Introduction to # Senior Two # Only an efficient learning method can quickly grasp the important and difficult points of knowledge. An effective reading method is to master the method according to the law. Don't recite it as soon as you come. Find the rules first, then remember the rules, and then learn the rules, so that you can master the knowledge quickly. The second channel of senior high school has compiled the compulsory key knowledge points of mathematics in the first volume of senior high school for you, hoping to help you!

1. The first volume of senior high school mathematics is a compulsory key knowledge point.

Definition: a function with the shape of y = x a (a is constant), that is, a function with the base as the independent variable and the exponent as the dependent variable is called a power function.

Domain and Value Domain:

When a is a different numerical value, the different situations of the domain of the power function are as follows: if a is any real number, the domain of the function is all real numbers greater than 0; If a is a negative number, then X must not be 0, but the definition domain of the function must also be determined according to the parity of Q, that is, if Q is even at the same time, then X cannot be less than 0, then the definition domain of the function is all real numbers greater than 0; If q is an odd number at the same time, the domain of the function is all real numbers that are not equal to 0. When x is different, the range of power function is different as follows: when x is greater than 0, the range of function is always a real number greater than 0. When x is less than 0, only when q is odd and the range of the function is non-zero real number. Only when a is a positive number will 0 enter the value range of the function.

Nature:

For the value of a nonzero rational number, it is necessary to discuss their respective characteristics in several cases:

First of all, we know that if a=p/q, q and p are integers, then x (p/q) = the root of q (p power of x), if q is odd, the domain of the function is r, if q is even, the domain of the function is [0, +∞). When the exponent n is a negative integer, let a=-k, then x = 1/(x k), obviously x≠0, and the domain of the function is (-∞, 0)∩(0, +∞). So it can be seen that the limitation of X comes from two points. First, it can be used as a denominator, but it cannot be used as a denominator.

Rule out two possibilities: 0 and negative number, that is, for x>0, then A can be any real number;

The possibility of 0 is ruled out, that is, for all real numbers of x0, q cannot be even;

The possibility of being negative is ruled out, that is, for all real numbers with x greater than or equal to 0, a cannot be negative.

To sum up, we can draw that when a is different, the different situations of the power function domain are as follows:

If a is any real number, the domain of the function is all real numbers greater than 0;

If a is a negative number, then X must not be 0, but the definition domain of the function must also be determined according to the parity of Q, that is, if Q is even at the same time, then X cannot be less than 0, then the definition domain of the function is all real numbers greater than 0; If q is an odd number at the same time, the domain of the function is all real numbers that are not equal to 0.

2. The first volume of high school mathematics must be a key knowledge point.

First, use samples to estimate the summary of the overall knowledge points.

1. histogram of frequency distribution

(1) Generally, our estimation of the population can be divided into two types: one is to estimate the distribution of the population by using the frequency distribution of samples; The other is to estimate the digital characteristics of the population with the digital characteristics of the sample.

(2) Making a histogram of frequency distribution.

① Find the interval (that is, the difference between the median value and the minimum value of a set of data).

② Determine the interval and number of groups.

③ Grouped data.

④ Column frequency distribution table.

⑤ Draw the histogram of frequency distribution.

(3) In the histogram of frequency distribution, the vertical axis represents the frequency/group distance, and the frequency of data falling in each group is represented by the area of each small rectangle. The total area of each small rectangle is equal to 1.

2. Frequency distribution line graph and overall density curve

(1) frequency distribution line chart: connect the midpoint of each small rectangle in the frequency distribution histogram to get the frequency distribution line chart.

(2) Overall density curve: With the increase of sample size, the number of groups increases and the distance between groups decreases, and the corresponding frequency line graph will be closer to a smooth curve, that is, the overall density curve.

3. Advantages of stem leaf diagram

There are two outstanding advantages of using stem-leaf diagram to represent data:

First, the original data information is not lost in the statistical diagram, and all data information can be obtained from the stem and leaf diagram;

Second, the data in the stem and leaf diagram can be recorded and added at any time, which is convenient for recording and representing.

4. Sample variance and standard deviation

note:

Two similarities and differences

Similarities and differences of (1) model, median and average

Mode, median and average are all quantities that describe the trend in a set of data sets, and average is the most important quantity.

② Because the average is related to each sample data, any change of sample data will cause the change of the average, which is a property that neither the median nor the mode has.

(3) Mode examines the frequency of each data, and its size is only related to some data in this group of data. When a group of data has a lot of repeated data, its pattern can often better reflect the problem.

④ Changes in some data may not affect the median. The median may or may not appear in the given data. When the individual data in a group of data changes greatly, the median can be used to describe its centralized trend.

(2) Similarities and differences between standard deviation and variance

Standard deviation and variance describe the fluctuation of a set of data around the average. The greater the standard deviation and variance, the greater the deviation of the data. The smaller the standard deviation and variance, the smaller the deviation of the data. Because the unit of variance is different from the original data, and the degree of deviation after squaring may be exaggerated, although variance and standard deviation are the same in describing the dispersion of sample data, standard deviation is generally used in solving practical problems.

Three characteristics

Using frequency distribution histogram to estimate the digital characteristics of samples;

(1) Median: In the frequency distribution histogram, the left and right histogram areas of the median are equal, so the median can be estimated.

(2) Average value: the estimated value of the average value is equal to the sum of the areas of each small rectangle multiplied by the abscissa of the midpoint at the bottom of the rectangle.

(3) Mode: the abscissa of the midpoint of the rectangle.

3. The first volume of senior high school mathematics is a compulsory key knowledge point.

The Basis of Proof of 1. Inequality

(2) the essence of inequality

(3) Important inequality:

①| a |≥0； a2≥0； (a-b)2≥0(a、b∈R)

②a2+b2≥2ab(a, b∈R, marked with "=" if and only if a=b).

2. Proof method of inequality

(1) comparison method: prove a & gtb(a0(a-b