ABSTRACT
« Full Study function and construction of its graph.”
INTRODUCTION
Studying the properties of a function and plotting its graph is one of the most wonderful applications of derivatives. This method of studying function has been repeatedly subjected to careful analysis. The main reason is that in applications of mathematics one had to deal with more and more complex functions that appear when studying new phenomena. Exceptions to the rules developed by mathematics appeared, cases appeared when the rules created were not suitable at all, functions appeared that did not have a derivative at any point.
The purpose of studying the course of algebra and beginnings of analysis in grades 10-11 is the systematic study of functions, disclosure of the applied meaning common methods mathematics related to the study of functions.
The development of functional concepts in the course of studying algebra and the beginnings of analysis at the senior level of education helps high school students to obtain a visual understanding of the continuity and discontinuities of functions, learn about the continuity of any elementary function in the field of its application, learn to construct their graphs and generalize information about the basic elementary functions and realize their role in the study of phenomena of reality, in human practice.
Increasing and decreasing function
Solving various problems from the fields of mathematics, physics and technology leads to the establishment of a functional relationship between the variables involved in this phenomenon.
If such a functional dependence can be expressed analytically, that is, in the form of one or more formulas, then it becomes possible to study it by means of mathematical analysis.
This refers to the possibility of clarifying the behavior of a function when changing one or another variable size(where the function increases, where it decreases, where it reaches a maximum, etc.).
The application of differential calculus to the study of a function is based on a very simple connection that exists between the behavior of a function and the properties of its derivative, primarily its first and second derivatives.
Let's consider how we can find intervals of increasing or decreasing function, that is, intervals of its monotonicity. Based on the definition of a monotonically decreasing and increasing function, it is possible to formulate theorems that allow us to relate the value of the first derivative of a given function to the nature of its monotonicity.
Theorem 1.1. If the function y
=
f
(
x
)
, differentiable on the interval(
a
,
b
)
, increases monotonically on this interval, then at any point
(
x
) >0;
if it decreases monotonically, then at any point in the interval (
x
)<0.
Proof. Let the functiony = f ( x ) monotonically increases by( a , b ) , This means that for anyone small enough > 0 the following inequality holds:
f ( x - ) < f ( x ) < f ( x + ) (Fig. 1.1).
Rice. 1.1
Consider the limit
.
If > 0, then 
> 0 if< 0, то
< 0.
In both cases, the expression under the limit sign is positive, which means the limit is positive, that is ( x )>0 , which was what needed to be proven. The second part of the theorem, related to the monotonic decrease of the function, is proved in a similar way.
Theorem 1.2. If the function y = f ( x ) , continuous on the segment[ a , b ] and is differentiable at all its interior points, and, in addition, ( x ) >0 for anyone x ϵ ( a , b ) , then this function increases monotonically by( a , b ) ; If
( x ) <0 for anyone xϵ ( a , b ), then this function decreases monotonically by( a , b ) .
Proof. Let's take ϵ ( a , b ) And ϵ ( a , b ) , and< . According to Lagrange's theorem
( c ) = .
But ( c )>0 and > 0, which means ( > 0, that is
(. The result obtained indicates a monotonic increase in the function, which is what needed to be proven. The second part of the theorem is proved in a similar way.
Extrema of the function
When studying the behavior of a function, a special role is played by the points that separate from each other the intervals of monotonic increase from the intervals of its monotonic decrease.
Definition 2.1. Dot called the maximum point of the function
y
=
f
(
x
)
, if for any, however small ,
(
< 0
, а точка
is called a minimum point if (
> 0.
The minimum and maximum points are collectively called extremum points. The piecewise monotonic function of such points has a finite number on a finite interval (Fig. 2.1).
Rice. 2.1
Theorem 2.1 (necessary condition for the existence of an extremum). If differentiable on the interval( a , b ) function has at point from this interval is the maximum, then its derivative at this point is equal to zero. The same can be said about the minimum point .
The proof of this theorem follows from Rolle’s theorem, in which it was shown that at the points of minimum or maximum = 0, and the tangent drawn to the graph of the function at these points is parallel to the axisOX .
It follows from Theorem 2.1 that if the functiony = f ( x ) has a derivative at all points, then it can reach an extremum at those points where = 0.
However, this condition is not sufficient, since there are functions for which the specified condition is satisfied, but there is no extremum. For example, the functiony= at a point x = 0 the derivative is zero, but there is no extremum at this point. In addition, the extremum may be at those points where the derivative does not exist. For example, the functiony = | x | there is a minimum at the pointx = 0 , although the derivative does not exist at this point.
Definition 2.2. The points at which the derivative of a function vanishes or has a discontinuity are called critical points of this function.
Consequently, Theorem 2.1 is not sufficient for determining extreme points.
Theorem 2.2 (sufficient condition for the existence of an extremum). Let the function y = f ( x ) continuous on the interval( a , b ) , which contains its critical point , and is differentiable at all points of this interval, with the possible exception of the point itself . Then, if, when moving this point from left to right, the sign of the derivative changes from plus to minus, then this is a maximum point, and, conversely, from minus to plus - a minimum point.
Proof. If the derivative of a function changes its sign when passing a point from left to right from plus to minus, then the function moves from increasing to decreasing, that is, it reaches at the point its maximum and vice versa.
From the above, a scheme for studying a function at an extremum follows:
1) find the domain of definition of the function;
2) calculate the derivative;
3) find critical points;
4) by changing the sign of the first derivative, their character is determined.
The task of studying a function for an extremum should not be confused with the task of determining the minimum and maximum values of a function on a segment. In the second case, it is necessary to find not only the extreme points on the segment, but also compare them with the value of the function at its ends.
Intervals of convex and concave functions
Another characteristic of the graph of a function that can be determined using the derivative is its convexity or concavity.
Definition 3.1. Function y = f ( x ) is called convex on the interval( a , b ) , if its graph is located below any tangent drawn to it on a given interval, and vice versa, it is called concave if its graph is above any tangent drawn to it on a given interval.
Let us prove a theorem that allows us to determine the intervals of convexity and concavity of a function.
Theorem 3.1. If at all points of the interval( a , b ) second derivative of the function ( x ) is continuous and negative, then the functiony = f ( x ) is convex and vice versa, if the second derivative is continuous and positive, then the function is concave.
We carry out the proof for the interval of convexity of the function. Let's take an arbitrary pointϵ ( a , b ) and draw a tangent to the graph of the function at this pointy = f ( x ) (Fig. 3.1).
The theorem will be proven if it is shown that all points of the curve on the interval( a , b ) lie under this tangent. In other words, it is necessary to prove that for the same valuesx curve ordinatesy = f ( x ) less than the ordinate of the tangent drawn to it at the point .
Rice. 3.1
For definiteness, we denote the equation of the curve: = f ( x ) , and the equation of the tangent to it at the point :
- f ( ) = ( )( x - )
or
= f ( ) + ( )( x - ) .
Let's make up the difference And :
- = f(x) – f( ) - ( )(x- ).
Apply to differencef ( x ) – f ( ) Lagrange's mean value theorem:
- = ( )( x - ) - ( )( x - ) = ( x - )[ ( ) - ( )] ,
Where ϵ ( , x ).
Let us now apply Lagrange's theorem to the expression in square brackets:
- = ( )( - )( x - ) , Where ϵ ( , ).
As can be seen from the figure,x > , Then x - > 0 And - > 0 . Moreover, according to the theorem, ( )<0.
Multiplying these three factors, we get that , which was what needed to be proven.
Definition 3.2. The point separating the convex interval from the concave interval is called the inflection point.
From Definition 3.1 it follows that at a given point the tangent intersects the curve, that is, on one side the curve is located below the tangent, and on the other, above.
Theorem 3.2. If at the point second derivative of the function
y = f ( x ) is equal to zero or does not exist, and when passing through a point the sign of the second derivative changes to the opposite, then this point is an inflection point.
The proof of this theorem follows from the fact that the signs ( x ) on opposite sides of the point are different. This means that on one side of the point the function is convex, and on the other it is concave. In this case, according to Definition 3.2, the point is the inflection point.
The study of a function for convexity and concavity is carried out according to the same scheme as the study for an extremum.
4. Asymptotes of the function
In the previous paragraphs, methods for studying the behavior of a function using the derivative were discussed. However, among the questions related to the complete study of a function, there are also those that are not related to the derivative.
So, for example, it is necessary to know how a function behaves when a point on its graph moves infinitely away from the origin. This problem can arise in two cases: when the argument of a function goes to infinity and when, during a discontinuity of the second kind at the end point, the function itself goes to infinity. In both of these cases, a situation may arise when the function tends to some straight line, called its asymptote.
Definition . Asymptote of the graph of a functiony = f ( x ) is a straight line that has the property that the distance from the graph to this straight line tends to zero as the graph point moves indefinitely from the origin.
There are two types of asymptotes: vertical and oblique.
Vertical asymptotes include straight linesx
=
, which have the property that the graph of the function in their vicinity goes to infinity, that is, the condition is satisfied: 
.
Obviously, the requirement of the specified definition is satisfied here: the distance from the graph of the curve to the straight linex = tends to zero, and the curve itself goes to infinity. So, at points of discontinuity of the second kind, functions have vertical asymptotes, for example,y= at a point x = 0 . Consequently, determining the vertical asymptotes of a function coincides with finding discontinuity points of the second kind.
Oblique asymptotes are described by the general equation of a straight line on a plane, that isy = kx + b . This means that, unlike vertical asymptotes, here it is necessary to determine the numbersk And b .
So let the curve = f ( x ) has an oblique asymptote, that is, atx the points of the curve come as close as desired to the straight line = kx + b (Fig. 4.1). Let M ( x , y ) - a point located on a curve. Its distance from the asymptote will be characterized by the length of the perpendicular| MN | .
The reference points when studying functions and constructing their graphs are characteristic points - points of discontinuity, extremum, inflection, intersection with coordinate axes. Using differential calculus, it is possible to establish the characteristic features of changes in functions: increase and decrease, maximums and minimums, the direction of convexity and concavity of the graph, the presence of asymptotes.
A sketch of the graph of the function can (and should) be drawn after finding the asymptotes and extremum points, and it is convenient to fill out the summary table of the study of the function as the study progresses.
The following function study scheme is usually used.
1.Find the domain of definition, intervals of continuity and breakpoints of the function.
2.Examine the function for evenness or oddness (axial or central symmetry of the graph.
3.Find asymptotes (vertical, horizontal or oblique).
4.Find and study the intervals of increase and decrease of the function, its extremum points.
5.Find the intervals of convexity and concavity of the curve, its inflection points.
6.Find the intersection points of the curve with the coordinate axes, if they exist.
7.Compile a summary table of the study.
8.A graph is constructed, taking into account the study of the function carried out according to the points described above.
Example. Explore function
and build its graph.
7. Let’s compile a summary table for studying the function, where we will enter all the characteristic points and the intervals between them. Taking into account the parity of the function, we obtain the following table:
Chart Features |
||||
[-1, 0[ |
Increasing |
Convex |
||
(0; 1) – maximum point |
||||
]0, 1[ |
Descending |
Convex |
||
The point of inflection forms with the axis Ox obtuse angle |
Constructing a graph of a function using singular points includes the study of the function itself: determining the range of permissible values of the argument, determining the range of variation of the function, determining whether the function is even or odd, determining the breakpoints of the function, finding intervals of constant sign of the function, finding asymptotes of the graph of the function. Using the first derivative, you can determine the intervals of increase (decrease) of the function and the presence of extremum points. Using the second derivative, you can determine the intervals of convexity (concavity) of the function graph, as well as inflection points. At the same time, we believe that if at some point xo tangent to the graph of the function above the curve, then the graph of the function at this point has convexity; if the tangent is below the curve, then the graph of the function at this point has a concavity.
y(x) = x³/(x²+3)
1. Function study.
a) Range of permissible values of the argument: (-∞,+∞).
b) Area of change of the function: (-∞, +∞).
c) The function is odd, because y(-x) = -y(x), those. the graph of the function is symmetrical about the origin.
d) The function is continuous, there are no discontinuity points, therefore, there are no vertical asymptotes.
e) Finding the equation of oblique asymptote y(x) = k∙x + b, Where
k = /x And b = 
In this example, the asymptote parameters are respectively equal:
k = , because the highest degree of the numerator and denominator are the same, equal to three, and the ratio of the coefficients at these highest degrees is equal to one. When x→ + ∞ the third remarkable limit was used to calculate the limit.
b = = = 0, when calculating the limit at x→ + ∞ used the third remarkable limit. So, the graph of this function has a slanted asymptote y=x.
2.
y´= /(x²+3)² - the derivative is calculated using the quotient differentiation formula.
a) Determine the zeros of the derivative and the discontinuity point, equating the numerator and denominator of the derivative to zero, respectively: y´=0, If x=0. The 1st derivative has no discontinuity points.
b) We determine the intervals of constant sign of the derivative, i.e. intervals of monotonicity of the function: at -∞
3. Studying a function using the 2nd derivative.
Using the formula for differentiating quotients and making algebraic transformations, we obtain: y´´ = /(x²+3)³
a) Determine the zeros of the 2nd derivative and intervals of constant sign: y´´ = 0, If x=0 And x= + 3 . The 2nd derivative has no discontinuity points.
b) Let us determine the intervals of constancy of the 2nd derivative, i.e. intervals of convexity or concavity of the graph of a function. At -∞
Example: Explore a function and graph it y(x)=((x-1)²∙(x+1))/x
1.Function study.
a) Range of acceptable values: (-∞,0)U(0,+∞).
b) Area of change of the function: (-∞,+∞).
d) This function has a discontinuity point of the 2nd kind at x=0.
e) Finding asymptotes. Because the function has a discontinuity point of the 2nd kind at x=0, then consequently the function has a vertical asymptote x=0. This function has no oblique or horizontal asymptotes.
2.Studying a function using the 1st derivative.
Let's transform the function by performing all the algebraic operations. As a result, the form of the function will be significantly simplified: y(x)=x²-x-1+(1/x). It’s very easy to take the derivative from the sum of the terms and we get: y´ = 2x – 1 –(1/x²).
a) Determine the zeros and discontinuity points of the 1st derivative. We bring the expressions for the 1st derivative to a common denominator and, equating the numerator and then the denominator to zero, we obtain: y´=0 at x=1, y´ - does not exist when x=0.
b) Let us determine the intervals of monotonicity of the function, i.e. intervals of constant sign of the derivative. At -∞<x<0
And 0
3.
y´´= 2 + 2/x³. Using the 2nd derivative, we determine the intervals of convexity or concavity of the function graph, as well as, if any, inflection points. Let us present the expression for the second derivative to the common denominator, and then, equating the numerator and denominator to zero in turn, we obtain: y´´=0 at x=-1, y´´- does not exist when x=0.
At -∞
rice. 4 fig. 5
Example: Explore a function and graph it y(x) = ln (x²+4x+5)
1.Function study.
a) Range of permissible argument values: the logarithmic function exists only for arguments strictly greater than zero, therefore, x²+4x+5>0 – this condition is satisfied for all values of the argument, i.e. O.D.Z. – (-∞, +∞).
b) Area of change of the function: (0, +∞). Let's transform the expression under the logarithm sign and equate the function to zero: ln((x+2)²+1) =0. Those. the function goes to zero when x=-2. The graph of the function will be symmetrical with respect to the straight line x=-2.
c) The function is continuous and has no breakpoints.
d) The graph of the function has no asymptotes.
2.Studying a function using the 1st derivative.
Using the rule for differentiating a complex function, we get: y´= (2x+4)/(x²+4x+5)
a) Let us determine the zeros and discontinuity points of the derivative: y´=0, at x=-2. The first derivative has no discontinuity points.
b) We determine the intervals of monotonicity of the function, i.e. intervals of constant sign of the first derivative: at -∞<x<-2
derivative y´<0,
therefore, the function decreases; when -2
3.Study of the function in terms of the 2nd derivative.
Let's represent the first derivative in the following form: y´=2∙(x+2)/(1+(x+2)²). y´´=2∙(1-(x+2)²/(1+(x+2)²)².
a) Let us determine the intervals of constant sign of the second derivative. Since the denominator of the 2nd derivative is always non-negative, the sign of the second derivative is determined only by the numerator. y´´=0 at x=-3 And x=-1.
At -∞
Example: Explore a Function and Plot a Graph y(x) = x²/(x+2)²
1.Function study.
a) Range of permissible values of the argument (-∞, -2)U(-2, +∞).
b) Area of change of function².
a) Let us determine the zeros and intervals of constant sign of the second derivative. Because Since the denominator of the fraction is always positive, the sign of the second derivative is completely determined by the numerator. At -∞
