Physical meaning of derivative. Tasks! Velocity as a derivative Acceleration as a derivative

Physical meaning of derivative. The Unified State Exam in mathematics includes a group of problems for solving which requires knowledge and understanding of the physical meaning of the derivative. In particular, there are problems where the law of motion of a certain point (object) is given, expressed by an equation, and it is required to find its speed at a certain moment in time of movement, or the time after which the object will acquire a certain given speed.The tasks are very simple, they can be solved in one action. So:

Let the law of motion of a material point x (t) along the coordinate axis be given, where x is the coordinate of the moving point, t is time.

Velocity at a certain moment in time is the derivative of the coordinate with respect to time. This is the mechanical meaning of the derivative.

Likewise, acceleration is the derivative of speed with respect to time:

Thus, the physical meaning of the derivative is speed. This could be the speed of movement, the rate of change of a process (for example, the growth of bacteria), the speed of work (and so on, there are many applied problems).

In addition, you need to know the derivative table (you need to know it just like the multiplication table) and the rules of differentiation. Specifically, to solve the specified problems, knowledge of the first six derivatives is necessary (see table):

Let's consider the tasks:

x (t) = t 2 – 7t – 20

where x t is the time in seconds measured from the beginning of the movement. Find its speed (in meters per second) at time t = 5 s.

The physical meaning of a derivative is speed (speed of movement, rate of change of a process, speed of work, etc.)

Let's find the law of speed change: v (t) = x′(t) = 2t – 7 m/s.

At t = 5 we have:

Answer: 3

Decide for yourself:

The material point moves rectilinearly according to the law x (t) = 6t 2 – 48t + 17, where x- distance from the reference point in meters, t- time in seconds measured from the start of movement. Find its speed (in meters per second) at time t = 9 s.

The material point moves rectilinearly according to the law x (t) = 0.5t 3 – 3t 2 + 2t, where xt- time in seconds measured from the start of movement. Find its speed (in meters per second) at time t = 6 s.

A material point moves rectilinearly according to the law

x (t) = –t 4 + 6t 3 + 5t + 23

Where x- distance from the reference point in meters,t- time in seconds measured from the start of movement. Find its speed (in meters per second) at time t = 3 s.

A material point moves rectilinearly according to the law

x(t) = (1/6)t 2 + 5t + 28

where x is the distance from the reference point in meters, t is the time in seconds, measured from the beginning of the movement. At what point in time (in seconds) was its speed equal to 6 m/s?

Let's find the law of speed change:

In order to find at what point in timetthe speed was 3 m/s, it is necessary to solve the equation:

Answer: 3

Decide for yourself:

The material point moves rectilinearly according to the law x (t) = t 2 – 13t + 23, where x- distance from the reference point in meters, t- time in seconds measured from the start of movement. At what point in time (in seconds) was its speed equal to 3 m/s?

A material point moves rectilinearly according to the law

x (t) = (1/3) t 3 – 3t 2 – 5t + 3

Where x- distance from the reference point in meters, t- time in seconds measured from the start of movement. At what point in time (in seconds) was its speed equal to 2 m/s?

I would like to note that you should not focus only on this type of tasks on the Unified State Exam. They may completely unexpectedly introduce problems that are the opposite of those presented. When the law of change of speed is given and the question will be about finding the law of motion.

Hint: in this case, you need to find the integral of the speed function (this is also a one-step problem). If you need to find the distance traveled at a certain point in time, you need to substitute time into the resulting equation and calculate the distance. However, we will also analyze such problems, don’t miss it!Good luck to you!

Sincerely, Alexander Krutitskikh.

P.S: I would be grateful if you tell me about the site on social networks.

Until now, we have associated the concept of derivative with the geometric representation of the graph of a function. However, it would be a gross mistake to limit the role of the concept of derivative to the task of determining the slope of the tangent to a given curve. An even more important task from a scientific point of view is calculating the rate of change of any quantity. f(t), changing over time t. It was from this side that Newton approached differential calculus. In particular, Newton sought to analyze the phenomenon of speed, considering time and the position of a moving particle as variables (in Newton's words, “fluents”). When a particle moves along the x axis, then its movement is completely defined, since the function is given x = f(t), indicating the position of particle x at any time t. "Uniform motion" with constant speed b along the x axis is determined by the linear function x = a + bt, where a is the position of the particle at the initial moment (at t = 0).

The motion of a particle on a plane is described by two functions

x = f(t), y = g(t),

which determine its coordinates as a function of time. In particular*, two linear functions correspond to uniform motion

x = a + bt, y = c + dt,

where b and d are two “components” of constant speed, and a and c are the coordinates of the initial position of the particle (at t = 0); the trajectory of a particle is a straight line, the equation of which is

(x - a) d - (y - c) b = 0

is obtained by eliminating t from the two relations above.

If a particle moves in the vertical plane x, y under the influence of gravity alone, then its motion (this is proven in elementary physics) is determined by two equations

Where a, b, c, d are constants depending on the state of the particle at the initial moment, and g is the acceleration due to gravity, equal to approximately 9.81 if time is measured in seconds and distance in meters. The trajectory obtained by eliminating t from the two given equations is a parabola

unless b≠0; otherwise, the trajectory is a segment of the vertical axis.

If a particle is forced to move along some given curve (similar to how a train moves along rails), then its movement can be determined by the function s (t) (a function of time t) equal to the length of the arc s, calculated along this curve from some starting point P 0 to the position of the particle at point P at time t. For example, if we are talking about a unit circle x 2 + y 2 = 1, then the function s = ct determines on this circle a uniform rotational motion with a speed With.

* Exercise. Draw trajectories of plane movements given by the equations: 1) x = sin t, y = cos t; 2) x = sin 2t, y = cos 3t; 3) x = sin 2t, y = 2 sin 3t; 4) in the parabolic motion described above, assume the initial position of the particle (at t = 0) at the origin of coordinates and consider b>0, d>0. Find the coordinates of the highest point of the trajectory. Find the time t and x value corresponding to the secondary intersection of the trajectory with the x axis.

The first goal that Newton set himself was to find the speed of a particle moving unevenly. For simplicity, let us consider the motion of a particle along a certain straight line, specified by the function x = f(t). If the movement were uniform, that is, performed at a constant speed, then this speed could be found by taking two moments of time t and t 1 and the corresponding positions of the particles f(t) And f (t 1) and forming an attitude

For example, if t is measured in hours and x in kilometers, then when t 1 - t = 1 difference x 1 - x will be the number of kilometers traveled in 1 hour, and v- speed (in kilometers per hour). When saying that speed is a constant quantity, they only mean that the difference ratio

does not change for any values of t and t 1. But if the movement is uneven (which occurs, for example, in the free fall of a body, the speed of which increases as it falls), then relation (3) does not give the value of the speed at the moment t, but represents what is usually called the average speed in the period of time from t to t 1. To get speed at time t, you need to calculate the limit average speed as t 1 tends to t. Thus, following Newton, we define speed as follows:

In other words, speed is the derivative of the distance traveled (the particle's coordinates on a straight line) with respect to time, or the "instantaneous rate of change" of the path with respect to time - as opposed to average rate of change, determined by formula (3).

The rate of change of the speed itself called acceleration. Acceleration is simply the derivative of the derivative; it is usually denoted by the symbol f"(t) and is called second derivative from the function f (t).

Algebra is generous. She often gives more than what is asked of her.

J. d'Alembert

Interdisciplinary connections are a didactic condition and a means of deep and comprehensive mastery of the fundamentals of science at school.
In addition, they help improve students' scientific knowledge, develop logical thinking and their creative abilities. The implementation of interdisciplinary connections eliminates duplication in the study of material, saves time and creates favorable conditions for the development of general educational skills of students.
Establishing interdisciplinary connections in a physics course increases the effectiveness of polytechnic and practical training.
The motivational side is very important in teaching mathematics. A mathematical problem is perceived better by students if it arises as if before their eyes, and is formulated after considering some physical phenomena or technical problems.
No matter how much a teacher talks about the role of practice in the progress of mathematics and the importance of mathematics for the study of physics and the development of technology, but if he does not show how physics influences the development of mathematics and how mathematics helps practice in solving its problems, then the development of a materialistic worldview will be harmed serious damage. But in order to show how mathematics helps in solving its problems, we need problems that are not invented for methodological purposes, but actually arise in various areas of practical human activity

Historical information

Differential calculus was created by Newton and Leibniz at the end of the 17th century based on two problems:

about finding a tangent to an arbitrary line;
on finding speed under an arbitrary law of motion.

Even earlier, the concept of a derivative was encountered in the works of the Italian mathematician Nicolo Tartaglia (about 1500 - 1557) - the tangent appeared here during the study of the issue of the angle of inclination of a gun, at which the greatest range of the projectile is ensured.

In the 17th century, based on the teachings of G. Galileo on motion, the kinematic concept of the derivative was actively developed.

The famous scientist Galileo Galilei devotes an entire treatise on the role of derivatives in mathematics. Various presentations began to be found in the works of Descartes, the French mathematician Roberval, and the English scientist L. Gregory. L'Hopital, Bernoulli, Lagrange, Euler, and Gauss made great contributions to the study of differential calculus.

Some applications of derivative in physics

Derivative- the basic concept of differential calculus, characterizing rate of change of function.

Determined as the limit of the ratio of the increment of a function to the increment of its argument as the increment of the argument tends to zero, if such a limit exists.

Thus,

So, to calculate the derivative of the function f(x) at the point x 0 by definition, you need:

Let us consider several physical problems in which this scheme is used.

Instantaneous velocity problem. Mechanical meaning of derivative

Let us recall how the speed of movement was determined. A material point moves along a coordinate line. The x coordinate of this point is a known function x(t) time t. Over the period of time from t 0 to t 0+ the point's displacement is x(t 0 + ) – x(t 0) – and its average speed is: .
Usually the nature of the movement is such that at small values, the average speed remains practically unchanged, i.e. the movement can be considered uniform with a high degree of accuracy. In other words, the value of the average speed at tends to some well-defined value, which is called the instantaneous speed v(t 0) material point at a moment in time t 0.

So,

But by definition
Therefore, it is believed that the instantaneous speed at the moment of time t 0

Reasoning similarly, we find that the derivative of speed with respect to time is acceleration, i.e.

The problem of the heat capacity of a body

For the temperature of a body weighing 1 g to increase from 0 degrees to t degrees, the body needs to provide a certain amount of heat Q. Means, Q there is a temperature function t, to which the body is heated: Q = Q(t). Let the body temperature rise from t 0 to t. The amount of heat expended for this heating is equal to The ratio is the amount of heat that is required on average to warm the body by 1 degree when the temperature changes by degrees. This ratio is called the average heat capacity of a given body and is denoted from Wed.
Because the average heat capacity does not give an idea of the heat capacity for any temperature T, then the concept of heat capacity at a given temperature is introduced t 0(at this point t 0).
Heat capacity at temperature t 0(at a given point) is called the limit

Problem on the linear density of a rod

Let's consider a non-uniform rod.

For such a rod, the question arises about the rate of change of mass depending on its length.

Average linear density the mass of the rod is a function of its length X.

Thus, the linear density of a non-uniform rod at a given point is determined as follows:

By considering similar problems, one can obtain similar conclusions for many physical processes. Some of them are shown in the table.

Function	Formula	Conclusion
m(t) – dependence of the mass of consumed fuel on time.		Derivative masses over time There is speed fuel consumption.
T(t) – dependence of the temperature of the heated body on time.		Derivative temperature over time There is speed body heating.
m(t) – dependence of mass during the decay of a radioactive substance on time.		Derivative mass of radioactive substance over time There is speed radioactive decay.
q(t) – dependence of the amount of electricity flowing through the conductor on time		Derivative amount of electricity over time There is current strength.
A(t) – dependence of work on time		Derivative work on time There is power.

Practical tasks:

A projectile fired from a cannon moves according to the law x(t) = – 4t 2 + 13t (m). Find the speed of the projectile at the end of 3 seconds.

The amount of electricity flowing through the conductor, starting at time t = 0 s, is given by the formula q(t) = 2t 2 + 3t + 1 (Kul) Find the current strength at the end of the fifth second.

The amount of heat Q (J) required to heat 1 kg of water from 0 o to t o C is determined by the formula Q(t) = t + 0.00002t 2 + 0.0000003t 3. Calculate the heat capacity of water if t = 100 o.

The body moves rectilinearly according to the law x(t) = 3 + 2t + t 2 (m). Determine its speed and acceleration at times 1 s and 3 s.

Find the magnitude of the force F acting on a point of mass m, moving according to the law x(t) = t 2 – 4t 4 (m), at t = 3 s.

A body whose mass is m = 0.5 kg moves rectilinearly according to the law x(t) = 2t 2 + t – 3 (m). Find the kinetic energy of the body 7 s after the start of movement.

Conclusion

One can point out many more technical problems, for the solution of which it is also necessary to find the rate of change of the corresponding function.
For example, finding the angular velocity of a rotating body, the linear coefficient of expansion of bodies when heated, the rate of a chemical reaction at a given time.
Due to the abundance of problems leading to the calculation of the rate of change of a function or, in other words, to the calculation of the limit of the ratio of the increment of a function to the increment of the argument, when the latter tends to zero, it turned out to be necessary to isolate such a limit for an arbitrary function and study its basic properties. This limit was called derivative of a function.

So, using a number of examples, we showed how various physical processes are described using mathematical problems, how analysis of solutions allows us to draw conclusions and predictions about the course of processes.
Of course, the number of examples of this kind is huge, and quite a large part of them are quite accessible to interested students.

“Music can uplift or soothe the soul,
Painting is pleasing to the eye,
Poetry is to awaken feelings,
Philosophy is to satisfy the needs of the mind,
Engineering is to improve the material side of people's lives,
And mathematics can achieve all these goals.”

That's what the American mathematician said Maurice Cline.

References :

Abramov A.N., Vilenkin N.Ya. and others. Selected questions of mathematics. 10th grade. – M: Enlightenment, 1980.
Vilenkin N.Ya., Shibasov A.P. Behind the pages of a mathematics textbook. – M: Enlightenment, 1996.
Dobrokhotova M.A., Safonov A.N.. Function, its limit and derivative. – M: Enlightenment, 1969.
Kolmogorov A.N., Abramov A.M. and others. Algebra and the beginnings of mathematical analysis. – M: Education, 2010.
Kolosov A.A. A book for extracurricular reading on mathematics. – M: Uchpedgiz, 1963.
Fikhtengolts G.M. Fundamentals of mathematical analysis, part 1 – M: Nauka, 1955.
Yakovlev G.N. Mathematics for technical schools. Algebra and the beginnings of analysis, part 1 - M: Nauka, 1987.

Until now, we have associated the concept of derivative with the geometric representation of the graph of a function. However, it would be a grave mistake to limit the role of the concept of derivative to only the problem of

determining the slope of the tangent to a given curve. An even more important task, from a scientific point of view, is to calculate the rate of change of any quantity that changes over time. It was from this side that Newton approached differential calculus. In particular, Newton sought to analyze the phenomenon of speed by considering time and the position of a moving particle as variables (in Newton's words, “fluents”). When a particle moves along the x-axis, then its movement is completely defined, since a function is given that indicates the position of the particle x at any time t. “Uniform motion” with a constant speed along the x axis is determined by a linear function where a is the position of the particle at the initial moment

The motion of a particle on a plane is described by two functions

which determine its coordinates as a function of time. In particular, two linear functions correspond to uniform motion

where two “components” of constant speed, and a and c are the coordinates of the initial position of the particle (with the particle’s trajectory being a straight line, the equation of which is

is obtained by eliminating the two relations above.

If a particle moves in the vertical plane x, y under the influence of gravity alone, then its motion (this is proven in elementary physics) is determined by two equations

where are constants depending on the state of the particle at the initial moment, the acceleration due to gravity is approximately 9.81 if time is measured in seconds and distance in meters. The trajectory obtained by eliminating these two equations is a parabola

unless otherwise the trajectory is a segment of the vertical axis.

If a particle is forced to move along a given curve (similar to how a train moves on rails), then its movement can be determined by a function (a function of time equal to the length of the arc calculated along a given curve from a certain starting point to the position of the particle at point P at the moment of time. For example, if we are talking about a unit circle, then the function determines uniform rotational motion on this circle with speed c.

Exercise. Draw trajectories of plane motions given by the equations: in the parabolic motion described above, assume the initial position of the particle (at the origin and consider Find the coordinates of the highest point of the trajectory. Find the time and x value corresponding to the secondary intersection of the trajectory with the axis

The first goal that Newton set himself was to find the speed of a particle moving unevenly. For simplicity, let us consider the movement of a particle along a certain straight line, specified by the function. If the movement were uniform, i.e., performed at a constant speed, then this speed could be found by taking two moments of time and the corresponding positions of the particles and making the ratio

For example, if measured in hours, and ; in kilometers, then the difference will be the number of kilometers traveled in 1 hour, the speed (kilometers per hour). When saying that speed is a constant quantity, they only mean that the difference ratio

does not change for any values. But if the movement is uneven (which occurs, for example, in the free fall of a body, the speed of which increases as it falls), then relation (3) does not give the value of the speed at the moment and represents what is commonly called the average speed in the time interval from to To obtain the speed at the moment you need to calculate the limit of the average

speed when tending Thus, together with Newton, we define speed as follows:

In other words, the speed is the derivative of the “path traveled” (the coordinates of the particle on a straight line) with respect to time, or the “instantaneous rate of change” of the path with respect to time - as opposed to the average rate of change determined by formula (3).

The rate of change of velocity itself is called acceleration. Acceleration is simply the derivative of the derivative; it is usually denoted by the symbol and is called the second derivative of the function

Galileo noticed that the vertical distance x covered during a free fall of a body over time is expressed by the formula