Curve sketching
Under Kurvendiskussion is understood in mathematics, the study of the graph of a function on its geometric properties, such as intersections with the coordinate axes, high and low points, turning points, possibly saddle and flat points, asymptotes, behavior at infinity, etc. This information allows it to make a sketch of the graph, from which all these are characteristic of the functional properties are read directly.
However, it is not the goal of curve sketching to support the people to produce the most accurate possible drawing of the graph of the function: everyone can function plotter (such as a graphing calculator, a smart phone with appropriate software, a spreadsheet or computer algebra software) better.
The aim of the curve discussion is rather
- To determine the coordinates of the characteristic points of the graph exactly ( from a function plot can be read only approximate values );
Figure 2: Plot of the function f in the cut-out: 1.995
In addition, a curve very similar discussion can also perform at functions that depend on many variables ( ie, for example, and instead of only ).
Visualization of such a function in 2D or 3D is not possible.
In a curve sketching the set of all real numbers is almost always assumed to be basic amount.
The maximum definition of a function, therefore, the set of real numbers for which the function value is defined.
For all rational functions ( polynomial functions ) the maximum domain of definition is the same.
In fractional rational functions are all real numbers except the zeros of the denominator for maximum definition area.
Examples:
The denominator of the fraction must not be equal to 0.
Therefore, the values 0 and 3 are excluded.
The cube root, that is, the calculation expression of the square root must be greater or equal to 0.
This condition is fulfilled exactly for numbers whose absolute value is less than or equal to 5.
The natural logarithm function is defined only for positive argument.
So you have to apply.
The zero setting of a function, and thus to find the intersection points of the graph with the axis, the amount of the solution of the equation is calculated.
The exact procedure depends on which function is investigated.
Is the function, for example represented by a fraction term the counter is set to 0 to obtain the zeroes.
To determine the intersection point of the graph to the axis, is used for the value 0.
The Y -axis intercept is then at consequently.
In answering the question whether the graph of the given function is symmetric in any way, several cases must be considered.
For all rational functions, this condition means that only even exponents occur.
The graph of a rational integral function is exactly then point symmetric about the origin, if only odd exponents occur.
Axisymmetrically include the graph of the quadratic function.
The axis of symmetry is obtained in this case from the coordinate of the ( parabolic ) vertex.
Or ( equivalently )
The graphs of all cubic functions are point- symmetric.
Center of symmetry, the ( single ) turning point (see below), respectively.
The extreme points - that is, high and low points - to determine a continuously differentiable function of the first derivative is equal to 0 is set, that is, the amount of the solution of the equation is calculated.
All solutions of this equation are possible extreme points.
However, the condition is only a necessary condition for extreme points.
A body with a slope of 0 could also be a saddle point.
The standard sample is at the position 0 is valid, although, the point is not an extreme point, but a saddle point.
To demonstrate the extreme point property is thus requires the sufficient conditions mentioned below.
It lights up a vividly that the tangent must pass to a function graph in an extreme point parallel to the axis.
The slope of such a tangent must therefore have the value 0.
Precise applies:
At the edge of the domain and at points where the given function is not differentiable, the requirements of this condition are not met.
Relative Extrema in such places so can not be determined by setting to zero the derivative in general.
The following sufficient condition often provides a convenient way to provide evidence for a relative extremum and at the same time to determine the type (maximum or minimum).
Since the second derivative is needed, it is called sometimes the test.
Applies in one place at the same time
So at this point has a relative minimum.
On the other hand it is also true
So at this point has a relative maximum.
Example:
By setting to zero the first derivative of () is obtained.
Insertion into the second derivative yields (condition for a relative maximum ).
The graph of has exactly one extreme point, namely a high point with the coordinate 4
Occasionally applies both.
In this case, further studies are needed to determine whether an extreme point or not.
In practice this means that as long as must be inferred, to a derivation of even order - fourth, sixth, ... order - is present, which is at this point other than 0.
The simplest example is.
The first three discharges at the location, so each has a value of 0, and only the fourth derivative allows for the detection of the extremum.
The positive sign indicates that there is a relative minimum is at the location.
In rare cases fail even this more general criterion, namely when all the derivatives at the point are equal to 0.
Another method for detecting the extreme point feature does not require the calculation of the second derivative.
It examines whether the first derivative at the point considered changes its sign.
This method, sometimes referred to as change of sign method ( VZW ) method, can be clearly interpreted as follows:
Iterates to the function graph in the vicinity of a high point from left to right, so the appearance of this curve can be described as follows:
The same, only in reverse, applies to low points.
Applies in one place at the same time
So at this point has a relative minimum.
Applies in one place at the same time
So at this point has a relative maximum.
Example:
Zero the derivative ( ) returns point with horizontal tangent and therefore as a candidate for an extreme point.
In test Considering VZW values that are smaller or larger than 0.
Shows that a trough is present.
One problem with the VZW tests is that the sign of the derivative must not be determined only for a single point, but for a whole interval.
In school mathematics is therefore often determined only for a single point of the interval, the sign and concludes that this sign throughout the interval applies.
This procedure is allowed when the function in the interval is continuously differentiable.
Historical side note: The determination of the extrema of the tangent slope was first proposed by Fermat in a letter to Descartes - before there was the dissipation term.
As turning points are referred to those points where the given function graph between left and right curvature changes.
The type of curvature can - under certain conditions - see the sign of the second derivative.
A positive sign suggesting links curvature, negative sign on right curvature.
The standard method for determining the turning points is therefore used the second derivative equal to 0, the solutions of the equation (see necessary condition ) come as a turning points in question.
With one of the sufficient conditions listed below then you can prove that there is a genuine turning point.
At Bavarian vocational high schools turning points are defined as extreme points of the first derivative.
The latter condition is not sufficient, so that further studies should be conducted.
A sufficient condition frequently used for the detection of turning points based on the third derivative:
Then the graph of at the site has a turning point.
Is at the position adjacent to the second derivative and third derivative is 0, so the latter criterion fails.
In this case to investigate whether the second derivative changes in the sign.
For this purpose is selected to a value smaller than and greater than the zero of the second derivative ( that is, the coordinate of the point of inflection ).
These two values are used in the second derivative.
If the value of the second derivative at this point is greater than 0, then there is a turning point with the transition to a "left curve ", it is less than 0, then there is a turning point with the transition to a " right bend ".
Or at the same time
Then the graph of at the site has a turning point.
A turning point with a horizontal tangent at the same time is called a saddle point or terrace point.
Therefore applies to him, and, as in the example of the function using the equation
In place.
However, this is not a sufficient criterion, it can also be, and without a saddle point occurs, as shown in the following example:
Only when it is, a saddle point is proven;
in more general terms: There is a turning point, when the degree of the first derivative different from 0 is odd;
is the level just as it is an extremum.
In cases like
However the third derivative useless, since it disappears at the point 0.
This further helps the sign change criterion.
A pole is fractional-rational functions if and only at a point before when the denominator has a zero at and the numerator has a zero of order at a lower or no zero at.
, Both the numerator and the denominator polynomial in a zero point, and is the order of the zero in the numerator is not less than that of the denominator, there is a continuous gap definition recoverable.
In high school mathematics, there are other types of non- defined locations that are neither gaps nor liftable poles.
In the case of fractional rational functions at a location in front of an ever -recoverable definition gap, if not only one zero of the denominator, but also a zero of the counter at least equal to a large degree.
In this case, you can cancel out the corresponding linear factor.
Example: if a recoverable definition gap at the site.
By shortening the factor arises:
Another way to test whether there is an ever -recoverable definition gap at the point is to calculate the threshold.
If this limit exists and is finite, there is an ever -recoverable gap.
To find out the behavior at infinity, the value of the function is examined, when grows beyond all limits, so to go:
The same for.
The test function is:
The graph of the function is shown in black in the image, also the first (red) and second (blue) are registered derivation:
Through trial and error ( for example, by making a table of values ) or according to the Gaussian inference with the knowledge that every integer zero point must be the absolute divider member 8 can find a zero.
If there were no such easily recognizable zero, then the formula of Cardano could apply for 3rd degree equations or Newtonian approximation methods.
To zero belongs to the linear factor.
To find the other zeros of a polynomial division is performed by this linear factor and the result set equal to 0.
A little faster obtains the result with the Horner scheme.
In this manner reduces the degree of the equation by 1
The new equation has no solution.
is thus the only real zero.
The first derivative function is
This has zero points and from.
This means that there may be extreme points.
The second derivative function
Has on the above points, the function values
Or
Therefore, the function graph has (positive first derivative equal to 0, the second derivative ) at a high point (first derivative equal to 0, the second derivative is negative ) and at a low point.
The coordinates of the two extreme points are found by applying the coordinates in.
To determine the turning points, the second derivative is set equal to zero:
The only solution of this equation.
To confirm that there is a genuine turning point, one can use the third derivative.
because of
Secured the turning point property.
The coordinate of the turning point is given by
Pole does not exist for polynomials.
As a polynomial of odd order (highest exponent in ) goes against the function or if against or goes.
Given is the function with the equation
The function is only defined where the denominator is not 0.
The investigation of the denominator to zero points yields:
The quadratic equation has in a double solution.
Only thus will the denominator 0 The domain is therefore
The set of real numbers, except the first, the denominator can - decomposed into linear factors - as
Be written.
The condition for zeros is.
For this purpose, it is sufficient that the counter is 0, as long as not yet, the denominator becomes 0.
Investigation of the counter to zeros yields:
The counter has in a simple zero and a double at.
Both locations are in the domain.
So has the zeros as well.
The counter can - decomposed into linear factors - as
Be written.
In the context of school mathematics it is often value such that, with each of the index for " zero " is written to.
In place of the denominator has a double root, without at the same time the counter is 0.
Thus, there is a pole in front.
If the counters are also 0, it must be a pole for the order of the zero of the denominator is greater than the order of the numerator be zero.
If the denominator of a fractional rational function at a point is equal to 0, the function at this point is not defined.
If the denominator is 0, but the counter is not 0, the function has a pole at this point ( " infinity point ").
The graph of the function is examined at this point symmetry.
Often only one examination is performed on the axis of symmetry for the axis (condition) and in point symmetry to the origin of the coordinate system (condition).
The example in
Replaced by.
After multiplying arises
Since neither with nor with matches the graph of neither axis is symmetrical to the axis or point symmetry to the origin.
Somewhat more difficult, the study designed to axial symmetry with respect to an arbitrary axis or on point symmetry with respect to an arbitrary point.
Ways and can be excluded axis of symmetry.
As a center of symmetry ( point symmetry ) would at most the intersection of the asymptotes (see below ), ie the point in question.
However, since, for example, the points do not lie symmetrical with respect to, the graph is not point-symmetric.
To form are the derivatives of
The representation in linear factors is more convenient because it simplifies the bracketing and shortening.
This results first
For the first derivative.
Then the second
And the third derivative
Formed.
For this must be.
It is sufficient to examine the zeros of the counter:
Has the solution.
The second clip has no real solutions.
lies in the domain.
The function value at this point, since this has a zero.
The second derivative at this point, so it is a low point at (2 / 0).
The turning point is determined by the second derivative is equal to 0.
This approach results.
Substituting this value in the third derivative yields.
Thus, there is indeed a turning point before.
The calculation of eventually results in the coordinate.
The desired turning point is thus.
At the pole, ie, at, is a vertical asymptote.
Since the degree of the counter ( 3) to 1 is greater than the denominator of ( 2), to go to to.
The difference indicates that the graph will approach asymptotically a linear function ( straight line).
The linear equation follows by polynomial division:
For going against the last term tends to 0, the equation of the asymptote is therefore
General:
In mathematics education is discussed at least since the 90s, the extent to which the curve discussion by the availability of graphics-capable calculators and dedicated software ( function plotter ) is obsolete.
One criticism is that the curve discussion is a purely computational routine that conveys little understanding.
On the other hand, it is precisely because of relatively popular as a relatively safe vorzubereitendes examination Thread to weaker students.
In the central Abitur examinations, it has been enforced, therefore, that such schematic tasks are very rarely found.
Popular are logged dressed tasks or tasks in which the context knowledge is retrieved, for example, about relationships between the derivative function and output function.
A didactic method is described in zoning.
Domain
Intersection points with the coordinate axes
Symmetry properties
Axis of symmetry with respect to the y-axis
Point symmetry to the origin
Axis of symmetry with respect to an arbitrary axis
Point symmetry with respect to an arbitrary center
Extreme points
Necessary condition
Sufficient condition: value of the second derivative
Sufficient conditions: sign of the first derivative
Turning points
Necessary condition
Sufficient condition: value of the third derivative
Sufficient conditions: sign of the second derivative
Special case: saddle points
Poles
Gap
Behavior at infinity
Overview of criteria
Example: Quite Rational Function
Zeros
Extreme points
Turning points
Poles and behavior at infinity
Example: Off - rational function
Domain
Zeros
Poles
Symmetry
Derivations
Extreme points
Turning points
Asymptotes
Didactic questions