USING A FUNCTION DERIVATIVE IN ECONOMIC TASKS

The article focuses on the application of mathematical methods in economics, in particular discussing economic problems that are easily solved using derivatives. The purpose of the article is to show students the way and opportunity to use mathematical methods to solve economic problems. To this end, the article discusses and analyzes several economic tasks in detail, which will be interesting and easy for students to master. I considered the derivative of a function as the rate of change and introduced the definition: The instantaneous rate of change of the function f with respect to x at a point is called the derivative if it exists. With the help of this definition I have discussed and explained Task 1: Suppose that the increase in production of a certain product over a period of time is described by a function And population growth is described by the following function: Where is the number of years from the initial period, then the production of these products per capita is given by the function: Find the growth rate of product production. By solving this task I came to the conclusion that after a year the production of products per capita increases. In the following tasks I used the method of finding the extremum values of a function using a derivative, namely I equated the first-order derivative to 0, found the critical points, and with the help of the second-order derivative I determined the extremes of the function. I discussed task 2: For the production of X volume of products, the firm plans a cost that is calculated by the formulan . For what volume of production will the average cost be the smallest? Find the numerical value of this small expense. Solving this problem, I came to the conclusion that the given function of the average cost takes on the least value when the output volume is a unit , and this value is equal to: which is the marginal cost when producing the volume output. I discussed Task 3: How many products should be sold in order for a firm to profit maximally if the derivative cost function is known: And return function: Here I came to the conclusion: if 600 units of the product are sold, then the firm's profit will be maximum and it will be numerically equal At the end of the article I discussed such a task 4 of applied optimization. What is the minimum amount of material needed to make a 2 liter cylindrical jar? Where I came to the conclusion: the smallest amount of material will be spent to get a cylindrical vessel if we take the height 2 times the radius of the base.