The section contains reference material on the main elementary functions and their properties. A classification of elementary functions is given. Below are links to subsections that discuss the properties of specific functions - graphs, formulas, derivatives, antiderivatives (integrals), series expansions, expressions through complex variables.
Reference pages for basic functions
Classification of elementary functions
Algebraic function is a function that satisfies the equation:
,
where is a polynomial in the dependent variable y and the independent variable x. It can be written as:
,
where are polynomials.
Algebraic functions are divided into polynomials (entire rational functions), rational functions and irrational functions.
Entire rational function, which is also called polynomial or polynomial, is obtained from the variable x and a finite number of numbers using arithmetic operations addition (subtraction) and multiplication. After opening the brackets, the polynomial is reduced to canonical form:
.
Fractional rational function, or simply rational function, is obtained from the variable x and a finite number of numbers using the arithmetic operations of addition (subtraction), multiplication and division. The rational function can be reduced to the form
,
where and are polynomials.
Irrational function is an algebraic function that is not rational. As a rule, an irrational function is understood as roots and their compositions with rational functions. A root of degree n is defined as the solution to the equation
.
It is designated as follows:
.
Transcendental functions are called non-algebraic functions. These are exponential, trigonometric, hyperbolic and their inverse functions.
Overview of basic elementary functions
All elementary functions can be represented as a finite number of addition, subtraction, multiplication and division operations performed on an expression of the form:
z t .
Inverse functions can also be expressed in terms of logarithms. The basic elementary functions are listed below.
Power function :
y(x) = x p ,
where p is the exponent. It depends on the base of the degree x.
Back to power function is also a power function:
.
For an integer non-negative value of the exponent p, it is a polynomial. For an integer value p - a rational function. At rational meaning- irrational function.
Transcendental functions
Exponential function :
y(x) = a x ,
where a is the base of the degree. It depends on the exponent x.
The inverse function is the logarithm to base a:
x = log a y.
Exponent, e to the x power:
y(x) = e x ,
This is an exponential function whose derivative is equal to the function itself:
.
The base of the exponent is the number e:
≈ 2,718281828459045...
.
The inverse function is the natural logarithm - the logarithm to the base of the number e:
x = ln y ≡ log e y.
Trigonometric functions:
Sine: ;
Cosine: ;
Tangent: ;
Cotangent: ;
Here i is the imaginary unit, i 2 = -1.
Inverse trigonometric functions:
Arcsine: x = arcsin y,
;
Arc cosine: x = arccos y,
;
Arctangent: x = arctan y,
;
Arc tangent: x = arcctg y,
.
Workshop
According to mathematical analysis
For evening students
Wow course
(Part I)
Educational and methodological manual
Moscow, 2006
UDC 512.8:516
BBK S42
Reviewers:
Candidate of Physical and Mathematical Sciences, Associate Professor Karolinskaya S.N. (Moscow Aviation Institute named after S. Ordzhonikidze);
Ph.D., Associate Professor Krasnoslobodtseva T.P. (MITHT named after M.V. Lomonosov).
Skvortsova M.I., Mudrakova O.A., Krotov G.S., Workshop on mathematical analysis for 1st year evening students (Part I), Educational and methodological manual - M.: MITHT im. M.V. Lomonosov, 2006 – 44 p.: ill. 29 .
Approved by the Library and Publishing Commission of MITHT. M.V. Lomonosov as a teaching aid. Pos. ___/2006.
The manual consists of notes from 6 practical lessons on the course mathematical analysis for evening students of MITHT named after. M.V. Lomonosov. Part I includes the following sections: “Function and its basic properties”, “Limit of a function”, “Continuity and discontinuity points of a function”.
Each lesson is devoted to a separate topic. Notes for 5 lessons contain summary corresponding theory, typical examples and tasks for independent decision(with answers). Lesson notes No. 6 provide a sample option test work(with solutions) conducted in this lesson.
The manual is intended for evening students of chemical universities.
© MITHT im. M.V. Lomonosova, 2006
Lesson 1.
| |
Lesson 2. Polar coordinate system. Plotting graphs of functions using the method of shifting and stretching along coordinate axes…………………………………………….
Lesson 3. Function limit. Continuity of function. Calculation of the limits of continuous, rational and some irrational functions…………......
Lesson 4. First and second wonderful limits. Calculating the limits of a power-exponential function. Infinitely small and infinitely large
values………………………………………………….
Lesson 5. Points of continuity and points of discontinuity of a function. Classification of break points. Investigation of a function for continuity………………………………
Lesson 6. Test No. 1 on the topic "Calculation of the limits of functions. Study of functions for continuity"…………………………………………………………………….
Literature……………………………………………….
Lesson 1.
The concept of function. Basic elementary functions, their properties and graphs.
Definition 1. The dependence of a variable on a variable is called function, if each value corresponds to a single value.
We write: And we talk, which is a function of . In this case it is called independent variable(or argument), and – dependent variable.
Definition 2. Function Domain(denoted by ) are all the values that . Multiple Function Values(denoted by ) are all the values that .
Definition 3. The function is called increasing (decreasing) on the numerical interval if for any of , such that , the inequality holds:
.
Definition 4. The function is called monotonous on the interval if it only decreases or only increases by .
Definition 5. The function is called even (odd), if it is symmetric about zero and for any of:
.
The length of the segment on the coordinate axis is determined by the formula:
The length of a segment on the coordinate plane is found using the formula:
To find the length of a segment in a three-dimensional coordinate system, use the following formula:
The coordinates of the middle of the segment (for the coordinate axis only the first formula is used, for the coordinate plane - the first two formulas, for a three-dimensional coordinate system - all three formulas) are calculated using the formulas:
Function– this is a correspondence of the form y= f(x) between variable quantities, due to which each considered value of some variable quantity x(argument or independent variable) corresponds to specific value another variable, y(dependent variable, sometimes this value is simply called the value of the function). Note that the function assumes that one argument value X only one value of the dependent variable can correspond at. However, the same value at can be obtained with different X.
Function Domain– these are all the values of the independent variable (function argument, usually this X), for which the function is defined, i.e. its meaning exists. The area of definition is indicated D(y). By and large, you are already familiar with this concept. The domain of a function is also called the domain acceptable values, or ODZ, which you have long been able to find.
Function Range- this is all possible values dependent variable of this function. Designated E(at).
Function increases on the interval in which a larger value of the argument corresponds to a larger value of the function. The function is decreasing on the interval in which a larger value of the argument corresponds to a smaller value of the function.
Intervals of constant sign of a function- these are the intervals of the independent variable over which the dependent variable retains its positive or negative sign.
Function zeros– these are the values of the argument at which the value of the function is equal to zero. At these points, the function graph intersects the abscissa axis (OX axis). Very often, the need to find the zeros of a function means the need to simply solve the equation. Also, often the need to find intervals of constancy of sign means the need to simply solve the inequality.
Function y = f(x) are called even X
This means that for any opposite values of the argument, the values of the even function are equal. Schedule even function always symmetrical relative to the ordinate axis of the op-amp.
Function y = f(x) are called odd, if it is defined on a symmetric set and for any X from the domain of definition the equality holds:
This means that for any opposite values of the argument, the values of the odd function are also opposite. The graph of an odd function is always symmetrical about the origin.
The sum of the roots of even and odd functions(points of intersection of the abscissa axis OX) is always equal to zero, because for every positive root X has a negative root - X.
It is important to note: some function does not have to be even or odd. There are many functions that are neither even nor odd. Such functions are called functions general view , and for them none of the equalities or properties given above is satisfied.
Linear function is a function that can be given by the formula:
Schedule linear function is a straight line and in the general case looks like this (an example is given for the case when k> 0, in this case the function is increasing; for the occasion k < 0 функция будет убывающей, т.е. прямая будет наклонена в другую сторону - слева направо):
Graph of a quadratic function (Parabola)
The graph of a parabola is given by a quadratic function:
A quadratic function, like any other function, intersects the OX axis at the points that are its roots: ( x 1 ; 0) and ( x 2 ; 0). If there are no roots, then the quadratic function does not intersect the OX axis; if there is only one root, then at this point ( x 0 ; 0) the quadratic function only touches the OX axis, but does not intersect it. The quadratic function always intersects the OY axis at the point with coordinates: (0; c). Schedule quadratic function(parabola) may look like this (the figure shows examples that are far from exhaustive possible types parabolas):
Wherein:
- if the coefficient a> 0, in function y = ax 2 + bx + c, then the branches of the parabola are directed upward;
- if a < 0, то ветви параболы направлены вниз.
The coordinates of the vertex of a parabola can be calculated using the following formulas. X tops (p- in the pictures above) parabolas (or the point at which the quadratic trinomial reaches its largest or smallest value):
Igrek tops (q- in the figures above) parabolas or the maximum if the branches of the parabola are directed downwards ( a < 0), либо минимальное, если ветви параболы направлены вверх (a> 0), value quadratic trinomial:
Graphs of other functions
Power function
Here are some examples of graphs of power functions:
Inversely proportional is a function given by the formula:
Depending on the sign of the number k back schedule proportional dependence may have two fundamental options:
Asymptote is a line that the graph of a function approaches infinitely close to but does not intersect. Asymptotes for graphs inverse proportionality shown in the figure above are the coordinate axes to which the graph of the function approaches infinitely close, but does not intersect them.
Exponential function with base A is a function given by the formula:
a The graph of an exponential function can have two fundamental options (we also give examples, see below):
Logarithmic function is a function given by the formula:
Depending on whether the number is greater or less than one a The graph of a logarithmic function can have two fundamental options:
Graph of a function y = |x| as follows:
Graphs of periodic (trigonometric) functions
Function at = f(x) is called periodic, if there is such a non-zero number T, What f(x + T) = f(x), for anyone X from the domain of the function f(x). If the function f(x) is periodic with period T, then the function:
Where: A, k, b are constant numbers, and k not equal to zero, also periodic with period T 1, which is determined by the formula:
Most examples periodic functions These are trigonometric functions. Here are the graphs of the main trigonometric functions. The following figure shows part of the graph of the function y= sin x(the entire graph continues indefinitely left and right), graph of the function y= sin x called sinusoid:
Graph of a function y=cos x called cosine. This graph is shown in the following figure. Since the sine graph continues indefinitely along the OX axis to the left and right:
Graph of a function y= tg x called tangentoid. This graph is shown in the following figure. Like the graphs of other periodic functions, this schedule repeats indefinitely along the OX axis to the left and right.
And finally, the graph of the function y=ctg x called cotangentoid. This graph is shown in the following figure. Like the graphs of other periodic and trigonometric functions, this graph repeats indefinitely along the OX axis to the left and right.
Successful, diligent and responsible implementation of these three points will allow you to show up on the CT excellent result, the maximum of what you are capable of.
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National Research University
Department of Applied Geology
Abstract on higher mathematics
On the topic: “Basic elementary functions,
their properties and graphs"
Completed:
Checked:
teacher
Definition. The function given by the formula y=a x (where a>0, a≠1) is called an exponential function with base a.
Let us formulate the main properties of the exponential function:
1. The domain of definition is the set (R) of all real numbers.
2. Range - the set (R+) of all positive real numbers.
3. For a > 1, the function increases along the entire number line; at 0<а<1 функция убывает.
4. Is a function of general form.
, on the interval xО [-3;3]
, on the interval xО [-3;3] A function of the form y(x)=x n, where n is the number ОR, is called a power function. The number n can take on different values: both integer and fractional, both even and odd. Depending on this, the power function will have a different form. Let's consider special cases that are power functions and reflect the basic properties of this type of curve in the following order: power function y=x² (function with an even exponent - a parabola), power function y=x³ (function with an odd exponent - cubic parabola) and function y=√x (x to the power of ½) (function with a fractional exponent), function with a negative integer exponent (hyperbola).
Power function y=x²
1. D(x)=R – the function is defined on the entire numerical axis;
2. E(y)= and increases on the interval
Power function y=x³
1. The graph of the function y=x³ is called a cubic parabola. The power function y=x³ has the following properties:
2. D(x)=R – the function is defined on the entire numerical axis;
3. E(y)=(-∞;∞) – the function takes all values in its domain of definition;
4. When x=0 y=0 – the function passes through the origin of coordinates O(0;0).
5. The function increases over the entire domain of definition.
6. The function is odd (symmetrical about the origin).
, on the interval xО [-3;3] Depending on the numerical factor in front of x³, the function can be steep/flat and increasing/decreasing.
Power function with negative integer exponent:
If the exponent n is odd, then the graph of such a power function is called a hyperbola. A power function with an integer negative exponent has the following properties:
1. D(x)=(-∞;0)U(0;∞) for any n;
2. E(y)=(-∞;0)U(0;∞), if n is an odd number; E(y)=(0;∞), if n is an even number;
3. The function decreases over the entire domain of definition if n is an odd number; the function increases on the interval (-∞;0) and decreases on the interval (0;∞) if n is an even number.
4. The function is odd (symmetrical about the origin) if n is an odd number; a function is even if n is an even number.
5. The function passes through the points (1;1) and (-1;-1) if n is an odd number and through the points (1;1) and (-1;1) if n is an even number.
, on the interval xО [-3;3] Power function with fractional exponent
A power function with a fractional exponent (picture) has a graph of the function shown in the figure. A power function with a fractional exponent has the following properties: (picture)
1. D(x) ОR, if n is an odd number and D(x)= 
, on the interval xО 
, on the interval xО [-3;3]
The logarithmic function y = log a x has the following properties:
1. Domain of definition D(x)О (0; + ∞).
2. Range of values E(y) О (- ∞; + ∞)
3. The function is neither even nor odd (of general form).
4. The function increases on the interval (0; + ∞) for a > 1, decreases on (0; + ∞) for 0< а < 1.
The graph of the function y = log a x can be obtained from the graph of the function y = a x using a symmetry transformation about the straight line y = x. Figure 9 shows a graph of the logarithmic function for a > 1, and Figure 10 for 0< a < 1.
; on the interval xО 
; on the interval xО The functions y = sin x, y = cos x, y = tan x, y = ctg x are called trigonometric functions.
The functions y = sin x, y = tan x, y = ctg x are odd, and the function y = cos x is even.
Function y = sin(x).
1. Domain of definition D(x) ОR.
2. Range of values E(y) О [ - 1; 1].
3. The function is periodic; the main period is 2π.
4. The function is odd.
5. The function increases on intervals [ -π/2 + 2πn; π/2 + 2πn] and decreases on the intervals [π/2 + 2πn; 3π/2 + 2πn], n О Z.
The graph of the function y = sin (x) is shown in Figure 11.