1. rounding difference method
2. Split terminology method
3. Variable substitution method
1) triangle substitution
2) Root power substitution
3) Reverse substitution
4. Post-formula integration
Organized
6. Sum-difference product method
7. According to the part (reverse, right, force, finger and three) integration
8. Power reduction method
Second, the calculation method of definite integral
1. Use function parity
2. Use function periodicity
3. Reference indefinite integral calculation method
Third, definite integral and limit
Limit of 1. product sum type
2. Use integral mean value theorem or differential mean value theorem to find the limit.
3. Lobida rule
4. Equivalent infinitesimal
Fourthly, the assignment of definite integral and the application of inequality.
1. Don't calculate the integral, compare the integral values.
1) comparison theorem: if it is in the same interval [a, b], there is always.
F(x)>=g(x), then >; =()dx
2) A) comparison by using the inequality satisfied by the integrand function.
B) when 0
2. Estimate the value of definite integral of specific function.
Integral estimation theorem: Let f(x) be continuous on [a, b], with the maximum value of m and the minimum value of m, then
m(b-a)& lt； = & lt=M(b-a)
3. Proof of inequality of definite integral of specific function
1) integral evaluation theorem
2) Scaling method
3) Cauchy integral inequality
≤ %
4. Proof of definite integral inequality of abstract function
1) Lagrange mean value theorem and boundedness of derivatives
2) integral mean value theorem
3) Constant variation method
4) Taylor formula expansion method is adopted.
Fifth, the derivative method of variable limit integral
Summary of knowledge points of higher mathematics integral 2 A. Function function
The definition and properties of (1) function (definition range, monotonicity, parity, periodicity, etc. )
(2) Power function (linear function, quadratic function, polynomial function and rational function)
(3) Exponent and Logarithm (formula operation and function properties of exponents and logarithms)
(4) trigonometric function and inverse trigonometric function (operation formula and function properties)
(5) Compound function and inverse function
*(6) Parameter function, polar coordinate function and piecewise function
(7) Translation and transformation of functional images
B. Limit and continuity Limit and continuity
Definition of (1) limit and left and right limit
(2) Limit algorithm and rational function to find the limit.
(3) Two important limitations
(4) Application of limit asymptote
(5) the definition of continuity
(6) Three kinds of discontinuous points (moving point, jumping point and infinite point)
(7) Maximum theorem, intermediate theorem and zero theorem.
C. derivative derivative
The Definition, Geometric Meaning and Unilateral Derivative of (1)
(2) the relationship between limit, continuity and derivability
(3) derivative rule (***2 1)
(4) Derivation of composite function
(5) Higher derivative
(6) Derivative and higher derivative of implicit function.
(7) Find the inverse function
*(8) Derivation of parametric function and polar coordinates
D. Application of derivatives
(1) Differential Mean Value Theorem (MVT)
(2) Geometrical Applications-Tangent, Normal and Relative Change Rate
(3) Physical application-finding velocity and acceleration (one-dimensional and two-dimensional motion)
(4) Find the extreme value, maximum value, increase or decrease and concavity of the function.
*(5) Find the limit with Robida's law
(6) Difference and linear estimation, four kinds of estimation are approximate.
(7) Approximation of Euler's Law
E. indefinite integral
The relationship between (1) indefinite integral and derivative
(2) Indefinite integral formula (18)
(3) U substitution method of indefinite integral.
*(4) Partial integral indefinite integral.
*(5) The undetermined coefficient method of indefinite integral.
F. definite integral
The definition and geometric significance of (1) riemann sum (left, right and middle trapezoid) and definite integral.
(2) Newton-Leibniz formula and the properties of definite integral
* (3) Find the accumulation function of the derivative
*(4) Integral of abnormal function
Application of Integral Application of Definite Integral
(1) integral mean value theorem (MVT)
(2) The area is determined by definite integral and polar coordinates.
(3) Calculating volume and cross-sectional volume by definite integral
(4) Find the arc length
(5) Physical application of definite integral
I. Differential equations
(1) Differential equations and Logistic differential equations of separable variables
(2) Slope land
* J. Infinite series Infinite series
Definition of (1) infinite series and series
(2) Three convergence methods-ratio, integral and comparative convergence.
(3) Four series-harmonic series, geometric series, P series and staggered series.
(4) Series of functions-power series (convergence radius), Taylor series and McLaughlin series.
(5) Operation of series and Lagrange remainder and Lagrange error.
note:
(1) Q&A mainly focuses on the comprehensive application of knowledge points. Generally, there are 3-4 questions in each question and answer, which may involve derivative, integral or differential equation at the same time, and the answer generally keeps 3 decimal places.
(2) Calculus BC course has more contents and more difficult topics than AB course. The content and difficulty of AB course are roughly equivalent to BC's 1/2, and the redundant content has been marked with *.
Summary of knowledge points of calculus theorem in higher mathematics integral 3: ——————.
If the function f(x) is continuous on [a, b] and the original function f(x) exists, then F(x) is integrable on [a, b], and
B (upper limit) ∫a (lower limit) F (x) DX = F (b)-F (a)
This is the Newton-Leibniz formula.
The significance of Newton-Leibniz formula lies in the connection between indefinite integral and definite integral, and it also provides a perfect and satisfactory method for the operation of definite integral.
Common formulas of calculus:—
Clever use of integral formula requires clever use of derivative, which is a reciprocal operation. I will provide you with some possible trigonometric formulas.
The basic theorem of calculus:—
The basic theorem of (1) calculus reveals the relationship between derivative and definite integral, and also provides an effective method to calculate definite integral.
(2) It is often difficult to find definite integral according to the definition of definite integral, but it is more convenient to find definite integral by using the basic theorem of calculus.
Question type:
It is known that f(x) is a quadratic function, f (- 1) = 2, f ′ (0) = 0, and f (x) dx =-2.
(1) Find the analytical formula of f(x);
(2) Find the maximum and minimum values of f(x) on [- 1, 1].
Solution:
(1) let f(x)=ax2+bx+c(a≠0),
Then f'(x)=2ax+b
Summary of knowledge points of advanced mathematics integration 4 "Complex variable function and integral transformation" is an important basic course and an important tool course for engineering majors such as electrical technology, automation and signal processing. This course includes two parts: complex variable function and integral transformation. The study of complex variable function and integral transformation will lay the foundation for studying engineering mechanics, electrotechnics, electromagnetics, vibration mechanics and radio technology in the future.
Second, the teaching process, methods and teaching results
1, proposition analysis
The proposition meets the basic requirements of the outline, with wide coverage of knowledge points and moderate difficulty. It pays attention to students' mastery of basic concepts, basic theories and basic skills and their comprehensive application ability. The proposition is concise and accurate, and the amount of questions is moderate.
2. Answer analysis
The vast majority of students have a good learning attitude and a high enthusiasm for learning. They can carefully prepare for the exam and master the relevant basic knowledge points and the operation of related topics. Judging from the students' exams, the overall effect is still relatively good.
3. Technical performance analysis
Average score of total students 104.
4. Teaching effect
The overall situation is ideal, and students generally feel that they have a certain understanding of the relevant theories of this course and have basically mastered the relevant knowledge of this course.
Third, the existing shortcomings and improvement measures
In the future teaching, we should especially strengthen the combination of teaching content and major, make students more interested in learning this course, properly handle teaching materials, adjust the order of explanation, grasp key knowledge points, and strengthen the training of students in class. Correct students' homework in time after class, and comment and answer students' various difficult questions in time.
Four. Suggestions on educational reform
There are relatively few class hours, and concepts and theories cannot be explained in depth; We should increase the class hours and exercise teaching appropriately, so that students can master the course more firmly.
90 ~ 100 (excellent) 80 ~ 89 (good) 167226 excellent rate 70 ~ 79 (medium) 13 15% 60 ~ 69 (and) 0 ~ 59 (and)