The first chapter function
1, real
As we all know, the concept of the number of filled our living space. Integers, fractions and rational numbers collectively referred to as zero. Irrational number in Elementary Mathematics have been met. Such as \ (\ SQRT2 \) , \ (\ sqrt3 \) , \ (pi \) , \ (LG5 \) and so on.
All rational and irrational numbers collectively referred to as a real number. The real number axis correspondence point on the body, but the number of axes and filled with no voids. It can be seen, the coordinates of each point of a certain identification number axis real number; the other hand, must be a real number for each point of coordinates of a number line.
2, section
In discussing certain issues, we often limited to a portion of the real scope of consideration, in order to concisely show the real part, where the introduction of the concept of the interval.
Definition: Interval between all is a real number between two real numbers, adding two real numbers breakpoint interval.
Finite interval range is divided into two major categories and unlimited range.
1, finite interval
(1), open interval
set a, b are two real numbers, and \ (A <B \) , satisfies the inequality \ (a <x <b \ ) an effective number of open interval called the entire x, denoted \ ((a, b) \) .
(2), closed interval
set a, b are two real numbers, and \ (a <B \) , satisfies the inequality \ (a ≤ x ≤ b \ ) of a number of practical all known closed interval, denoted \ ([a, b] \) .
(. 3), half-open interval
set a, b are two real numbers, and \ (A <B \) , satisfies the inequality \ (a <x ≤ b \) or \ (a ≤ x <b \ ) is a whole number x is called the effective half-open interval, respectively denoted \ ((a, b] \ ) and \ ([a, B) \) .
have to be mentioned here is referred to the length of the distance interval between the two sections of the endpoints.
Length of each section as described above are \ (BA \) .
2, infinite horizon
Ends is not limited, i.e., satisfies the inequality \ (- ∞ <x <+ ∞ \) section composed of a number of practical, denoted \ ((- ∞, + ∞) \) ;
the left is not shown, and the right end of the limit, i.e., satisfies the inequality \ (- ∞ <x <b \) or \ - (∞ <x ≤ b \) interval referred to a practical number constituting do \ ((- ∞, b) \) or \ ((- infinity, b] \) .
which represents real numbers less than or less equal to b, all.
the right is not limited, but the left is limited, i.e., satisfies the inequality \ (a <x <+ ∞ \) or \ (a ≤ x <+ ∞ \) section composed of a number of practical denoted \ ((a, + ∞) \) or \ ([a, + ∞) \)
3, neighborhood
[Delta] is provided with a two real numbers and δ> 0, satisfies the inequality \ (| XA | <[delta] \) [delta] collectively referred neighborhood of a point x of a practical number, calling a neighborhood centered , δ is the radius of the neighborhood. Thereby clearly \ (a-δ <x <
a + δ \) is visible, i.e. the neighborhood is a center point, a length of the open interval 2δ \ (([delta]-a, a + [delta]) \) .
3, constants and variables
Natural phenomena, we often encounter two different amounts, one is always in the process of keeping the amount unchanged, namely to maintain a certain amount of value; there is a constantly changing during the course of the amount, to the amount of different values, i.e., the amount of so-called two constants and variables .
Definitions : in a process called the amount of value remains unchanged constant , changing values called variables .
The amount is a constant or variable, is not absolute, it depends on where the occasion to study this phenomenon. The study area of a circle, with a radius r of his determination value, then r is a constant. If the radius of several different research area of the circle, r that is a variable.
The amount of x, each of which is a number value, and therefore a number of available point axis to represent it. If x is constant, with a point on the number line to represent, if x is a variable, the number of axes with a fixed point is represented.
4, function
4.1, function
in nature, movement of each thing around it are associated with each other things, and mutual restraint, such as the area of a circle s depends on its radius r, which is the relationship between s and r by the formula \ (s = πr ^ 2 \) OK.
As another example, in a free fall, the fall time t at the distance s changes, their dependencies by the formula \ (s = \ frac {1 } {2} gt ^ 2 \) is determined, where g is the gravitational acceleration .
In mathematics, for this process of change between the same variables to determine the relationship is called a function . Definitions Let x and y are two variables , which allows, when x takes the value within a certain range, the variable y is determined dependence also has a certain value determined corresponding thereto. X of y is called a function . Denoted \ (= the y-f (the X-) \) . Wherein x is called the independent variable , y is called the dependent variable , independent variable x allowable range is called the domain of the function . \ (F (x) \) also indicates the x value corresponding to the relative value of the function, the function value of all collective referred shrunk range function . y is a function of x may also be referred to as a \ (y = G (x) \) , \ (y = [Phi] (x) \) ,
\ (Y = F. (X) \) .
4.2, the function representation
showing a correspondence relationship function may be expressed in a variety of ways, usually analytical method , list method and image method .
(1), analytical method
analytical method that is a function of the relationship between two variables is represented by analytic equation, i.e. equation mathematically represented. The \ (^ Y = 2x 2 \) , \ (Y = SiNx \) and the like, also known analytical method analysis.
(2), the list method
list method that is a function of the relationship between two variables shows tables.
(3), an image process
image method that is a function of the relationship between two variables representing image.
It must be noted, is not necessarily a function of two variables is given by an analytical formula given by different analytical formula for different domain. The \ [y = F (X) = X +. 1 (X <0), 0 (X = 0),. 1-X (X> 0) \]
4.3 Compound Function
Definition: Let y is a function of u \ [ = F y (u) \] , and u is a function of x \ [u = [mu] (x) \] , is referred to as x y a composite function , referred to as \ [y = f [φ ( x)] \ ] where u is called an intermediate variable.
We usually function without intermediate variable is called a simple function.
5, the characteristic function
Single-valued and multivalued functions 5.1
Definition: with function \ [y = F (x) \] , if a value for the argument x, the dependent variable y is determined that only a corresponding value, such a function is called as a single-valued function . Otherwise call this function as a multi-valued function .
5.2 function of parity
Definitions: For the function \ [Y = F (X) \] , if \ [f (-x) = - f (x) \] called the function is an odd function; if \ [f (-x) = f (x) \] claimed that the function is an even function.
Obviously, even function symmetric pattern in the y-axis. The odd symmetry in the graphics function and origin.
5.3 Periodic function
Definitions: For the function \ [Y = F (X) \] , if there exists a real number T ≠ 0, there \ [f (x + T) = f (x) \] called the function cycle T as periodic functions, otherwise known as \ [f (x) \] non-periodic function.
5.4 monotone decreasing function of
Definitions: For the function \ [Y = F (X) \] , if there are any two points in the interval (A, B) \ [x_1 \] , \ [x_2 \] , when \ (x_1 \) \ (< \) $ x_2 \ (when there is \) $ F (x_1) <F (x_2) \ [called the function is monotonically increasing in the interval (a, b); when $ x_1 $ <$ x_2 $, there \] F (x_1) \ [> \] F (x_2) \ [this function is called in the interval (a, b) is a monotonically decreasing. Clearly, that is a monotonically increasing function of the rising edge of the horizontal axis, i.e., monotonically decreasing function is lowered in the horizontal axis direction. Similarly, we can define a monotonically increasing function on an infinite interval or monotonically decreasing, on the whole section called a monotonic function of monotonically increasing or monotonically decreasing function. ### 5.5 Bounded function definition: For the function \] Y = F (x) \ [, if there exists a positive number M, for any x in the domain, there is always \] | F (x) | ≤M \ [called \] F (X) \ [function on a bounded domain. When this number M does not exist, called \] F (X) \ [unbounded functions on the domain ## 6, the inverse function is defined: For the function \] Y = F (X) \ [, if y as the independent variable, x as the dependent variable, x is written by the expression y \] x = [mu] (y) \ [is called \] F (x)\ [Inverse function, called \] f (the X-) \ [direct function. Graphics Graphics difficult to directly know the function of an inverse function with respect to the straight line \] Y = X \ [symmetry. ## 7, ### 7.1 Basic elementary function elementary function power function, an exponential function, logarithmic functions, trigonometric and inverse trigonometric functions collectively referred to as basic elementary, they are 1, the power function \] Y = X ^ [mu] \ [([mu] is a real number) 2, exponential function \] Y = A X ^ \ [(A> 0, A ≠. 1). 3, a logarithmic function \] Y = log_ax \ [(A> 0, A ≠ 1,, x> 0) 4, a trigonometric function \] Y = SiNx, Y = cosx, TGX = Y, Y = ctgx \ [. 5, an inverse trigonometric function \] Y = arcsinx, arccosx = Y, Y = arctgx, arcctgx Y = \ [addition function \] Y = C \ [(C is a constant) constant function is called, its pattern is a straight line parallel to the x-axis. ### 7.2 elementary function is defined: substantially constituted by a finite four elementary function arithmetic and step through a limited number of compliance, and can be represented by analytic function expressions are referred to as elementary function. Finally, we have to point out that there is only one independent variable function is called ** ** functions of one variable, the function has two or more independent variables are called multi-function ** **. Chapter # ## limit 1, #### the number of columns limit the number of columns 1.1 Definition: In accordance with certain rules arranged in a number of $ x_1, x_2, x_3, ... , x_n $ is called ** ** series, denoted \ {$ x_n $ \}, the number of columns for each item number, called number sequence, \] x_n\ [Called the general terms of the number of columns. We can put the number of columns \ {$ x_n $ \} $ x_n $ is considered a positive integer argument function values of n \] x_n = F (n), n = l, 2,3, ... \ [Therefore, the number of columns is a function whose domain is all positive integers. 1.2 When the number of columns #### to limit the number of columns \ {$ x_n $ \}, while when n → ∞, $ x_n $ can be infinitely close to one of the constants a, then we say that the number of columns \ {$ x_n $ \ }, when n → ∞ when the limit is a. Definition: with the number of columns \ {$ x_n $ \}, if for any given predetermined small positive number [epsilon], there is always a positive integer N, so that for all n> N when there is \] | x_n-A | < [epsilon] \ [called a number column \ {$ x_n $ \} limits, or that the series converges to a, denoted \] \ lim_ {n-\ rightarrow + \ infty} x_n = a \ [\]
\ lim_ {n-\ + rightarrow \ infty x_n} a =
\ [or when \] n-→ ∞ \ [time, \] x_n → a \ [. If the sequence does not limit the number of said columns it is divergent. This definition of the limits of the number of columns is called Sequence Limit "ε-N" definition. Here ε is an arbitrary positive number given, it is mainly used to reflect \] x_n \ [closeness and constant a; N is a natural number, which is previously related to a given ε, when ε is reduced, in general, N will increase accordingly. In addition, a [epsilon], its corresponding N is not unique. ** Theorem 1 If the sequence \ {\] x_n} $$ convergence, it is the only limit. **
Theorem 2 If the sequence { \ [x_n \] } converges, it must be bounded. I.e., for all n (n = 1,2, ...) , you can always find a positive number M, so that
\ [| x_n | ≤M \]
with a number of columns bound by the number of convergence can be derived must be unbounded diverging column , that is without limits the number of boundary column does not exist.
2 function Limit
We know that the number of columns of the argument can be regarded as a function of n
\ [x_n = f (n)
\] series can be viewed as a special type of limit function which takes a positive argument n is an integer discretely increases infinitely . N is the number of columns is only one trend, i.e. n → ∞.
General function \ [y = f (x) \] argument x is continuously changed, the change trend of the following two situations:
(1), the independent variable x infinitely close to a predetermined number \ (x_0 \) , denoted as \ [x → x_0 \] ;
(2), the absolute value of the argument x increases indefinitely, referred to as \ [x → ∞ \] .
2.1 \ [n-→ ∞ \] limit function of time
For the function f (x), regardless of the absolute value of x is first set up what always makes sense. If | x | unlimited increase, whether the corresponding function value infinitely close to a certain constant, i.e., when | x | infinitely increases, f (x) and the absolute value of difference of a constant a may be less than pre-specified arbitrarily small positive number ε, referred to at this time we put a limit function f (x) of.