The plane of the coordinate system is called the coordinate plane Axis The common origin of is called the origin of rectangular coordinate system. The X and Y axes divide the coordinate plane into four quadrant The one on the top right is called first quadrant The other three parts are called Beta Quadrant 、 third quadrant and Delta Quadrant 。 The quadrant is bounded by the number axis, and the points on the horizontal and vertical axes do not belong to any quadrant. stay Rectangular coordinate system Can be drawn according to the point coordinates Inverse proportional function 、 Positive proportional function 、 Linear function 、 Quadratic function And so on. It is said that one day, French philosopher and mathematician Descartes He was in bed when he was ill, and his illness was very serious. However, he still pondered over a question: geometry is intuitive, and Algebra equation It is relatively abstract. Can we combine geometric figures with algebraic equations, that is, can we use geometric figures to express equations? To achieve this goal, the key is how to hook the points that make up the geometric figure with each group of "numbers" that meet the equation. He pondered hard and tried to figure out how to link "points" and "numbers". Suddenly, he saw a spider on the corner of the roof, pulling the silk down. After a while, the spider climbed up along the silk again, pulling the silk from side to side. The "performance" of spiders made Descartes' thinking suddenly clear. He thought that the spider could be regarded as a point. It could move up, down, left and right in the room. Could you use one for each position of the spider Number of groups Are you sure? He also thought that the two adjacent walls in the room intersected three lines with the ground. If the corner of the wall on the ground is taken as the starting point and the three lines intersected are taken as three number axes, then the position of any point in the space can use these three number axes to find three numbers in order. Conversely, given a group of three ordered numbers, you can also find a point P corresponding to it in space. In the same way, a group of numbers (x, y) can represent a point on the plane, and a point on the plane can also be represented by a group of two ordered numbers, which is the prototype of the coordinate system. The creation of the rectangular coordinate system has built a bridge in algebra and geometry. It enables geometric concepts to be expressed in numbers, and geometric figures can also be expressed in algebraic form. Thus, on the basis of creating a rectangular coordinate system, Descartes created a Algebra To study Geometry Analytic geometry, a branch of mathematics of, he boldly envisioned that if geometric figures were regarded as the trajectory of moving points, they could be regarded as consisting of points with some common characteristics. For example, we can regard a circle as having the same distance from a moving point to a fixed point Track of points If we regard the point as the basic element of the geometric figure and the number as the composition Solution of the equation So algebra and geometry become a family. Two number axes
① They are perpendicular to each other; ② the origin coincides with each other; ③ the orientation is generally right and upward; ④ the unit length is the same. [1]
Rectangular coordinate system
Draw two number axes perpendicular to each other and coincident with the origin in the plane to form a rectangular coordinate system. The horizontal number axis is called the x-axis or horizontal axis, and the right direction is usually the positive direction; The vertical number axis is the y-axis or vertical axis, and the upward direction of orientation is the positive direction; The intersection of the two coordinate axes is the origin of the plane rectangular coordinate system. [1]
Coordinates of points
We use a pair of ordinal pairs to represent points on the plane, and the logarithms are called coordinates. The representation method is (a, b), where a is the value on the horizontal axis corresponding to the point, and b is the value on the vertical axis corresponding to the point. [1]
After the establishment of the plane rectangular coordinate system, the plane is divided into four parts by the coordinate axis, namely the first quadrant, the second quadrant, the third quadrant and the fourth quadrant. [1] (The area of the positive half axis of the two axes is the first quadrant, and the quadrants are arranged in counterclockwise order)
For the quadratic equation with one variable, when K>0, the two branches are located in the first quadrant and the third quadrant respectively, and Y decreases with the increase of X in each quadrant; When K<0, the two branches are located in the second quadrant and the fourth quadrant respectively. In each quadrant, Y increases with the increase of X.
When the absolute value of X increases infinitely or approaches zero, the two branches of the inverse ratio are infinitely close to the X axis and Y axis, but never intersect with the X axis and Y axis.
And the famous Cardioid 。 [2] Cardioid line is the track formed by a fixed point on a circle when it rolls around another circumference that is tangent to it and has the same radius. It is named for its shape like a heart shape. [3]
Cardiac line is also Clam thread One of. stay Mandelbrot set The figure in the middle is a heart line. The English name of cardioid was published by de Castillon in 1741 in Philosophical Transactions of the Royal Society; It means "like the heart". In the plane rectangular coordinate system xOy, let the beginning of ∠β be the positive half axis of the x axis, and let P (x, y) be any point on the end of ∠β that does not coincide with the origin O, let r=OP, and let ∠β=∠α, then: [4]
sin r=y/r
cos r=x/r
tan r=y/x
cot r=x/y
csc r=r/y
sec r=r/x
1. S=(1/2) * bottom * height
2. Helen formula: √ [p (p-a) (p-b) (p-c)] where p=1/2 (a+b+c), s=1/2 perimeter * inscribed circle radius
3、 s=1/2absinC,s=1/2acsinB ,s=1/2bcsinA [5]
It is applicable to the area whose integrand Ω does not contain a circle, and pay attention to the transformation And integral upper and lower limits (1) The first two projection method is used to calculate a vertical bar integral in the vertical direction, and then calculate the bottom surface integral.
① Regional conditions: there is no limit to the integral region Ω;
② Function condition: no limit on f (x, y, z).
(2) Two before one method (section method): first calculate the bottom integral, then calculate the vertical integral.
① Regional conditions: the integral region Ω is enclosed by a plane or other surface (excluding cylinder, cone and sphere);
② Function condition: f (x, y,) is a function of only one variable. [6]
