Mathematics originates from Antiquity. Thanks to her, architecture, construction and military science 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 branches.
In order to be educated, children from the first grade begin to gradually merge into this environment. It is very important to understand mathematics, as it, to one degree or another, occurs to every person throughout his life. This article will analyze one of the key elements - finding and applying derivatives. Not every person can imagine how widely this concept is used. Consider more than 10 applications of derivatives in certain fields or sciences.
Application of the derivative to the study of a function
The derivative is such a limitthe ratio of the increment of a function to the increment of its argument when the exponent of the argument tends to zero. The derivative is an indispensable thing in the study of a function. For example, it can be used to determine the increase and decrease of the latter, extrema, convexity and concavity. Differential calculus is included in the compulsory curriculum for 1st and 2nd 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 domain of definition is set along the abscissa axis, in other words, these are numerical values on the OX axis. Often the scope is already set, but if it is not, then the value of the x argument should be evaluated. For example, 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.
Increase and decrease
The use of the derivative to study functions for monotonicity can be considered from two positions. A monotonic function is a category that has only positive values of the derivative, or only negative values. In simple words, the function only increases or only decreases over the entire interval under study:
- Increase parameter. Functionf(x) will increase if the derivative of f`(x) is greater than zero.
- Descending parameter. The function f(x) will decrease if the derivative of f`(x) is less than zero.
Tangent and Slope
The application of the derivative to the study of a function is also determined by the tangent (straight line directed at an angle) to the graph of the function at a given point. Tangent at a point (x0) - a line that passes through a point and belongs to the function whose coordinates are (x0, f(x0)) and having slope f`(x0).
y=f(x0) + f`(x0)(x - x0) - the equation of the tangent to the given point of the graph of the function.
Geometric meaning of the derivative: the derivative of the function f(x) is equal to the slope 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 OX axis (abscissa) in the positive direction. This corollary is fundamental to the application of the derivative to the graph of a function.
Extremum points
Applying a derivative to a study involves finding high and low points.
In order to find and determine the minimum and maximum points, you must:
- Find the derivative of the function f(x).
- Set the resulting equation to zero.
- Find the roots of the equation.
- Find high and low points.
To find extremesfeatures:
- Find the minimum and maximum points using the method above.
- Substitute these points into the original equation and calculate ymax and ymin
The maximum point of the function is the largest value of the function f(x) on the interval, in other words xmax.
The minimum point of the function is the smallest value of the function f(x) on the interval, in other words xname
Extremum points are the same as the maximum and minimum points, and the extremum of the function (ymax. and yminimum) - function values that correspond to extremum points.
Convexity and concavity
You can determine the convexity and concavity by resorting to the use of the derivative for plotting:
- A function f(x) examined on the interval (a, b) is concave if the function is located below all of its tangents within this interval.
- The function f(x) studied on the interval (a, b) is convex if the function is located above all its tangents inside this interval.
The point that separates convexity and concavity is called the inflection point of the function.
To find inflection points:
- Find critical points of the second kind (second derivative).
- Inflection points are those critical points that separate two opposite signs.
- Calculate function values at function inflection points.
Partial derivatives
Applicationthere are derivatives of this type in problems where more than one unknown variable is used. Most often, such derivatives are encountered when plotting a function graph, 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 when calculating partial derivatives is to choose one variable and treat the rest as constants. Therefore, when calculating the partial derivative, the constant 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 OX and OY axes, that is, it characterizes the steepness of the depressions and bulges of the constructed surface.
Derivative in physics
The use of the derivative in physics is widespread and important. Physical meaning: the derivative of the path with respect to time is the speed, and the acceleration is the derivative of the speed with respect to time. From the physical meaning, many branches can be drawn to various branches of physics, while completely preserving the meaning of the derivative.
With the help of the derivative, the following values are found:
- Speed in kinematics, where the derivative of the distance traveled is calculated. If the second derivative of the path or the first derivative of the speed is found, then the acceleration of the body is found. In addition, it is possible to find the instantaneous velocity of a material point, but for this it is necessary to know the increment ∆t and ∆r.
- In electrodynamics:calculation of the instantaneous strength of the 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 strength in the conductor.
Derivative in chemistry and biology
Chemistry: The derivative is used to determine the rate of a chemical reaction. The chemical meaning of the derivative: function p=p(t), in this case p is the amount of a substance that enters into a chemical reaction in time t. ∆t - time increment, ∆p - substance quantity increment. The limit of the ratio of ∆p to ∆t, at which ∆t tends to zero, is called the rate of a chemical reaction. The average value of a chemical reaction is the ratio ∆p/∆t. When determining the speed, it is necessary to know exactly all the necessary parameters, conditions, to 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 a derivative is used to calculate the average reproduction rate. Biological meaning: we have a function y=x(t). ∆t - time increment. Then, with the help of some transformations, we obtain the function y`=P(t)=x`(t) - the vital activity of the population of time t (average reproduction rate). This use 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 decidetasks such as finding population, calculating values in seismography, calculating radioactivity of nuclear geophysical indicators, calculating interpolation.
In economics, an important part of 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 values. For example, the highest and lowest labor productivity, costs, profits. Basically, these values are calculated from function graphs, where they find extrema, determine the monotonicity of the function in the desired area.
Conclusion
The role of this differential calculus is involved, as 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. It’s not for nothing that we were taught in high school and university to build complex graphs, explore and work on functions. As you can see, without derivatives and differential calculations, it would be impossible to calculate vital indicators and quantities. Mankind has learned to model various processes and explore them, to solve complex mathematical problems. Indeed, mathematics is the queen of all sciences, because this science underlies all other natural and technical disciplines.
