Advanced Mathematics-Summary notes on functions and limits

functions and Limits 1 functions and mappings

1) Understand the fundamental principle of function--a case of mapping, a mapping of real numbers to real sets

2) Mapping law, which is the law of function, the rule between the independent variable and the strain amount

3) Features of the function:

a) The difficulty of boundedness is greatest, and the inequality needs to be constructed

b) monotonicity proving inequalities using monotonicity

c) Parity

d) Periodic attention to periodic changes, corresponding functions are equivalent to the 2 series limit

1) Definition: The sequence {Xn} is in the centroid neighborhood of a (the Xn element increases with the increase of N), there is a positive integer n,n, when n > N, for any e > 0, satisfies

Xn-a < e A is constant

A is the limit of the sequence.

That is, all the elements in the series, as the n subscript increases, getting closer to a

2) Understanding 3 function limits with geometric models

1) The function limit that tends to the finite value, defines: The function is defined in a centroid neighborhood ((A-r,a + R)), when exists R > 0, for all e> 0, on the interval (a-r,a + R), satisfies

F (x)-a < E A is constant

It is said that a is the limit of f (x) on (a-r,a + R)

2) tend to the infinite function limit, defined: The function is greater than a certain "positive x" has a definition (function interval is (X, Infinity)), for all e > 0, when the |x| > x (function defined) satisfies

|f (x)-a |< e A is constant

3) The function limit is based on the function model, that is, a two-dimensional change process, the independent variable is the dynamic , the response variable reflects the phenomenon of change, gradually tending to an exact value a

I. The independent variable is the dynamic force: also means that, in order to find the function limit, it is necessary to clarify the function definition field, and the function is defined in the domain of the definition

4) The sequence limit is based on the sequence model, that is, a one-dimensional change process, the dynamic is subscript n, the data item reflects the change phenomenon, gradually tending to an exact value a

I. Change motive is subscript N: also means that there will be a subscript n as the demarcation point, the value of the data item of subscript >n and the limit a more close to the 4 limit of existence criterion two important limits

1) Clamping Force guidelines

The difficulty lies in how to construct inequality at both ends, 1* maximum items < < number of items * Maximum

2) Bounded monotone series

I. Proving the difficulty of bounded-and-constructed inequalities

Ii. Monotony: The general situation does not need to use the derivation formula, but the simple preceding paragraph-latter the preceding paragraph/latter can determine the monotonicity

3) Critical limits

a) or

I. is the limit of the type

Ii. = precondition is the limit of the satisfying type

Iii. = precondition is the limit of the satisfying type

Iv. difficulties in, the limit of the tectonic type limit

b) = 1 5 The algorithm of the limit

1) There is a poor sum of infinitely small add = infinitesimals

2) Infinitesimal * Bounded quantity = Infinity (0) often used, when the limit is calculated

3) limit existence It's a major premise. Using the limit algorithm should be a validation

A) subtraction also satisfies

b)

c) b 6 infinitely small Infinity

1) Infinity, infinity is a process of change , not an exact value.

2) Infinity can be expressed in 0

3) The sufficient and necessary condition of the function f (x) with limit A is f (x) = A + A, A is infinitesimal 7 Taylor Formula * * * *

1) Taylor formula: Decomposition of a function into a polynomial consisting of exponential functions

2) The aim is to approximate the value, which means that there is always an error and there is no precise value. The more the Taylor formula expands, the more accurate the value

3) Taylor formula is used to approximate the approximation with infinitesimal/large approximation, in the limit,

, you can use the Taylor formula. ( note The premise , if not satisfied, it should be equipped with)

4) The principle of infinitesimal substitution is the deformation of Taylor's formula

a) McLoughlin formula with Lagrangian residual term

b) McLoughlin formula with Peano type remainder

just a symbol that represents a higher-order item than n

5) The comparison between Infinity and infinity in the substitution of the infinite-back formula 7

1) High order Infinity Small 0

A) = 0, recorded as

2) low-order infinitesimals

A) =, also known as the limit does not exist

3) Same Order infinitesimals

a) = C

4) equivalent infinitesimals

A) =1 as the continuity and discontinuity of the ~ 8 function

1) Continuity

A) The function is continuous at the point

I. Limit must exist

II. defined in place

Iii. "=" Established

b) Judging whether the function is continuous from these three conditions in order to judge, whether the interruption is also the case, as long as a condition is not satisfied, the intermittent

c) left continuous

I. Function in the right end of the interval (continuous on the interval) continuous

D) Right Continuous

I. Function in the left end of the interval (continuous on the interval) continuous

2) Discontinuity point

A) generally exist in

I. Points with a denominator of 0

Ii. points for which the function has no definition

b) The first type of discontinuity: the left and right limits of the discontinuity exist.

I. can go to the discontinuity point

Present, but not defined or, but equal to the left and right limits

c) Second type of discontinuity: In addition to the first type of discontinuity, the second type of discontinuity.

I. Infinite discontinuity point

II. Jumping breaks

The operation of 9 continuous function and continuity of elementary function

1) Continuous function

A) function is continuous, then the inverse function of the function is also continuous, monotonic is also consistent

b) Two functions in the point continuous, then their and, poor, product, the trader is continuous

c) The sub-functions that make up the compound function are continuous, so the compound function is also continuous

2) Continuity of elementary functions

A) basic Elementary functions are contiguous in their defined fields

b) all Elementary functions are contiguous in their defined intervals

I. Defining the interval is the 10 laws of law contained in the defined fields * * * *

1) Acting on the type, if not both of these types, you need to construct

2)

A) conditions

I. when the function f (x) g (x) tends to 0 is not

II. at the heart of point A, all exist, and are not defined at a point

III. existence, or for

b)

c) does not exist, but may still exist

3) Use the Indeterminate form rule to find the limit

A) first check whether the limit is undefined type

b) after derivation, algebraic formula is more complex, should be Jianhuan type

c) after multiple derivation, algebraic formula and the same as the original, should be Jianhuan type

d) conversion, parametric indexation

e) to,

I. -Pass

Ii. substitution, re--pass