knowledge structure
Derivative concept and calculation of differential
Derivative of the function, respectively, and differential velocity function reflects the relative speed change caused by small changes in the independent variables caused (conversion rate) and the absolute change in the size of the (linear approximation) two issues.
Common forms of functions: explicit function (including piecewise function, the inverse function), an implicit function, parametric function and an integrator changeable function.
The concept of the derivative
Guide function in the left limit point is equal to the right limit equal to the function value.
This function close to the guide function value close to this point function value point.
Definite integral and limit functions are constants.
Function can be turned at one point, but after adding the absolute value of the function at that point can not be turned, necessary and sufficient condition is a function of the value function is zero at this point, the guide function is non-zero function values at that point.
Derivative function periodic function is a periodic function with a period unchanged.
Defined by the formula given in relation to the function at any point derivative of the function in the interval into at x = definitional equation of the derivative 0 can conductivity available functions, find the relation of the derivative, i.e. the establishment of a differential equations, solving equations can be obtained function expression.
Calculate the derivative
Derivative compound function: obtaining the derivative of a function of the inner and outer layers, and then by multiplying. The inner guide is a function that can be, but not necessarily sufficient for a compound that can function in a guide.
Derivative piecewise function: segmentation points define a calculation to use.
N If the calculation method adopted by the first derivative of the first derivation, and sometimes needs to be done before the derivative few bands identical deformation necessary, they will be into the same type of function, facilitate to find the law to write a general expression. Available mathematical induction to prove the correctness of its conclusions, if necessary.
Euler's formula can be interchangeable trigonometric and exponential functions. In many calculus handle the exponential function more convenient.
Find the product of two functions of the n-order derivative, if one of the functions of each of the first derivative is not zero in only a few of several, Leibniz formula is used more convenient.
When a function is specified point Taylor expansion is relatively easy to obtain, the choice of which the first derivative is calculated Taylor number of stages n is the most convenient.
General Procedure Rational seeking higher-order derivative of the function: If the fractional rational function is false, then the use of the polynomial division and a polynomial into fractions and Fraction, Fraction then removed once the female component in the form of partial fractions (which may be complex system) and then seek higher order derivatives.
Application of the Mean Value Theorem derivative
Mean Value Theorem
Common applications value theorem: Theorem proof, zero is determined to prove equality and inequality, calculated limit is determined extremum, the approximate calculation.
Taylor Formula I with suitable for research Peano term local state functions, such as the limit, etc. extremum; Taylor Formula I with a suitable Lagrangian function term integrity of states, such as equation or inequality approximate calculation.
By conventional methods Rolle theorem equation:
The equation to be written permit only a period equals zero, then constructing auxiliary functions. Comprises the following steps:
(i) to f (ξ) = 0 is rewritten as f (x) = 0;
(ii) based on f (x) to construct the auxiliary function F (x), commonly used methods are:
① direct observation using algorithms derivative hash differential;
② The obtained helper definite integral;
③ helper differential equation solution.
(Iii) verification of helpers in a given interval to meet the conditions Rolle theorem, we can draw conclusions.
Zero conventional method (Equation root) is the presence of a claimant function: Intermediate Value Theorem continuous function, Rolle theorem. First, consider the general value theorem, if a constant function with the letter or the sign function is not easy to determine an even number of zeros in the interval in question so that the function is not different at both ends of the point number, with the Rolle theorem.
The only proof of monotonicity common root, there are a plurality of common root Rolle theorem.
Utilization of the Mean Inequality method is:
① the characteristics of inequality, and select the appropriate interval appropriate function.
② the derivative of the equation for estimating portion (enlarged or reduced), to give the results demonstrated.
Taylor formula shows the relation between the values of their respective function derivative values, when a second order function and a proposition relating to the second order derivative of the above should be considered Taylor formula. Lagrange mean value theorem is a zero-order Taylor formula.
Applications of the Derivative
Constant proof function expression is a common method constant a:
① its derivative identically zero description, the description of the function in a given function value of a point;
② described in the specified range of maximum and minimum values are is A;
③ by contradiction.
Inequality methods:
① defined using derivative;
② Mean Value Theorem use;
③ using monotonic function;
④ using Taylor's formula;
⑤ by the function value of the best extrema;
⑥ using the function of the irregularities.
