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]。
[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。
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