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Complete investigation of the function and differential calculus

Having received extensive knowledge in working with functions, we are armed with a sufficient set of tools that allow us to conduct 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 wide variety of problems in the fields of physics, mathematics and engineering, it becomes necessary to investigate the functional dependence between the variables participating 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.

Carry out a full study of the function - it is to find out and identify 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 for which a complete investigation of the function is carried out. Examples of lists of mathematical research conducted are reduced to finding almost identical moments. An approximate analysis plan assumes 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.

In considering only some of the items in this plan, it 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 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 decrease, increasing the function, they also received the name of intervals of monotonicity.

For this it is sufficient to determine the sign of the first derivative 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 determine the sections of the graph, called convexities, and also the concavities of the function. It is proved that if in the course of the 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 change in the 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 on the above points, but the use of differential calculus greatly simplifies this process. In this case, 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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