Function- this is a thing that connects two (or more) variables with each other. In other words, the function helps to find one variable if we know the value of the second variable. For example, if we have 100 rubles in our pocket, and a chocolate bar costs 50 rubles, then we can buy 2 chocolate bars. If we have 200 rubles in our pocket, then we can buy 4 chocolates. In this case, the first variable is the amount we have in our pocket, and the second variable is the number of chocolates we can buy. The cost of a chocolate bar is 50 rubles, it does not depend on how much money we have, so this value is constant.

You can create a function for this case: y = 50X, Where at- money in your pocket, X- number of chocolates.

Naturally, functions can be more complex. But to solve OGE tasks in mathematics, it is enough to know what the graphs of basic functions look like.

1. Function of the form y = kx + b (straight line)

In this function k And b these are numbers. The function can be written in in different forms: y = x,y = 2x, y = 3x – 4, y = -9x +44, y= etc. The main feature is the presence of x ( X) to the first power (that is, all cases when we do not divide by X).
Number k in this case, it determines in which direction the line is inclined. If k > 0 , then the function increases to the right. If k < 0 , then the function increases to the left.


Number b y. If b >0 , then the graph intersects the y-axis above the origin, If b < 0 - below.

2. Function of the form y = ax 2 + bx +c (parabola)

In this function a, b, c– numbers. The function can be written in different forms: y = x 2, y = 3x 2 + 8, y = 2x 2 -4x + 10,y = -x 2 – 9x +1,y =– 7, etc. The main feature is the presence of x squared ( x 2).

Number A is responsible for which direction (up or down) the branches of the parabola are directed (I also call it a happy smiley and a sad smiley). If a > 0 , then a cheerful smiley, If a < 0 - sad.

Number b is responsible for which direction (right or left) the starting point of the parabola (inflection point) is shifted relative to the axis y. If b > 0 , then the graph is shifted to the left, If b < 0 – to the right.

Number c – this is the point of intersection of the graph with the axis y. If c >0 , then the graph intersects the axis y above the origin, If c < 0 - below.

3. Function of the form y = k/x + b (hyperbola)

This function is similar in appearance to the straight line function, with the exception that X is in the denominator. This is exactly her distinctive feature. Number k is responsible for arranging the function in quarters, If k > 0 , then the branches of the hyperbola are located in the first and third quarters, If k < 0 , then the branches are located in the second and fourth quarters.

Number A is responsible for shifting the entire function down ( A < 0 ) or up ( a > 0 ).

4. Function of the form y = a (direct)

In this case the function looks like straight, parallel to the axis X . For example at= 2, this is a straight line that runs parallel to the axis X and intersects the axis at at point 2.

5. Function of the form y = √x

This type is rarely found in tasks, but it is better to remember. It's practically a parabola, but rotated clockwise by 90 0, and also lacks its lower half. If it’s not clear, just look at the picture:

The derivative of a function $y = f(x)$ at a given point $x_0$ is the limit of the ratio of the increment of a function to the corresponding increment of its argument, provided that the latter tends to zero:

$f"(x_0)=(lim)↙(△x→0)(△f(x_0))/(△x)$

Differentiation is the operation of finding the derivative.

Table of derivatives of some elementary functions

Basic rules of differentiation

1. The derivative of the sum (difference) is equal to the sum (difference) of the derivatives

$(f(x) ± g(x))"= f"(x)±g"(x)$

Find the derivative of the function $f(x)=3x^5-cosx+(1)/(x)$

The derivative of a sum (difference) is equal to the sum (difference) of derivatives.

$f"(x) = (3x^5)"-(cos x)" + ((1)/(x))" = 15x^4 + sinx - (1)/(x^2)$

2. Derivative of the product

$(f(x) g(x)"= f"(x) g(x)+ f(x) g(x)"$

Find the derivative $f(x)=4x cosx$

$f"(x)=(4x)"·cosx+4x·(cosx)"=4·cosx-4x·sinx$

3. Derivative of the quotient

$((f(x))/(g(x)))"=(f"(x) g(x)-f(x) g(x)")/(g^2(x)) $

Find the derivative $f(x)=(5x^5)/(e^x)$

$f"(x)=((5x^5)"·e^x-5x^5·(e^x)")/((e^x)^2)=(25x^4·e^x- 5x^5 e^x)/((e^x)^2)$

4. The derivative of a complex function is equal to the product of the derivative external function to the derivative of the inner function

$f(g(x))"=f"(g(x)) g"(x)$

$f"(x)=cos"(5x)·(5x)"=-sin(5x)·5= -5sin(5x)$

Physical meaning of the derivative

If a material point moves rectilinearly and its coordinate changes depending on time according to the law $x(t)$, then the instantaneous speed of this point is equal to the derivative of the function.

The point moves along the coordinate line according to the law $x(t)= 1.5t^2-3t + 7$, where $x(t)$ is the coordinate at time $t$. At what point in time will the speed of the point be equal to $12$?

1. Speed ​​is the derivative of $x(t)$, so let’s find the derivative of the given function

$v(t) = x"(t) = 1.5 2t -3 = 3t -3$

2. To find at what point in time $t$ the speed was equal to $12$, we create and solve the equation:

Geometric meaning of derivative

Recall that the equation of a straight line that is not parallel to the coordinate axes can be written in the form $y = kx + b$, where $k$ is the slope of the straight line. The coefficient $k$ is equal to the tangent of the angle of inclination between the straight line and the positive direction of the $Ox$ axis.

The derivative of the function $f(x)$ at the point $х_0$ is equal to the slope $k$ of the tangent to the graph at this point:

Therefore, we can create a general equality:

$f"(x_0) = k = tanα$

In the figure, the tangent to the function $f(x)$ increases, therefore the coefficient $k > 0$. Since $k > 0$, then $f"(x_0) = tanα > 0$. The angle $α$ between the tangent and the positive direction $Ox$ is acute.

In the figure, the tangent to the function $f(x)$ decreases, therefore, the coefficient $k< 0$, следовательно, $f"(x_0) = tgα < 0$. Угол $α$ между касательной и положительным направлением оси $Ох$ тупой.

In the figure, the tangent to the function $f(x)$ is parallel to the $Ox$ axis, therefore, the coefficient $k = 0$, therefore, $f"(x_0) = tan α = 0$. The point $x_0$ at which $f "(x_0) = 0$, called extremum.

The figure shows a graph of the function $y=f(x)$ and a tangent to this graph drawn at the point with the abscissa $x_0$. Find the value of the derivative of the function $f(x)$ at point $x_0$.

The tangent to the graph increases, therefore, $f"(x_0) = tan α > 0$

In order to find $f"(x_0)$, we find the tangent of the angle of inclination between the tangent and the positive direction of the $Ox$ axis. To do this, we build the tangent to the triangle $ABC$.

Let's find the tangent of the angle $BAC$. (The tangent of an acute angle in a right triangle is the ratio of the opposite side to the adjacent side.)

$tg BAC = (BC)/(AC) = (3)/(12)= (1)/(4)=$0.25

$f"(x_0) = tg BAC = 0.25$

Answer: $0.25$

The derivative is also used to find the intervals of increasing and decreasing functions:

If $f"(x) > 0$ on an interval, then the function $f(x)$ is increasing on this interval.

If $f"(x)< 0$ на промежутке, то функция $f(x)$ убывает на этом промежутке.

The figure shows the graph of the function $y = f(x)$. Find among the points $х_1,х_2,х_3...х_7$ those points at which the derivative of the function is negative.

In response, write down the number of these points.

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The figure shows a graph of the derivative of the function f(x), defined on the interval (-1;17). Find the intervals of decrease of the function f(x). In your answer, indicate the length of the largest of them. f(x)

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The figure shows a graph of the derivative of the function f(x), defined on the interval (-9; 2). At what point on the segment -8; -4 function f(x) takes highest value? On the segment -8; -4 f(x)

The function y = f(x) is defined on the interval (-5; 6). The figure shows a graph of the function y = f(x). Find among the points x 1, x 2, ..., x 7 those points at which the derivative of the function f(x) is equal to zero. In response, write down the number of points found. Answer: 3 Points x 1, x 4, x 6 and x 7 are extremum points. At point x 4 there is no f (x)

Literature 4 Algebra and beginning analysis class. Textbook for general education institutions, basic level / Sh. A. Alimov and others, - M.: Education, Semenov A. L. Unified State Examination: 3000 problems in mathematics. – M.: Publishing House “Exam”, Gendenshtein L. E., Ershova A. P., Ershova A. S. A visual guide to algebra and the beginnings of analysis with examples for grades 7-11. – M.: Ilexa, Electronic resource Open Unified State Exam task bank.