This syllabus requires that concepts and theories be divided into two levels: "understanding" and "understanding". Methods and operations are divided into three levels: understanding, mastering and mastering skillfully.
Review the exam content
I. Function, Limit and Continuity
function
1. knowledge range
The concept of (1) function
Definition of function, representation of function, implicit function of piecewise function
(2) the nature of the function
Monotonicity parity bounded periodicity
(3) Inverse function
Definition of inverse function image of inverse function
(4) Basic elementary functions
Power function exponential function logarithmic function trigonometric function inverse trigonometric function
(5) Four operations and compound operations of functions
(6) Elementary function
2. Requirements
(1) Understand the concept of function. Find the expression, domain and function value of a function. Can find the definition domain and function value of piecewise function, and can make a simple image of piecewise function.
(2) Understand monotonicity, parity, boundedness and periodicity of functions.
(3) Knowing the relationship between a function and its inverse function (domain, range, image), we can find the inverse function of a monotonous function.
(4) Master the four operations and compound operations of functions.
(5) Master the properties and images of basic elementary functions.
(6) Understand the concept of elementary function.
(7) Establish the functional relationship of simple practical problems.
(2) Limit
1. knowledge range
The concept of (1) sequence limit
Definition of sequence limit
(2) The nature of sequence limit
Uniqueness and boundedness of monotone bounded sequence, four algorithms, compression theorem, limit existence theorem
(3) the concept of function limit
Definition of the limit of a function at one point The geometric meaning of the limit of a function when the left and right limits and their relationship with the limit tend to infinity.
(4) the nature of function limit
Pinch Theorem of Uniqueness Four Operations
(5) Infinite quantity and infinite quantity
Definition of infinitesimal and infinitesimal The relationship between infinitesimal and infinitesimal The infinitesimal nature of infinitesimal order.
(6) Two important limitations
2. Requirements
(1) Understand the concept of limit (the descriptions of ","and "are not required in the definition of limit). Will find the left and right limits of a function at a point, and understand the necessary and sufficient conditions for the existence of a function at a point limit.
(2) Understand the related properties of limit and master four algorithms of limit.
(3) Understand the concepts of infinitesimal and infinitesimal, and master the nature of infinitesimal and the relationship between infinitesimal and infinitesimal. Infinitely small order comparisons (high order, low order, same order and equivalence) will be made. Will use equivalent infinitesimal substitution to find the limit.
(4) Master the method of finding the limit with two important limits.
(3) continuity
1. knowledge range
The concept of (1) function continuity
The definition of function continuity at a discontinuous point and the classification of necessary and sufficient condition functions of left continuous and right continuous functions at a discontinuous point.
(2) The property that the function is continuous at one point
Composite function continuity of inverse function Four operations of continuity of continuous function
(3) Properties of continuous functions on closed intervals
Boundedness theorem, maximum theorem, minimum theorem, intermediate theorem (including zero theorem)
(4) continuity of elementary function
2. Requirements
(1) Understand the concept of continuity and discontinuity of a function at one point, understand the relationship between continuity and limit existence of a function at one point, and master the method of judging the continuity of a function (including piecewise functions) at one point.
(2) Find the discontinuity of the function and determine its type.
(3) Grasp the properties of continuous functions on closed intervals, and use the intermediate value theorem to derive some simple propositions.
(4) To understand the continuity of elementary function in its defined interval, we will use continuity to find the limit.
Second, the differential calculus of unary function
(a) derivative and differential
1. knowledge range
The concept of (1) derivative
The definition of derivative The function of left derivative and right derivative can be derived at one point. It is necessary and sufficient that the relationship between geometric meaning and physical meaning of derivative and its derivability and continuity.
(2) Derivation rules and basic formulas of derivatives.
Four operations of the basic formula of inverse function derivative
(3) Derivation method
Derivation of compound function, derivative of implicit function, derivative of logarithm, derivative of piecewise function determined by parameter equation
(4) Higher derivative
Definition and calculation of higher derivative
(5) Difference
Definition of differential, relationship between differential and derivative, differential law, invariance of first-order differential form
2. Requirements
(1) Understand the concept of derivative and its geometric meaning, understand the relationship between derivability and continuity, and master the method of finding the derivative of a function at a point by definition.
(2) Find the tangent equation and the normal equation of a point on the curve.
(3) Mastering the basic formula of derivative, four algorithms and derivative methods of compound function, we can find the derivative of inverse function.
(4) Mastering the derivative method of implicit function, logarithmic derivative method and derivative method of function determined by parameter equation will find the derivative of piecewise function.
(5) By understanding the concept of higher derivative, we can find the first derivative of a simple function.
(6) Understand the concept of function differentiation, grasp the law of differentiation, understand the relationship between differentiability and derivability, and find the first-order differentiation of functions.
(2) The application of differential mean value theorem and derivative.
1. knowledge range
(1) differential mean value theorem
Rolle theorem Lagrange mean value theorem
(2) the law of lobida.
(3) Judgment method of function increase or decrease.
(4) Extreme value and extreme point, maximum value and minimum value of the function
(5) Bump and inflection point of the curve
(6) Horizontal asymptote and vertical asymptote of the curve
2. Requirements
(1) Understand Rolle theorem, Lagrange mean value theorem and their geometric significance. Rolle theorem will be used to prove the existence of the root of the equation. Will use Lagrange mean value theorem to prove simple inequalities.
(2) Master the method of finding the limit of all kinds of infinitives with Robida's law.
(3) Mastering the method of judging monotonicity of function by derivative and finding monotone increase and decrease interval of function, and proving simple inequality by monotonicity of function.
(4) Understand the concept of function extremum. Mastering the method of finding the extreme value, maximum value and minimum value of a function will solve simple application problems.
(5) Will judge the convexity of the curve and find the inflection point of the curve.
(6) Find the horizontal asymptote and vertical asymptote of the curve.
(7) Graphics that can make simple functions.
3. Integral calculus of unary function
indefinite integral
1. knowledge range
(1) indefinite integral
Definition of primitive function and indefinite integral; Existence Theorem of Primitive Function and Properties of Indefinite Integral
(2) Basic integral formula
(3) Substitution integration method
1 substitution method (rounding difference method) The second substitution method.
(4) Component integration
(5) Integrals of some simple rational functions
2. Requirements
(1) Understand the concepts of original function and indefinite integral and their relationship, master the properties of indefinite integral, and understand the existence theorem of original function.
(2) Master the basic formula of indefinite integral.
(3) Master method of substitution of indefinite integral 1 and master the second method of substitution (limited to triangular method of substitution and simple radical method of substitution).
(4) Mastering the partial integral of indefinite integral.
(5) The indefinite integral of a rational function with one variable can be found.
(2) definite integral
1. knowledge range
The concept of (1) definite integral
Definition of definite integral and its geometric meaning integrable condition
(2) Properties of definite integral
(3) Calculation of definite integral
Variable upper bound integral Newton-Leibniz formula substitution integral method partial integral
(4) Generalized integral of infinite interval
(5) Application of definite integral
The area of a plane figure, the volume of a rotating body, and the work done by a variable force when an object moves along a straight line.
2. Requirements
(1) Understand the concept of definite integral and its geometric meaning, and understand the conditions of function integrability.
(2) Master the basic properties of definite integral.
(3) Understand that the variable upper bound integral is a variable upper bound function, and master the method of finding the derivative of the variable upper bound definite integral.
(4) Master Newton-Leibniz formula.
(5) Master the substitution integral method of definite integral and partial integral.
(6) Understand the concept of infinite interval generalized integral and master its calculation method.
(7) Grasp the area of the plane figure calculated by lower integral in rectangular coordinate system and the volume of the rotating body generated by the rotation of the plane figure around the coordinate axis.
Will use definite integral to find the work done by time-varying force moving along a straight line.
4. Vector Algebra and Spatial Analytic Geometry
(A) Vector Algebra
1. knowledge range
The concept of (1) vector
The coordinates of the projection vector of the modular unit vector on the coordinate axis represent the direction cosine of the normal vector.
(2) Linear operation of vectors
Multiplication of addition vector and subtraction vector of vector.
(3) Quantity product of vectors
Necessary and sufficient conditions for perpendicular included angle between two vectors.
(4) Necessary and sufficient conditions for two vectors of cross product to be parallel.
2. Requirements
(1) Understand the concept of vector, master the coordinate representation of vector, and find the projection of unit vector, direction cosine and vector on the coordinate axis.
(2) Master the linear operation of vectors, and the calculation method of vector product and cross product.
(3) Grasp the necessary and sufficient conditions for two vectors to be parallel and vertical.
(2) Plane and straight line
1. knowledge range
(1) common plane equation
General equation of point method equation
(2) the positional relationship between two planes (parallel, vertical and inclined)
(3) Distance from point to plane
(4) Spatial linear equation
Parameter equation of standard equation (also known as symmetric equation or point-to-point equation)
(5) the positional relationship between two straight lines (parallel and vertical)
(6) the positional relationship between a straight line and a plane (parallel, vertical and straight lines are all on the plane)
2. Requirements
(1) You can find the point equation and the general equation of the plane. The perpendicularity and parallelism of the two planes will be determined. You will find the angle between two planes.
(2) Find the distance from a point to a plane.
(3) By understanding the general equation of straight line, we can find the standard equation and parameter equation of straight line. Make sure that the two lines are parallel and vertical.
(4) Determine the relationship between straight line and plane (vertical, parallel, straight line on plane).
(3) Simple quadric surface
1. knowledge range
The spherical generatrix is parallel to the coordinate axis.
2. Requirements
Understand the equations and graphs of spherical surface, cylinder whose generatrix is parallel to the coordinate axis, paraboloid of revolution, conical surface and ellipsoid.
Five, multivariate function calculus
Differential calculus of multivariate functions
1. knowledge range
(1) multivariate function
Definition of multivariate function, geometric meaning of bivariate function, and concepts of limit and continuity of bivariate function.
(2) Partial derivative and total differential
Partial derivative full differential second-order partial derivative
(3) Partial derivative of composite function
(4) Partial derivative of implicit function
(5) unconditional extremum and conditional extremum of binary function
2. Requirements
(1) Understand the concept of multivariate function and the geometric meaning of bivariate function. You can find the expression and domain of quadratic function. Understand the concepts of limit and continuity of binary functions (calculation is not required).
(2) Understand the concept of partial derivative, understand the geometric meaning of partial derivative, understand the concept of total differential, and understand the necessary and sufficient conditions for the existence of total differential.
(3) Master the calculation method of the first and second partial derivatives of binary functions.
(4) Mastering the solution of the first-order partial derivative of composite function.
(5) Can find the total differential of binary function.
(6) Master the calculation method of the first-order partial derivative of the implicit function determined by the equation.
(7) Will find the unconditional extreme value of binary function. Lagrange multiplier method will be used to find the conditional extreme value of binary function.
(2) Double integral
1. knowledge range
The concept of (1) double integral
The Definition of Double Integral and Its Geometric Significance
(2) The properties of double integral
(3) Calculation of double integral
(4) Application of double integral
2. Requirements
(1) Understand the concept and properties of double integral.
(2) Master the calculation method of double integral in rectangular coordinate system and polar coordinate system.
(3) We will use double integral to solve simple application problems (limited to the volume of the bounded area surrounded by closed surfaces and the mass of plane thin plates).
Six, infinite series
(A) the series of several terms
1. knowledge range
(1) polynomial series
Polynomial series concept series convergence and divergence series basic properties and necessary conditions for series convergence.
(2) The method of judging the convergence and divergence of positive series.
Comparative discrimination method
(3) arbitrary term series
Leibniz discriminant method for convergence of absolute convergence conditions of staggered series
2. Requirements
(1) Understand the concept of convergence and divergence of series. Master the necessary conditions of series convergence and understand the basic properties of series.
(2) Master the ratio discrimination method of positive series. You can use the comparison and discrimination method of positive series.
(3) Master the convergence of geometric series, harmonic series and series.
(4) In order to understand the concepts of absolute convergence and conditional convergence of series, Leibniz discriminant method will be used.
(2) Power series
1. knowledge range
The concept of (1) power series
Convergence radius convergence interval
(2) Basic properties of power series
(3) Expand a simple elementary function into a power series.
2. Requirements
(1) Understand the concept of power series.
(2) Understand the basic properties of power series in its convergence interval (sum, difference, item-by-item derivation, item-by-item integration).
(3) Master the method of finding the convergence radius and convergence interval of power series (without discussing the endpoints).
(4) Some simple elementary functions will be expanded into power series by Maclaurin formula.
Seven, ordinary differential equations
(1) first order differential equation
1. knowledge range
The concept of (1) differential equation
Definition of differential equations, general solutions, initial conditions and special solutions
(2) separable variable equation
(3) First order linear equation
2. Requirements
(1) Understand the definition of differential equation, and understand the order, solution, general solution, initial condition and special solution of differential equation.
(2) Master the solution of separable variable equation.
(3) Master the solution of the first-order linear equation.
(2) Price reduction equation
1. knowledge range
Equation of type (1)
(2) Type equation
2. Requirements
(1) will solve the type equation by reducing the order.
(2) The reduced order method will be used to solve the type equation.
③ Second order linear differential equation
1. knowledge range
Structure of Solutions of (1) Second Order Linear Differential Equations
(2) Second-order homogeneous linear differential equation with constant coefficients
(3) Second-order non-homogeneous linear differential equation with constant coefficients
2. Requirements
(1) Understand the structure of solutions of second-order linear differential equations.
(2) Master the solution of second-order homogeneous linear differential equation with constant coefficients.
(3) Master the solution of second-order non-homogeneous linear differential equation with constant coefficients.
Examination form and examination paper structure
Total score of test paper: 150.
Examination time: 150 minutes
Examination methods: closed book and written test.
Test paper content ratio:
Function, limit and continuity are about 15%.
The differential calculus of unary function accounts for about 25%
The integral of unary function is about 20%
Multivariate function calculus (including vector algebra and spatial analytic geometry) is about 20%
Infinite series is about 10%
Ordinary differential equation is about 10%
Proportion of test questions:
The multiple-choice question is about 15%
Fill in the blanks about 25%
About 60% of the answers.
Test difficulty ratio:
Good question, about 30%
About 50% of moderately difficult questions
The difficulty is increased by about 20%.
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