[从头学数学] 第204节 不等式选讲

剧情提要：
[机器小伟]在[工程师阿伟]的陪同下进入了[九转金丹]之第七转的修炼。
这次要研究的是[不等式选讲]。

正剧开始：

星历2016年05月07日 12:11:48, 银河系厄尔斯星球中华帝国江南行省。
[工程师阿伟]正在和[机器小伟]一起研究[不等式选讲]。

<span style="font-size:18px;">	if (1) {    
        var r = 20;          
        config.setSector(1,1,1,1);            
        config.graphPaper2D(0, 0, r);          
        config.axis2D(0, 0,180);            
              
        //坐标轴设定      
        var scaleX = 2*r, scaleY = 2*r;        
        var spaceX = 2, spaceY = 2;         
        var xS = -10, xE = 10;        
        var yS = -8, yE = 8;        
        config.axisSpacing(xS, xE, spaceX, scaleX, 'X');          
        config.axisSpacing(yS, yE, spaceY, scaleY, 'Y');          
                  
        var transform = new Transform();          
        //存放函数图像上的点      
        var a = [], b = [], c = [], d = [];        
                
        //需要显示的函数说明      
        var f1 = 'y=(x+3)*(x+7)-(x+4)*(x+6)', 
			f2 = 'y=', f3 = '', f4 = '';      
          

        //函数描点      
        for (var x = xS; x <= xE; x+=1) {        
                a.push([x, taskFun(x)]);    
        }        
                
        //存放临时数组      
        var tmp = [];      
                
        //显示变换      
        if (a.length > 0) {      
            a = transform.scale(transform.translate(a, 0, 0), scaleX/spaceX, scaleY/spaceY);       
            //函数1      
            tmp = [].concat(a);          
            shape.pointDraw(tmp, 'red');          
            tmp = [].concat(a);          
            shape.multiLineDraw(tmp, 'pink');        
                    
            plot.setFillStyle('red');      
            plot.fillText(f1, -270, -90, 400);        
        }

		if (b.length > 0) {    
            b = transform.scale(transform.translate(b, 0, 0), scaleX/spaceX, scaleY/spaceY);                
            //函数2    
           // tmp = [].concat(b);        
           // shape.pointDraw(tmp, 'blue');        
            tmp = [].concat(b);        
            shape.multiLineDraw(tmp, '#22CCFF');     
                
            plot.setFillStyle('blue');    
            plot.fillText(f2, -270, -120, 400);     
    
        } 
    }  
}  

function taskFun(x) {
	return (x+3)*(x+7)-(x+4)*(x+6);
}</span>

<span style="font-size:18px;">	if (1) {    
        var r = 20;          
        config.setSector(1,1,1,1);            
        config.graphPaper2D(0, 0, r);          
        config.axis2D(0, 0,180);            
              
        //坐标轴设定      
        var scaleX = 2*r, scaleY = 2*r;        
        var spaceX = 2, spaceY = 2;         
        var xS = -10, xE = 10;        
        var yS = -8, yE = 8;        
        config.axisSpacing(xS, xE, spaceX, scaleX, 'X');          
        config.axisSpacing(yS, yE, spaceY, scaleY, 'Y');          
                  
        var transform = new Transform();          
        //存放函数图像上的点      
        var a = [], b = [], c = [], d = [];        
                
        //需要显示的函数说明      
        var f1 = 'y=0.38+0.04*x*x+4/(x*x)', 
			f2 = 'y=', f3 = '', f4 = '';      
          
		var y = 0;
		
        //函数描点      
        for (var x = 0.1; x <= xE; x+=0.1) {   
				y = taskFun(x);
                a.push([x, y]); 
				d.push([x, y]);
				
        }        
         
		

        //存放临时数组      
        var tmp = [];      
           
		d.sort(function(p1, p2) {
				return (p1[1]-p2[1]);
		});
		
		var x0 = d[0][0], y0 = d[0][1];
		
		
		
        //显示变换      
        if (a.length > 0) {      
            a = transform.scale(transform.translate(a, 0, 0), scaleX/spaceX, scaleY/spaceY);
			
            //函数1      
            tmp = [].concat(a);          
            shape.pointDraw(tmp, 'red');          
            tmp = [].concat(a);          
            shape.multiLineDraw(tmp, 'pink');        
                    
            plot.setFillStyle('red');      
            plot.fillText(f1, -270, -90, 400);        
        }
		

		b.push([x0, y0]);
		
		if (b.length > 0) {    
            b = transform.scale(transform.translate(b, 0, 0), scaleX/spaceX, scaleY/spaceY);                
			//最值点
			tmp = [].concat(b);        
            shape.pointDraw(tmp, 'blue');        
            //tmp = [].concat(b);        
            //shape.multiLineDraw(tmp, '#22CCFF');     
                
            plot.setFillStyle('blue');    
            plot.fillText(x0.toFixed(3)+', '+y0.toFixed(3), b[0][0], -b[0][1]-20, 100);  
    
        } 
    }  
}  

function taskFun(x) {
	return 0.38+0.04*x*x+4/(x*x);
}</span>

<span style="font-size:18px;">	if (1) {    
        var r = 20;          
        config.setSector(1,5,1,1);            
        config.graphPaper2D(0, 0, r);          
        config.axis2D(0, 0,180, 3);            
              
        //坐标轴设定      
        var scaleX = 2*r, scaleY = 2*r;        
        var spaceX = 0.05, spaceY = 0.05;         
        var xS = -10, xE = 10;        
        var yS = -8, yE = 8;        
        config.axisSpacing(xS, xE, spaceX, scaleX, 'X');          
        config.axisSpacing(yS, yE, spaceY, scaleY, 'Y');          
                  
        var transform = new Transform();          
        //存放函数图像上的点      
        var a = [], b = [], c = [], d = [];        
                
        //需要显示的函数说明      
        var f1 = 'y=(1-2*x)*(1-2*x)*x', 
			f2 = 'y=', f3 = '', f4 = '';      
          
		var y = 0;
		
        //函数描点      
        for (var x = 0; x <= 0.5; x+=0.01) {   
				y = taskFun(x);
                a.push([x, y]); 
				d.push([x, y]);
				
				
        }            
		

        //存放临时数组      
        var tmp = [];      
		
        //显示变换      
        if (a.length > 0) {      
            a = transform.scale(transform.translate(a, 0, 0), scaleX/spaceX, scaleY/spaceY);
			
            //函数1      
            tmp = [].concat(a);          
            shape.pointDraw(tmp, 'red');          
            tmp = [].concat(a);          
            shape.multiLineDraw(tmp, 'pink');        
                    
            plot.setFillStyle('red');      
            plot.fillText(f1, 30, -100, 400);        
        }
		
		
		
		d.sort(function(p1, p2) {
				//求最大值加负号，最小值不加
				return -(p1[1]-p2[1]);
		});
		
		var x0 = d[0][0], y0 = d[0][1];
		
		
		b.push([x0, y0]);
		
		if (b.length > 0) {    
            b = transform.scale(transform.translate(b, 0, 0), scaleX/spaceX, scaleY/spaceY);                
			//最值点
			tmp = [].concat(b);        
            shape.pointDraw(tmp, 'blue');        
            //tmp = [].concat(b);        
            //shape.multiLineDraw(tmp, '#22CCFF');     
                
            plot.setFillStyle('blue');    
            plot.fillText(x0.toFixed(3)+', '+y0.toFixed(3), b[0][0], -b[0][1]-20, 100);  
    
        } 
    }  
}  

function taskFun(x) {
	//var a = x, b = a+1, c = b+1;
	return (1-2*x)*(1-2*x)*x;
}

>>> 2/27
0.07407407407407407</span>

<span style="font-size:18px;">	if (1) {    
        var r = 20;          
        config.setSector(1,1,1,1);            
        config.graphPaper2D(0, 0, r);          
        config.axis2D(0, 0,180);            
              
        //坐标轴设定      
        var scaleX = 2*r, scaleY = 2*r;        
        var spaceX = 1, spaceY = 2;         
        var xS = -100, xE = 100;        
        var yS = -100, yE = 100;        
        config.axisSpacing(xS, xE, spaceX, scaleX, 'X');          
        config.axisSpacing(yS, yE, spaceY, scaleY, 'Y');          
                  
        var transform = new Transform();          
        //存放函数图像上的点      
        var a = [], b = [], c = [], d = [];        
                
        //需要显示的函数说明      
        var f1 = 'y=(Math.abs(x-1)+Math.abs(x+2))-5', 
			f2 = 'y=', f3 = '', f4 = '';      
          
		var y = 0;
		
        //函数描点      
        for (var x = xS; x <= xE; x+=1) {   
				y = taskFun(x);
                a.push([x, y]); 
				d.push([x, y]);
				
				
        }            
		

        //存放临时数组      
        var tmp = [];      
		
        //显示变换      
        if (a.length > 0) {      
            a = transform.scale(transform.translate(a, 0, 0), scaleX/spaceX, scaleY/spaceY);
			
            //函数1      
            tmp = [].concat(a);          
            shape.pointDraw(tmp, 'red');          
            tmp = [].concat(a);          
            shape.multiLineDraw(tmp, 'pink');        
                    
            plot.setFillStyle('red');      
            plot.fillText(f1, 30, -100, 400);        
        }
		
		
		
		d.sort(function(p1, p2) {
				//求最大值加负号，最小值不加
				return (p1[1]-p2[1]);
		});
		
		var x0 = d[0][0], y0 = d[0][1];
		
		
		b.push([x0, y0]);
		
		if (b.length > 0) {    
            b = transform.scale(transform.translate(b, 0, 0), scaleX/spaceX, scaleY/spaceY);                
			//最值点
			tmp = [].concat(b);        
            shape.pointDraw(tmp, 'blue');        
            //tmp = [].concat(b);        
            //shape.multiLineDraw(tmp, '#22CCFF');     
                
            plot.setFillStyle('blue');    
            plot.fillText(x0.toFixed(3)+', '+y0.toFixed(3), b[0][0], -b[0][1]-20, 100);  
    
        } 
    }  
}  

function taskFun(x) {
	//var a = x, b = a+1, c = b+1;
	return (Math.abs(x-1)+Math.abs(x+2))-5;
}</span>

<span style="font-size:18px;">0.16666666666666669  --  0.25  --  0.5
0.25  --  0.3611111111111111  --  0.6666666666666666
0.3  --  0.4236111111111111  --  0.75
0.33333333333333337  --  0.4636111111111111  --  0.8
0.35714285714285715  --  0.4913888888888889  --  0.8333333333333334
0.375  --  0.511797052154195  --  0.8571428571428571
0.3888888888888889  --  0.527422052154195  --  0.875
0.4  --  0.5397677311665408  --  0.8888888888888888
0.40909090909090906  --  0.5497677311665408  --  0.9

def tmp():

    for n in range(2, 11):
        a_1 = 1/2-1/(n+1);

        a_2 = 0;
        for i in range(2, n+1):
            a_2 += 1/(i*i);

        a_3 = (n-1)/n;

        print(a_1, ' -- ', a_2, ' -- ', a_3);
		
n = 10000时：0.4999000099990001  --  0.6448340718480534  --  0.9999</span>

<span style="font-size:18px;">	if (1) {
		var mathText = new MathText();
		
		//希腊字母表(存此用于Ctrl C/V
			//ΑΒΓΔΕΖΗΘΙΚΛΜΝΞΟΠΡΣΤΥΦΧΨΩ
			//αβγδεζηθικλμνξοπρστυφχψω
		
		
		var s = [
			/*
			'V=[PI]*r^[2]*h',
			'S=2*[PI]*r^[2]+2*[PI]*r*h',
			' = 2*[PI]*r^[2]+2V/r',
			'>=3*(V/r*V/r*2[pi]r^[2])^[1/3]',
			'=3*(2[PI]V^[2])^[1/3]',
			'4*[PI]*r-2V/r^[2] = 0 => r = (V/(2[PI]))^[1/3]',
			'h = (4V/[PI])^[1/3]',*/
			
			' ',
			
			'V = 1/3SH',
			'r = R(2[PI]-θ)/2[PI]',
			'H = R(1-((2[PI]-θ)/2[PI])^[2])^[1/2]'
			
			
		];
		
		var x =40, y=40;
		var r1 = 40;
			
		var len = s.length;
		for (var i = 0; i < len; i++) {
		
			if (s[i] == '') {
				if (x < 100) {
					x += 300;
					y-=r1*3;
				}
				else {
					x = 20;
					y += r1;
				}
			}
			else {			
				mathText.print(s[i], x, y, '@');
				y+=r1;
			}
		}		
	
	}</span>

<span style="font-size:18px;">>>> 
338350  --  171700

def tmp():
    a1 = list(range(1, 101));
    a2 = list(range(100, -1, -1));
    sum_1 = 0;
    sum_2 = 0;

    for i in range(len(a1)):
        sum_1 += a1[i]*a1[i];
        sum_2 += a1[i]*a2[i];

    print(sum_1, ' -- ', sum_2);</span>

本节到此结束，欲知后事如何，请看下回分解。