We are studying the derivative. Is it really that important in life? “Differential calculus is a description of the world around us, made in mathematical language. The derivative helps us to successfully solve not only mathematical problems, but also practical problems in various fields of science and technology.”
Concept in the language of chemistry Designation Concept in the language of mathematics Number of in-va at time t 0 p \u003d p (t 0) Function Time interval t \u003d t- t 0 Increment of the argument Change in the number of in-va p \u003d p (t 0 + t) – p(t 0) Function increment Average chemical reaction rate p/t Ratio of function increment to argument increment V (t) = p (t) Solution:
A population is a set of individuals of a given species occupying a certain area of the territory within the range of the species, freely interbreeding with each other and partially or completely isolated from other populations, and is also an elementary unit of evolution.
Solution: Concept in the language of biology Designation Concept in the language of mathematics Number at time t 1 x \u003d x (t) Function Time interval t \u003d t 2 - t 1 Argument increment Change in population size x \u003d x (t 2) - x (t 1) Function increment Rate of population change x/t Ratio of function increment to argument increment Relative increment at a given moment Lim x/t t 0 Derivative Р = x (t)
Algorithm for finding the derivative (for the function y=f(x)) Fix the value of x, find f(x). Give the argument x an increment Dx, (move x+Dx to a new point), find f(x+Dx). Find the increment of the function: Dy= f(x+Dx)-f(x) Compose the ratio of the increment of the function to the increment of the argument Calculate the limit of this ratio (this limit is f `(x).)
FGOU SPO
Novosibirsk Agricultural College
Essay
in the discipline "mathematics"
"Application of the derivative in science and technology"
S. Razdolnoe 2008
Introduction
1. Theoretical part
1.1 Problems leading to the concept of a derivative
1.2 Derivative definition
1.3 General rule for finding the derivative
1.4 Geometric meaning of the derivative
1.5 Mechanical meaning of the derivative
1.6 Second order derivative and its mechanical meaning
1.7 Definition and geometric meaning of the differential
2. Investigation of functions with the help of the derivative
Conclusion
Literature
Introduction
In the first chapter of my essay, we will talk about the concept of a derivative, the rules for its application, about the geometric and physical meaning of the derivative. In the second chapter of my essay, we will talk about the use of the derivative in science and technology and about solving problems in this area.
1. Theoretical part
1.1 Problems leading to the concept of a derivative
When studying certain processes and phenomena, the problem often arises of determining the speed of these processes. Its solution leads to the concept of a derivative, which is the basic concept of differential calculus.
The method of differential calculus was created in the 17th and 18th centuries. The names of two great mathematicians, I. Newton and G.V. Leibniz.
Newton came to the discovery of differential calculus when solving problems about the speed of a material point at a given moment of time (instantaneous speed).
As is known, uniform movement is a movement in which a body travels equal lengths of the path in equal intervals of time. The distance traveled by a body in a unit of time is called speed uniform movement.
However, most often in practice we are dealing with uneven movement. A car driving on the road slows down at the crossings and speeds it up in those sections where the path is clear; the aircraft slows down when landing, etc. Therefore, most often we have to deal with the fact that in equal time intervals the body passes path segments of different lengths. Such a movement is called uneven. Its speed cannot be characterized by a single number.
Often, to characterize uneven motion, the concept is used average speed movement during the time ∆t٫ which is determined by the relation where ∆s is the path traveled by the body during the time ∆t.
So, with a body in free fall, the average speed of its movement in the first two seconds is
In practice, such a characteristic of movement as average speed says very little about movement. Indeed, at 4.9 m / s, and for the 2nd - 14.7 m / s, while the average speed for the first two seconds is 9.8 m / s. The average speed during the first two seconds does not give any idea of how the movement occurred: when the body moved faster, and when slower. If we set the average speeds of movement for each second separately, then we will know, for example, that in the 2nd second the body moved much faster than in the 1st. However, in most cases much faster than we are not satisfied with. After all, it is easy to understand that during this 2nd second the body also moves in different ways: at the beginning it is slower, at the end it is faster. And how does it move somewhere in the middle of this 2nd second? In other words, how to determine the instantaneous speed?
Let the motion of the body be described by the law for a time equal to ∆t. At the moment t0 the body has passed the path, at the moment - the path. Therefore, during the time ∆t, the body has traveled a distance and the average speed of the body over this period of time will be.
The shorter the time interval ∆t, the more accurately it is possible to establish with what speed the body is moving at the moment t0, since a moving body cannot significantly change its speed in a short period of time. Therefore, the average speed as ∆t tends to zero approaches the actual speed of movement and, in the limit, gives the speed of movement at a given time t0 (instantaneous speed).
Thus ,
Definition 1. Instant Speed of rectilinear motion of the body at a given time t0 is called the limit of the average speed over the time from t0 to t0+ ∆t, when the time interval ∆t tends to zero.
So, in order to find the speed of rectilinear non-uniform motion at a given moment, it is necessary to find the limit of the ratio of the increment of the path ∆to the increment of time ∆t under the condition i.e. Leibniz came to the discovery of differential calculus while solving the problem of constructing a tangent to any curve given by his equation.
The solution to this problem is of great importance. After all, the speed of a moving point is directed along a tangent to its trajectory, so determining the speed of a projectile on its trajectory, the speed of any planet in its orbit, is reduced to determining the direction of the tangent to the curve.
The definition of a tangent as a straight line that has only one common point with a curve, which is valid for a circle, is unsuitable for many other curves.
The following definition of a tangent to a curve not only corresponds to the intuitive idea of it, but also allows you to actually find its direction, i.e. calculate the slope of the tangent.
Definition 2. Tangent to the curve at the point M is called the straight line MT, which is the limiting position of the secant MM1, when the point M1, moving along the curve, indefinitely approaches the point M.
1.2 Derivative definition
Note that when determining the tangent to the curve and the instantaneous speed of non-uniform motion, essentially the same mathematical operations are performed:
1. The given value of the argument is incremented and a new value of the function is calculated corresponding to the new value of the argument.
2. Determine the function increment corresponding to the selected argument increment.
3. The increment of the function is divided by the increment of the argument.
4. Calculate the limit of this ratio, provided that the increment of the argument tends to zero.
Solutions of many problems lead to limit transitions of this type. It becomes necessary to make a generalization and give a name to this passage to the limit.
The rate of change of the function depending on the change of the argument can obviously be characterized by a ratio. This relationship is called average speed function changes on the interval from to. Now we need to consider the limit of a fraction. The limit of this ratio as the increment of the argument tends to zero (if this limit exists) is some new function of. This function is denoted by the symbols y', called derivative this function, since it is obtained (produced) from the function The function itself is called primitive function with respect to its derivative
Definition 3. derivative functions at a given point name the limit of the ratio of the increment of the function ∆y to the corresponding increment of the argument ∆x, provided that ∆x→0, i.e.
1.3 General rule for finding the derivative
The operation of finding the derivative of some function is called differentiation functions, and the branch of mathematics that studies the properties of this operation is differential calculus.
If a function has a derivative at x=a, then it is said to be differentiable at this point. If a function has a derivative at every point in a given interval, then it is said to be differentiable On this interval .
The definition of the derivative not only fully characterizes the concept of the rate of change of a function when the argument changes, but also provides a way to actually calculate the derivative of a given function. To do this, you must perform the following four actions (four steps) indicated in the definition of the derivative itself:
1. Find a new function value by presenting a new argument value instead of x to this function: .
2. The increment of the function is determined by subtracting the given value of the function from its new value: .
3. Compose the ratio of the increment of the function to the increment of the argument: .
4. Go to the limit at and find the derivative: .
Generally speaking, a derivative is a “new” function derived from a given function according to a specified rule.
1.4 Geometric meaning of the derivative
The geometric interpretation of the derivative, first given at the end of the 17th century. Leibniz is as follows: the value of the derivative of the function at the point x is equal to the slope of the tangent drawn to the graph of the function at the same point x, those.
The equation of a tangent, like any straight line passing through a given point in a given direction, has the form - current coordinates. But the tangent equation will also be written as follows: . The normal equation will be written in the form
1.5 Mechanical meaning of the derivative
The mechanical interpretation of the derivative was first given by I. Newton. It consists in the following: the speed of movement of a material point at a given moment of time is equal to the derivative of the path with respect to time, i.e. Thus, if the law of motion of a material point is given by an equation, then in order to find the instantaneous speed of a point at some particular moment in time, you need to find the derivative and substitute the corresponding value of t into it.
1.6 Second order derivative and its mechanical meaning
We get (an equation from what was done in the textbook Lisichkin V.T. Soloveychik I.L. "Mathematics" p. 240):
Thus, the acceleration of the rectilinear motion of the body at a given moment is equal to the second derivative of the path with respect to time, calculated for a given moment. This is the mechanical meaning of the second derivative.
1.7 Definition and geometric meaning of the differential
Definition 4. The main part of the increment of a function, linear with respect to the increment of the function, linear with respect to the increment of the independent variable, is called differential functions and is denoted by d, i.e. .
Function differential geometrically represented by the increment of the ordinate of the tangent drawn at the point M ( x ; y ) for given values of x and ∆x.
calculation differential – .
Application of the differential in approximate calculations – , the approximate value of the increment of the function coincides with its differential.
Theorem 1. If the differentiable function increases (decreases) in a given interval, then the derivative of this function is not negative (not positive) in this interval.
Theorem 2. If the derivative function is positive (negative) in some interval, then the function in this interval is monotonically increasing (monotonically decreasing).
Let us now formulate the rule for finding intervals of monotonicity of the function
1. Calculate the derivative of this function.
2. Find points where is zero or does not exist. These points are called critical for function
3. With the points found, the domain of the function is divided into intervals, on each of which the derivative retains its sign. These intervals are intervals of monotonicity.
4. Examine the sign on each of the found intervals. If on the considered interval, then on this interval increases; if, then it decreases on such an interval.
Depending on the conditions of the problem, the rule for finding monotonicity intervals can be simplified.
Definition 5. A point is called a maximum (minimum) point of a function if the inequality holds, respectively, for any x from some neighborhood of the point.
If is the maximum (minimum) point of the function, then we say that (minimum) at the point. Maximum and minimum functions unite name extremum functions, and the maximum and minimum points are called extremum points (extreme points).
Theorem 3.(necessary sign of an extremum). If and the derivative exists at this point, then it is equal to zero: .
Theorem 4.(sufficient sign of an extremum). If the derivative when x passes through a changes sign, then a is the extremum point of the function .
The main points of the study of the derivative:
1. Find the derivative.
2. Find all critical points from the domain of the function.
3. Set the signs of the derivative of the function when passing through the critical points and write out the extremum points.
4. Calculate the function values at each extreme point.
2. Investigating Functions with the Derivative
Task #1 . Log volume. Logs of the correct form without wood defects with a relatively small difference in the diameters of the thick and thin ends are called industrial roundwood. When determining the volume of industrial round timber, a simplified formula is usually used, where is the length of the log, is the area of its average section. Find out whether the real volume ends or underestimates; estimate the relative error.
Solution. The shape of a round business timber is close to a truncated cone. Let be the radius of the larger, smaller end of the log. Then its almost exact volume (the volume of a truncated cone) can, as is known, be found by the formula. Let be the volume value calculated by the simplified formula. Then;
Those. . This means that the simplified formula gives an underestimation of the volume. Let's put it now. Then. This shows that the relative error does not depend on the length of the log, but is determined by the ratio. Since when increases on the interval . Therefore, which means that the relative error does not exceed 3.7%. In the practice of forest science, such an error is considered quite acceptable. With greater accuracy, it is practically impossible to measure either the diameters of the ends (because they are somewhat different from circles), or the length of the log, since they measure not the height, but the generatrix of the cone (the length of the log is tens of times greater than the diameter, and this does not lead to large errors). Thus, at first glance, an incorrect, but simpler formula for the volume of a truncated cone in a real situation turns out to be quite legitimate. Repeatedly carried out with the help of special methods of verification showed that with the mass accounting of the industrial forest, the relative error when using the considered formula does not exceed 4%.
Task #2 . When determining the volumes of pits, trenches of buckets and other containers that have the shape of a truncated cone, a simplified formula is sometimes used in agricultural practice, where is the height, are the areas of the bases of the cone. Find out whether the real volume is overestimated or underestimated, estimate the relative error under the condition natural for practice: (- base radii, .
Solution. Denoting through the true value of the volume of the truncated cone, and through the value calculated by the simplified formula, we get: , i.e. . This means that the simplified formula gives an overestimation of the volume. Repeating further the solution of the previous problem, we find that the relative error will be no more than 6.7%. Probably, such accuracy is acceptable when rationing excavation work - after all, the pits will not be ideal cones, and the corresponding parameters in real conditions are measured very roughly.
Task #3 . In special literature, to determine the angle β of rotation of the spindle of a milling machine when milling couplings with teeth, a formula is derived where. Since this formula is complex, it is recommended to discard its denominator and use a simplified formula. At what (- an integer,) can this formula be used if an error in is allowed when determining the angle?
Solution. The exact formula after simple identical transformations can be reduced to the form. Therefore, when using an approximate formula, an absolute error is allowed, where. We study the function on the interval . In this case, 0.06, i.e. the corner belongs to the first quarter. We have: . Note that on the interval under consideration, and hence the function is decreasing on this interval. Since further, for all considered. Means, . Since it is a radian, it is enough to solve the inequality. Solving this inequality by selection, we find that, . Since the function is decreasing, it follows that
Conclusion
The use of the derivative is quite broad and can be fully covered in this type of work, but I have tried to cover the main points. In our time, in connection with scientific and technological progress, in particular with the rapid evolution of computing systems, differential calculus is becoming more and more relevant in solving both simple and super-complex problems.
Literature
1. V.A. Petrov "Mathematical analysis in production tasks"
2. Soloveichik I.L., Lisichkin V.T. "Mathematics"
In this paper, I will consider the applications of the derivative in various sciences and industries. The work is divided into chapters, each of which deals with one of the aspects of the differential calculus (geometric, physical meaning, etc.)
1. The concept of a derivative
1-1. Historical information
The differential calculus was created by Newton and Leibniz at the end of the 17th century on the basis of two problems:
1) about finding a tangent to an arbitrary line
2) about the search for speed with an arbitrary law of motion
Even earlier, the concept of a derivative was encountered in the works of the Italian mathematician Tartaglia (circa 1500 - 1557) - here a tangent appeared in the course of studying the issue of the angle of inclination of the gun, which ensures the greatest range of the projectile.
In the 17th century, on the basis of G. Galileo's theory of motion, the kinematic concept of the derivative was actively developed. Various presentations began to appear in the works of Descartes, the French mathematician Roberval, and the English scientist L. Gregory. Lopital, Bernoulli, Lagrange, Euler, Gauss made a great contribution to the study of differential calculus.
1-2. The concept of a derivative
Let y \u003d f (x) be a continuous function of the argument x, defined in the interval (a; b), and let x 0 be an arbitrary point of this interval
Let's give the argument x an increment?x, then the function y = f(x) will receive an increment?y = f(x + ?x) - f(x). The limit to which the ratio?y /?x tends when?x > 0 is called the derivative of the function f(x).
y"(x)=
1-3. Rules of differentiation and table of derivatives
| C" = 0 | (x n) = nx n-1 | (sin x)" = cos x |
| x" = 1 | (1 / x)" = -1 / x2 | (cos x)" = -sin x |
| (Cu)"=Cu" | (vx)" = 1 / 2vx | (tg x)" = 1 / cos 2 x |
| (uv)" = u"v + uv" | (a x)" = a x log x | (ctg x)" = 1 / sin 2 x |
| (u / v)"=(u"v - uv") / v 2 | (ex)" = ex | (arcsin x)" = 1 / v (1- x 2) |
| (log a x)" = (log a e) / x | (arccos x)" = -1 / v (1- x 2) | |
| (ln x)" = 1 / x | (arctg x)" = 1 / v (1+ x 2) | |
| (arcctg x)" = -1 / v (1+ x 2) |
2. The geometric meaning of the derivative
2-1. Tangent to curve
Let we have a curve and a fixed point M and a point N on it. A tangent to the point M is a straight line, the position of which tends to be occupied by the chord MN, if the point N is indefinitely approached along the curve to M.
Consider the function f(x) and the curve y = f(x) corresponding to this function. For some value x, the function has the value y = f(x). These values on the curve correspond to the point M(x 0 , y 0). Let's introduce a new argument x 0 + ?x, its value corresponds to the value of the function y 0 + ?y = f(x 0 + ?x). The corresponding point is N(x 0 + ?x, y 0 + ?y). Draw a secant MN and denote? the angle formed by a secant with the positive direction of the Ox axis. It can be seen from the figure that ?y / ?x = tg ?. If now? x will approach 0, then the point N will move along the curve, the secant MN will rotate around the point M, and the angle? - change. If at? x > 0 the angle? tends to some ?, then the straight line passing through M and making the angle ? with the positive direction of the abscissa axis will be the desired tangent. At the same time, its slope coefficient:
That is, the value of the derivative f "(x) for a given value of the argument x is equal to the tangent of the angle formed with the positive direction of the Ox axis by the tangent to the graph of the function f (x) at the point M (x, f (x)).
A tangent to a space line has a definition similar to that of a tangent to a plane curve. In this case, if the function is given by the equation z = f(x, y), the slopes at the OX and OY axes will be equal to the partial derivatives of f with respect to x and y.
2-2. Tangent plane to surface
The tangent plane to the surface at the point M is the plane containing the tangents to all spatial curves of the surface passing through M - the point of contact.
Take the surface given by the equation F(x, y, z) = 0 and some ordinary point M(x 0 , y 0 , z 0) on it. Consider on the surface some curve L passing through M. Let the curve be given by the equations
x = ?(t); y = ?(t); z = ?(t).
Let us substitute these expressions into the equation of the surface. The equation will turn into an identity, since the curve lies entirely on the surface. Using the invariance property of the form of the differential, we differentiate the resulting equation with respect to t:
The equations of the tangent to the curve L at the point M have the form:
Since the differences x - x 0, y - y 0, z - z 0 are proportional to the corresponding differentials, the final equation of the plane looks like this:
F" x (x - x 0) + F" y (y - y 0) + F" z (z - z 0)=0
and for the particular case z = f(x, y):
Z - z 0 \u003d F "x (x - x 0) + F" y (y - y 0)
Example: Find the equation of the tangent plane at the point (2a; a; 1,5a) of the hyperbolic paraboloid
Solution:
Z" x \u003d x / a \u003d 2; Z" y \u003d -y / a \u003d -1
The equation of the desired plane:
Z - 1.5a = 2(x - 2a) - (Y - a) or Z = 2x - y - 1.5a
3. Using the derivative in physics
3-1. Material point speed
Let the dependence of the path s on time t in a given rectilinear motion of a material point be expressed by the equation s = f(t) and t 0 is some moment of time. Consider another time t, denote?t = t - t 0 and calculate the path increment: ?s = f(t 0 + ?t) - f(t 0). The ratio?s /?t is called the average speed of movement for the time?t elapsed from the initial moment t 0 . The speed is called the limit of this ratio when? t\u003e 0.
The average acceleration of uneven motion in the interval (t; t + ?t) is the value =?v / ?t. The instantaneous acceleration of a material point at time t will be the limit of the average acceleration:
That is, the first time derivative (v "(t)).
Example: The dependence of the path traveled by the body on time is given by the equation s \u003d A + Bt + Ct 2 + Dt 3 (C \u003d 0.1 m / s, D \u003d 0.03 m / s 2). Determine the time after the start of movement, after which the acceleration of the body will be equal to 2 m / s 2.
Solution:
v(t) = s "(t) = B + 2Ct + 3Dt 2 ; a(t) = v"(t) = 2C + 6Dt = 0.2 + 0.18t = 2;
1.8 = 0.18t; t = 10 s
3-2. The heat capacity of a substance at a given temperature
To increase different temperatures T by the same value, equal to T 1 - T, per 1 kg. a given substance needs a different amount of heat Q 1 - Q, and the ratio
for this substance is not constant. Thus, for a given substance, the amount of heat Q is a non-linear function of temperature T: Q = f(T). Then?Q = f(t + ?T) - f(T). Attitude
is called the average heat capacity on the segment, and the limit of this expression at? T > 0 is called the heat capacity of the given substance at temperature T.
3-3. Power
A change in the mechanical motion of a body is caused by forces acting on it from other bodies. In order to quantitatively characterize the process of energy exchange between interacting bodies, the concept of the work of a force is introduced in mechanics. To characterize the rate of doing work, the concept of power is introduced:
4. Differential calculus in economics
4-1. Function research
Differential calculus is a mathematical apparatus widely used for economic analysis. The basic task of economic analysis is to study the relationships of economic quantities written as functions. In what direction will government revenue change if taxes are increased or if import duties are introduced? Will the firm's revenue increase or decrease when the price of its products increases? In what proportion can additional equipment replace retired workers? To solve such problems, the connection functions of the variables included in them must be constructed, which are then studied using the methods of differential calculus. In economics, it is often required to find the best or optimal value of an indicator: the highest labor productivity, maximum profit, maximum output, minimum costs, etc. Each indicator is a function of one or more arguments. Thus, finding the optimal value of the indicator is reduced to finding the extremum of the function.
According to Fermat's theorem, if a point is an extremum of a function, then the derivative either does not exist in it or is equal to 0. The type of an extremum can be determined by one of the sufficient conditions for an extremum:
1) Let the function f(x) be differentiable in some neighborhood of the point x 0 . If the derivative f "(x) when passing through the point x 0 changes sign from + to -, then x 0 is the maximum point, if from - to +, then x 0 is the minimum point, if it does not change sign, then there is no extremum.
2) Let the function f (x) be twice differentiable in some neighborhood of the point x 0, and f "(x 0) \u003d 0, f "" (x 0) ? 0, then at the point x 0 the function f (x 0) has a maximum , if f ""(x 0)< 0 и минимум, если f ""(x 0) > 0.
In addition, the second derivative characterizes the convexity of the function (the graph of the function is called convex up [down] on the interval (a, b) if it is located on this interval not higher [not lower] than any of its tangents).
Example: choose the optimal volume of production by the firm, the profit function of which can be modeled by the dependence:
?(q) = R(q) - C(q) = q 2 - 8q + 10
Solution:
?"(q) = R"(q) - C"(q) = 2q - 8 = 0 > q extr = 4
For q<
q extr = 4 >?"(q)< 0 и прибыль убывает
For q > q extr = 4 > ?(q) > 0 and the profit increases
When q = 4, the profit takes the minimum value.
What is the optimal output for the firm? If the firm cannot produce more than 8 units of output in the period under review (p(q = 8) = p(q = 0) = 10), then the optimal solution would be to produce nothing at all, but to receive income from renting premises and / or equipment. If the firm is able to produce more than 8 units, then the optimal for the firm will be to produce at the limit of its production capacity.
4-2. Elasticity of demand
The elasticity of the function f (x) at the point x 0 is called the limit
Demand is the quantity of a good demanded by the buyer. Price elasticity of demand E D is a measure of how demand responds to price changes. If ¦E D ¦>1, then the demand is called elastic, if ¦E D ¦<1, то неэластичным. В случае
E D =0 спрос называется совершенно
неэластичным, т. е. изменение цены не приводит
ни к какому изменению спроса. Напротив,
если самое малое снижение цены побуждает
покупателя увеличить покупки от 0 до предела
своих возможностей, говорят, что спрос
является совершенно эластичным. В зависимости
от текущей эластичности спроса, предприниматель
принимает решения о снижении или повышении
цен на продукцию.
4-3. Limit Analysis
An important section of the methods of differential calculus used in economics is the methods of limiting analysis, i.e., a set of methods for studying changing values of costs or results with changes in production, consumption, etc. based on an analysis of their limiting values. The limiting indicator (s) of a function is its derivative (in the case of a function of one variable) or partial derivatives (in the case of a function of several variables)
In economics, averages are often used: average labor productivity, average costs, average income, average profit, etc. But it is often required to find out by what amount the result will increase if costs are increased or vice versa, how much the result will decrease if costs are reduced. It is impossible to answer this question with the help of average values. In such problems, it is required to determine the limit of the ratio of the increase in the result and costs, i.e., to find the marginal effect. Therefore, to solve them, it is necessary to use the methods of differential calculus.
5. Derivative in approximate calculations
etc.................
Description of the presentation on individual slides:
1 slide
Description of the slide:
Lesson topic: Application of the derivative in various fields of knowledge Mathematics teacher MBOU "School No. 74" Zagumennova Marina Vladimirovna
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The purpose of the lesson: Learn the main areas of application of the derivative in various fields of science and technology; Consider, using examples of solving practical problems, how the derivative is used in chemistry, physics, biology, geography, and economics.
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“There is not a single area of mathematics, however abstract it may be, that will not someday be applicable to the phenomena of the real world.” N.I. Lobachevsky
4 slide
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Differentiation rules Derivative of a sum About a constant factor Derivative of a product Derivative of a fraction Derivative of a complex function (u+v)"= u" + v' (Cu)"=Cu' (uv)"=u"v+uv' (u/v)" =(u"v-uv")/v2 hꞌ(x)=gꞌ(f(x))f ꞌ(x)
5 slide
Description of the slide:
Derivative in physics Problem. The movement of the car during braking is described by the formula s(t) = 30t - 5t2, (s is the stopping distance in meters, t is the time in seconds from the beginning of braking to the complete stop of the car). Find how many seconds the car is in motion from the moment it starts braking until it comes to a complete stop. What is the distance traveled by the car from the beginning of braking until it comes to a complete stop? Solution: Since the speed is the first derivative of the movement in time, then v = S'(t) = 30 - 10t, because when braking, the speed is zero, then 0=30–10t; 10t=30; t=3(sec). Stopping distance S(t) = 30t - 5t2 = 30∙3-5∙32 = 90-45 = 45(m). Answer: deceleration time 3s, braking distance 45m.
6 slide
Description of the slide:
This is interesting Steamship "Chelyuskin" in February 1934 successfully passed the entire northern sea route, but in the Bering Strait was trapped in the ice. The ice carried the Chelyuskin to the north and crushed it. Here is a description of the disaster: “The strong metal of the hull did not succumb immediately,” the head of the expedition, O.Yu., reported on the radio. Schmidt. - It was visible how the ice floe was pressed into the side, and how the sheathing sheets above it bulged, bending outward. The ice continued its slow but irresistible advance. The swollen iron sheets of the hull plating were torn at the seam. Rivets flew with a crack. In an instant, the port side of the ship was torn off from the bow hold to the aft end of the deck ... ”Why did the disaster happen?
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The ice pressure force Р is decomposed into two: F and R. R is perpendicular to the board, F is directed tangentially. The angle between P and R - α - the angle of the side to the vertical. Q is the force of ice friction against the board. Q = 0.2 R (0.2 is the coefficient of friction). If Q< F, то F увлекает напирающий лед под воду, лед не причиняет вреда, если Q >F, then friction prevents the ice floe from sliding, and the ice can crush and push through the side. 0.2R< R tgα , tgα >0.2; Q< F, если α >1100. The inclination of the ship's sides to the vertical at an angle α > 1100 ensures safe navigation in ice.
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Derivative in chemistry A derivative in chemistry is used to determine the rate of a chemical reaction. This is necessary for: process engineers in determining the effectiveness of chemical production, chemists developing drugs for medicine and agriculture, as well as doctors and agronomists who use these drugs to treat people and to apply them to the soil. To solve production problems in the medical, agricultural and chemical industries, it is simply necessary to know the reaction rates of chemicals.
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Problem in chemistry Let the amount of a substance that entered into a chemical reaction be given by the dependence: р(t) = t2/2 + 3t –3 (mol). Find the rate of a chemical reaction after 3 seconds. Reference: The rate of a chemical reaction is the change in the concentration of reactants per unit time or the derivative of the concentration of reactants with respect to time (in the language of mathematics, concentration would be a function, and time would be an argument)
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Solution Concept in the language of chemistry Notation Concept in the language of mathematics Amount of substance at time t0 p = p(t0) Function Time interval ∆t = t – t0 Argument increment Change in the amount of substance ∆p = p(t0+ ∆t) – p(t0) Function increment Average rate of a chemical reaction ∆p/∆t The ratio of the increment of the function to the increment of the argument V (t) = p'(t)
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Derivative in biology Problem in biology: Based on the known dependence of the population size x(t), determine the relative growth at time t. Reference: A population is a collection of individuals of a given species, occupying a certain area of the territory within the range of the species, freely interbreeding with each other and partially or completely isolated from other populations, and is also an elementary unit of evolution.
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Solution Concept in the language of biology Designation Concept in the language of mathematics Number at time t x = x(t) Function Time interval ∆t = t – t0 Increment of the argument Population change ∆x = x(t) – x(t0) Increment of the function Rate of change population size ∆x/∆t The ratio of the increment of the function to the increment of the argument Relative growth at a given moment lim∆x/∆t ∆t → 0 Derivative Р = x" (t)
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Derivative in geography The derivative helps to calculate: Some values in seismography Features of the electromagnetic field of the earth Radioactivity of nuclear geophysical indicators Many values in economic geography Derive a formula for calculating the population in the territory at time t.
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Geography problem Derive a formula for calculating the population in a limited area at time t.
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Solution Let y=y(t) be the population. Consider the population growth for ∆t = t – t0 ∆у = k∙y∙∆t, where k = kр – kс is the population growth rate, (kр is the birth rate, kс is the death rate). ∆у/∆t = k∙y as ∆t → 0 we get lim ∆у/∆t = у’. Population growth - y’ = k∙y. ∆t → 0 Conclusion: the derivative in geography is combined with many of its branches (seismography, location and population) as well as with economic geography. All this makes it possible to more fully study the development of the population and countries of the world.
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Derivative in Economics The derivative solves important questions: In what direction will the government's income change with an increase in taxes or with the introduction of customs duties? Will the firm's revenue increase or decrease when the price of its products increases? To solve these questions, it is necessary to construct the connection functions of the input variables, which are then studied by the methods of differential calculus. Also, using the extremum of a function in the economy, one can find the highest labor productivity, maximum profit, maximum output and minimum costs.
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Problem in economics No. 1 (production costs) Let y be production costs, and x be the quantity of production, then x1 is the increase in production, and y1 is the increase in production costs.
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Project activity in mathematics lessons
Project theme: Application of the derivative
Members: 1st year students of the State Educational Institution "SKSiS"
Fundamental question : How to measure speed speed?
Problematic issues
Who worked on the question of "differentiation"?
How is the derivative used in the study of a function?
How does the derivative help biologists, chemists?
What problems in physics are solved with the help of a derivative?
How is the derivative used in economics?
What is the relationship between derivative and geography?
Target: The study of the use of the derivative for solving problems in the principles of analysis, physics, economics, biology, chemistry and geography; deepening and expanding knowledge on the topic "Derivative".
Tasks:
Find information about the history of the origin of the derivative, study it and systematize it.
Investigation of functions for monotonicity, extrema, convexity-concavity using a derivative.
Selection of tasks from different branches of biology, which are solved using the derivative
Find out what processes the derivative in geography regulates. Consider tasks in geography that are solved using the derivative
Select problems from different sections of physics that are solved using the derivative.
Select economic problems that are solved using the derivative.
Consider the application of the rules for calculating the derivative to solving practical problems with economic content.
"I warn you to beware of dropping dx - this is a mistake that is often made and that hinders progress"
G.W. Leibniz
Using the derivative to solve problems requires students to think outside the box. It should be noted that knowledge of non-standard methods and techniques for solving problems contributes to the development of new, non-standard thinking, which can also be successfully applied in other areas of human activity (economics, physics, chemistry, biology, etc.). This proves the relevance of this work. In the work on the project, certain stages of student activity are necessarily observed. Each of them contributes to the formation of personal qualities.
Preparatory stage
At this stage, the students and I are immersed in the project: activity is motivated, the topic, problem and goals are defined. The topic of the project should be not only close and interesting, but also accessible to the student. This stage of the project is the shortest in terms of time, but it is very important for achieving the expected results.A conversation is held during the demonstration of an introductory presentation; actualization of existing knowledge on the topic, discussion of the general plan of the project, planning work on the project. Determining the direction of information search in different sources.
The topic "Derivative" is one of the most important sections of the course of mathematical analysis, since this concept is the main one in differential calculus and serves as the initial base for the construction of integral calculus. But often, students, faced with this concept for the first time, do not understand why it is necessary to study it. They do not see the practical application of this topic. Therefore, this project "Application of the derivative" is aimed at ensuring that students find out why it is necessary to study the derivative, where you can use the knowledge related to the derivative in life, as well as in other subjects.
Stage of planning and organization of activities.
At this stage, we define groups by areas of activity, highlight the goals and objectives of each group. Suggested topics for group selection:
Group 1 - "Historical information of differential calculus";
Group 2 - "The geometric meaning of the derivative"
group 4 - "Application of the derivative in solving physical problems";
Group 3 - "Finding the best solution in applied, including socio-economic, tasks"
Group 4 - "Application of the derivative in chemistry and biology"
Group 5 - "The use of the derivative in solving problems with geographical content."
The group included students with different learning abilities. Each group was given the task to analyze the chosen topic, find information. The work of groups is planned: responsibilities are distributed between students, sources of information are determined, methods of collecting and analyzing information, ways of presenting the results of activities (in our case, presentations and booklets.).
Search stage.
At this stage, there is a search and collection of information on their chosen topic, the solution of intermediate tasks. Analysis and generalization of the collected material. Written presentation of the results and intermediate control, by the teacher, of the results obtained. Consultations were held on the programs PowerPoint, Publisher, Word, for students who had problems in practical work to formalize the results. Formulation of conclusions.
Stage of presentation of results, report.
The presentation stage is necessary to complete the work, to analyze what has been done, self-assessment and assessment from the outside, and demonstrate the results. The form of presenting the results in our project: an oral report with a demonstration of materials designed in the form of a presentation, booklet, abstract.
Evaluation of results, reflection
One of the final stages of work on the project is the evaluation of the results, reflection. The project is defended in a lesson or in a circle lesson.
The appendices contain the works of students prepared as part of the project activities in the form of presentations and a booklet.
When assessing the work of students on the project, the content is taken into account (completeness of the disclosure of the topic, presentation of aspects of the topic, presentation of the strategy for solving the problem, the logic of presenting information, the use of various resources), the degree of independent work of the group (coordinated work in the group, distribution of roles in the group, author's originality), design of the presentation product (grammar, suitable dictionary, absence of spelling errors and typos), protection (quality of the report, volume and depth of knowledge on the topic, culture of speech, manner of speaking in front of the audience, answers to questions).
