17. Derivative and its application

The Concept of (1) Derivative and Its Geometric Significance

① Understand the actual background of the concept of derivative.

② Understand the geometric meaning of derivative.

(2) the operation of derivative

① According to the definition of derivative, we can find the derivative of function (c is a constant).

② Using the derivative formula of basic elementary function and the four algorithms given below, we can find the derivative of simple function and simple composite function (only for the composite function with the shape of f(ax+b)).

Derivative formula and derivative operation formula of common basic elementary functions;

Commonly used derivative algorithm:

Rule number two

Rule three.

(3) The application of derivative in function research.

① Understand the relationship between monotonicity and derivative of function; The monotonicity of functions can be studied by using derivatives, and the monotone interval of functions can be found (in which polynomial functions are generally not more than three times).

(2) Understand the necessary and sufficient conditions for the function to obtain the extreme value at a certain point; Will use derivatives to find the maximum and minimum values of functions (in which polynomial functions generally do not exceed three times); Will find the maximum and minimum value of the function in the closed interval (where the polynomial function generally does not exceed three times).

(4) Optimization in life.

Will use derivatives to solve some practical problems. ..

(5) Basic theorems of definite integral and calculus

① Understand the actual background, basic ideas and concepts of definite integral.

The significance of understanding the basic theorem of calculus.

18. Reasoning and proof

(1) Rational reasoning and deductive reasoning

① Understand the meaning of sensible reasoning, make simple reasoning through induction and analogy, and understand the role of sensible reasoning in mathematical discovery.

Understand the importance of deductive reasoning, master the basic models of deductive reasoning, and use them for some simple reasoning.

Understand the connection and difference between perceptual reasoning and deductive reasoning.

(2) Direct proof and indirect proof

① Understand the two basic methods of direct proof-analysis and synthesis; Understand the thinking process and characteristics of analytical methods and comprehensive methods.

② Understand a basic method of indirect proof ── reduction to absurdity; Understand the thinking process and characteristics of reduction to absurdity.

(3) Mathematical induction

Knowing the principle of mathematical induction, we can prove some simple mathematical propositions by mathematical induction.

19. Extension of number system and introduction of complex number

The concept of (1) complex number

Understand the basic concepts of complex numbers.

Understanding the necessary and sufficient conditions for the equality of complex numbers.

Understand the algebraic representation of complex numbers and its geometric significance.