Senior one mathematics knowledge point key daquan
The domain of (1) exponential function is the set of all real numbers, where a is greater than 0. If a is not greater than 0, there will be no continuous interval in the definition domain of the function, so we will not consider it.
(2) The range of exponential function is a set of real numbers greater than 0.
(3) The function graph is concave.
(4) If a is greater than 1, the exponential function increases monotonically; If a is less than 1 and greater than 0, it is monotonically decreasing.
(5) We can see an obvious law, that is, when a tends to infinity from 0 (of course, it can't be equal to 0), the curves of the functions tend to approach the positions of monotonic decreasing functions of the positive semi-axis of Y axis and the negative semi-axis of X axis respectively. The horizontal straight line y= 1 is the transition position from decreasing to increasing.
(6) Functions always infinitely tend to a certain direction on the X axis and never intersect.
(7) The function always passes (0, 1).
Obviously the exponential function is unbounded.
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definition
Generally, for the function f(x)
(1) If any x in the function definition domain has f(-x)=-f(x), then the function f(x) is called odd function.
(2) If any x in the function definition domain has f(-x)=f(x), the function f(x) is called an even function.
(3) If f(-x)=-f(x) and f(-x)=f(x) are true for any x in the function definition domain, then the function f(x) is both a odd function and an even function, which is called an even-even function.
(4) If f(-x)=-f(x) and f(-x)=f(x) cannot be established for any x in the function definition domain, then the function F (x) is neither a odd function nor an even function, which is called an even-even function.
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 X.
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, when a is different, the different situations of the domain of 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 greater than 0, the range of the 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.
Since x is greater than 0, it is meaningful to any value of a, so the following gives the respective situations of power function in the first quadrant.
You can see:
(1) All graphs pass (1, 1).
(2) When a is greater than 0, the power function monotonically increases, while when a is less than 0, the power function monotonically decreases.
(3) When a is greater than 1, the power function graph is concave; When a is less than 1 and greater than 0, the power function graph is convex.
(4) When a is less than 0, the smaller A is, the greater the inclination of the graph is.
(5)a is greater than 0, and the function passes (0,0); A is less than 0, and the function has only (0,0) points.
(6) Obviously the power function is unbounded.
Definition:
The angle between the positive direction of the X axis and the upward direction of the straight line is called the inclination angle of the straight line. In particular, when a straight line is parallel or coincident with the X axis, we specify that its inclination angle is 0 degrees.
Scope:
The range of inclination angle is 0 ≤ α.
Understand:
(1) Pay attention to "two directions": the direction of the straight line and the positive direction of the X axis;
(2) When a straight line is parallel to or coincident with the X axis, its inclination angle is 0 degrees.
Meaning:
① The inclination angle of the straight line reflects the inclination degree of the straight line to the positive direction of the X axis;
② In the plane rectangular coordinate system, every straight line has a certain inclination;
③ The same inclination does not necessarily represent the same straight line.
Formula:
k=tanα
α ∈ (0,90) at k & gt0°.
α ∈ (90, 180) at k<0.
When k=0, α = 0.
When α = 90, K does not exist.
Ax+by+c=0(a≠0) with an inclination of a,
TanA=-a/b，
A=arctan(-a/b)
When a≠0,
The tilt angle is 90 degrees, that is, perpendicular to the X axis.
People's education edition, a compulsory knowledge point of mathematics in senior one.
Structural characteristics of 1, column, cone, platform and ball
(1) prism:
Definition: Geometry surrounded by two parallel faces, the other faces are quadrangles, and the common edges of every two adjacent quadrangles are parallel to each other.
Classification: According to the number of sides of the bottom polygon, it can be divided into three prisms, four prisms and five prisms.
Representation: Use the letter of each vertex, such as a five-pointed star, or use the letter at the opposite end, such as a five-pointed star.
Geometric features: the two bottom surfaces are congruent polygons with parallel corresponding sides; The lateral surface and diagonal surface are parallelograms; The sides are parallel and equal; The section parallel to the bottom surface is a polygon that is congruent with the bottom surface.
② Pyramid
Definition: One face is a polygon, the other faces are triangles with a common vertex, and the geometric figure enclosed by these faces.
Classification: According to the number of sides of the bottom polygon, it can be divided into three pyramids, four pyramids and five pyramids.
Representation: Use the letters of each vertex, such as a pentagonal pyramid.
Geometric features: the side and diagonal faces are triangles; The section parallel to the bottom surface is similar to the bottom surface, and its similarity ratio is equal to the square of the ratio of the distance from the vertex to the section to the height.
(3) Prism:
Definition: Cut the part between the pyramid, the section and the bottom with a plane parallel to the bottom of the pyramid.
Classification: According to the number of sides of the bottom polygon, it can be divided into triangular, quadrangular and pentagonal shapes.
Representation: Use the letters of each vertex, such as a pentagonal pyramid.
Geometric features: ① The upper and lower bottom surfaces are similar parallel polygons; ② The side is trapezoidal; ③ The sides intersect with the vertices of the original pyramid.
(4) Cylinder:
Definition: Geometry surrounded by surfaces that rotate on one side of a rectangle and on the other three sides.
Geometric features: ① The bottom is an congruent circle; ② The bus is parallel to the shaft; ③ The axis is perpendicular to the radius of the bottom circle; ④ The side development diagram is a rectangle.
(5) Cone:
Definition: Geometry surrounded by the surface formed by the circle rotating with the right-angled side of the right-angled triangle as the rotation axis.
Geometric features: ① the bottom is round; (2) The generatrix intersects with the apex of the cone; ③ The side spread diagram is a fan.
(6) frustum of a cone:
Definition: Cut the part between the cone, the section and the bottom with a plane parallel to the bottom of the cone.
Geometric features: ① The upper and lower bottom surfaces are two circles; (2) The side generatrix intersects with the vertex of the original cone; (3) The side development diagram is an arch.
(7) Sphere:
Definition: Geometry formed by taking the straight line where the diameter of the semicircle is located as the rotation axis and the semicircle surface rotates once.
Geometric features: ① the cross section of the ball is round; ② The distance from any point on the sphere to the center of the sphere is equal to the radius.
2. Three views of space geometry
Define three views: front view (light is projected from the front of the geometry to the back); Side view (from left to right) and top view (from top to bottom)
Note: the front view reflects the position relationship of the object, that is, it reflects the height and length of the object;
The top view reflects the position relationship between the left and right of the object, that is, the length and width of the object;
The side view reflects the up-and-down and front-and-back positional relationship of the object, that is, it reflects the height and width of the object.
3. Intuition of space geometry-oblique two-dimensional drawing method.
Characteristics of oblique mapping;
(1) the original line segment parallel to the X axis is still parallel to X, with the same length;
② The line segment originally parallel to the Y axis is still parallel to Y, and its length is half of the original.
Summarize the knowledge points of senior one mathematics.
The Meaning and Expression of 1. Set
1, the meaning of set: a set is the sum of some different things, and people can realize these things and judge whether a given thing belongs to the whole.
The research objects are collectively called elements, and the whole composed of some elements is called set, which is called set for short.
2. Three characteristics of elements in a set:
(1) Determinism of elements: If the set is certain, then whether an element belongs to this set is certain: yes or no.
(2) Mutual dissimilarity of elements: the elements in a given set are affirmative and unrepeatable.
(3) The disorder of elements: the position of elements in the set can be changed, and changing the position does not affect the set.
3. Representation of sets: {...}
(1) indicates the set in capital letters: A={ basketball players in our school}, B={ 1, 2,3,4,5}.
(2) Representation of sets: enumeration and description.
Enumeration: Enumerate the elements in the set one by one {a, B, c...}
B, description method:
① Interval method: describe the common attributes of the elements in the set, and write them in braces to represent the set.
{x？ r | x—3 & gt； 2}，{ x | x—3 & gt； 2}
② Language description: Example: {A triangle that is not a right triangle}
③ venn diagram: Draw a closed curve, which represents the set.
4, the classification of the set:
(1) finite set: a set with finite elements.
(2) Infinite set: a set containing infinite elements.
(3) Empty set: a set without any elements.
5, the relationship between elements and sets:
(1) If an element is in a set, it belongs to the set, that is, a? A
(2) If the element is not in the set, it does not belong to the set, that is, a ￠ a
Note: Commonly used digit sets and their symbols:
The set of nonnegative integers (i.e. natural number set) is recorded as n.
Positive integer set N- or N+
Integer set z
Rational number set q
Real number set r
6, the basic relationship between sets
( 1)。 "Inclusion" relation (1)- subset
Definition: If any element of set A is an element of set B, we say that these two sets have an inclusion relationship, and set A is a subset of set B. ..
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