Mathematical problems find their application in many sciences. These include not only physics, chemistry, technology and economics, but also medicine, ecology and other disciplines. One of the important concepts that must be mastered in order to find solutions to important dilemmas is the derivative of a function. The physical meaning of it is not at all as difficult to explain as it may seem to the uninitiated at the heart of the matter. It is enough to find suitable examples of this in real life and ordinary everyday situations. In fact, any motorist copes with a similar task every day when he looks at the speedometer, determining the speed of his car in a particular instant of a fixed time. Indeed, it is in this parameter that the essence of the physical meaning of the derivative lies.
How to find the speed
Any fifth grader can easily determine the speed of a person on the road, knowing the distance traveled and the travel time. To do this, the first of the given values should be divided into the second. But not every young mathematician knows that at the moment he finds the ratio of increments of function and argument. Indeed, if we imagine the movement in the form of a graph, putting off the path along the ordinate axis, and the time along the abscissa, this will be so.
However, the speed of a pedestrian or any other object, which we determine over a large section of the path, considering the movement uniform, may very well change. In physics, many forms of motion are known. It can be accomplished not only with constant acceleration, but also slow down and increase arbitrarily. It should be noted that in this case the line describing the movement will no longer be a straight line. Graphically, it can accept the most complex configurations. But for any of the points in the graph, we can always draw a tangent represented by a linear function.
To clarify the parameter of movement change depending on time, it is necessary to reduce the measured segments. When they become infinitesimal, the calculated speed will be instantaneous. This experience helps us define the derivative. Its physical meaning also logically follows from such reasoning.
In terms of geometry
It is known that the greater the speed of the body, the steeper the graph of the dependence of displacement on time, and hence the angle of inclination of the tangent to the graph at some specific point. An indicator of such changes can be the tangent of the angle between the abscissa axis and the tangent line. It just determines the value of the derivative and is calculated by the ratio of the lengths of the opposite to the adjacent leg in a right-angled triangle formed by a perpendicular, omitted from a point on the abscissa axis.
This is the geometric meaning of the first derivative. The physical is revealed in the fact that the size of the opposite leg in our case represents the distance traveled, and the adjacent - time. Moreover, their ratio is speed. And again we come to the conclusion that the instantaneous speed, determined by the tendency of both intervals to infinitesimal, is the essence of the concept of a derivative, indicating its physical meaning. The second derivative in this example will be the acceleration of the body, which, in turn, demonstrates the degree of change in speed.
Examples of finding derivatives in physics
A derivative is an indicator of the rate of change of any function, even when it comes to movement in the literal sense of the word. To clearly demonstrate this, we give a few specific examples. Suppose, the current strength, depending on time, varies according to the following law: I = 0.4t 2 . It is required to find the value of the speed with which this parameter changes at the end of the 8th second of the process. Note that the sought quantity itself, as can be judged from the equation, is constantly increasing.
The solution requires finding the first derivative, the physical meaning of which was considered earlier. Here dI / dt = 0.8 t . Next, we find it at t = 8 , we find that the speed at which the current changes is equal to 6.4 A / s . Here it is believed that the current is measured in amperes, and time, respectively, in seconds.
Everything is changeable
The visible surrounding world, consisting of matter, constantly undergoes changes, being in motion of various processes proceeding in it. You can use a variety of parameters to describe them. If they are combined by dependency, they are mathematically written as a function that clearly shows their changes. And where there is movement (in whatever form it is expressed), there also exists a derivative, the physical meaning of which we are considering at the moment.
On this occasion, the following example. Suppose body temperature changes according to the law T = 0.2 t 2 . You should find the speed of its heating at the end of the 10th second. The problem is solved in a manner similar to that described in the previous case. That is, we find the derivative and substitute the value for t = 10 , we get T = 0.4 t = 4. Therefore, the final answer is 4 degrees per second, that is, the heating process and the change in temperature, measured in degrees, takes place at such a speed.
The solution of practical problems
Of course, in real life everything is much more complicated than in theoretical problems. In practice, the value of the values is usually determined during the experiment. In this case, instruments are used that give indications during measurements with a certain error. Therefore, in calculations, one has to deal with approximate values of parameters and resort to rounding off uncomfortable numbers, as well as other simplifications. Having taken this into account, we will again proceed to the problems of the physical meaning of the derivative, given that they are only a kind of mathematical model of complex processes occurring in nature.
Eruption
Imagine an eruption of a volcano. How dangerous can it be? To clarify this issue, many factors must be considered. We will try to take into account one of them.
Stones having an initial velocity from the moment of 120 m / s exit to the outside are thrown vertically up from the mouth of the “fiery monster” . It is necessary to calculate what they can reach the maximum height.
To find the desired value, we compose the equation for the dependence of the height H, measured in meters, on other values. These include initial speed and time. The acceleration value is considered known and approximately equal to 10 m / s 2 .
Private derivative
Let us now consider the physical meaning of the derivative function on the other hand, because the equation itself may contain not one but several variables. For example, in the previous problem, the dependence of the height of the stones ejected from the vent of the volcano was determined not only by changing the temporal characteristics, but also by the value of the initial velocity. The latter was considered a constant, fixed value. But in other tasks with completely different conditions, everything could be different. If there are several quantities on which the complex function depends, the calculations are performed according to the formulas below.
The physical meaning of the frequent derivative should be determined, as in the usual case. This is the rate of change of the function at a certain point with the growth of the variable parameter. It is calculated in such a way that all other components are taken as constants, only one is considered as a variable. Further, everything happens according to the usual rules.
An irreplaceable adviser on many issues
Understanding the physical meaning of the derivative, examples of solving complicated and complex problems, the answer to which allow us to find such knowledge, are simple to cite. If we have a function that describes the fuel consumption depending on the speed of the car, we can calculate at what parameters of the latter the gas mileage will be the smallest.
In medicine, one can foresee how the human body will react to the medicine prescribed by the doctor. Taking the drug affects a variety of physiological parameters. These include changes in blood pressure, heart rate, body temperature, and more. All of them depend on the dose of the drug taken. These calculations help to predict the course of treatment, both in favorable manifestations and in undesirable accidents that can fatally affect changes in the patient's body.
Undoubtedly, it is important to understand the physical meaning of the derivative in technical matters, in particular in electrical engineering, electronics, design and construction.
Braking distances
Consider the next task. Moving at a constant speed, the car, approaching the bridge, was forced to slow down 10 seconds before entering, as the driver noticed a traffic sign prohibiting movement at a speed of more than 36 km / h. Did the driver break the rules if the stopping distance can be described by the formula S = 26t - t 2 ?
Having calculated the first derivative, we find the formula for the velocity, we obtain v = 28 - 2t. Next, substitute the value t = 10 into the indicated expression.
Since this value was expressed in seconds, the speed is 8 m / s, which means 28.8 km / h. This makes it possible to understand that the driver began to slow down in time and did not violate the rules of movement, and therefore the limit of the speed indicated on the sign.
This proves the importance of the physical meaning of the derivative. An example of solving this problem demonstrates the breadth of the use of this concept in various spheres of life. Including in domestic situations.
Derivative in the economy
Until the 19th century, economists mainly operated on average values, whether it was labor productivity or the price of manufactured products. But from some point on, to make effective forecasts in this area, limit values have become more necessary. These include marginal utility, income, or cost. Understanding this gave an impetus to the creation of a completely new tool in economic research, which has existed and has been developing for over a hundred years.
To make such calculations, where such concepts are dominant, at least and maximum, it is simply necessary to understand the geometric and physical meaning of the derivative. Among the creators of the theoretical basis of these disciplines are such prominent English and Austrian economists as W. S. Jevons, C. Menger and others. Of course, limit values in economic calculations are not always convenient to use. And, for example, quarterly reports do not necessarily fit into the existing scheme, but nevertheless, the application of such a theory in many cases is useful and effective.