Analytic expression of function
(1). The analytic formula of the function is a representation of the function. When the functional relationship between two variables is needed, the corresponding law between them and the definition domain of the function are needed.
(2) The main methods for finding analytic expressions of functions are:
1) matching method
2) undetermined coefficient method
3) Alternative methods
4) Parameter elimination method
Function (small) value (see p36 for definition)
1 Use the property of quadratic function (collocation method) to find the (small) value of the function.
2 Use the image to find the (small) value of the function
3 use the monotonicity of the function to judge the (small) value of the function:
If the function y=f(x) monotonically increases in the interval [a, b] and monotonically decreases in the interval [b, c], then the function y=f(x) has a value f (b) at x=b;
If the function y=f(x) monotonically decreases in the interval [a, b] and monotonically increases in the interval [b, c], then the function y=f(x) has a minimum value f (b) at x=b;
2. Senior three, the first volume of mathematics requires a knowledge point induction.
Surface area volume formula of space geometry;
1, cylinder: surface area: 2πRr+2πRh volume: πR2h(R is the radius of the upper and lower bottom circles of the cylinder, and h is the height of the cylinder).
2. Cone: surface area: πR2+πR[(h2+R2)] Volume: πR2h/3(r is the radius of the low circle of the cone, and H is its height.
3, A side length, S=6a2, V=a3.
4. Cuboid A- length, B- width, C- height S=2(ab+ac+bc)V=abc.
5, prism S-h- height v = s-h-
6, pyramid S-h- height v = s-h-/3
7.S 1 and S2-upper and lower h-height v = h [s1+S2+(s1S2)1/2]/3.
8. s1-upper bottom area, S2-lower bottom area, s0-middle h-high, V=h(S 1+S2+4S0)/6.
9, cylinder R- base radius, H- height, C- base perimeter S- base area, S- side, S- surface area C=2πrS base =πr2, S- side =Ch, S- table =Ch+2S base, V = S- base h=πr2h.
10, hollow cylinder r- outer circle radius, r- inner circle radius h- height v = π h (r 2-r 2)
1 1, r- bottom radius h- height v = π r 2h/3.
12, R- upper bottom radius, R- lower bottom radius, H- height V=πh(R2+Rr+r2)/3 13, ball R- radius d- diameter v = 4/3 π r 3 = π d 3/6.
14, ball gap H- ball gap height, R- ball radius, A- ball gap bottom radius V = π h (3A2+H2)/6 = π h2 (3R-H)/3.
15, table r 1 and R2- radius h- height V=πh[3(r 12+r22)+h2]/6.
16, ring R- ring radius D- ring diameter R- ring section radius D- ring section diameter V = 2π 2RR2 = π 2d2/4.
17, barrel D- barrel belly diameter D- barrel bottom diameter H- barrel height V=πh(2D2+d2)/ 12, (the bus is round with the center of the barrel) v = π h (2d2+DD+3d2/4)/1.
3. The first volume of senior three mathematics needs a knowledge point induction.
The positional relationship between two planes:
(1) The definition that two planes are parallel to each other: there is no common point between two planes in space.
(2) the positional relationship between two planes:
The two planes are parallel-have nothing in common; Two planes intersect-there is a straight line.
First, parallel
Theorem for determining the parallelism of two planes: If two intersecting lines in one plane are parallel to the other plane, then the two planes are parallel.
Parallel theorem of two planes: if two parallel planes intersect with the third plane at the same time, the intersection lines are parallel.
B, crossroads
dihedral angle
(1) Half-plane: A straight line in a plane divides this plane into two parts, and each part is called a half-plane.
(2) dihedral angle: The figure composed of two half planes starting from a straight line is called dihedral angle. The range of dihedral angle is [0, 180].
(3) The edge of dihedral angle: This straight line is called the edge of dihedral angle.
(4) Dihedral facet: These two half planes are called dihedral facets.
(5) Plane angle of dihedral angle: Take any point on the edge of dihedral angle as the endpoint, and make two rays perpendicular to the edge in two planes respectively. The angle formed by these two rays is called the plane angle of dihedral angle.
(6) Straight dihedral angle: A dihedral angle whose plane angle is a right angle is called a straight dihedral angle.
Esp。 The two planes are perpendicular.
Definition of two planes perpendicular: two planes intersect, and if the angle formed is a straight dihedral angle, the two planes are said to be perpendicular to each other. Write it down as X.
A theorem to determine the perpendicularity of two planes: If one plane passes through the perpendicular of the other plane, then the two planes are perpendicular to each other.
Verticality theorem of two planes: If two planes are perpendicular to each other, a straight line perpendicular to the intersection in one plane is perpendicular to the other plane.
note:
Solution of dihedral angle: direct method (making plane angle), triple vertical theorem and inverse theorem, area projection theorem, normal vector method of space vector (pay attention to the complementary relationship between the obtained angle and the required angle) polyhedron.
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.
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.
Positive 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
esp:
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.
4. The first volume of senior three mathematics needs a knowledge point induction.
Images and properties of linear functions;
1. Practice and graphics: Through the following three steps.
(1) list;
(2) tracking points;
(3) The connection can be the image of a function-a straight line. So the image of a function only needs to know two points and connect them into a straight line. (Usually find the intersection of the function image with the X and Y axes)
2. Nature:
Any point P(x, y) on the (1) linear function satisfies the equation: y = kx+b.
(2) The coordinate of the intersection of the linear function and the Y axis is always (0, b), and the image of the proportional function always intersects the origin of the X axis at (-b/k, 0).
3. Quadrant where K, B and function images are located:
When k>0, the straight line must pass through the first and third quadrants, and Y increases with the increase of X;
When k < 0, the straight line must pass through the second and fourth quadrants, and y decreases with the increase of x.
When b>0, the straight line must pass through the first and second quadrants;
When b=0, the straight line passes through the origin.
When b<0, the straight line must pass through three or four quadrants.
Especially, when b=O, the straight line passing through the origin o (0 0,0) represents the image of the proportional function.
At this time, when k>0, the straight line only passes through the first and third quadrants; When k < 0, the straight line only passes through the second and fourth quadrants.
5. Senior three, the first volume of mathematics requires a knowledge point induction.
Parity of 1. function
(1) If f(x) is an even function, then f (x) = f (-x);
(2) If f(x) is odd function and 0 is in its domain, then f(0)=0 (which can be used to find parameters);
(3) The parity of the judgment function can be defined in equivalent form: f (x) f (-x) = 0 or (f (x) ≠ 0);
(4) If the analytic formula of a given function is complex, it should be simplified first, and then its parity should be judged;
(5) odd function has the same monotonicity in the symmetric monotone interval; Even functions have opposite monotonicity in symmetric monotone interval;
2. Some questions about compound function.
Solution of the domain of (1) composite function: If the domain is known as [a, b], the domain of the composite function f[g(x)] can be solved by the inequality a≤g(x)≤b; If the domain of f[g(x)] is known as [a, b], find the domain of f(x), which is equivalent to x∈[a, b], and find the domain of g(x) (that is, the domain of f(x)); When learning functions, we must pay attention to the principle of domain priority.
(2) The monotonicity of the composite function is determined by "the same increase but different decrease";
3. Function image (or symmetry of equation curve)
(1) Prove the symmetry of the function image, that is, prove that the symmetry point of any point on the image about the symmetry center (symmetry axis) is still on the image;
(2) Prove the symmetry of the image C 1 and C2, that is, prove that the symmetry point of any point on C 1 about the symmetry center (symmetry axis) is still on C2, and vice versa;
(3) curve C 1: f (x, y) = 0, and the equation of symmetry curve C2 about y=x+a(y=-x+a) is f(y-a, x+a)=0 (or f(-y+a,-x+a) =
(4) Curve C 1:f(x, y)=0 The C2 equation of the symmetrical curve about point (a, b) is: f(2a-x, 2b-y) = 0;
(5) If the function y=f(x) is constant to x∈R, and f(a+x)=f(a-x), then the image y=f(x) is symmetrical about the straight line x=a;
(6) The images of functions y=f(x-a) and y=f(b-x) are symmetrical about the straight line x=;
4. The periodicity of the function
(1)y=f(x) for x∈R, f(x+a)=f(x-a) or f (x-2a) = f (x) (a >: 0) is a constant, then y=f(x) is a period of 2a.
(2) If y=f(x) is an even function, and its image is symmetrical about the straight line x=a, then f(x) is a periodic function with a period of 2 ~ a;
(3) If y=f(x) odd function, and its image is symmetrical about the straight line x=a, then f(x) is a periodic function with a period of 4 ~ a;
(4) If y=f(x) is symmetric about points (a, 0) and (b, 0), then f(x) is a periodic function with a period of 2;
(5) If the image of y=f(x) is symmetrical (a ≠ b) about straight lines x = a and x = b, then the function y = f (x) is a periodic function with a period of 2;
(6) When y=f(x) equals x∈R, f(x+a)=-f(x) (or f(x+a)=, then y = f (x) is a periodic function with a period of 2;
5. Equation
(1) equation k=f(x) has a solution k∈D(D is the range of f(x));
(2)a≥f(x) considers A ≥ [f (x)] max;
A≤f(x) considers a ≤ [f (x)] min;
(3)(a & gt； 0，a≠ 1，b & gt0，n∈R+)；
logaN =(a & gt； 0，a≠ 1，b & gt0，b≠ 1)；
(4)logab symbols are memorized by the formula of "same positive but different negative";
a Logan = N(a & gt； 0，a≠ 1，N & gt0);
Step 6 draw pictures
When judging whether the corresponding relationship is a mapping, we should grasp two points:
The elements in (1)A must all have images and;
(2) All elements in B may not have original images, and different elements in A may have the same images in B;
7. Monotonicity of functions
(1) can skillfully use definitions to prove monotonicity of functions, find inverse functions and judge parity of functions;
(2) According to monotonicity, the problem of finding the range of a class of parameters can be solved by using the sign-preserving property of linear functions on intervals.