An Overview of Calculus in Higher Mathematics
I. Function, Limit and Continuity
Examination requirements
1. Understand the concept of function and master the expression of function, and you will establish the functional relationship of application problems.
2. Understand the boundedness, monotonicity, periodicity and parity of functions.
3. Understand the concepts of compound function and piecewise function, inverse function and implicit function.
4. Grasp the nature and graphics of basic elementary functions and understand the concept of elementary functions.
5. Understand the concepts of sequence limit and function limit (including left limit and right limit).
6. Understand the nature of limit and two criteria for the existence of limit, master four algorithms of limit, and master the method of finding limit by using two important limits.
7. Understand the concept and basic properties of infinitesimal, master the comparison method of infinitesimal, and understand the concept of infinitesimal and its relationship with infinitesimal.
8. Understanding the concept of function continuity (including left continuity and right continuity) will distinguish the types of function discontinuity points.
9. Understand the properties of continuous function and continuity of elementary function, understand the properties of continuous function on closed interval (boundedness, maximum theorem, mean value theorem), and apply these properties.
Second, the differential calculus of unary function
Examination requirements
1. Understand the concept of derivative and the relationship between derivability and continuity, understand the geometric and economic significance of derivative (including the concepts of allowance and elasticity), and find the tangent equation and normal equation of plane curve.
2. Mastering the derivation formula of basic elementary function, four arithmetic rules of derivation and the derivation rule of compound function, we can obtain the derivation of piecewise function, inverse function and implicit function.
3. If you understand the concept of higher derivative, you will find the higher derivative of simple function.
4. Understand the concept of differential, the relationship between derivative and differential, and the invariance of first-order differential form, and you will find the differential of function.
5. Understand Rolle theorem, Lagrange mean value theorem, Taylor theorem and Cauchy mean value theorem, and master the simple application of these four theorems.
6. Will use the Lobida rule to find the limit.
7. Master the method of judging monotonicity of function, understand the concept of function extreme value, and master the solution and application of function extreme value, maximum value and minimum value.
8. The concavity and convexity of the function graph can be judged by the derivative (note: in the interval, let the function have the second derivative. When, the figure is concave; When the graph is convex, you will find the inflection point and asymptote of the function graph.
9. Graphics that can describe simple functions.
3. Integral calculus of unary function
Examination requirements
1. Understand the concepts of original function and indefinite integral, master the basic properties and basic integral formula of indefinite integral, and master the substitution integral method and integration by parts of indefinite integral.
2. Understand the concept and basic properties of definite integral, understand the mean value theorem of definite integral, understand the function of upper limit of integral and find its derivative, and master Newton-Leibniz formula, method of substitution and integration by parts of definite integral.
3. Will use definite integral to calculate the area of plane figure, the volume of rotating body and the average value of function, and will use definite integral to solve simple economic application problems.
4. Understand the concept of generalized integral and calculate generalized integral.
Four, multivariate function calculus
Examination requirements
1. Understand the concept of multivariate function and the geometric meaning of bivariate function.
2. Understand the concepts of limit and continuity of binary function and the properties of binary continuous function in bounded closed region.
3. Knowing the concepts of partial derivative and total differential of multivariate function, we can find the first and second partial derivatives of multivariate composite function and the total differential and partial derivative of multivariate implicit function.
4. Understand the concepts of extreme value and conditional extreme value of multivariate function, master the necessary conditions of extreme value of multivariate function, understand the sufficient conditions of extreme value of binary function, find the extreme value of binary function, find the conditional extreme value by Lagrange multiplier method, find the maximum value and minimum value of simple multivariate function, and solve the simple application problem.
5. Understand the concept and basic properties of double integral, and master the calculation method of double integral (rectangular coordinates. Polar coordinates), understand the simple abnormal double integral of unbounded region and calculate it.
Five, infinite series
Examination requirements
1. Understand the convergence and divergence of series. The concept of the sum of convergent series.
2. Understand the basic properties of series and the necessary conditions of convergence and divergence of series, master the conditions of convergence and divergence of geometric series and series, and master the comparative judgment method and ratio judgment method of convergence and divergence of positive series.
3. Understand the concepts of absolute convergence and conditional convergence of arbitrary series and the relationship between absolute convergence and convergence, and understand Leibniz discriminant method of staggered series.
4. Will find the convergence radius, convergence interval and convergence domain of power series.
5. Knowing the basic properties of power series in its convergence interval (continuity of sum function, item-by-item derivation, item-by-item integration), we can find the sum function of simple power series in its convergence interval.
6. Understand the Maclaurin expansion of power x, sin x, cos x, ln( 1+x) and E (1+x).
Six, ordinary differential equations and difference equations
Examination requirements
1. Understand differential equations and their concepts such as order, solution, general solution, initial condition and special solution.
2. Master the solutions of differential equations, homogeneous differential equations and first-order linear differential equations with separable variables.
3. Second-order homogeneous linear differential equations with constant coefficients can be solved.
4. Knowing the properties and structure theorems of solutions of linear differential equations, you can solve second-order non-homogeneous linear differential equations with constant coefficients with polynomial, exponential function, sine function and cosine function as free terms.
5. Understand the concepts of difference and difference equation and their general and special solutions.
6. Understand the solution method of the first-order linear difference equation with constant coefficients.
7. Can use differential equations to solve simple economic application problems.
References:
Baidu Encyclopedia-Three Outline of Postgraduate Mathematics