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Definition. Let the function \(y = f(x)\) be defined in a certain interval containing the point \(x_0\) inside it. Let's give the argument an increment \(\Delta x \) such that it does not leave this interval. Let's find the corresponding increment of the function \(\Delta y \) (when moving from the point \(x_0 \) to the point \(x_0 + \Delta x \)) and compose the relation \(\frac(\Delta y)(\Delta x) \). If there is a limit to this ratio at \(\Delta x \rightarrow 0\), then the specified limit is called derivative of a function\(y=f(x) \) at the point \(x_0 \) and denote \(f"(x_0) \).
$$ \lim_(\Delta x \to 0) \frac(\Delta y)(\Delta x) = f"(x_0) $$
The symbol y is often used to denote the derivative." Note that y" = f(x) is new feature, but naturally associated with the function y = f(x), defined at all points x at which the above limit exists. This function is called like this: derivative of the function y = f(x).
Geometric meaning of derivative is as follows. If it is possible to draw a tangent to the graph of the function y = f(x) at the point with abscissa x=a, which is not parallel to the y-axis, then f(a) expresses the slope of the tangent:
\(k = f"(a)\)
Since \(k = tg(a) \), then the equality \(f"(a) = tan(a) \) is true.
Now let’s interpret the definition of derivative from the point of view of approximate equalities. Let the function \(y = f(x)\) have a derivative at a specific point \(x\):
$$ \lim_(\Delta x \to 0) \frac(\Delta y)(\Delta x) = f"(x) $$
This means that near the point x the approximate equality \(\frac(\Delta y)(\Delta x) \approx f"(x)\), i.e. \(\Delta y \approx f"(x) \cdot\Delta x\). The meaningful meaning of the resulting approximate equality is as follows: the increment of the function is “almost proportional” to the increment of the argument, and the coefficient of proportionality is the value of the derivative at a given point x. For example, for the function \(y = x^2\) the approximate equality \(\Delta y \approx 2x \cdot \Delta x \) is valid. If we carefully analyze the definition of a derivative, we will find that it contains an algorithm for finding it.
Let's formulate it.
1. Fix the value of \(x\), find \(f(x)\)
2. Give the argument \(x\) an increment \(\Delta x\), go to new point\(x+ \Delta x \), find \(f(x+ \Delta x) \)
3. Find the increment of the function: \(\Delta y = f(x + \Delta x) - f(x) \)
4. Create the relation \(\frac(\Delta y)(\Delta x) \)
5. Calculate $$ \lim_(\Delta x \to 0) \frac(\Delta y)(\Delta x) $$
This limit is the derivative of the function at point x.
If a function y = f(x) has a derivative at a point x, then it is called differentiable at a point x. The procedure for finding the derivative of the function y = f(x) is called differentiation functions y = f(x).
Let us discuss the following question: how are continuity and differentiability of a function at a point related to each other?
Let the function y = f(x) be differentiable at the point x. Then a tangent can be drawn to the graph of the function at point M(x; f(x)), and, recall, the angular coefficient of the tangent is equal to f "(x). Such a graph cannot “break” at point M, i.e. the function must be continuous at point x.
These were “hands-on” arguments. Let us give a more rigorous reasoning. If the function y = f(x) is differentiable at the point x, then the approximate equality \(\Delta y \approx f"(x) \cdot \Delta x \) holds. If in this equality \(\Delta x \) tends to zero, then \(\Delta y\) will tend to zero, and this is the condition for the continuity of the function at a point.
So, if a function is differentiable at a point x, then it is continuous at that point.
The reverse statement is not true. For example: function y = |x| is continuous everywhere, in particular at the point x = 0, but the tangent to the graph of the function at the “junction point” (0; 0) does not exist. If at some point a tangent cannot be drawn to the graph of a function, then the derivative does not exist at that point.
One more example. The function \(y=\sqrt(x)\) is continuous on the entire number line, including at the point x = 0. And the tangent to the graph of the function exists at any point, including at the point x = 0. But at this point the tangent coincides with the y-axis, i.e., it is perpendicular to the abscissa axis, its equation has the form x = 0. Such a straight line does not have an angle coefficient, which means that \(f"(0)\) does not exist.
So, we got acquainted with a new property of a function - differentiability. How can one conclude from the graph of a function that it is differentiable?
The answer is actually given above. If at some point it is possible to draw a tangent to the graph of a function that is not perpendicular to the abscissa axis, then at this point the function is differentiable. If at some point the tangent to the graph of a function does not exist or it is perpendicular to the abscissa axis, then at this point the function is not differentiable.
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. If C is a constant number and f=f(x), g=g(x) are some differentiable functions, then the following are true differentiation rules:
If given a set of numbers X and the method is indicated f, according to which for each value XЄ X only one number is assigned at. Then it is considered given function y = f(X), in which domain X(usually denoted D(f) = X). A bunch of Y all values at, for which there is at least one value XЄ X, such that y = f(X), such a set is called set of meanings functions f(most often denoted E(f)= Y).
Or dependence of one variable at from another X, at which each value of the variable X from a certain set D corresponds to a single variable value at, called function.
The functional dependence of the variable y on x is often emphasized by the notation y(x), which is read as a letter from x.
Domain functions at(X), i.e. the set of values of its argument X, denoted by the symbol D(y), which is read by de from igrek.
Range of values functions at(X), i.e., the set of values that the function y takes is denoted by the symbol E(at), which is read from the game.
The main ways to specify a function are:
A) analytical(using formula y = f(X)). This method also includes cases when the function is specified by a system of equations. If a function is given by a formula, then its domain of definition consists of all those values of the argument for which the expression written on the right side of the formula has values.
b) tabular(using a table of corresponding values X And at). In this way it is often asked temperature regime or exchange rates, but this method is not as clear as the next one;
V) graphic(using a graph). This is one of the most visual ways of specifying a function, since changes are immediately “read” from the graph. If the function at(X) is given by the graph, then its domain of definition D(y) is the projection of the graph onto the x-axis, and the range of values E(at) - projection of the graph onto the ordinate axis (see figure).
G) verbal. This method is often used in problems, or more precisely in describing their conditions. Usually this method is replaced by one of the above.
Functions y = f(X), xЄ X, And y = g(X), xЄ X, are called identically equal on a subset M WITH X, if for each x 0 Є M equality is true f(X 0) = g(X 0).
Graph of a function y = f(X) can be represented as a set of such points ( X; f(X)) on the coordinate plane, where X- arbitrary variable, from D(f). If f(X 0) = 0, where X 0 then the point with coordinates ( x 0 ; 0) is the point at which the graph of the function y = f(X) intersects with the O axis x. If 0Є D(f), then point (0; f(0)) is the point at which the graph of the function at = f(x) intersects with the O axis at.
Number X 0 of D(f) functions y = f(X) is the zero of the function, then when f(X 0) = 0.
Interval M WITH D(f) This interval of sign constancy functions y = f(X), if either for an arbitrary xЄ M right f(X) > 0, or for an arbitrary XЄ M right f(X) < 0.
Eat devices, which draw graphs of dependencies between quantities. These are barographs - devices for recording the dependence of atmospheric pressure on time, thermographs - devices for recording the dependence of temperature on time, cardiographs - devices for graphically recording the activity of the heart. The thermograph has a drum that rotates evenly. The paper wound on the drum touches the recorder, which, depending on the temperature, rises and falls and draws a certain line on the paper.
From representing a function with a formula, you can move on to representing it with a table and graph.
When studying mathematics, it is very important to understand what a function is, its domains of definition and meaning. Using the study of extremum functions, you can solve many problems in algebra. Even problems in geometry sometimes come down to considering the equations of geometric figures on a plane.
Description of the video lesson
A function is the dependence of the variable yrek on the variable x, in which each value of the variable x corresponds to a single value of the variable yrek.
X is called the independent variable or argument. The y is called the dependent variable, the value of the function, or simply the function.
If the dependence of the variable yrek on the variable x is a function, then it is briefly written as follows: yrek is equal to eff of x. This symbol also denotes the value of the function corresponding to the value of argument x.
Let the function be given by the formula y = three x square minus five. Then we can write that ef of x is equal to three x squared minus five. Let's find the values of the function eff for values of x equal to two and minus five. They will be equal to seven and seventy.
Note that in the notation yrek equals ef from x, instead of ef, you can use other letters: zhe, phi, and so on.
All x values form the domain of definition of the function. All values that the player takes form the range of values of the function.
A function is considered given if its domain of definition and the rule are indicated, according to which each value of x is associated with a single value of i.
If the function yrek equals ef of x is given by a formula and its domain of definition is not indicated, then it is considered that the domain of definition of the function consists of all values of the variable x for which the expression ef of x makes sense...
The graph of a function is the set of all points of the coordinate plane, the abscissas of which are equal to the values of the argument, and the ordinates are equal to the corresponding values of the function.
The figure shows a graph of the function yreq equals ef from x, the domain of which is the segment from one to five. Using a graph, you can find, for example, that the function of the number one is equal to minus three, the function of two is equal to two, the function of the number four is equal to minus two, and the function of the number five is equal to minus four. Lowest value function is equal to minus four, and the greatest is two. Moreover, any number from minus four to two, including these numbers, is the value of this function. Thus, the range of values of the function yrek equals ef from x is the segment from minus four to two.
Previously, we have already studied some types of functions:
The graph of the function ig = ka x plus be is a straight line. The domain of this function is the set of all numbers. The range of values of this function when ka is not equal to zero is the set of all numbers, and when ka is equal to zero its range of values consists of one number b.e.
The graph of the function yr divided by x is called a hyperbola.
The figure shows a graph of the function y = ka divided by x, for ka greater than zero. The domain of this function is the set of all numbers except zero. This set is also its range of values...
Functions describe many real processes and patterns. For example, direct proportionality is the dependence of body mass on its volume at constant density; dependence of the circumference of a circle on its radius. Inverse proportionality is the dependence of the current strength in a section of the circuit on the resistance of the conductor at constant voltage; the dependence of the time it takes a uniformly moving body to travel a given path on the speed of movement.
We also studied the functions given by the formulas ygr is equal to x square, y is equal to x cube, y is equal to the square root of x.
Consider the function given by the formula ig equals modulus x.
Since the expression modulus x makes sense for any x, the domain of definition of this function is the set of all numbers. By definition, the module x is equal to x if x is greater than or equal to zero, and minus x if x less than zero. Therefore, the function ig = modul x can be specified by the following system.
The graph of the function under consideration in the interval from zero to plus infinity, including zero, coincides with the graph of the function y = x, and in the interval from minus infinity to zero - with the graph of the function y = minus x. The graph of the function yg = modulus x consists of two rays that emanate from the origin and are the bisectors of the first and second coordinate angles.
1) Function domain and function range.
The domain of a function is the set of all valid real values argument x(variable x), for which the function y = f(x) determined. The range of a function is the set of all real values y, which the function accepts.
In elementary mathematics, functions are studied only on the set of real numbers.
2) Function zeros.
Function zero is argument value, at which the value of the function is equal to zero.
3) Intervals of constant sign of a function.
Intervals of constant sign of a function are sets of argument values on which the function values are only positive or only negative.
4) Monotonicity of the function.
An increasing function (in a certain interval) is a function for which higher value the argument from this interval corresponds to a larger value of the function.
A decreasing function (in a certain interval) is a function in which a larger value of the argument from this interval corresponds to a smaller value of the function.
5) Even (odd) function.
An even function is a function whose domain of definition is symmetrical with respect to the origin and for any X from the domain of definition the equality f(-x) = f(x). Schedule even function symmetrical about the ordinate axis.
An odd function is a function whose domain of definition is symmetrical with respect to the origin and for any X from the domain of definition the equality is true f(-x) = - f(x). Schedule odd function symmetrical about the origin.
6) Limited and unlimited functions.
A function is called bounded if there is a positive number M such that |f(x)| ≤ M for all values of x. If such a number does not exist, then the function is unlimited.
7) Periodicity of the function.
A function f(x) is periodic if there is a non-zero number T such that for any x from the domain of definition of the function the following holds: f(x+T) = f(x). This smallest number is called the period of the function. All trigonometric functions are periodic. (Trigonometric formulas).
19. Basic elementary functions, their properties and graphs. Application of functions in economics.
Linear function is called a function of the form , where x is a variable, a and b are real numbers.
Number A called slope straight line, it is equal to the tangent of the angle of inclination of this straight line to the positive direction of the abscissa axis. The graph of a linear function is a straight line. It is defined by two points.
1. Domain of definition - the set of all real numbers: D(y)=R
2. The set of values is the set of all real numbers: E(y)=R
3. The function takes a zero value when or.
4. The function increases (decreases) over the entire domain of definition.
5. A linear function is continuous over the entire domain of definition, differentiable and .
A function of the form, where x is a variable, coefficients a, b, c are real numbers, is called quadratic
