Chapter 1 Derivative and Its Application
average rate of change
Derivative (or instantaneous rate of change)
Derivative function (derivative):
Geometric meaning of derivative: the derivative (x0) of function y = f (x) at point x0 is the slope of the tangent of curve y = f (x) at point (x0, f(x0)), that is, k = (x0).
Application: Find the tangent equation and judge whether a given point is a tangent point.
The operation of derivative:
(1) derivatives of several commonly used functions;
① (c)' = 0 (c is a constant); ②()′=(x > 0，)； ③(sinx)′= cosx；
④(cosx)′=-sinx； ⑤(ex)′= ex； ⑥ (ax)' = axlna (a > 0, and a ≠1);
⑦; 8 (a > 0, and a ≠ 1).
(2) Derivative algorithm:
①[u(x)v(x)]′= u′(x)v′(x)； ②[u(x)v(x)]′= u′(x)v(x)+u(x)v′(x)；
③.
Let the function have a derivative at one point and the function have a derivative at the corresponding point, then the composite function also has a derivative at one point, and or. The derivative of compound function to independent variable is equal to the derivative of known function to intermediate variable, multiplied by the derivative of intermediate variable to independent variable.
The concept of definite integral, its geometric meaning, the integral representation of edge graph area, the determination of upper function, the selection of lower function and the division of interval. Fundamental theorem of calculus.
Physical application: automobile driving distance and displacement; The problem of changing force to do work.
Monotonicity of function
(1) Let the function be differentiable in a certain interval (a, b), and if it is, it is increasing function in this interval; If so, it is a decreasing function in this interval;
(2) If it is constant within a certain interval, it is constant.
On the other hand, if the differentiable function is known to increase monotonically in a certain interval, it is not always zero; Differentiable functions decrease monotonically in a certain interval and are not always zero.
Steps to find monotonicity:
Determine the domain of the function (essential, otherwise it is easy to make mistakes);
Address inequality;
Determine and point out the monotone interval (interval form, non-range form) of the function, which is separated by ","and cannot be connected by ",".
Extreme value and maximum value
For a differentiable function, if the extreme value is in, then.
Maximal Theorem: A continuous function must have a maximum and a minimum in a closed interval.
If there is a unique extreme point in the open interval, it is the maximum point.
Steps to find the extreme value:
Determine the domain of the function (essential, otherwise it is easy to make mistakes);
Address inequality;
Check the symbols on both sides of the root (usually through a list) to determine the maximum, minimum or non-extreme point.
When finding the maximum value, the step compares each extreme value with the function value at the endpoint on the basis of finding the extreme value, and it is forbidden to directly say that so-and-so is the maximum value or the minimum value.
Keep pressing the questions ""and "",and pay attention to whether the "=" in the parameter value can be obtained.
Example 1, tangent equation is
Example 2 let the function get the extreme value.
( 1);
(2) If it is true for any one, find the value range of c..
(Answer: (1)a=-3, b = 4；; (2))
Example 3 Setting Function
(1) Find the monotone interval and extreme value of the function.
(2) If it existed at that time, try to determine the value range of A. 。
(Answer: (1) monotonically increases on (a, 3a) and monotonically decreases on (-∞, a) and (3a,+∞); (2) The value range of A is)
Chapter II Reasoning and Proof
Distinguish concepts: perceptual reasoning and deductive reasoning
Step specification of comprehensive analysis method
Steps of reduction to absurdity: ① Put forward reverse design; ② Pushing out contradictions; ③ Affirm the conclusion.
Procedure specification of mathematical induction: (1) induction basis; (2) Recursive step
(Finally, it must be explained that when n=k+ 1, the conclusion holds. According to (1)(2), the conclusion is essential for (or other) holding. )
The example 1 is proved by comprehensive analysis.
Example 2 is known.
Example 3, the value of finding, the general formula of guessing, and proof.
(a:)
Chapter 3: The extension of number system and the introduction of complex numbers.
The concept of complex number has three forms: algebraic form, point Z(a, b) on the complex plane, and vector.
Distinguish between real number, imaginary number, pure imaginary number and complex number
Four operations of complex numbers and their geometric significance
Modulus of complex number
The necessary and sufficient condition of example 1 () is _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _
Example 2 If a complex number meets the conditions, then the maximum value is ().
Article 3, paragraph 2, paragraph 4, paragraph 3, paragraph 4, item 4
Example 3 When a real number is a value, it is a complex number.
(1) is a real number;
(2) it is an imaginary number;
(3) pure imaginary number;
(4) The corresponding point is in the second quadrant.
Example 4. Called a real number. (1) If, find; (2) If, the value of.