ ∈ X such that x′< x′′
выполняется неравенство:
f (x′) ≤ f(x′′)( f (x′) ≥ f(x′′) )
.
This implies that a strictly increasing function is also nondecreasing. A strictly decreasing function is also nonincreasing.
Definition of a monotonic function
The function is called monotonous if it is non-decreasing or non-increasing.
To study the monotonicity of a function on some set X, you need to find the difference of its values at two arbitrary points belonging to this set. If , then the function is strictly increasing; if , then the function does not decrease; if , then strictly decreases; if , then does not increase.
If on some set the function is positive: , then to determine monotonicity, one can examine the quotient of dividing its values at two arbitrary points of this set. If , then the function is strictly increasing; if , then the function does not decrease; if , then strictly decreases; if , then does not increase.
Theorem
Let the function f (x) does not decrease over the interval (a,b), where .
If it is bounded from above by the number M : , then there is a finite left limit at the point b : . If f (x) not bounded above, then .
If f (x) is bounded from below by the number m : , then there is a finite right limit at the point a : . If f (x) not bounded below, then .
If the points a and b are at infinity, then in the expressions the limit signs mean that .
This theorem can be formulated more compactly.
Let the function f (x) does not decrease over the interval (a,b), where . Then there are one-sided limits at points a and b:
;
.
A similar theorem for a non-increasing function.
Let the function not increase on the interval , where . Then there are one-sided limits:
;
.
Consequence
Let the function be monotonic on the interval . Then at any point from this interval, there are one-sided finite limits of the function :
and .
Proof of the theorem
The function does not decrease
b - final number
Function limited from above
1.1.1. Let the function be bounded from above by the number M : for .
.
;
.
Since the function does not decrease, then for . Then
at .
Let's transform the last inequality:
;
;
.
Because , then . Then
at .
at .
"Definitions of one-sided limits of a function at a finite point").
The function is not limited from above
1. Let the function not decrease on the interval .
1.1. Let the number b be finite: .
1.1.2. Let the function be unbounded from above.
Let us prove that in this case there is a limit .
.
at .
Let's denote . Then for any exists , so that
at .
This means that the limit on the left at point b is (see "Definitions of one-sided infinite limits of a function at the end point").
b early plus infinity
Function limited 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 least upper bound, the following conditions are satisfied:
;
for any positive there is an argument for which
.
Since the function does not decrease, then for . Then at . Or
at .
So we have found that for any there exists a number , so that
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 plus infinity: .
1.2.2. Let the function be unbounded from above.
Let us prove that in this case there is a limit .
Since the function is not bounded from above, then for any number M there is an argument , for which
.
Since the function does not decrease, then for . Then at .
So, for any there is a number , so that
at .
This means that the limit at is (see "Definitions of One-Sided Infinite Limits at Infinity").
The function does not increase
Now consider the case when the function is not increasing. You can, as above, consider each option separately. But we will 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 function values:
.
Here B can be either a finite number or a point at infinity. According to the definition of the exact infimum, the following conditions are satisfied:
;
for any neighborhood of point B there is an argument for which
.
By the condition of the theorem, . That's why .
Since the function does not increase, then for . Since , then
at .
Or
at .
Further, we note that the inequality defines the left punctured neighborhood of the point b .
So, we have found that for any neighborhood of the point , there is such a punctured left neighborhood of the point b that
at .
This means that the limit on the left at point b is :
(see the universal definition of the limit of a function according to Cauchy).
Limit at point a
Now let's show that there is a limit at the point a and find its value.
Let's consider a function. By the condition of the theorem, the function is monotonic for . Let's replace the variable x with - x (or do the substitution and then replace the variable t with x ). Then the function is monotone for . Multiplying the inequalities by -1 and changing their order, we conclude 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 doesn't increase, then it doesn't 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's express f in terms of g :
.
Take an arbitrary positive number . Let there be an epsilon neighborhood of point A . Epsilon neighborhood is defined for both finite and infinite values of A (see "Neighborhood of a point"). Since there is a limit (1), then, according to the definition of a 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 .
Let's replace x with -x and take into account that:
at .
The last two inequalities define a punctured right neighborhood of the point a . Then
at .
Let a be an infinite number, . We repeat the discussion.
at ;
at ;
at ;
at .
So, we have found that for any there exists such that
at .
It means that
.
The theorem has been proven.
We will call the function y=f(x) BOUNDED UP (BOTTOM) on the set A from the domain D(f), if there is such a number M , that for any x from this set the condition
Using logical symbols, the definition can be written as:
f(x) – bounded from above on the set
(f(x) – bounded from below on the set
Functions bounded in absolute value or simply bounded are also introduced into consideration.
We will call a function BOUNDED on the set A from the domain of definition if there exists a positive number M such that
In the language of logical symbols
f(x) – limited on the set
A function that is not bounded is called unbounded. We know that definitions given through negation have little content. To formulate this assertion as a definition, we use the properties of the quantifier operations (3.6) and (3.7). Then the denial of boundedness of the function in the language of logical symbols will give:
f(x) – limited on the set
The result obtained allows us to formulate the following definition.
A function is called UNLIMITED on the set A, which belongs to the domain of the function, if on this set for any positive number M there is such a value of the argument x , that the value will still exceed the value of M, that is, .
As an example, consider the function
It is defined on the entire real axis. If we take the segment [–2;1] (set A), then on it it will be bounded both from above and from below.
Indeed, to show that it is bounded from above, we must consider the predicate
and show that there is (exists) M such that for all x taken on the segment [–2;1], it will be true
It is not difficult to find such an M. We can assume M = 7, the existence quantifier implies finding at least one value of M. The presence of such M confirms the fact that the function on the segment [–2;1] is bounded from above.
To prove its boundedness from below, we need to consider the predicate
The value of M, which ensures the truth of this predicate, is, for example, M = -100.
It can be proved that the function will be bounded modulo as well: for all x from the segment [–2;1], the values of the function coincide with the values of , therefore, as M, we can take, for example, the previous value M = 7.
Let us show that the same function, but on the interval , will be unbounded, that is,
To show that such x exist, consider the statement
Searching for the required values of x among the positive values of the argument, we get
This means that no matter what positive Mwe take, the values of x that ensure the fulfillment of the inequality
are obtained from the ratio.
Considering a function on the entire real axis, one can show that it is unbounded in absolute value.
Indeed, from the inequality
That is, no matter how large the positive M is, or will ensure the fulfillment of the inequality .
EXTREME FUNCTION.
The function has at the point With local maximum (minimum) if there is such a neighborhood of this point that for x¹ With this neighborhood satisfies the inequality
especially that the extremum point can only be an internal point of the gap, and f(x) must be defined in it. Possible cases of the absence of an extremum are shown in Figs. 8.8.
If a function increases (decreases) on some interval and decreases (increases) on some interval , then the point With is the local maximum (minimum) point.
The absence of a maximum of the function f(x) at a point With can be formulated like this:
_______________________
f(x) has a maximum at c
This means that if the point c is not a local maximum point, then no matter what the neighborhood that includes the point c as an interior one, there is at least one value of x not equal to c, for which . Thus, if there is no maximum at point c, then there may not be an extremum at all at this point, or it may be a minimum point (Fig. 8.9).
The concept of an extremum gives a comparative assessment of the value of a function at any point in relation to nearby ones. A similar comparison of function values can be made for all points of some interval.
The GREATEST (MINIMUM) value of a function on a set is its value at a point from this set such that – for . The greatest value of the function is reached at the inner 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 of its maxima (minimums), as well as values taken at the ends of the interval. It will be the largest (smallest) value of the function. This rule will be specified later.
The problem of finding the largest and smallest values of a function on an open interval is not always easily solved. For example, the function
in the interval (Fig. 8.11) does not have them.
Let's make sure, for example, that this function does not have the greatest value. Indeed, given the monotonicity of the function , it can be argued that no matter how close we set the values of x to the left of unity, there will be other x in which the values of the function will be greater than its values at the given fixed points, but still less than unity.
