Complete investigation of the function and differential calculus

Having received extensive knowledge in working with functions, wearmed with a sufficient set of tools that allow a full study of a specifically specified mathematical regularity in the form of a formula (function). Of course, one could go the simplest, but painstaking way. For example, specify the boundaries of the argument, select an interval, calculate the values ​​of the function on it, and plot the graph. With powerful modern computer systems, this problem is solved in a matter of seconds. But to remove from their arsenal a full study of the function of mathematics is in no hurry, since it is by these methods that it is possible to evaluate the correctness of the operation of computer systems in solving similar problems. With the mechanical construction of the graph, we can not guarantee the accuracy of the interval specified above in the choice of the argument.

And only after a full investigation of the function is carried out, one can be sure that all the nuances of "behavior" are taken into account not at a sampling interval, but over the entire range of the argument.

To solve a variety of tasks in the fields ofphysics, mathematics, and technology, it becomes necessary to investigate the functional relationship between the variables involved in the phenomenon under consideration. The latter, given analytically by one or a set of several formulas, allows us to carry out research using methods of mathematical analytics.

To conduct a full investigation of a function is to find out and determine the areas on which it increases (decreases), where it reaches a maximum (minimum), as well as other features of its schedule.

There are certain schemes by whicha complete investigation of the function is performed. Examples of lists of mathematical research conducted are reduced to finding almost identical moments. An approximate plan of analysis involves the following studies:

- find the domain of the function definition, investigate the behavior within its boundaries;

- We find the points of discontinuity with the classification by means of unilateral limits;

- we carry out the definition of asymptotes;

- we find extremum points and intervals of monotonicity;

- We determine the points of inflection, intervals of concavity and convexity;

- we perform the construction of the graph on the basis of the results obtained during the research.

When considering only certain points of thisIt should be noted that the differential calculus turned out to be a very successful tool for investigating the function. There are rather simple connections existing between the behavior of the function and the characteristics 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 decrease, increasing the function, they also received the name of intervals of monotonicity.

For this it is sufficient to determine the sign of the firstderivative on a certain interval. If it is constantly greater than zero on a segment, then we can safely judge the monotonic increase of the function in this range, and vice versa. The negative values ​​of the first derivative characterize the function as monotonically decreasing.

Using the calculated derivative, we determineThe sections of the graph, called convexities, and also the concavities of the function. It is proved that if in the course of calculations the derivative of the function is continuous and negative, then this indicates convexity, the continuity of the second derivative and its positive value indicate the concavity of the graph.

Finding the moment when there is a sign changethe second derivative or areas where it does not exist indicates the definition of the inflection point. It is the boundary on the intervals of convexity and concavity.

A full investigation of the function does not end withthe above points, but the use of differential calculus greatly simplifies this process. At the same time, the results of the analysis have a maximum degree of reliability, which makes it possible to construct a graph that completely corresponds to the properties of the functions being studied.

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