1. Five Necessary Knowledge Points of Mathematics in Senior Three.
1. logarithmic function log.a(MN)=logaM+logN
loga(M/N)=logaM-logaN
logaM^n=nlogaM(n=R)
logbN = logaN/logab(a & gt； 0, b>0, N & gt0a and b are not equal to 1)
Second, the area and volume of simple geometry
Rectangular edge of prism =c*h (perimeter of bottom multiplied by height)
Vertebral side of S regular prism = 1/2 * c * h' (base circumference and half inclined height)
Let the perimeters of the upper and lower bottom surfaces of a regular prism be c' and c, respectively, and the inclined height be h', and s =1/2 * (c+c') * h.
S cylindrical edge =c*l
S frustum edge = 1/2*(c+c')*l= upright *(r+r')*l
S- cone edge =1/2 * c * l = r * r * l.
S ball =4* u * r 3
V cylinder =S*h
V-cone =( 1/3)*S*h
V-ball =(4/3)* Wu * R 3
Third, the positional relationship and distance formula of two straight lines
(1) formula for the distance between two points on the number axis |AB|=|x2-x 1|
(2) The distance formula between two points A (X 1, Y 1) and (X2, Y2) on the plane.
|ab|=sqr[(x2-x 1)^2+(y2-y 1)^2]
(3) The distance formula from point P(x0, y0) to straight line L: Ax+By+C=0 is d=|Ax0+By0+C|/sqr.
(A^2+B^2)
(4) The distance between two parallel straight lines L 1: = AX+BY+C = 0, L2 = AX+BY+C2 = 0 d=|C 1-
C2|/sqr(A^2+B^2)
Basic relations and inductive formulas of trigonometric functions with the same angle
Sin(2*k* Wu +a)=sin(a)
Cos(2*k* u +a)=cosa
Tan(2* u +a)=tana
sin(-a)=-sina，cos(-a)=cosa，tan(-a)=-tana
Sin(2* Uighur -a)=-sina, cos(2* Uighur -a)=cosa, tan(2* Uighur -a)=-tana.
Sin (Wu +a)=- SiNa
Sin(Wu-a)= Sina
Cos (Wu +a)=-cosa
Cos (u -a)=-cosa
Tan (Wu+A) = Tana
Quadruple angle formula and its deformation and application
1, double angle formula
sin2a=2*sina*cosa
cos2a=(cosa)^2-(sina)^2=2*(cosa)^2- 1= 1-2*(sina)^2
tan2a=(2*tana)/[ 1-(tana)^2]
2. The deformation of the double-angle formula
(cosa)^2=( 1+cos2a)/2
(sina)^2=( 1-cos2a)/2
Tan (a/2)= Sina/(1+COSA) = (1-COSA)/Sina.
Five, sine theorem and cosine theorem
Sine theorem:
a/sinA=b/sinB=c/sinC
Cosine theorem:
a^2=b^2+c^2-2bccosA
b^2=a^2+c^2-2accosB
c^2=a^2+b^2-2abcosC
cosA=(b^2+c^2-a^2)/2bc
cosB=(a^2+c^2-b^2)/2ac
cosC=(a^2+b^2-c^2)/2ab
Tan (Wu A) =-Tana
Sin (Wu /2+a)=cosa
Sin (Wu /2-a)=cosa
cos(δ/2+a)=-Sina
Cos (Wu /2-a)= Sina
Tan (Wu /2+a)=-cota
Tan (Wu /2-a)=cota
(sina)^2+(cosa)^2= 1
Sina /cosa=tana
Cosine formula of sum and difference of two angles
cos(a-b)=cosa*cosb+sina*sinb
cos(a-b)=cosa*cosb-sina*sinb
Sine formula of sum and difference of two angles
sin(a+b)=sina*cosb+cosa*sinb
sin(a-b)=sina*cosb-cosa*sinb
Tangent formula of sum and difference of two angles
tan(a+b)=(tana+tanb)/( 1-tana * tanb)
tan(a-b)=(tana-tanb)/( 1+tana * tanb)
2. Five compulsory knowledge points in senior three mathematics.
Cluster sampling Cluster sampling is also called cluster sampling. It is to combine all the units in the population into several sets that do not cross each other and do not repeat each other, which are called groups; A sampling method in which samples are sampled in groups.
When cluster sampling is applied, each cluster is required to have good representativeness, that is, the differences between units within the cluster are large and the differences between groups are small.
merits and demerits
The advantages of cluster sampling are convenient implementation and saving money;
The disadvantage of cluster sampling is that the sampling error caused by large differences between different groups is often greater than that caused by simple random sampling.
Implementation steps
Firstly, the population is divided into group I, and then several groups are extracted from group I clock to investigate all individuals or units in these groups. The sampling process can be divided into the following steps:
Firstly, the label of clustering is determined.
Second, divide the whole (n) into several non-overlapping parts, and each part is a group.
Third, according to the sample size, determine the number of groups to be extracted.
Fourthly, by using simple random sampling or systematic sampling method, the determined group number is extracted from the I group.
For example, investigate the situation of middle school students suffering from myopia and make statistics in the last class; Conduct product inspection; All products produced by 1h are sampled and inspected every 8 hours, etc.
Difference from stratified sampling
Cluster sampling and stratified sampling are similar in form, but quite different in fact.
Stratified sampling requires large differences between layers, small differences between individuals or units within layers, small differences between groups and large differences between individuals or units within groups;
The sample of stratified sampling consists of several units or individuals extracted from each layer, while cluster sampling is either cluster sampling or cluster sampling is not.
systematic sampling
definition
When there are many individuals in the group, it is more troublesome to adopt simple random sampling. At this time, the population can be divided into several balanced parts, and then an individual can be extracted from each part according to predetermined rules to get the required samples. This kind of sampling is called systematic sampling.
step
Generally speaking, if you want to extract a sample with a capacity of n from a population with a capacity of n, you can carry out systematic sampling according to the following steps:
(1) First number the n individuals in the population. Sometimes you can directly use your own number, such as student number, admission ticket number, house number, etc.
(2) Determine the segment interval k and the number of segments. When N/n(n is the sample size) is an integer, take k = n/n;
(3) determining the first individual number L (L ≤ K) by simple random sampling in the first paragraph;
(4) Sampling according to certain rules. Usually, the interval k is added with L to get the second number of individuals (l+k), and then K is added to get the third number of individuals (l+2k), and so on until the whole sample is obtained.
3. Five compulsory knowledge points in senior three mathematics.
One deduction deduces the sum of the first n terms of geometric series by dislocation subtraction: sn = a1+a1q+a1q 2+…+a1,
Use q: qsn = a1q+a1q 2+a1q 3+…+a1qn,
Subtract the two expressions to get (1-q) sn = a1-a1qn, ∴Sn=(q≠ 1).
Two preventive measures
(1) From an+ 1=qan, q≠0, we can't immediately assert that {an} is a geometric series, so we should verify a 1≠0.
(2) When using the first n terms and formulas of geometric series, we must pay attention to the classification and discussion of q= 1 and q≠ 1 to prevent mistakes caused by ignoring the special situation of q= 1.
Three methods
The judgment methods of geometric series are as follows:
(1) Definition: If an+ 1/an=q(q is a non-zero constant) or an/an- 1=q(q is a non-zero constant and n≥2 and n∈N_), {an} is a geometric series.
(2) Term formula: In series {an}, an≠0 and a = an an+2 (n ∈ n _), then series {an} is a geometric series.
(3) General formula method: If the general formula of series can be written as an = cqn (both c and q are constants not equal to 0, n∈N_), then {an} is a geometric series.
Note: The first two methods can also be used to prove that a series is a geometric series.
4. Five compulsory knowledge points in senior three mathematics.
1, the concept of set is the most primitive undefined concept in mathematics, which can only give a descriptive explanation: when some formulaic and different objects are gathered together, they are called a set. The objects that make up a set are called elements. The set usually uses uppercase letters A, B, C, … Elements usually use lowercase letters A, B, C, …
A set is a definite whole, so it can also be described as a set composed of all objects with certain attributes.
2. The relationship between elements and sets There are two kinds of relationships between elements and sets: element A belongs to set A and is marked as A ∈ A; Element a does not belong to set a, so it is recorded as a? Answer.
3. Characteristics of elements in the set
(1) Determinism: Let A be a given set and X be a specific object, then X is either an element of A or not, and one and only one of the two situations must be true. For example, A={0, 1, 3, 4}, then 0 ∈ a, 6? Answer.
(2) Reciprocity: "The elements of a set list must be different from each other", that is, "any two elements of a given set are different".
(3) Disorder: A set has nothing to do with the arrangement order of its elements, for example, set {a, b, c} and set {c, b, a} are the same set.
4. Classification of sets
According to the number of elements it contains, set division can be divided into two categories:
Finite set: A set containing a finite number of elements. For example, "the set composed of solutions of equation 3x+ 1=0" and "the set composed of 2,4,6,8" have countable elements, so these two sets are finite.
Infinite set: a set containing infinite elements, such as "the distance between two fixed points on a plane is equal to all points" and "all triangles". The elements that make up the above set are uncountable, so it is an infinite set.
In particular, we call a set that does not contain any elements an empty set, and we remember f wrong, such as {x? R|+ 1=0} .
5. Representation of a specific set
For the convenience of writing, we stipulate that commonly used data sets are represented by specific letters. Here are several common number sets, please keep them in mind.
(1) The set of all non-negative integers is usually called the set of non-negative integers (or the set of natural numbers), and is recorded as n.
(2) A set of zeros in a non-negative integer set, also known as a positive integer set, is denoted as N_ or N+.
(3) The set of all integers is usually referred to as integer set z for short.
(4) The set of all rational numbers is usually referred to as rational number set for short, and is denoted as Q. ..
(5) The set of all real numbers is usually referred to as the set of real numbers for short, and is recorded as R. ..
5. Five compulsory knowledge points in senior three mathematics.
1. The contents of set and function intersect and complement the set, and there is a power exponential pair function. Parity and increase and decrease are the most obvious observation images.
When the compound function appears, the law of property multiplication is distinguished. To prove it in detail, we must grasp the definition.
Exponential function and logarithmic function are reciprocal functions. Cardinality is not a positive number of 1, and 1 increases or decreases on both sides.
The domain of the function is easy to find. Denominator cannot be equal to 0, even roots must be non-negative, and zero and negative numbers have no logarithm;
The tangent function angle is not straight, and the cotangent function angle is uneven; The real number sets of other functions have intersection in many cases.
Two mutually inverse function have that same monotone property; The images are symmetrical with Y=X as the symmetry axis;
Solve the very regular inverse solution of substitution domain; The domain of inverse function, the domain of original function.
The nature of power function is easy to remember, and the index reduces the score; Keywords exponential function, odd mother and odd son odd function,
Even function with odd mother and even son, even mother non-parity function; In the first quadrant of the image, the function is increased or decreased to see the positive and negative.
2. Trigonometric function
Trigonometric functions are functions, and quadrant symbols are labeled. Function image unit circle, periodic parity increase and decrease.
The same angle relation is very important, and both simplification and proof are needed. At the vertex of the regular hexagon, cut the chord from top to bottom;
The numb 1 records that triangle connecte the vertices in the center; The sum of the squares of the downward triangle, the reciprocal relationship is diagonal,
Any function of a vertex is equal to the division of the last two. The inductive formula is good, negative is positive and then big and small,
It is easy to look up the table when it becomes a tax corner, and it is essential to simplify the proof. Half of the integer multiple of two, odd complementary pairs remain unchanged,
The latter is regarded as an acute angle, and the sign is judged as the original function. The cosine of the sum of two angles is converted into a single angle, which is convenient for evaluation.
Cosine product minus sine product, angular deformation formula. Sum and difference products must have the same name, and the complementary angle must be renamed.
The calculation proves that the angle is the first, pay attention to the name of the structural function, the basic quantity remains unchanged, and it changes from complexity to simplicity.
Guided by the principle of reverse order, the product of rising power and falling power and difference. The proof of conditional equality, the idea of equation points out the direction.
Universal formula is unusual, rational formula is ahead. The formula is used in the right and wrong direction, and the deformation is used skillfully;
1 add cosine to think of cosine, 1 subtract cosine to think of sine, power-on angle is halved, and power-on and power-off is a norm;
The inverse function of trigonometric function is essentially to find the angle, first to find the value of trigonometric function, and then to determine the range of angle value;
Using right triangle, the image is intuitive and easy to rename, and the equation of simple triangle is transformed into the simplest solution set;
3. Inequality
The solution to inequality is to use the properties of functions. The unreasonable inequality of the opposite side is transformed into a rational inequality.
From high order to low order, step-by-step transformation should be equivalent. The mutual transformation between numbers and shapes helps to solve problems.
The method of proving inequality is powerful in real number property. Difference is compared with 0, and quotient is compared with 1.
Comprehensive method with good direct difficulty analysis and clear thinking. Non-negative common basic expressions, positive difficulties are reduced to absurdity.
There are also important inequalities and mathematical induction. Graphic function help, drawing modeling construction method.
4. "series"
Arithmetic ratio two series, the sum of n terms in the general formula. Two finites seek the limit, and four operations are the other way around.
The problem of sequence is changeable, and the equation is simplified as a whole calculation. It is difficult to sum series, but it is skillful to eliminate dislocation and transform.
Learn from each other's strong points and calculate the sum formula of split terms. Inductive thinking is very good, just do a program to think about it:
Counting two and seeing three associations, guessing proves indispensable. There is also mathematical induction to prove that the steps are programmed:
First verify and then assume, from k to k plus 1, the reasoning process must be detailed and affirmed by the principle of induction.