Analysis of Mathematics Knowledge Points in Senior Two of People's Education Press (1)
1. In middle school, we only study straight cylinders, cones and frustums. So the definition of rotation of cylinder, cone and frustum is actually the definition of straight cylinder, straight cone and straight frustum. This definition of intuitive images is easy to understand and their properties can be easily deduced.
In the definition of ball, we should pay attention to distinguish the concepts of ball and sphere, and the ball is solid.
Equilateral cylinder and equilateral cone are special cylinders and cones, which are defined by their axial sections and are widely used in practice. Pay attention to the difference between them and general cylinders and cones.
2. Properties of cylinders, cones, circles and spheres
The nature of (1) cylinder should emphasize two points: first, the connecting line is perpendicular to the bottom of the cylinder; The second is the three-segment nature-the segment parallel to the bottom is a circle that is congruent with the bottom; The shaft section is a rectangle composed of more than one lower bottom circle diameter and a generatrix; The section parallel to the axis is a rectangle composed of many chords and generatrix with a bottom circle.
(2) The nature of the cone should emphasize three points.
(1) Properties of the section circle parallel to the bottom:
The ratio of the circular area of the section to the circular area of the bottom surface is equal to the square ratio of the distance from the vertex to the section and from the vertex to the bottom surface.
(2) The cross section passing through the apex of the cone and intersecting with its bottom surface is an isosceles triangle composed of two generatrix lines and chords of the bottom circle, and its area is:
It is easy to know that the vertex angle of the section triangle is not greater than the vertex angle of the shaft section (as shown in figure 10-20). In fact, from BC≥AB and VC=VB=VA, we can get ∠AVB≤BVC.
Because the apex angle of the cross-sectional triangle is not greater than the apex angle of the shaft cross-section.
Therefore, the vertex angle θ of the important official section is less than or equal to 90, and when there is 0 90, the area of the shaft section is not, because if 90 is less than or equal to α 0.
③ The generatrix L, the height h and the radius of the bottom circle of the cone form a diameter triangle. The calculation problem of cone generally boils down to solving this right triangle, especially the relation.
l2=h2+R2
(3) The nature of frustum comes from the fact that frustum is a truncated cone. But the college entrance examination still emphasizes the following points:
(1) The generatrix of the frustum of a cone is at the point of * * *, so the cross section determined by any two generatrix is an isosceles trapezoid, but the cross section intersecting the upper and lower bottom surfaces is not necessarily a trapezoid, let alone an isosceles trapezoid.
(2) If the cross section parallel to the bottom surface divides the height of the frustum of a cone into two sections from the upper and lower bottom surfaces, and the cross section area is S, then
Where S 1 and S2 are the upper and lower bottom regions, respectively.
Generalization of section properties of.
(3) The generatrix L, the height H of the frustum of a cone and the radii R and R of the upper and lower bottom circles form a right-angled trapezoid, and there are
l2=h2+(R-r)2
The calculation of frustum of a cone often boils down to solving this right-angled trapezoid.
(4) The properties of the ball, especially the properties of its cross section.
① The cross section obtained by intercepting the ball with any plane is a circular surface, and the connecting line between the center of the ball and the center of the cross section is perpendicular to this cross section.
(2) If R and R represent the radius of the ball and the radius of the cross-section circle respectively, and D represents the distance from the center of the ball to the cross-section, then
R2=r2+d2
That is, the radius of the ball, the radius of the cross-section circle and the distance from the center of the ball to the cross-section form a right triangle. The calculation of a ball often boils down to solving this right triangle.
3. Surface areas of cylinders, cones, frustums and spheres
(1) Cylinders, cones, frustums and polyhedrons can all be unfolded on the same plane.
① The side development diagram of cylinder, cone and frustum is the basic basis for finding its side area.
The side development diagram of a cylinder is a rectangle composed of the perimeter of the base map and the length of the bus.
(2) The cone-side development diagram is a sector consisting of two generatrix lengths and the perimeter of the bottom circle, and the central angle of the sector is
(3) The side development diagram of the frustum of a cone is a sector ring composed of two bus lengths and the circumferences of the upper and lower bottom surfaces, and the central angle of the sector ring is
This formula is beneficial to the reciprocity of space geometry and its side expansion diagram.
Obviously, when r=0, this formula is the formula of the fan-shaped central angle of the cone-side expansion diagram, so the formula of the fan-shaped central angle of the cone-side expansion diagram is a special case of the cone-side correlation angle.
(2) The lateral formula of cylinder, cone and frustum is
S side =π(r+R)l
When r=R, the S side =2πRl, that is, the lateral area formula of the cylinder.
When r=0, the S-side =rRl, that is, the area formula of the cone.
Pay attention to this relationship of lateral area.
(3) Sphere is a figure that can't be unfolded on the plane, so the method of finding its area is completely different from that of column, cone and platform.
Derivation requires knowledge of advanced mathematics such as calculus, which cannot be regarded as a proof in textbooks.
The common method to find the measurement properties of irregular circles is "subdivision-summation-limit". This method is self-evident after learning the relevant contents of Calculus, and it is omitted here.
4. The method of drawing the direct view of cylinder, cone, frustum and sphere-orthometric height measurement.
(1) requirements for isometric drawing:
(1) When drawing the three axes of X, Y and Z, the Z axis is drawn in the vertical direction, and the X axis and Y axis respectively form 120 with the Z axis.
(2) The method of taking the line segment length on the projection map is: take the real length of the line segment on or parallel to the three axes.
Here, it is different from oblique survey method and drawing direct view method, so we should pay attention to their differences.
(2) The difference between cylinder, cone and frustum orthographic drawings is mainly the horizontal figure.
When drawing a horizontal plane circle with isometric drawing, draw the horizontal position with X axis and 120 with Y axis. On the projection diagram, the line segments on the X-axis and Y-axis, or the line segments parallel to the X-axis and Y-axis, are all taken as real lengths, and the drawing method of the line segments on the Z-axis or parallel to the Z-axis is the same as that of oblique survey.
5. The shortest distance between two points on a geometric surface.
The surfaces of cylinders, cones and platforms can be developed in a plane, and the shortest distance between two points in these geometric surfaces is the length of the line segment between two points in their plane development diagrams.
Because the sphere cannot be unfolded on a plane, it is a brand-new method to find the spherical distance between two points on the sphere. The shortest distance is the lower arc length of the great circle passing through these two points.
Analysis of Mathematics Knowledge Points in Senior Two of People's Education Press (2)
simplerandom sampling
1. Population and sample
In statistics, the whole research object is called population.
Call each research object an individual.
The total number of individuals in a group is called the total capacity.
In order to study the related properties of the population, a part is generally randomly selected from the population:
Research, we call it a sample. The number of individuals is called sample size.
2. Simple random sampling, also called pure random sampling. That is, as a whole, it does not go through any grouping, classification, queuing, etc. , completely with.
Machine-based measurement unit extraction. The characteristics are: the probability of each sample unit being extracted is the same (the probability is equal), and each unit of the sample is completely independent, and there is no certain correlation and exclusion between them. Simple random sampling is the basis of other sampling forms. This method is usually only used when the difference between the whole units is small and the number is small.
3. Common methods of simple random sampling:
Draw lots; Random number table method; Computer simulation method; Direct extraction with statistical software.
In the sample size design of simple random sampling, the main considerations are: ① population variation; ② Allowable error range; ③ Degree of probability assurance.
4. Draw lots:
(1) Number each object in the investigation team;
(2) Prepare the lottery tool and implement the lottery.
(3) Measure or investigate each individual in the sample.
Please investigate the favorite sports activities of students in your school.
5. Random number table method:
Example: Select 10 students from the class to participate in an activity by using a random number table.
systematic sampling
1. System sampling (equidistant sampling or mechanical sampling):
Sort the units of the population, then calculate the sampling distance, and then sample according to this fixed sampling distance. The first sample was selected by simple random sampling.
K (sampling distance) =N (overall size) /n (sample size)
Prerequisite: For the variables studied, the arrangement of individuals in the group should be random, that is, there is no regular distribution related to the variables studied. You can start sampling from different samples and compare the characteristics of several samples under the conditions allowed by the investigation. If there are obvious differences, it shows that the distribution of samples in the population follows a certain cycle law, and this cycle coincides with the sampling distance.
2. Systematic sampling, namely equidistant sampling, is one of the most commonly used sampling methods in practice. Because it has low requirements for sampling frames and simple implementation. More importantly, if there are some auxiliary variables related to the survey indicators available, and the whole unit is queued according to the size of the auxiliary variables, systematic sampling can greatly improve the estimation accuracy.
group sampling
1. stratified sampling (type sampling):
First of all, according to some characteristics or signs (gender, age, etc.), all units in the group are divided into several types or levels. ), and then extract a sub-sample from each type or level by simple random sampling or systematic sampling. Finally, these sub-samples are combined to form a total sample.
Two methods:
1. Firstly, the population is divided into several layers by stratification variables, and then extracted from each layer according to the proportion of each layer in the population.
2. Firstly, the population is divided into several layers by stratification variables, and then the elements in each layer are arranged neatly in hierarchical order. Finally, samples are extracted by systematic sampling.
2. Stratified sampling is to divide people with strong heterogeneity into sub-populations with strong homogeneity, and then draw samples from different sub-populations to represent sub-populations, and all samples represent people again.
Stratification standard:
(1) Take the main variables or related variables to be analyzed and studied in the investigation as the standard of stratification.
(2) Ensure that the variables with strong homogeneity in each layer, strong interlayer heterogeneity and outstanding overall internal structure are used as stratified variables.
(3) Take those variables with obvious stratification as hierarchical variables.
3. The proportion of stratification:
(1) Proportional stratified sampling: a method to extract sub-samples according to the proportion of units of various types or levels to the total units.
(2) Non-proportional stratified sampling: If the proportion of some levels in the population is too small, the sample size will be very small. At this time, this method is mainly used to facilitate special research or comparison of different levels of subpopulations. If we want to infer the population from the sample data, we need to first weight the data of each layer, adjust the proportion of each layer in the sample, and restore the data to the actual proportion structure of each layer in the population.
Estimate the digital characteristics of the population with the digital characteristics of the sample.
1, average:
2, sample standard deviation:
3. When estimating the population with samples, if the sampling method is reasonable, then the samples can reflect the information of the population, but the information obtained from the samples will be biased. In random sampling, this deviation is inevitable.
Although the distribution, mean and standard deviation we get from the sample data are not the real distribution, mean and standard deviation of the population, but only an estimate, this estimate is reasonable, especially when the sample size is large, and they do reflect the information of the population.
4.( 1) If the same constant is added or subtracted from each data in a set of data, the standard deviation remains unchanged.
(2) If each data in a set of data is multiplied by a constant k, the standard deviation becomes k times the original value.
(3) The influence of values and minimum values in a group of data on standard deviation and the application of intervals;
The scientific truth in "removing a point and removing a lowest point"
Linear correlation of two variables
1, concept:
(1) regression linear equation (2) regression coefficient
2. Least square method
3. Application of linear regression equation
(1) describes the dependency between two variables; Linear regression equation can be used to quantitatively describe the quantitative relationship between two variables.
(2) Using regression equation to forecast; Substituting the predictor (independent variable x) into the regression equation to estimate the predictor (dependent variable y), the allowable interval of individual y value can be obtained.
(3) Use regression equation for statistical control, specify the change of Y value, and achieve the purpose of statistical control by controlling the range of X. If the regression equation between NO2 concentration in the air and traffic flow is obtained, the NO2 concentration in the air can be controlled by controlling traffic flow.
4. Matters needing attention in the application of linear regression
(1) regression analysis should have practical significance;
(2) Before regression analysis, make a scatter plot;
(3) Don't extend the tropic of cancer.