Mathematics has its origins since antiquity. Thanks to it, architecture, construction and military affairs gave a new round of development, the achievements that were obtained with the help of mathematics led to the movement of progress. To this day, mathematics remains the main science that is found in all other sectors.
To be educated, children from the first grade begin to gradually pour into this environment. It is very important to understand mathematics, since it, to one degree or another, is encountered by every person throughout his life. This article will look at one of the key elements - finding and using derivatives. Not everyone can imagine how widely this concept is used. Consider more than 10 applications of derivatives in specific fields or sciences.
Applying a derivative to a function study
The derivative is such a limit of the ratio of the increment of a function to an increase in its argument when the exponent of the argument tends to zero. Derivative - an indispensable thing in the study of function. For example, using it, you can determine the increase and decrease of the latter, extremes, convexity and concavity. Differential calculi are included in the compulsory curriculum for first and second year students of mathematical universities.
Scope and function zeros
The first stage of any study of the graph begins with finding out the domain of definition, in more rare cases, the value. The definition area is set along the abscissa axis, in other words, these are numerical values on the OX axis. Often the domain of definition is already set, but if it is not set, then you should evaluate the value of the argument x. Suppose if for some values of the argument the function does not make sense, then this argument is excluded from the scope.
The zeros of the function are found in a simple way: the function f (x) should be equated to zero and the resulting equation should be solved with respect to one variable x. The obtained roots of the equation are the zeros of the function, that is, in these x the function is 0.
Ascending and descending
The use of a derivative for studying functions of monotonicity can be considered from two perspectives. A monotonic function is a category that has only positive values of the derivative, or only negative ones. In simple words - the function only increases or only decreases over the entire studied interval:
- Ascending parameter. The function f (x) will increase if the derivative f` (x) is greater than zero.
- Decrease parameter. The function f (x) will decrease if the derivative f` (x) is less than zero.
Tangent and angular coefficient
The application of the derivative to the investigation of a function is also determined by the tangent (a line directed at an angle) to the graph of the function at a given point. The tangent at the point (x 0 ) is the line that passes through the point and belongs to a function whose coordinates (x 0 , f (x 0 )) and have an angular coefficient f` (x 0 ).
y = f (x 0 ) + f` (x 0 ) (x - x 0 ) is the equation of the tangent to a given point in the graph of the function.
The geometric meaning of the derivative: the derivative of the function f (x) is equal to the angular coefficient of the formed tangent to the graph of this function at a given point x. The angular coefficient, in turn, is equal to the tangent of the angle of inclination of the tangent to the axis OX (abscissa) in the positive direction. This corollary is fundamental to the application of a derivative to a function graph.
Extremum points
Applying the derivative to the study involves finding the maximum and minimum points.
In order to find and determine the minimum and maximum points, it is necessary:
- Find the derivative of the function f (x).
- Equate the resulting equation to zero.
- Find the roots of the equation.
- Determine the maximum and minimum points.
To find function extremes:
- Find the minimum and maximum points by the method above.
- Substitute these points in the original equation and calculate y naib. and y name
The maximum point of the function is the largest value of the function f (x) on the gap, in other words x naib.
The minimum point of the function is the smallest value of the function f (x) on the interval, in other words x
The points of the extremum are the same as the points of the maximum and minimum, and the extremum of the function (y most and least) are the values of the functions that correspond to the points of the extremum.
Bulges and concavities
You can determine the convexity and concavity by resorting to the use of the derivative for graphing:
- The function f (x) studied on the interval (a, b) is concave if the function is located below all its tangents that are inside this interval.
- The function f (x) studied on the interval (a, b) is convex if the function is located above all of its tangents inside this interval.
The point that separates the convexity and concavity is called the inflection point of the function.
To find the inflection points:
- Find critical points of the second kind (second derivative).
- Inflection points are those critical points that separate two opposite signs.
- Calculation of function values at inflection points of a function.
Private derivatives
The use of derivatives of this type is in problems where more than one unknown variable is used. Most often, such derivatives are encountered when plotting a function, to be more precise, surfaces in space where instead of two axes there are three, therefore, three quantities (two variables and one constant).
The basic rule in calculating partial derivatives is to choose one variable, and consider the rest as constants. Therefore, when calculating the partial derivative, the constant value becomes as if a numerical value (in many tables of derivatives they are denoted as C = const). The meaning of such a derivative is the rate of change of the function z = f (x, y) along the axis OX and OY, that is, it characterizes the steepness of the depressions and convexities of the constructed surface.
Derivative in Physics
The use of a derivative in physics is widespread and significant. Physical meaning: the derivative of the path in time is speed, and the acceleration is the derivative of speed in time. From the physical sense, many branches can be drawn into various branches of physics, while completely preserving the meaning of the derivative.
Using the derivative, the following quantities are found:
- Speed in kinematics, where the derivative of the distance traveled is calculated. If there is a second derivative of the path or the first derivative of speed, then the acceleration of the body is found. In addition, it is possible to find the instantaneous velocity of a material point, however, for this it is necessary to know the increment ∆t and ∆r.
- In electrodynamics: calculating the instantaneous strength of an alternating current, as well as the EMF of electromagnetic induction. By calculating the derivative, you can find the maximum power. The derivative of the amount of electric charge is the current in the conductor.
Derivative in chemistry and biology
Chemistry: A derivative is used to determine the rate of a chemical reaction. The chemical meaning of the derivative: the function p = p (t), in this case p is the amount of a substance that enters into a chemical reaction in time t. ∆t is the increment of time, ∆p is the increment of the amount of substance. The limit of the ratio ∆p to ∆t at which ∆t tends to zero is called the rate of the chemical reaction. The average value of the chemical reaction is the ∆p / ∆t ratio. When determining the speed, it is necessary to know exactly all the necessary parameters, conditions, know the aggregate state of the substance and the flow medium. This is a rather large aspect in chemistry, which is widely used in various industries and human activities.
Biology: the concept of derivative is used in calculating the average rate of reproduction. Biological meaning: we have the function y = x (t). ∆t is the time increment. Then, using some transformations, we obtain the function y` = P (t) = x` (t) - the vital activity of a population of time t (average reproduction rate). This application of the derivative allows you to keep statistics, track the rate of reproduction, and so on.
Derivative in Geography and Economics
The derivative allows geographers to solve problems such as finding the population, calculate values in seismography, calculate the radioactivity of nuclear-geophysical indicators, calculate interpolation.
In economics, an important part of the calculations is the differential calculus and the calculation of the derivative. First of all, this allows us to determine the limits of the necessary economic quantities. For example, the greatest and least labor productivity, costs, profits. Basically, these values are calculated from the graphs of the functions where the extremes are found, determine the monotonicity of the function in the desired area.
Conclusion
The role of this differential calculus is involved, as was noted in the article, in various scientific structures. The use of derivative functions is an important element in the practical part of science and production. No wonder we were taught in high school and university to build complex graphics, to research and work on functions. As we see, without derivatives and differential calculus it would be impossible to calculate vital indicators and quantities. Mankind has learned to model various processes and investigate them, solve complex mathematical problems. Indeed, mathematics is the queen of all sciences, because this science underlies all other natural and technical disciplines.