Presentations concept derivative historical background. Presentation on the topic: Derivative. Application of derivatives in various fields of science
The history of "Derivative". Slide number 3. I. Historical background. David Gilbert. The general concept of derivative was made independently almost simultaneously. End XVI- The middle of the 17th century was marked by the enormous interest of scientists in explaining movement and finding the laws to which it obeys. Questions about determining and calculating the speed of movement and its acceleration have become more acute than ever. The solution to these questions led to the establishment of a connection between the problem of calculating the speed of a body and the problem of drawing a tangent to a curve describing the dependence of the distance traveled on time. English physicist and mathematician I. Newton. German philosopher and mathematician G. Leibniz.
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“Derivative of a function at a point” - Programmed control. Theory issues. 0. Find the value of the derivative at point xo. 1) Find the angular coefficient of the tangent to the graph of the function f(x)=Cosх at point x=?/4. A. At the point. X.
“Prime function” - Repetition. Repeating and generalizing lesson (algebra 11th grade). Complete the task. Prove that the function F is an antiderivative of a function f on the set R. The main property of an antiderivative. Find the general form of the antiderivative for the function. Formulate: Definition of antiderivative. Rules for finding an antiderivative.
“Derivative of an exponential function” - www.thmemgallery.com. Grade 11. Rules of differentiation. Theorem 1. The function is differentiable at each point of the domain of definition, and. Derivative of an exponential function. Application of the derivative when studying a function. Theorem 2. Tangent equation. Derivatives of elementary functions. The natural logarithm is the logarithm to base e:
“Calculation of derivatives” - Oral warm-up, repetition of the rules for calculating derivatives (slide No. 1) 3. Practical part. Today's lesson will use presentations. 2. Activation of knowledge. The operation of finding the derivative is called differentiation. Slide No. 1. Student self-esteem. Main stages of the lesson Organizational moment.
“Geometric meaning of derivative” - B. Geometric meaning function increments. S. So, the geometric meaning of the relation at. A. Slide 10. K – angular coefficient of the straight line (secant). Determination of the derivative of a function (To the textbook by A.N. Kolmogorov “Algebra and the beginnings of analysis 10-11”). The purpose of the presentation is to ensure maximum clarity of the topic.
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From history: In the history of mathematics, several stages in the development of mathematical knowledge are traditionally distinguished: Formation of the concept of a geometric figure and number as an idealization of real objects and sets of homogeneous objects. The advent of counting and measurement, which made it possible to compare different numbers, lengths, areas and volumes. Invention of arithmetic operations. Accumulation of knowledge about properties empirically (by trial and error) arithmetic operations, about methods for measuring areas and volumes of simple figures and bodies. The Sumerian-Babylonian, Chinese and Indian mathematicians of antiquity made great progress in this direction. Appearance in ancient Greece a deductive mathematical system that showed how to obtain new mathematical truths based on existing ones. The crowning achievement of ancient Greek mathematics was Euclid's Elements, which served as the standard of mathematical rigor for two millennia. Mathematicians from Islamic countries not only preserved ancient achievements, but were also able to synthesize them with the discoveries of Indian mathematicians, who advanced further than the Greeks in number theory. IN XVI-XVIII centuries European mathematics is being revived and moving far ahead. Her conceptual basis During this period, there was confidence that mathematical models are a kind of ideal skeleton of the Universe, and therefore the discovery of mathematical truths is at the same time the discovery of new properties of the real world. The main success along this path was the development of mathematical models of dependence (function) and accelerated motion (analysis of infinitesimals). All natural sciences were rebuilt on the basis of newly discovered mathematical models, and this led to colossal progress. In the 19th and 20th centuries, it became clear that the relationship between mathematics and reality was far from being as simple as it previously seemed. There is no generally accepted answer to a kind of "fundamental question in the philosophy of mathematics": to find the reason for the "incomprehensible effectiveness of mathematics in natural sciences" In this, and not only in this respect, mathematicians were divided into many debating schools. Several dangerous trends have emerged: excessively narrow specialization, isolation from practical problems, etc. At the same time, the power of mathematics and its prestige, supported by the effectiveness of its application, are higher than ever before
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Differentiability The derivative f"(x0) of a function f at a point x0, being a limit, may not exist or exist and be finite or infinite. A function f is differentiable at a point x0 if and only if its derivative at this point exists and is finite: For a function f differentiable at x0 in a neighborhood of U(x0) has the following representation: f(x) = f(x0) + f"(x0)(x − x0) + o(x − x0)
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Remarks Let's call Δx = x − x0 the increment of the function argument, and Δy = f(x0 + Δx) − f(x0) the increment of the function value at the point x0. Then Let the function have a finite derivative at each point. Then the derivative function is defined. A function that has a finite derivative at a point is continuous at it. The reverse is not always true. If the derivative function itself is continuous, then the function f is called continuously differentiable and is written:
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Geometric and physical meaning of the derivative Geometric meaning of the derivative. On the graph of the function, the abscissa x0 is selected and the corresponding ordinate f(x0) is calculated. An arbitrary point x is selected in the vicinity of the point x0. A secant line is drawn through the corresponding points on the graph of function F (the first light gray line C5). The distance Δx = x - x0 tends to zero, as a result the secant turns into a tangent (gradually darkening lines C5 - C1). The tangent of the angle α of the slope of this tangent is the derivative at the point x0.
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Derivatives of higher orders The concept of a derivative of an arbitrary order is defined recursively. We assume that If a function f is differentiable at x0, then the first-order derivative is determined by the relation Let now the nth-order derivative f(n) be defined in some neighborhood of the point x0 and differentiable. Then
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Methods of writing derivatives Depending on the goals, scope and mathematical apparatus used, use various ways records of derivatives. Thus, the nth order derivative can be written in the notation: Lagrange f(n)(x0), while for small n primes and Roman numerals are often used: f(1)(x0) = f"(x0) = fI( x0),f(2)(x0) = f""(x0) = fII(x0),f(3)(x0) = f"""(x0) = fIII(x0),f(4)(x0 ) = fIV(x0), etc. This notation is convenient because of its brevity and is widely used; Leibniz, a convenient visual notation of the ratio of infinitesimals: Newton, which is often used in mechanics for the time derivative of the coordinate function (for the spatial derivative, the notation is more often used Lagrange). The order of the derivative is indicated by the number of points over the function, for example: - the first order derivative of x with respect to t at t = t0, or - the second derivative of f with respect to x at the point x0, etc. Euler, using a differential operator (strictly speaking, a differential expression, while the corresponding functional space has not been introduced), and therefore convenient in questions related to functional analysis: Of course, we must not forget that they all serve to designate the same objects:
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Examples: Let f(x) = x2. Then Let f(x) = | x | . Then if then f"(x0) = sgnx0, where sgn denotes the sign function. If x0 = 0, then f"(x0) does not exist
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Rules of differentiation The operation of finding the derivative is called differentiation. When performing this operation, you often have to work with quotients, sums, products of functions, as well as “functions of functions,” that is, complex functions. Based on the definition of derivative, we can derive differentiation rules that make this work easier. (the derivative of a sum is equal to the sum of its derivatives) (from here, in particular, it follows that the derivative of the product of a function and a constant is equal to the product of the derivative of this function and a constant) If the function is given parametrically: then,
Derivative of a function Teacher of GAPOU RO "RKTM" Kolykhalina K.A. Increment of argument, increment of function Let x be an arbitrary point lying in some neighborhood of a fixed point x0. The difference x-x0 is called the increment of the independent variable (or increment of the argument) at the point x0 and is denoted ∆x. ∆х = x – x0 – increment of the independent variable. The increment of a function f at point x0 is the difference between the values of the function at an arbitrary point and the value of the function at a fixed point. f(х) – f(х0)=f(х0+∆х) – f(х0) – increment of function f∆f=f(x0+∆x) – f(x0) Determination of the derivative Derivative of the function y= f(x) at the point x =x0 is the limit of the ratio of the increment of the function ∆y at this point to the increment of the argument ∆x, as the increment of the argument tends to zero. Algorithm for calculating the derivative The derivative of the function y= f(x) can be found by following diagram: 1. Let's give the argument x an increment ∆x≠0 and find the incremented value of the function y+∆y= f(x+∆x). 2. Find the increment of the function ∆y= f(x+∆x) - f(x). 3. We compose the relation 4. We find the limit of this relation at ∆x⇾0, i.e. (if this limit exists). Determination of the derivative of a function at a given point. Its geometric meaning
k – angular coefficient of the straight line (secant)
Tangent
Geometric meaning of derivative
The derivative of a function at a given point is equal to the slope of the tangent drawn to the graph of the function at this point.
Physical meaning of the derivative 1. The problem of determining the speed of movement of a material particle Let a point move along a certain straight line according to the law s= s(t), where s is the distance traveled, t is time, and it is necessary to find the speed of the point at the moment t0. By the moment of time t0, the distance traveled is equal to s0 = s(t0), and by the moment (t0 +∆t) - the path s0 + ∆s=s(t0 +∆t). Then over the interval ∆t the average speed will be. The smaller ∆t, the better the average speed characterizes the movement of the point at the moment t0. Therefore, under speed of the point at time t0 should be understood as the limit of the average speed for the period from t0 to t0 +∆t, when ∆t⇾0, i.e. 2. PROBLEM ABOUT THE RATE OF A CHEMICAL REACTION Let some substance enter into a chemical reaction. The amount of this substance Q changes during the reaction depending on time t and is a function of time. Let the amount of substance change by ∆Q during time ∆t, then the ratio will express average speed chemical reaction in time ∆t, and the limit of this ratio is the rate of the chemical reaction in this moment time t.3. THE PROBLEM OF DETERMINING THE RADIOACTIVE DECAY RATE
If m is the mass of a radioactive substance and t is time, then the phenomenon of radioactive decay at time t, provided that the mass of the radioactive substance decreases over time, is characterized by the function m = m(t).
The average decay rate over time ∆t is expressed by the ratio
and the instantaneous decay rate at time t
Physical meaning of the derivative of a function at a given point
Derivatives of basic elementary functions Basic rules of differentiation Let u=u(x) And v=v(x) – differentiable functions at point x. 1) (u v) = u v 2) (uv) = uv +uv (cu) = cu 3) , If v 0
The branch of mathematics that studies the derivatives of functions and their applications is called differential calculus. This calculus arose from solving problems of drawing tangents to curves, calculating the speed of movement, and finding the largest and smallest values of a function.
A number of problems in differential calculus were solved in ancient times by Archimedes, who developed a method for drawing tangent lines. Archimedes constructed a tangent to the spiral that bears his name. Archimedes (c. 287 - 212 BC) is a great scientist. The discoverer of many facts and methods of mathematics and mechanics, a brilliant engineer.
The problem of finding the rate of change of a function was first solved by Newton. The problem of finding the rate of change of a function was first solved by Newton. He called the function fluent, i.e. current value. Derivative - flux. He called the function fluent, i.e. current value. Derivative - flux. Newton came to the concept of derivative based on questions of mechanics. Isaac Newton (1643 – 1722) – English physicist and mathematician.
Based on Fermat's results and some other conclusions, Leibniz published the first paper on differential calculus in 1684, which outlined the basic rules of differentiation. Leibniz Gottfried Friedrich (1646 - 1716) - great German scientist, philosopher, mathematician, physicist, lawyer, linguist
Application of derivative: Application of derivative: 1) Power is the derivative of work with respect to time P = A" (t). 2) Current strength is the derivative of charge with respect to time I = g" (t). 3) Force is the derivative of the work with respect to displacement F = A" (x). 4) Heat capacity is the derivative of the amount of heat with respect to temperature C = Q" (t). 5) Pressure is the derivative of force with respect to the area P = F"(S) 6) The circumference is the derivative of the area of the circle with respect to the radius l approx = S" cr (R). 7) The growth rate of labor productivity is the derivative of labor productivity with respect to time. 8) Are you successful in your studies? Derivative of knowledge growth.
Application of derivatives in physics Problem: Two bodies move rectilinearly according to the laws: S 1 (t) = 3.5t 2 – 5t + 10 and S 2 (t) = 1.5t 2 + 3t –6. At what point in time will the speeds of the bodies be equal? Problem: Two bodies move rectilinearly according to the laws: S 1 (t) = 3.5t 2 – 5t + 10 and S 2 (t) = 1.5t 2 + 3t –6. At what point in time will the speeds of the bodies be equal?
Application of derivatives in economics Problem: An enterprise produces X units of some homogeneous product per month. It has been established that the dependence of the financial savings of an enterprise on the volume of output is expressed by the formula Problem: The enterprise produces X units of some homogeneous products per month. It has been established that the dependence of the financial savings of an enterprise on the volume of output is expressed by the formula Explore the potential of the enterprise. Explore the potential of the enterprise. 15
Ministry of Education of the Saratov Region
State autonomous professional educational institution Saratov region "Engels Polytechnic"
APPLICATION OF DERIVATIVE IN VARIOUS FIELDS OF SCIENCE
Performed: Verbitskaya Elena Vyacheslavovna
mathematics teacher at GAPOU SO
"Engels Polytechnic"
Introduction
The role of mathematics in various fields of natural science is very great. No wonder they say “Mathematics is the queen of sciences, physics is its right hand, chemistry is leftist.”
The subject of the study is derivative.
The leading goal is to show the significance of the derivative not only in mathematics, but also in other sciences, its importance in modern life.
Differential calculus is a description of the world around us, done in mathematical language. The derivative helps us to successfully solve not only math problems, but also practical tasks in various fields of science and technology.
The derivative of a function is used wherever there is an uneven process: uneven mechanical movement, alternating current, chemical reactions and radioactive decay of a substance, etc.
Key and thematic questions of this essay:
1. History of the derivative.
2. Why study derivatives of functions?
3. Where are derivatives used?
4. Application of derivatives in physics, chemistry, biology and other sciences.
I decided to write a paper on the topic “Application of derivatives in various fields of science” because I think this topic is very interesting, useful and relevant.
In my work, I will talk about the application of differentiation in various fields of science, such as chemistry, physics, biology, geography, etc. After all, all sciences are inextricably linked, which is very clearly seen in the example of the topic I am considering.
Application of derivatives in various fields of science
From the high school algebra course, we already know that the derivative is 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.
The act of finding a derivative is called differentiating it, and a function that has a derivative at a point x is called differentiable at that point. A function that is differentiable at each point of an interval is said to be differentiable in that interval.
Honor of discovery of fundamental laws mathematical analysis belongs to the English physicist and mathematician Isaac Newton and the German mathematician, physicist, and philosopher Leibniz.
Newton introduced the concept of derivative while studying the laws of mechanics, thereby revealing its mechanical meaning.
Physical meaning of the derivative: the derivative of the function y = f (x) at the point x 0 is the rate of change of the function f (x) at the point x 0.
Leibniz came to the concept of derivative by solving the problem of drawing a tangent to a derivative line, thereby explaining its geometric meaning.
The geometric meaning of the derivative is that the derivative function at the point x 0 is equal to the slope of the tangent to the graph of the function drawn at the point with the abscissa x 0 .
The term derivative and modern designations y ", f" were introduced by J. Lagrange in 1797.
The 19th century Russian mathematician Panfutiy Lvovich Chebyshev said that “of particular importance are those methods of science that make it possible to solve a problem common to all practical activities a person, for example, how to dispose of his means to achieve the greatest benefit.”
Representatives of a variety of specialties have to deal with such tasks nowadays:
Technological engineers try to organize production in such a way that as many products as possible are produced;
Designers are trying to develop a device for spaceship so that the mass of the device is minimal;
Economists try to plan the plant's connections with sources of raw materials so that fare turned out to be minimal.
When studying any topic, students have a question: “Why do we need this?” If the answer satisfies curiosity, then we can talk about the students’ interest. The answer for the topic "Derivative" can be obtained by knowing where derivatives of functions are used.
To answer this question, we can list some disciplines and their sections in which derivatives are used.
Derivative in algebra:
1. Tangent to the graph of a function 
Tangent to the graph of a function f, differentiable at the point x o, is a straight line passing through the point (x o; f(x о)) and having a slope f′(x o).
y = f(x o) + f′(x о) (x – x о)
2. Search for intervals of increasing and decreasing functions
Function y=f(x) increases over the interval X, if for any and 
inequality holds. In other words, a larger argument value corresponds to a larger function value.
Function y=f(x) decreases on the interval X, if for any and the inequality 
. In other words, a larger value of the argument corresponds to a smaller value of the function.
3. Search for extremum points of the function
The point is called maximum point functions y=f(x), if for everyone x from its neighborhood the inequality is valid. The value of the function at the maximum point is called maximum of the function and denote .
The point is called minimum point functions y=f(x), if for everyone x from its neighborhood the inequality is valid. The value of the function at the minimum point is called minimum function and denote .
The neighborhood of a point is understood as the interval 
, where is a sufficiently small positive number.
The minimum and maximum points are called extremum points , and the values of the function corresponding to the extremum points are called extrema of the function .
4. Finding the intervals of convexity and concavity of a function
convex, if the graph of this function within the interval lies no higher than any of its tangents (Fig. 1).
The graph of a function differentiable on the interval is on this interval concave, if the graph of this function within the interval lies not lower than any of its tangents (Fig. 2).
The inflection point of the graph of a function is the point separating the intervals of convexity and concavity.
5. Finding bending points of a function
Derivative in physics:
1. Velocity as a derivative of path 
2. Acceleration as a derivative of speed a = 
3. Decay rate of radioactive elements 
= -
λN
And also in physics, the derivative is used to calculate:
Velocities of a material point 
Instantaneous speed as the physical meaning of the derivative
Instantaneous AC current value
Instantaneous value of EMF of electromagnetic induction
Maximum power
Derivative in chemistry:
And in chemistry, differential calculus has found wide application for constructing mathematical models of chemical reactions and subsequent description of their properties.
A derivative in chemistry is used to determine a very important thing - the rate of a chemical reaction, one of the decisive factors that must be taken into account in many areas of scientific and industrial activity. V (t) = p ‘(t)
Derivative in biology:
A population is a collection of individuals of a given species, occupying a certain area of territory within the species’ range, freely interbreeding and partially or completely isolated from other populations, and is also an elementary unit of evolution.
Derivative in geography:
1. Some meanings in seismography
2. Features electromagnetic field land
3. Radioactivity of nuclear-geophysical indicators
4.Many meanings in economic geography
5. Derive a formula for calculating the population in a territory at time t.
y'= k y
The idea of Thomas Malthus's sociological model is that population growth is proportional to the number of people at a given time t through N(t). Malthus' model worked well to describe the population of the United States from 1790 to 1860. This model is no longer valid in most countries.
Derivative in electrical engineering:
In our homes, in transport, in factories: electric current works everywhere. Electric current is understood as the directed movement of free electrically charged particles.
Quantitative characteristics electric current is the current strength.
In an electric current circuit, the electric charge changes over time according to the law q=q (t). Current strength I is the derivative of charge q with respect to time.
Electrical engineering mainly uses alternating current.
An electric current that changes over time is called alternating. An AC circuit may contain various elements: heaters, coils, capacitors.
The production of alternating electric current is based on the law of electromagnetic induction, the formulation of which contains the derivative of the magnetic flux.
Derivative in economics:
Economics is the basis of life, and differential calculus, an apparatus for economic analysis, occupies an important place in it. The basic task of economic analysis is to study the relationships of economic quantities in the form of functions.
The derivative in economics solves important issues:
1. In what direction will state income change with an increase in taxes or with the introduction of customs duties?
2. Will the company's revenue increase or decrease if the price of its products increases?
To solve these questions, it is necessary to construct connection functions of the input variables, which are then studied by methods of differential calculus.
Also, using the extremum of the function (derivative) in the economy, you can find the highest labor productivity, maximum profit, maximum output and minimum costs.
CONCLUSION: derivative is successfully used in solving various applied problems in science, technology and life
As can be seen from the above, the use of the derivative of a function is very diverse, not only in the study of mathematics, but also in other disciplines. Therefore, we can conclude that studying the topic: “Derivative of a function” will have its application in other topics and subjects.
We were convinced of the importance of studying the topic “Derivative”, its role in the study of processes in science and technology, the possibility of constructing mathematical models based on real events, and solving important problems.
“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,
A mathematics can achieve all these goals.”
That's what the American mathematician said Maurice Kline.
Bibliography:
1. Bogomolov N.V., Samoilenko I.I. Mathematics. - M.: Yurayt, 2015.
2. Grigoriev V.P., Dubinsky Yu.A., Elements of higher mathematics. - M.: Academy, 2014.
3. Bavrin I.I. Fundamentals of higher mathematics. - M.: Higher School, 2013.
4. Bogomolov N.V. Practical lessons mathematics. - M.: Higher School, 2013.
5. Bogomolov N.V. Collection of problems in mathematics. - M.: Bustard, 2013.
6. Rybnikov K.A. History of mathematics, Moscow University Publishing House, M, 1960.
7. Vinogradov Yu.N., Gomola A.I., Potapov V.I., Sokolova E.V. – M.: Publishing Center “Academy”, 2010
8. Bashmakov M.I. Mathematics: algebra and principles of mathematical analysis, geometry. – M.: Publishing Center “Academy”, 2016
Periodic sources:
Newspapers and magazines: “Mathematics”, “ Public lesson»
Use of Internet resources and electronic libraries.
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