Chapter I Functions

1, the domain of function and the evaluation of piecewise function.

2. Basic elementary functions: power function, exponential function, logarithmic function, trigonometric function and inverse trigonometric function.

Elementary function: A function that is composed of basic elementary functions and constants and can be expressed by formulas after four operations of finite degree and compound steps of finite degree functions is called elementary function.

3. Common economic functions (demand function, supply function, total cost function, total income function, total profit function and inventory function)

Chapter II Limit and Continuity

The definition and properties of 1 and infinitesimal.

1) A variable whose limit is zero is called infinitesimal.

Note: (1) infinitesimal is a variable, not a small number.

(2) Zero is the only infinitesimal quantity in a constant.

2) Properties of infinitesimal: the algebraic sum of finite infinitesimals is infinitesimal, the product of bounded function and infinitesimal is infinitesimal, the product of constant and infinitesimal is infinitesimal, and the product of finite infinitesimal is infinitesimal.

3) Relationship between function limit and infinitesimal: If and only if A is constant.

2. The definition of infinity.

In a certain change process, if the absolute value of f(x) increases infinitely, the function f(x) is said to be infinite in this change process.

Note: Infinity is a variable, not a number with great absolute value.

Infinity and infinitesimal are reciprocal.

4. Algorithm of limit.

See the textbook P48 Theorem 1, 2,3,4 and Inference 1, 2.

5. Two important limitations.

Will use the important limit to find the function limit.

6, will use equivalent infinitesimal instead of seeking limit.

7. Definition of continuity. See textbook P66.

The function f(x) is continuous at the point x0 and must satisfy three conditions at the same time:

1) is defined at point x0;

2) existence;

3) The limit value is equal to the function value, that is.

8. The necessary and sufficient condition for a function to be continuous at one point is: left and right continuity.

9. The relationship between the continuity of a function at a point and the limit at that point:

If a function is continuous at a certain point, there must be a limit at that point, but if a function has a limit at that point, it is not necessarily continuous at that point.

10, how to find the limit of continuous function

The limit of continuous function must exist, and the limit value is equal to the function value, that is,

1 1 1. For the continuity of a piecewise function at a piecewise point, if the expressions of the function on both sides of the piecewise point are different, we should discuss it according to the necessary and sufficient conditions for the function to be continuous at one point.

12, how to find the continuous interval?

The basic elementary function is continuous in its definition domain;

All elementary functions are continuous within their defined intervals.

13, the definition of discontinuity.

14, discontinuous type.

(A) the first discontinuity

1, you can go to the discontinuity.

(1) is not defined, but exists.

(2) There is a definition where the left and right limits exist and are equal, but.

2. Jump discontinuity: the left and right limits of a point exist, but they are not equal.

The first discontinuity is that the function exists at this point.

(2) the second kind of discontinuity (if at least one of the left and right limits does not exist, it is called the second kind of discontinuity. )

1, infinitely discontinuous.

2. The oscillation is discontinuous.

Related exercises are as follows:

P47 3 P53 2，3，4 P62 1，2 P65 1，2，3 P73 2，3，5，6

Chapter III Derivative, Differential, Marginal and Elasticity

The necessary and sufficient condition for 1. function to be differentiable at a point is that the left and right derivatives exist and are equal at that point.

2. Judge whether the split point is derivable: at the split point, the left and right derivatives should be calculated according to the definition. If the left and right derivatives at the split point exist and are equal, the split point is derivable.

3. Relationship between continuity and derivability: If a function is derivable at one point, then the function is continuous at one point. Otherwise.

4. The derivative of the function at this point geometrically represents the tangent slope of the curve at this point.

5. Tangent equation and normal equation

6. Derivation of implicit function and derivative of function expressed by parametric equation.

7. logarithmic derivative Law

8. Definition of differentiability.

9. The necessary and sufficient condition for a function to be differentiable at one point is that it is differentiable at one point.

Related exercises are as follows:

P9 1 7， 1 12， 15 P 100

The fourth chapter is the application of the mean value theorem and derivative.

10, the content of the mean value theorem.

1 1, Robida's law.

12, function monotonicity discrimination method: the steps to find the extreme value:

13, the steps to find the maximum (minimum) value:

14, definition and judgment method of function concavity and inflection point

15, the application of derivative in economy (maximum profit, maximum income, economic batch, maximum tax, etc. )

Related exercises are as follows:

142 2 147 1 162 1，2，4，5 168 3

The fifth chapter indefinite integral

1, the relationship between primitive function and indefinite integral: all primitive functions constitute indefinite integral. Namely.

Integral operation and differential operation have the following interrelationships:

1) or.

2) or.

2. Substitution integration method of indefinite integral.

The first kind of substitution method (integral differential method).

The second alternative.

Partial integral

Related exercises are as follows:

p 183 1 p 197 1 P203 1

Chapter VI Definite Integral

1, the properties of definite integral.

2. The mean value theorem of definite integral.

3. It is a function of the upper limit of integral (or definite integral with variable upper limit).

Its derivative is

4. Newton-Leibniz formula, also known as the basic formula of calculus.

5. The integral method of partial substitution of definite integral.

6. Economic application of definite integral (from marginal function to original function, from rate of change to total)

Related exercises are as follows:

p 2 19 2 P225 1 2 3 p 23 1 2 p 233 1 p 239 1 p 252 1 2 3 4 5

Chapter 10 Differential Equation

4 1 Basic concepts of differential equations (differential equations, angles of differential equations, special solutions, general solutions, order of differential equations, initial conditions, initial value problems, etc. )

42, the solution of the differential equation of separable variables

43, the solution of the first order linear differential equation

44. The solution of the second-order differential equation can be simplified.

45, the solution of the second order constant coefficient differential equation

Related exercises are as follows:

p384 1 3 4 P396 1 2 P405 4 6 7