Having gained extensive knowledge in working with functions, we have armed ourselves with a sufficient set of tools that allow us to conduct a complete study of a specific mathematical pattern in the form of a formula (function). Of course, one could go the simplest, but painstaking way. For example, ask yourself the bounds of the argument, select the interval, calculate the values of the function on it and build a graph. With powerful modern computer systems, this task can be solved in a matter of seconds. But they are not in a hurry to remove from their arsenal a complete study of the functions of mathematics, since it is by these methods that one can evaluate the correct operation of computer systems in solving such problems. With mechanical plotting, we cannot guarantee the accuracy of the above interval in the choice of argument.
And only after a complete study of the function is carried out, you can be sure that all the nuances of the “behavior” of such a function are taken into account, not on the sample interval, but on the entire range of the argument.
To solve a wide variety of problems in the fields of physics, mathematics, and technology, it becomes necessary to study the functional dependence between the variables involved in the phenomenon under consideration. The latter, defined analytically by one or a set of several formulas, allows the study to be carried out by methods of mathematical analytics.
To conduct a full study of the function is to find out and determine the areas in which it increases (decreases), where it reaches a maximum (minimum), as well as other features of its schedule.
There are certain schemes by which a complete study of the function is performed. Examples of lists of mathematical studies are reduced to finding almost the same points. A sample analysis plan involves the following studies:
- we find the domain of definition of the function, investigate the behavior within its boundaries;
- we carry out the finding of break points with classification using one-sided limits;
- we carry out the determination of asymptotes;
- we find points of extremum and intervals of monotony;
- we determine the inflection points, intervals of concavity and convexity;
- we carry out the construction of a schedule based on the results obtained during the study.
When considering only some points of this plan, it is worth noting that differential calculus turned out to be a very successful tool for studying functions. There are rather simple connections that exist between the behavior of a function and the features of its derivative. To solve this problem, it is sufficient to calculate the first and second derivatives.
Consider the order of finding the intervals of decreasing, increasing function, they still got the name of the intervals of monotony.
To do this, it is enough to determine the sign of the first derivative on a certain interval. If it is constantly greater than zero in the interval, then we can safely judge the monotonous increase in function in this range, and vice versa. Negative values of the first derivative characterize the function as monotonically decreasing.
Using the calculated derivative, we determine the parts of the graph called convexities, as well as concavities of the function. It is proved that if in the course of calculations the derivative of the function was obtained continuous and negative, then this indicates convexity, the continuity of the second derivative and its positive value indicates the concavity of the graph.
Finding the moment when a change of sign occurs at the second derivative or sections where it does not exist, indicates the determination of the inflection point. It is it that is boundary on the intervals of convexity and concavity.
A full study of the function does not end with the above points, but the use of differential calculus greatly simplifies this process. Moreover, the analysis results have the maximum degree of reliability, which allows you to build a graph that fully corresponds to the properties of the studied functions.