Compulsory knowledge points of high school mathematics 1
There are only three positional relationships between two straight lines in space: parallel, intersecting and nonplanar.
1, according to whether * * * surface can be divided into two categories:
(1)*** plane: parallel intersection.
(2) Different planes:
Definition of non-planar straight lines: two different straight lines on any plane are neither parallel nor intersecting.
Judgment theorem of out-of-plane straight line: use the straight line between a point in the plane and a point out of the plane, and the straight line in the plane that does not pass through this point is the out-of-plane straight line.
The angle formed by two straight lines on different planes: the range is (0,90) esp. Space vector method
Distance between two straight lines in different planes: common vertical line segment (only one) esp. Space vector method
2, if from the perspective of the existence of public * * *, points can be divided into two categories:
(1) has only one thing in common-intersecting straight lines; (2) There is nothing in common-parallel or non-parallel.
The positional relationship between a straight line and a plane:
There are only three positional relationships between a straight line and a plane: within the plane, intersecting the plane and parallel to the plane.
(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.
Space vector method (finding the normal vector of a plane)
Provisions: a, when the straight line is perpendicular to the plane, the angle formed is a right angle; B, when the line is parallel or in the plane, 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.
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.
High school mathematics compulsory two knowledge points 2
one
The zero point of the 1. function
(1) Definition:
For the function y=f(x)(x∈D), let the real number x with f(x)=0 be called the zero point of the function y=f(x)(x∈D).
(2) the relationship between the zero point of the function and the root of the corresponding equation, and the intersection of the image of the function and the X axis:
Does the equation f(x)=0 have a real root? Does the image of the function y=f(x) intersect with the x axis? The function y=f(x) has zero.
(3) Determination of zero point of function (zero point existence theorem):
If the image of the function y=f(x) on the interval [a, b] is a continuous curve with f (a) f (b).
2. quadratic function y = ax2+bx+c (a >; 0) The relationship between the image and the zero point.
Step 3 split
For continuous intervals [a, b] and f (a) f (b)
4. The zero point of the function is not a point:
The zero point of the function y=f(x) is the real root of the equation f(x)=0, that is, the abscissa of the intersection of the image of the function y=f(x) and the x axis, so the zero point of the function is a number, not a point. When writing function zero, it must be a number, not a coordinate.
5. When judging the existence of functional zero, it must be emphasized that:
(1)f(x) is continuous on [a, b];
(2)f(a)f(b )& lt； 0;
(3) Zero exists in (a, b).
This is a sufficient condition for the existence of zero, but it is not a necessary condition.
6. For continuous functions in the definition domain, all function values between two adjacent zeros keep the same sign.
two
Related concepts of 1. geometric series
(1) Definition:
A series is called a geometric series if the ratio of each term to its previous term is equal to the same constant (non-zero) of the second term. This constant is called the common ratio of geometric series, and is usually expressed by the letter Q. The defined expression is an+ 1/an=q(n∈N_, q is a non-zero constant).
(2) Proportion:
If a, g and b are in geometric series, then g is called the proportional mean of a and b, that is, g is the proportional mean of a and b? A, g and b do geometric series? G2=ab。
2. Relevant formulas of geometric series.
(1) general formula: an=a 1qn- 1.
3. Common properties of geometric series {an}
(1) in the geometric series {an}, if m+n=p+q=2r(m, n, p, q, r∈N_), then am an = apaq = a.
Especially a1an = a2an-1= a3an-2 =.
(2) In the geometric series {an} and q, the sequences am, am+k, am+2k, am+3k, … are still geometric series, and the sequences Q: Sm, S2m-Sm, S3m-S2m, … are still geometric series (at this time Q ≦-1); an=amqn-m。
4. The characteristics of geometric series
(1) According to the definition of geometric series, any term of geometric series is non-zero and the common ratio q is also non-zero.
(2) If an+ 1=qan, q≠0 cannot immediately assert that {an} is a geometric series, we need to verify a 1≠0.
5. the first n terms of geometric series and Sn.
The first n terms and Sn of (1) geometric series are obtained by dislocation subtraction. Pay attention to the application of this thinking method in the summation of series.
(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.
Senior high school mathematics compulsory two knowledge points 3
1, prism
Definition of prism: two faces are parallel to each other, the other face is a quadrilateral, and the common sides of every two quadrilaterals are parallel to each other. The geometric shape enclosed by these faces is called a prism.
Properties of prism
(1) All sides are equal, and the sides are parallelogram.
(2) The sections parallel to the two bottom surfaces are congruent polygons.
(3) The cross section (diagonal plane) passing through two non-adjacent sides is a parallelogram.
2. Pyramid
Definition of Pyramid: One face is a polygon and the other faces are triangles with a common vertex. The geometry surrounded by these faces is called a pyramid.
The essence of the pyramid:
The sides of (1) intersect at one point. The sides are triangular.
(2) The section parallel to the bottom surface is a polygon similar to the bottom surface. And its area ratio is equal to the square of the ratio of the height of the truncated pyramid to the height of the far pyramid.
3. Right Pyramid
Definition of a regular pyramid: If the bottom of the pyramid is a regular polygon and the projection of the vertex at the bottom is the center of the bottom, such a pyramid is called a regular pyramid.
The nature of the regular pyramid:
(1) An isosceles triangle whose sides intersect at one point and are equal. The height on the base of each isosceles triangle is equal, which is called the oblique height of a regular pyramid.
(3) Some special right-angled triangles
A For a regular triangular pyramid with two adjacent sides perpendicular to each other, the projection of the vertex on the bottom surface can be obtained as the vertical center of the triangle on the bottom surface by the three perpendicular theorems.
B there are three pairs of straight lines with different planes in the tetrahedron. If two pairs are perpendicular to each other, the third pair is perpendicular. And the projection of the vertex on the bottom surface is the vertical center of the triangle on the bottom surface.
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