Valerii Gilev

The paper deals with the problems that lead to the concept of a derivative: the tangent to the curve and the instantaneous rate of change of the function. A generalization function is introduced, which is a derivative, but it is determined without using the theory of limits. The method by which the generalization function is defined is called the generalization method.

The paper considers the definition of convexity (up, down) of the graph of function using the method of tangents, chords and analytical method. The author formulates signs and related methods to study functions for bulge graphics using the second derivative and functions of generalization.

The paper considers the definition of monotonicity of the function using visual, verbal, analytical and geometric methods. The author formulates signs and appropriate ways to study functions on monotonicity using generalizations and first order derivative.

The paper solves the problem of elementary functions study on monotony and convexity graphics, without using derivative, but by definition. The solution is the method that is called the method of generalization. The result is the function, which determines the intervals of constant sign intervals of monotonicity and convexity of the graph of original function.