(1) The straight line is in the plane-there are countless things in common.
(2) A straight line intersects a plane-there is only one common point.
Angle between a straight line and a plane: the acute angle formed by the diagonal of a plane and its projection on the plane.
Esp。 Space vector method (finding the normal vector of a plane)
Terms:
When a straight line is perpendicular to the plane, the angle formed is a right angle.
B, the straight line is parallel to the plane or in the plane, and the angle is 0.
The included angle between the straight line and the plane is [0,90].
Minimum angle theorem: the angle formed by the diagonal line and the plane is the smallest angle between the diagonal line and any straight line in the plane.
Three Verticality Theorems and Inverse Theorems: If a straight line in a plane is perpendicular to the projection of a diagonal line in this plane, it is also perpendicular to this diagonal line.
Esp。 This line is perpendicular to the plane.
Definition of vertical line and plane: If straight line A is perpendicular to any straight line in the plane, we say that straight line A and plane are perpendicular to each other. The straight line A is called the perpendicular of the plane, and the plane is called the vertical plane of the straight line A. ..
Theorem for judging whether a straight line is perpendicular to a plane: If a straight line is perpendicular to two intersecting straight lines in a plane, then the straight line is perpendicular to the plane.
Theorem of the property that straight lines are perpendicular to a plane: If two straight lines are perpendicular to a plane, then the two straight lines are parallel.
③ The straight line is parallel to the plane-there is nothing in common.
Definition of parallelism between straight line and plane: If straight line and plane have nothing in common, then we say that straight line and plane are parallel.
Theorem for determining the parallelism between a straight line and a plane: If a straight line out of the plane is parallel to a straight line in this plane, then this straight line is parallel to this plane.
Theorem of parallelism between straight lines and planes: If a straight line is parallel to a plane and the plane passing through it intersects with this plane, then the straight line is parallel to the intersection line.
Conceptual knowledge points of mathematical functions
1. Constants and variables: in a certain change process, quantities that can take different values are called variables; The quantity whose value remains constant during a certain change is called a constant.
2. Function: When two variables X and Y are changing in a certain process, if Y has a unique fixed value corresponding to each fixed value of X in a certain range, Y is called a function of X, where X is the independent variable and Y is the dependent variable.
Determination of the value range of (1) independent variable
(1) the range of independent variables of algebraic functions are all real numbers.
② The range of the independent variable of the fractional function is a real number that makes the denominator not 0.
③ The independent variable of quadratic root function is a real number with non-negative root number. If the function involves practical problems, it should not only meet the above requirements, but also make the practical problems meaningful.
Knowledge points of mathematical sequence
The relationship between the general term of 1. series, the number of terms of series, recursive formula and recursive series, and the general term of series and the summation formula of the previous stage of series.
2. Arithmetic Series Middle School
(1) The value of arithmetic progression tolerance and the monotonicity of arithmetic progression.
(2) arithmetic progression.
(3) The new series composed of the sum (difference) of two arithmetic progression still becomes arithmetic progression.
(4) Still become arithmetic progression.
(5) When Shouzheng is delivered to arithmetic progression, the maximum sum in the preceding paragraph is the sum of all non-negative items; In the "first negative" increasing arithmetic progression, the minimum sum in the previous section is the sum of all non-positive terms;
(6) In finite arithmetic progression, the sum of odd terms must be related to the sum of even terms, which is determined by whether the total number of terms in a series is even or odd. If the total number of items is even, the sum of even items = the product of half of the total number of items and its tolerance; If the total number of terms is odd, the sum of odd terms and-even terms = the median term of this series.
(7) The median term of the arithmetic difference between two numbers only exists. When three numbers or four numbers become arithmetic progression, we often consider the "middle term relation" for transformation.
(8) The main methods to judge whether a series is arithmetic progression are: definition method, middle term method, general term method, summation method and mirror image method (that is to say, the necessary and sufficient conditions for a series to be arithmetic progression mainly include these five forms).
3. In geometric series:
The symbolic characteristics of (1) geometric series (all positive or all negative or one positive and one negative), the first term of geometric series, the common ratio, and the monotonicity of geometric series.
(2) The new series of product (quotient) of two geometric series counterparts still becomes geometric series.
(3) In the "first decreasing geometric series greater than 1", the maximum value of the previous product is the product of all terms greater than or equal to 1; In the "first less than 1" positive increasing geometric series, the minimum value of the previous product is the product of all terms less than or equal to 1;
(4) In a finite geometric series, the sum of odd terms must be related to the sum of even terms, which is determined by whether the total number of terms in a series is even or odd. If the total number of terms is even, the sum of even terms = the product of the sum of odd terms and the common ratio; If the total number of terms is odd, the sum of the products of odd terms and the first term plus the sum of common ratio and even terms.
(5) Not any two numbers always have equal ratio mean terms. Only when real numbers have the same sign, real numbers have equal ratio median terms. The equal ratio median term of two real numbers with the same sign not only exists, but also has a pair. That is to say, two real numbers either don't have an equal ratio median term (when they have different signs), and if they do, they must have a pair (when they have the same sign). When three or four numbers become arithmetic progression, the "middle term" is always given priority.
(6) There are four main methods to determine whether a sequence is a geometric series: definition method, median method, general method and summation method (that is, there are four necessary and sufficient conditions for a sequence to be a geometric series).
4. Links between arithmetic progression and geometric progression
(1) If the series becomes arithmetic progression, then the series (always meaningful) must become geometric progression.
(2) If a series becomes a geometric series, then the series must become an arithmetic series.
(3) If the series becomes arithmetic progression and geometric progression, then the series is a non-zero constant series; But it is only a necessary and sufficient condition that the sequence is a constant sequence, which makes it arithmetic progression and geometric progression.
(4) If two arithmetic progression have common terms, then the new series composed of their common terms is also arithmetic progression, and the tolerance of the new arithmetic progression is the least common multiple of the tolerance of the original two arithmetic progression.
If a arithmetic progression and a geometric progression have common items to form a new series in turn, then the discussion is often conducted by the method of "from special to general", focusing on geometric progression items, to explore which items in geometric progression are their common items to form a new series.
5. The common methods of sequence summation:
(1) formula method: ① sum formula of arithmetic sequence (three forms),
(2) Sum formula of equal proportion series (three forms),
(2) Group summation method: When it is difficult to sum directly by formula method, the "similar items" in the "summation formula" are often combined first and then summed by formula method.
(3) Reverse addition: In the summation of series, if the sum of two terms with the same distance from the beginning to the end in the summation formula has its * * property or the general term of the series is related to the number of combinations, it is often considered to choose reverse addition to give full play to its * * * property (this is also the derivation method of the summation formula of arithmetic series).
(4) Dislocation subtraction: If the general term of a series is formed by multiplying the general term of arithmetic progression by the general term of geometric progression, then the sum is often converted into "the sum of a new geometric progression" by dislocation subtraction (Note: after general dislocation subtraction, the number of terms in the new geometric progression is the difference between the number of terms in the original series minus one! This is also one of the derivation methods of geometric series summation formula.
(5) Split-term elimination method: If the general term of a series can be split into the form of the difference between two terms, and the adjacent terms after splitting are related, the split-term elimination method is often used to sum.
(6) Common terminology conversion method.