Math 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 important concepts, which must be mastered in order to find solutions to important dilemmas, is the derivative of a function. Its physical meaning is not at all as difficult to explain as it might seem to those uninitiated into the essence of the issue. It is enough just 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 at a specific instant of fixed time. After all, it is precisely this parameter that contains the essence of the physical meaning of the derivative.
How to find speed
Any fifth grader can easily determine the speed of a person on the road, knowing the distance traveled and travel time. To do this, the first one follows given values divide by the second. But not every young mathematician knows that in this moment finds the ratio of the increments of a function and its argument. Indeed, if you imagine the movement in the form of a graph, plotting the path along the ordinate axis and time along the abscissa, it will be exactly like this.
However, the speed of a pedestrian or any other object, which we determine over a large section of the path, considering the movement to be uniform, may well change. There are many forms of motion known in physics. It can be done not only with constant acceleration, but slow down and increase in an arbitrary manner. It should be noted that in in this case the line describing the movement will no longer be a straight line. Graphically, it can take on the most complex configurations. But for any of the points on the graph we can always draw the tangent represented by linear function.
To clarify the parameter of change in displacement depending on time, it is necessary to shorten the measured segments. When they become infinitesimal, the calculated speed will be instantaneous. This experience helps us define a derivative. Its physical meaning also logically follows from such reasoning.
From a geometry point of view
It is known that the greater the speed of the body, the steeper the graph of the dependence of displacement on time, and therefore the angle of inclination of the tangent to the graph at a certain point. An indicator of such changes can be the tangent of the angle between the abscissa axis and the tangent line. It is precisely this that determines the value of the derivative and is calculated by the ratio of the lengths of the opposite to adjacent leg V right triangle, formed by a perpendicular dropped from a certain point to the abscissa axis.
This is geometric meaning first derivative. The physical one is revealed in the fact that the value of the opposite side in our case represents the distance traveled, and the adjacent side represents time. In this case, their ratio is speed. Once again we come to the conclusion that instantaneous speed, determined when both intervals tend to infinitesimal, and is the essence of pointing to 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
The derivative is an indicator of the rate of change of any function, even when we are not talking about movement in literally words. To clearly demonstrate this, here are a few specific examples. Suppose the current strength, depending on time, changes according to next law: I= 0.4t 2 . It is required to find the value of the speed at which this parameter changes at the end of the 8th second of the process. Note that the desired value itself, as can be judged from the equation, is constantly increasing.
To solve, it is necessary to find the first derivative, the physical meaning of which was discussed earlier. Here dI/ dt = 0,8 t. Next we will find it at t=8 , we find that the rate at which the current changes occurs is equal to 6,4 A/ c. Here it is considered that the current strength is measured in amperes, and time, accordingly, in seconds.
Everything is changeable
Visible the world, consisting of matter, constantly undergoes changes, being in motion flowing in it various processes. To describe them you can use the most different parameters. If they are united by a dependence, then they are written mathematically in the form of 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 present moment.
On this occasion next example. Let's say body temperature changes according to the law T=0,2 t 2 . You should find the rate 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. This means that the final answer is 4 degrees per second, that is, the heating process and change in temperature, measured in degrees, occurs at exactly this speed.
Solving practical problems
Of course, in real life everything is much more complicated than in theoretical problems. In practice, the value of quantities is usually determined during an experiment. In this case, instruments are used that provide readings during measurements with a certain error. Therefore, when calculating, you have to deal with approximate values of the parameters and resort to rounding of inconvenient numbers, as well as other simplifications. Having taken this into account, let us again proceed to problems on the physical meaning of the derivative, taking into account that they are only some mathematical model complex processes occurring in nature.
Eruption
Let's imagine that a volcano is erupting. How dangerous can he be? To clarify this issue, many factors need to be considered. We will try to take one of them into account.
From the mouth of the “fire monster” stones are thrown vertically upward, having an initial speed from the moment they come out. It is necessary to calculate what maximum height they can reach.
To find the desired value, we will draw up an equation for the dependence of height H, measured in meters, on other values. These include initial speed and time. We consider the acceleration value to be known and approximately equal to 10 m/s 2 .
Partial derivative
Let us now consider the physical meaning of the derivative of a function from a slightly different angle, because the equation itself may contain not one, but several variables. For example, in previous task the dependence of the height of the rise of stones thrown out of the crater of a volcano was determined not only by a change in time characteristics, but also by the value initial speed. The latter was considered a constant, fixed value. But in other problems with completely different conditions, everything could be different. If the quantities on which it depends complex function, several, calculations are made 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 a function at a certain point as the parameter of the variable increases. It is calculated in such a way that all other components are taken as constants, only one is considered as a variable. Then everything happens according to the usual rules.
Understanding the physical meaning of the derivative, examples of solving confusing and complex problems, the answer to which such knowledge can be found is not difficult to give. If we have a function that describes fuel consumption depending on the speed of the car, we can calculate at what parameters of the latter the gasoline consumption will be the least.
In medicine, it is possible to predict how a person will react human body on a medicine prescribed by a doctor. Taking the drug affects a variety of physiological indicators. These include changes blood pressure, pulse, body temperature and much more. They all depend on the dose taken medicine. These calculations help to predict the course of treatment, both in favorable manifestations and in undesirable events 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
Let's consider the next problem. Moving with constant speed, the car, approaching the bridge, was forced to brake 10 seconds before the entrance, as the driver noticed road sign, prohibiting movement at speeds exceeding 36 km/h. Did the driver break the rules if his braking distance can be described by the formula S = 26t - t 2?
Having calculated the first derivative, we find a formula for speed, we get v = 28 - 2t. Next we substitute in specified expression value t=10.
Since this value was expressed in seconds, the speed turns out to be 8 m/s, which means 28.8 km/h. This makes it possible to understand that the driver began to brake on time and did not violate the traffic rules, and therefore the speed limit 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 use of this concept in various areas of life. Including in everyday situations.
Derivative in economics
Until the 19th century, economists mainly operated with averages, be it labor productivity or the price of manufactured products. But at some point, limit values became more necessary to make effective forecasts in this area. These may include marginal utility, income or costs. Understanding this gave impetus to the creation of a completely new tool in economic research, which has existed and developed for more than a hundred years.
To draw up such calculations, where such concepts as minimum and maximum dominate, it is simply necessary to understand the geometric and physical meaning of the derivative. Among the creators theoretical basis These disciplines include such prominent English and Austrian economists as W. S. Jevons, K. Menger and others. Of course, it is not always convenient to use limit values in economic calculations. And, for example, quarterly reports do not necessarily fit into the existing scheme, but still the application of such a theory in many cases is useful and effective.
Avogadro's law: at constant pressure and temperature, equal volumes of gases contain the same number of molecules.
Isothermal process
6.Formulate the main consequences of Avogadro’s law. What conditions are considered normal and what is the molar volume of gas under these conditions?
A corollary from Avogadro's law: one mole of any gas, under the same conditions, occupies the same volume. In particular, when normal conditions, i.e. at 0 ° C (273 K) and 101.3 kPa, the volume of 1 mole of gas is 22.4 liters. This volume is called the molar volume of the gas Vm.
7.What characterizes the relative density of one gas compared to another gas? How is gas density calculated and what is its physical meaning?
Mass ratio equal volumes two gases under identical conditions is called the density of one gas relative to the other, i.e.
8.Formulate the Boyle-Mariotte and Gay-Lussac laws and write down their mathematical expressions.
The Boyle-Mariotte law reflects the relationship between pressure p and volume V of a certain amount of gas at constant temperature: at constant temperature, the pressure produced by a given mass of gas is inversely proportional to the volume of gas: pV = const. In other words, when a gas transitions from a state with parameters p 1 and V 1 to a state with parameters p 2 and V 2 (at T, n = const), the following condition is satisfied: p 1 V 1 = p 2 V 2 .
This ratio is used in calculations.
Gay-Lussac's law connects the volume of a gas V with its temperature T (at p = const): at constant pressure, the volume of a gas changes in direct proportion to the absolute temperature:
P 
In calculations, the relation is usually used
9.Formulate a combined gas law and write down its mathematical expression. In what calculations is it used?
Based on the Boyle-Mariotte, Gay-Lussac and Avogadro principles, the combined gas is removed:
= const. For calculations the following ratio is used: . The physical meaning of the law is as follows: a change in any of the parameters p, V, T during the transition from state 1 to state 2 leads to a change in other parameters, but the relation 
- the value is constant. It can be seen that at T = const (T 1 = T 2) we obtain the Boyle-Mariotte law (p 1 V 1 = p 2 V 2), and at p = const (p 1 = p 2) - the Gay-Lussac-Charles law 
, i.e. these laws are a special case of the combined gas law. The combined value is used to calculate gas parameters during the transition from one state to another and, most often, one of these states corresponds to standard conditions. The standard conditions are taken to be a pressure of 101325 Pa (1 atm) and a temperature of 273.15 K (0 °C). For calculations, approximate values are usually used: 1 10 s Pa and 273 K.
10.Write the Clayperon-Mendeleev equation. What is the physical meaning of the universal gas constant? What values can it take and what does its value depend on?
The combined gas law is valid for any amount of gas. For ideal gas quantity 1 mole ratio 
denoted by R. This quantity is a fundamental physical constant and is called the universal (molar) gas constant. For 1 mole of gas pV m = RT, and for n moles pV = nRT. Taking n into account, the resulting equation takes the form
pV = 
RT.
The last equation is known as the Mendeleev-Clapeyron equation and is most often used in calculations. It establishes the relationship between pressure, volume, temperature and quantity of a substance. The Mendeleev-Clapeyron equation is valid for an ideal gas, but allows calculations of the parameters of real gases under physical conditions approaching normal ones, or more precisely at not too high pressures and not too low temperatures.
R=8.32*Pa*m 3 /mol*K
The derivative of a function is the brainchild differential calculus Newton and Leibniz - has a very definite physical meaning if you look at it more deeply.
