MindMap Gallery Differential median value theorem and derivative
This is a mind map about the differential median theorem and derivatives, and its main contents include: derivative application and differential median theorem. Easy to understand and remember knowledge points.
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Differential median value theorem and derivative
Differential median theorem
Conclusion: If x=a is f(x) extreme point Þf'(a)=0 or f'(a) does not exist
Fermat Lemma: Suppose f(x) can be derived at x0. If the function f(x) obtains the extreme value at x0, then f'(x)=0
Roll's theorem
Lagrangian median theorem
Cauchy's median theorem
Taylor's formula (McLaurin)
Peano type remaining
Lagrangian remaining items
Derivative application
The monotonicity of the function
Extreme value of function
Necessary conditions for extreme values
The first sufficient condition for extreme values
The second sufficient condition for extreme values
Maximum and minimum values of the function
Convexity of the function
Definition: f(x1 x2/2)<f(x1) f(x2)/2, then it is concave, otherwise it is convex
theorem
y=f(x) is continuous on [a,b], and second order can be derived on (a,b). If f''(x)>0, then the graph of the function is concave on [a,b]
y=f(x) is continuous on [a,b], and second order can be derived on (a,b). If f''(x) "0, then the graph of the function is convex on [a,b]
Inflection point: the dividing point between concave and convex on a continuous curve arc
Necessary conditions for turning point
The first sufficient condition for turning point
The second sufficient condition for turning point
Asymptomatic lines of curves
Horizontal asymptomatic
Direct asymptomatic line
Inclined asymptomatic line
Drawing of functions
Arc differential and curvature of curves
Formula: Arc Differential Curvature Radius of Curvature Center
Some proof calculations
The root of the equation: the zero point theorem can be used, the limit evaluation of both sides, the monotonicity of each interval
Function inequality: Using Lagrangian and function monotonicity
Prove using the median theorem: Lagrangian is the most common, and other Rohl Fermat also involves