knowledge structure
function
Seeking a complex function piecewise explicit expression of: introducing an intermediate variable, the function of each section of the outer sections of the domain, via an intermediate to give the corresponding variable range of the argument x, and then determine the specific expression to obtain the intermediate variable Composite function.
Micro segmentation function calculus often non-elementary function according to the class, in addition to the form of these functions indicates significant segment, the more common are: absolute value function, the floor function, the value of the most function, and to limit the form the function of such a study and a function limit continuity, conductivity, type integral nature, paying particular attention to the special nature of the segment point.
If undetermined function in its composite form comprising an algebraic equation, may generally be obtained related variable substitution equation, coefficients of equations obtained solution then pending. When doing variable substitution can not be effectively solved the equation association, it will be considered as recurrence relations of equations (infinite number), solving (sequentially substituting) limit the use of processing methods.
When the decision function properties, depending on the nature of the column needs to determine the function equation solving implicit expression, can be discussed.
limit
Elementary deformation method
Elementary functions, variable substitution, identical deformation limit equations to simplify the method, then the limit determined by the limit of four operations, complex algorithms.
Calculated using the existing limit of two principles
Squeeze principle used to find the Limits Limit of Sequence and the infinite, and if all of the fraction, and each denominator in different but equivalent to infinity, the equivalence can be maintained by the denominator to form scaling cm mother, Jane achieve differentiation through the effect, and then clip the principle of seeking to force limit.
If the number of columns by the recursion formulas Xn = f (Xn-1) is given, the equation X = f (X) have roots x = a (a is called f (X) Fixed Point), with limit theorem proving Squeeze more convenient.
If you are a formula of the fraction, and the same denominator highest power of the variable coefficient limit, you can change any of the other items in the denominator of the coefficient of the highest power in addition to terms and without altering the limit, use Squeeze Theorem will be very simple.
Monotone bounded principles used for recurrence sequence of number limit given by the relationship.
The method used proved to have bounded of: viewed from the recurrence relation of the number of columns; Release by known inequality; proved by induction; monotone proof; by the recursion formula are bounded to.
The method used has proved monotonicity: ratio method; difference method; recurrence formula obtained by the monotonicity; mathematical induction.
Columns Xn = f (Xn-1) Monotonicity and boundedness depends form the function f (X) is, within a certain range, if X> f (X), the number of columns in a monotonically decreasing; if X <f (X), the monotonically increasing number of columns.
Calculation defined derivative
Formula deformable when the limit is a function of the ratio of the increment in which differentiable point increment argument, the derivative is calculated limit available functions.
Differential integral value calculated by the formulas or
When the limit of the formula contained in a delta function, consider a differential mean value theorem.
If the column number is given by the definite integral of the form, and while difficult to calculate the definite integral, common integral theorem of the integral sign is removed, and then seek the limit.
Unlimited number of columns and the solution of problems limit
① do elementary transformation and the part of Jane (limited entry) requirements limit;
② on the part and make estimates, using Squeeze Theorem seek limits;
③ into definite integral calculation;
④ use of power series and calculations.
continuous
