[从头学数学] 第192节 导数及其应用

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]之第五转的修炼。
这次要研究的是[导数及其应用]。

正剧开始：

星历2016年04月23日 16:32:36, 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[导数及其应用]。

<span style="font-size:18px;">>>> 
[-3.000001001396413, -2.999998999442255]
[4.999998999721811, 5.000000996346898]

def taskFun(x):
    return x**2-7*x+15;

def derivatives(x):
    epsilon = 1e-6;
    leftValue = taskFun(x-epsilon);
    rightValue = taskFun(x+epsilon);
    value = taskFun(x);
    
    leftLimit = (value-leftValue)/epsilon;
    rightLimit = (rightValue-value)/epsilon;    

    return [leftLimit, rightLimit];
    
def tmp():
    print(derivatives(2));
    print(derivatives(6));</span>

<span style="font-size:18px;">>>> 
函数在0.6点的左、右导数分别是：[0.29068008811083956, 0.290679765257984]
函数在1.2点的左、右导数分别是：[0.1831169299526536, 0.18311682836724685]

def taskFun(x):
    return (3*x/(4*math.pi))**(1/3);

def derivatives(x):
    epsilon = 1e-6;
    leftValue = taskFun(x-epsilon);
    rightValue = taskFun(x+epsilon);
    value = taskFun(x);
    
    leftLimit = (value-leftValue)/epsilon;
    rightLimit = (rightValue-value)/epsilon;    

    return [leftLimit, rightLimit];
    
def tmp():
    a = [0.6, 1.2];
    for i in range(len(a)):
        print('函数在{0}点的左、右导数分别是：{1}'.format(a[i],derivatives(a[i])));</span>

<span style="font-size:18px;">>>> 
函数在10点的左、右导数分别是：[0.07947403424246602, 0.07947403823926891]
>>> 1.05**10*math.log(1.05)
0.07947403625517714

def taskFun(x):
    return (1+0.05)**x;

def derivatives(x):
    epsilon = 1e-6;
    leftValue = taskFun(x-epsilon);
    rightValue = taskFun(x+epsilon);
    value = taskFun(x);
    
    leftLimit = (value-leftValue)/epsilon;
    rightLimit = (rightValue-value)/epsilon;    

    return [leftLimit, rightLimit];

#例1
def tmp():
    a = [10];
    for i in range(len(a)):
        print('函数在{0}点的左、右导数分别是：{1}'.format(a[i],derivatives(a[i])));</span>

<span style="font-size:18px;">>>> 
函数在自变量取值为90处的左、右导数分别是：[52.83999450966803, 52.84000519623078]
函数在自变量取值为98处的左、右导数分别是：[1320.9993362579553, 1321.0006572990096]


def taskFun(x):
    return 5284/(100-x);

def derivatives(x):
    epsilon = 1e-6;
    leftValue = taskFun(x-epsilon);
    rightValue = taskFun(x+epsilon);
    value = taskFun(x);
    
    leftLimit = (value-leftValue)/epsilon;
    rightLimit = (rightValue-value)/epsilon;    

    return [leftLimit, rightLimit];

#例3
def tmp():
    a = [90, 98];
    for i in range(len(a)):
        print('函数在自变量取值为{0}处的左、右导数分别是：{1}'.format(a[i],derivatives(a[i])));</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 = 0.3;       
        var xS = -10, xE = 10;      
        var yS = -10, yE = 10;      
        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=log_[2]x的导数', f2 = '', f3 = '', f4 = '';    
		
		var epsilon = 0.000001;
		var derivative = 0;
        //函数描点    
        for (var x = xS; x <= xE; x+=1) {      
				derivative = (funTask(x+epsilon)-funTask(x))/epsilon;
                a.push([x, derivative]);      

  
  
        }      
              
        //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, 100, -90, 200);      
        }  
	}

}

function funTask(x) {
	return Math.log(x)/Math.log(2);
}

function funTask(x) {
	return 2*Math.pow(Math.E, x);
}</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 = 200;       
        var xS = -10, xE = 10;      
        var yS = -1000, yE = 1000;      
        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=2x^5-3x^2-4的导数', f2 = '10x^4-6x', f3 = '', f4 = '';    
		
		var epsilon = 0.000001;
		var derivative = 0;
        //函数描点    
        for (var x = xS; x <= xE; x+=0.2) {      
				//derivative = (funTask(x+epsilon)-funTask(x))/epsilon;
               // a.push([x, derivative]);      
				
				b.push([x, funTask_2(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, 100, -90, 200);      
        }  
		
		        //显示变换    
        if (b.length > 0) {    
            b = transform.scale(transform.translate(b, 0, 0), scaleX/spaceX, scaleY/spaceY);     
            //函数1    
            tmp = [].concat(b);        
            shape.pointDraw(tmp, 'blue');        
            tmp = [].concat(b);        
            shape.multiLineDraw(tmp, '#22ccFF');      
                  
            plot.setFillStyle('blue');    
            plot.fillText(f2, 100, -90, 200);      
        } 
	}

}

function funTask(x) {
	return 2*Math.pow(x, 5)-3*Math.pow(x, 2)-4;
}

function funTask_2(x) {
	return 10*Math.pow(x, 4)-6*x;
}
</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 = -10, yE = 10;      
        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=3cos(x)-4sin(x)的导数', f2 = '10x^4-6x', f3 = '', f4 = '';    
		
		var epsilon = 0.000001;
		var derivative = 0;
        //函数描点    
        for (var x = xS; x <= xE; x+=0.5) {      
				derivative = (funTask(x+epsilon)-funTask(x))/epsilon;
                a.push([x, derivative]);      
				
				//b.push([x, funTask_2(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, 100, -110, 200);      
        }  
		
		        //显示变换    
        if (b.length > 0) {    
            b = transform.scale(transform.translate(b, 0, 0), scaleX/spaceX, scaleY/spaceY);     
            //函数1    
            tmp = [].concat(b);        
            shape.pointDraw(tmp, 'blue');        
            tmp = [].concat(b);        
            shape.multiLineDraw(tmp, '#22ccFF');      
                  
            plot.setFillStyle('blue');    
            plot.fillText(f2, 100, -90, 200);      
        } 
	}

}

function funTask(x) {
	return 3*Math.cos(x)-4*Math.sin(x);
}

function funTask(x) {
	return Math.cos(x/3);
}</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 = 200;       
        var xS = -10, xE = 10;      
        var yS = -1000, yE = 1000;      
        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+2x^2+10x-20', f2 = '10x^4-6x', f3 = '', f4 = '';    
		
		var epsilon = 0.000001;
		var derivative = 0;
		//给定初始试探点
		var x0 = 4, xResult = 0;
		
        //函数描点    
        for (var x = xS; x <= xE; x++) {      
                a.push([x, funTask(x)]);   
        }      
		
		//牛顿法求零点
		derivative = (funTask(x0+epsilon)-funTask(x0))/epsilon;
		xResult = x0-funTask(x0)/derivative;
		while (Math.abs((xResult-x0)/x0)>epsilon) {
			x0 = xResult;
			derivative = (funTask(x0+epsilon)-funTask(x0))/epsilon;
			xResult = x0 -funTask(x0)/derivative;
		}
			
		b.push([xResult, 0]);
              
        //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, 100, -110, 200);      
        }  
		
		        //显示变换    
        if (b.length > 0) {  
			plot.setFillStyle('blue');    
            plot.fillText('零点坐标是：'+b[0][0].toFixed(3), 100, -90, 200);  
			
            b = transform.scale(transform.translate(b, 0, 0), scaleX/spaceX, scaleY/spaceY);     
            //函数1    
            tmp = [].concat(b);        
            shape.pointDraw(tmp, 'blue');        
  
                  
    
        } 
	}

}

function funTask(x) {
	return Math.pow(x, 3)+2*Math.pow(x, 2)+10*x-20;
}
</span>

这是一张勉强能用的手写式，如果要想做成印刷上的多行立体形态，只怕是不容易。

当然，那就是公式编辑器的事了。

<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 = [
			'lim_[Δt->0](h(2+Δt)-h(2))/(Δt) = k',
			'lim_[Δx->0]Δy/Δx=lim_[Δt->0](f(x_[0]+Δx)-f(x_[0]))/Δx',
			'f \'(x_[0])=lim_[Δx->0]Δy/Δx=lim_[Δx->0](f(x_[0]+Δx)-f(x_[0]))/Δx',
			'S=lim_[Δx->0]Ξ_[i=1]^[n]   f(ξ_[i])Δx=lim_[Δx->0]Ξ_[i=1]^[n]   1/nf(ξ_[i])=1/3',
			'[S]_[a]^[b]  f(x)dx = lim_[n->[inf]]Ξ_[i=1]^[n] (b-a)/nf(ξ_[i])',
			'[S]_[a]^[b]  f(x)dx = F(x) |_[a]^[b] = F(b)-F(a)',
			
		];
		
		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.printIntegral(s[i], x, y, '@');
				y+=r1;
			}
		}		
	
	}
	
/**
* @usage   数学表达式，代数式的书写
* @author  mw
* @date    2016年03月12日  星期六  11:05:12 
* @param
* @return
*
*/
function MathText() {
	//上标标记形式为...^[内容]...
	//分数不进行处理， 根式不进行处理，都转成指数式进行
	//特殊数学符号设想加\[alpha]进行转义，待续
	//可以进行指数上标代数式的书写
	//可扩展下标，待续
	
	
	this.setNormalFont = function() {
		plot.setFont("normal normal normal 24px Times Lt Std");	
	}
	
	this.setScriptFont = function() {
		plot.setFont("italic normal bold 16px Dark Courier ");
	}
	
	this.print = function(text, xPos, yPos, style, splitChar) {
		splitChar = splitChar ? splitChar : '|';
		xPos = xPos ? xPos : 0;
		yPos = yPos ? yPos : 0;
		style = style ? style : 'black';
		
		plot.save()
			.setStrokeStyle(style)
			.setFillStyle(style);
		
		
		var s = text ? text : '';
		
		if (s != '') {
			s = s.replace(/\/\//ig, '÷');
			s = s.replace(/\[>=\]/ig, '≥');
			s = s.replace(/\[<=\]/ig, '≤');
			s = s.replace(/\[!=\]/ig, '≠');
			s = s.replace(/\[PI\]/ig, 'π');
			s = s.replace(/\[+-\]/ig, '±');
			s = s.replace(/->/ig, '→');
			s = s.replace(/<-/ig, '←');
			s = s.replace(/<-/ig, '←');
			s = s.replace(/Ξ/g, '\u2211');
			
		}
		
		//字符串长度
		var len = s.length;
		//不同字体大小设置在此
		var r1 = 20;
		//单个字符暂存处
		var c;
		//文本显示位置
		var x = xPos, y = yPos;
		//正常文本暂存
		var s0 = '';
		//字符串打印长度
		var measure; 
		//记录上一个x位置,可记录三层
		var xMem = [x, x, x];
		//记录每一层的左括号位置
		var bracketPos = [x, x, x];
		//记录括号层次
		var bracketsLevel = 0;
		//记录根号层次
		var radicalLevel = 0;
		//记录每一层根号的起始位置和层次数的数组...[[start, end, level], ...]
		var radicalSpan = [];
		
		//设置正常字体
		this.setNormalFont();				
		
		for (var i = 0; i < len; i++) {
			if (s[i] == '_' && s[i+1] == '[') {
				//下标开始
				//下标标记形式为..._[内容]...
				
				if (s0 != '') { //先把正常字符打印出
					if (r1 != 20) { //字体字号大小还在上标状态
						r1 = 20;
						this.setNormalFont();					
					}
					
					measure = plot.measureText(s0);
					plot.fillText(s0, x, y, measure);
					s0 = '';

					x += measure;
				
				}
				
				var subScript = '';
				var j = 0;
				for (j = i+1; s[j]!=']'; j++) {
					if (s[j] != '[') {
						subScript+=s[j];
					}
				}
				
				if (r1 != 10) {//正常字体状态，需要改为上标字体
					r1 = 10;
					this.setScriptFont();
					
				}
			
				measure = plot.measureText(subScript);
				plot.fillText(subScript, x, y+8, measure);
					
				if (j < len-1 && s[j+1] == '^') {
				
				}
				else {
					x += 1.2*measure;
				}
				
				i = j;
			
			}
			else if (s[i] == '^'&&s[i+1] == '[') {
				//上标开始
				//上标标记形式为...^[内容]...
				
				if (s0 != '') { //先把正常字符打印出
					if (r1 != 20) { //字体字号大小还在上标状态
						r1 = 20;
						this.setNormalFont();					
					}
					
					measure = plot.measureText(s0);
					plot.fillText(s0, x, y, measure);
					s0 = '';

					x += measure;
				
				}
					
				var upperScript = '';
				var j = 0;
				for (j = i+1; s[j]!=']'; j++) {
					if (s[j] != '[') {
						upperScript+=s[j];
					}
				}
				
				

					
				
				//二次根式
				if (upperScript == '1/2' || upperScript == '0.5') {		
					var x1, y1;
					
					if (i > 0 && s[i-1] == ')') {
						x1 = bracketPos[bracketsLevel];
					}
					else {
						x1 = xMem[bracketsLevel];
					}
					
					
					/* 存疑代码
					
					if (radicalSpan == []) {
						radicalLevel = 0;
						radicalSpan.push([x1, x, radicalLevel]);
					}
					else {
						var len = radicalSpan.length;
						for (var k = 0; k < len; k++) {
							if (x1 < radicalSpan[k][0]) {
								radicalLevel = radicalSpan[k][2]+1;
								break;
							}
							
							if (k >= len-1) {
								radicalLevel = 0;
							}
						}
						radicalSpan.push([x1, x, radicalLevel]);
					}*/
					
					y1 = y-20-5*radicalLevel;			

					
					plot.save()
						.setLineWidth(1);
					plot.beginPath()
						.moveTo(x1-5, y+5)
						.lineTo(x1-8, y-3)
						.moveTo(x1-5, y+5)
						.lineTo(x1+5, y1)
						.moveTo(x1+5, y1)
						.lineTo(x, y1)
						.closePath()
						.stroke();
					plot.restore();
					
				}
				
				//向量符号
				else if (upperScript == '->') {
					var x1, y1;
					
					if (i > 0 && s[i-1] == ')') {
						x1 = bracketPos[bracketsLevel];
					}
					else {
						x1 = xMem[bracketsLevel];
					}
					
					y1 = y-18-5*radicalLevel;			

					
					plot.save()
						.setLineWidth(1);
					plot.beginPath()
						.moveTo(x1, y1)
						.lineTo(x+2, y1)
						.moveTo(x+2, y1)
						.lineTo(x-5, y1-3)
						.moveTo(x+2, y1)
						.lineTo(x-5, y1+3)
						.closePath()
						.stroke();
					plot.restore();
					
				
				
				
				}
				else {

					if (r1 != 10) {//正常字体状态，需要改为上标字体
						r1 = 10;
						this.setScriptFont();
						
					}
				
					measure = plot.measureText(upperScript);
					plot.fillText(upperScript, x, y-8, measure);
					
					if (j < len-1 && s[j+1] == '_') {
					
					}
					else {
						x += 1.2*measure;
					}
				}
				
				//直接跳跃过上标字符区段
				i = j;
			}
			else {
				c = s[i];
				
				if (c == ')') {
					s0 += c;
					bracketsLevel -= 1;
					
				}
				else if (c == '(') {
					//如果整个括号被开根式，根号在括号左边
					bracketPos[bracketsLevel] = x + plot.measureText(s0);					
					s0 += c;
					
					bracketsLevel+=1;
					//过了括号就是过了一道关，要刷新坐标
					xMem[bracketsLevel] = x + plot.measureText(s0);
					
					
				}
				else if (c == '+' || c == '-' || c == '*' || c == '/' || c == '÷'
					|| c == '=' || c == ' ') {
					
					if (c == '*') {
						if (i > 0 && 
							((s[i-1] == ')' && /[0-9]/.test(s[i-2]) ||
								/[0-9]/.test(s[i-1]))) &&
							((s[i+1] == '(' && /[0-9]/.test(s[i+2])) || 
								/[0-9]/.test(s[i+1]))) {
							//对于乘号前后都是数字的情况，把乘号改成叉号
							c = ' \u00D7 ';
						}
						else {
							//对于代数式中，乘号改为点号
							c = ' \u00B7 ';
						}
					}
					
					//如果是运算符后的数被开根式，根号在运算符右边
					if (c == '-' || c == '/') {
						s0 += ' '+c+' ';
					}
					else {
						s0 += c;
					}
				
					if (bracketsLevel < 3) {
						xMem[bracketsLevel] = x+plot.measureText(s0);
					}						
				}
				else if (c == splitChar) { //隔字符
					if (bracketsLevel < 3) {
						xMem[bracketsLevel] = x+plot.measureText(s0)-3;
					}
				
				}
				else {
					s0 += c;				
				}				
				
			}
		}
		
		if (s0 != '') { //先把正常字符打印出
			if (r1 != 20) { //字体字号大小还在上标状态
				r1 = 20;
				this.setNormalFont();				
			}
			measure = plot.measureText(s0);
			plot.fillText(s0, x, y, measure);
			x += measure;
		}
		
		plot.restore();
	}
	
	//集合符号，集合表达式的书写
	this.printSet = function(text, xPos, yPos, style, splitChar) {
		//隔字符
		splitChar = splitChar ? splitChar : '|';
		var s = text ? text : '';
		xPos = xPos ? xPos : 0;
		yPos = yPos ? yPos : 0;
		style = style ? style : 'black';
		
		if (s != '') {
			//希腊字母表(存此用于Ctrl C/V
			//ΑΒΓΔΕΖΗΘΙΚΛΜΝΞΟΠΡΣΤΥΦΧΨΩ
			//αβγδεζηθικλμνξοπρστυφχψω
			//
			
			s = s.replace(/\[B\]/ig, '\u2208'); //∈
			s = s.replace(/\[NB\]/ig, '\u2209'); //不属于
			s = s.replace(/\[S\]/ig, '\u2286'); //包含于(是子集)
			s = s.replace(/\[SS\]/ig, '\u2287'); //包含
			s = s.replace(/\[ST\]/ig, '\u228A'); //真包含于(是真子集)
			s = s.replace(/\[SST\]/ig, '\u228B'); //真包含
			
			s = s.replace(/\[UU\]/ig, '\u222A'); //并集 ，由于U表示全集，又常为下标，此处错开
			s = s.replace(/\[I\]/ig, '\u2229'); //交集
			s = s.replace(/\[C\]/ig, '\u2201'); //补集
			s = s.replace(/\[INF\]/ig, '\u221E'); //无穷大
			s = s.replace(/\[NULL\]/ig, '\u2205');//空集
			s = s.replace(/\[&\]/ig, '\u2227');//且
			s = s.replace(/\[\|\]/ig, '\u2228');//或
			s = s.replace(/\[~\]/ig, '﹁');//非
			
			s = s.replace(/\[ALL\]/ig, '\u2200');//全称量词 Universal quantifier
			s = s.replace(/\[Exist\]/ig, '\u2203');//存在量词 Existential quantifier
			

			

		}

		return this.print(s, xPos, yPos, style, splitChar);
	}
	
	//微积分符号
	this.printIntegral = function(text, xPos, yPos, style, splitChar) {
		//隔字符
		splitChar = splitChar ? splitChar : '|';
		var s = text ? text : '';
		xPos = xPos ? xPos : 0;
		yPos = yPos ? yPos : 0;
		style = style ? style : 'black';
		
		if (s != '') {
			//希腊字母表(存此用于Ctrl C/V
			//ΑΒΓΔΕΖΗΘΙΚΛΜΝΞΟΠΡΣΤΥΦΧΨΩ
			//αβγδεζηθικλμνξοπρστυφχψω
			//
			s = s.replace(/\[S\]/ig, '\u222B'); //定积分符号 一次
			s = s.replace(/\[SS\]/ig, '\u222C');//定积分符号 二次
			s = s.replace(/\[SSS\]/ig, '\u222C');//定积分符号 三次
			s = s.replace(/\[INF\]/ig, '\u221E'); //无穷大
			
		}
		return this.print(s, xPos, yPos, style, splitChar);
	}

}</span>

本节到此结束，欲知后事如何，请看下回分解。