Visualization Notes: 08/10/21

Function and Their Graphs 2

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set_verbose(-1); var('x'); f=5^(x/(1-x^2))
st=r'$\displaystyle{f(x)=%s}$'%latex(f)
st1=r'$\displaystyle{\lim_{x\to-1^{-}}f(x)=%s;\;}$'\
    %latex(limit(f,x=-1,dir='left'))
st2=r'$\displaystyle{\;\;\lim_{x\to-1^{+}}f(x)=%s}$'\
    %latex(limit(f,x=-1,dir='right'))
st3=r'$\displaystyle{\lim_{x\to1^{-}}f(x)=%s;\;}$'\
    %latex(limit(f,x=1,dir='left'))
st4=r'$\displaystyle{\;\;\lim_{x\to1^{+}}f(x)=%s}$'\
    %latex(limit(f,x=1,dir='right'))
for el in [st,st1+st2,st3+st4]:
    pretty_print(html(el))
p=plot(f,(-5,5),ymax=3)
p+=line([[-1,0],[-1,3]],color='slategray',
        linestyle='--',thickness=2)
p+=line([[1,0],[1,3]],color='slategray',
        linestyle='--',thickness=2)
p.show(frame=True,gridlines=True,figsize=(5,5))

Functions and Their Graphs

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var('x'); y=4/(x^2+2*x-3)
yb=10; cols=['steelblue','darkblue','purple']
ll=[r'$y=%s$'%latex(y.factor()),
    r'$y\prime=%s$'%latex(diff(y,x).factor()),
    r'$y\prime\prime=%s$'%latex(diff(y,x,x).factor())]
p=plot([y.factor(),diff(y,x).factor(),
        diff(y,x,x).factor()],(-5,3),
       color=cols,thickness=1.5,legend_label=ll)
p+=sum([point((x0,y(x=x0)),color=cols[0],size=25) 
        for x0 in [-4,-1,2]])
p+=line([[-3,-yb],[-3,yb]],color='crimson',linestyle='-.',
        thickness=3,legend_label=r'$x=-3$')
p+=line([[1,-yb],[1,yb]],color='red',linestyle='-.',
        thickness=3,legend_label=r'$x=1$')
p+=line([[-5,0],[3,0]],color='magenta',linestyle=':',
        thickness=3,legend_label=r'$y=0$')
p.set_legend_options(font_size=14,loc=3)
p.show(gridlines=True,figsize=(6,6),ymin=-yb,ymax=yb)

JS & Functions

<script src='https://code.highcharts.com/highcharts.js'></script> was added in the page head.

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%%html
<style>
 @import url('https://fonts.googleapis.com/css?family=Roboto|Aladin');
</style>
<p style="background-color:white; color:steelblue; 
          font-family:Aladin; font-size:120%;">
The code cell calls a function 
which performs iterations and returns the result:</p>
<button id="calculation1" width=600 
style="background-color:lightgray; text-align:left; 
       color:steelblue; font-family:Roboto; font-size:80%;"></button>
<button id="calculation2" width=600 
style="background-color:lightgray; text-align:left; 
       color:steelblue; font-family:Roboto; font-size:80%;"></button>
<script>
function F1(x){return Math.pow(x,3);};
var A1=-0.1,B1=0.1,A2=0,B2=2,E=Math.pow(10,-4);
function F2(x){return Math.log(20/Math.sqrt(.009))-
                      1.5*(Math.exp(x/3)+Math.exp(-x/3))-x;};
function findroot(f,a,b,e){
    var s1='It is not possible to use ',
        s2='the segment for this decision method';
    l=f.toString().length; 
    var s3='Iteration steps'+'<br/>f(x)='+
           f.toString().slice(22,l-2);
    if (Math.sign(f(a))==Math.sign(f(b))){return s1+s2;} 
    else {
        while (b-a>e){
            var ab=a+(b-a)/2; 
            s3=s3+'<br/>'+f(a)+'; '+f(ab)+'; '+f(b);
            if (Math.sign(f(a))!=Math.sign(f(ab)))
                {if (f(ab)==0){break}; b=ab;}
            else {a=ab;};};
        return s3+'<br/> f(x)=0 => x&#8776;'+ab+'<p/>';};}
document.getElementById('calculation1').innerHTML=findroot(F1,A1,B1,E);
document.getElementById('calculation2').innerHTML=findroot(F2,A2,B2,E);

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%%html
<div id='container2' 
     style='height:500px; width:500px; margin:0 auto'>
</div><script>
function f(x,z){
    return (Math.log(Math.sqrt(Math.PI))+Math.abs(2-x))/
           (3.3-1/z)+Math.pow(z,2/3)*Math.cos(Math.pow(x,2));};
var series=[]; var x0=6.7,hx=.1,n=4,z0=3.3,hz=.2,m=3;
for (var j=0; j<m; j++) {
    xz=Array(4).fill(z0+j*hz).map((z,i)=>[x0+i*hx,f(x0+i*hx,z)]); 
    series.push({name:'z='+(z0+j*hz),
                 marker:{radius:7-2*j},data:xz});};
Highcharts.chart('container2',{
    chart:{type:'line'},title:{text:'Function of Two Variables'},
    xAxis:{title:{text:'x'}},yAxis:{title:{text:'f(x,z)'}},
    credits:{enabled:false},series:series });

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%%html
<div id='container3' 
     style='height:500px; width:500px; margin:0 auto'></div>
<script>
function factorial(x){
    if (x===0){return 1;} return x*factorial(x-1);};
function sumf(n,x){var s=0;
  for (var k=1; k<n+1; k++){
      var m=1/factorial(k)+1/factorial(2*k+1);
      s+=Math.pow(Math.sign(-1),k+1)*Math.pow(x,2*k+1)*m;};
  return s;};
function controlf(x){
    return 2*x-x*Math.exp(-1*Math.pow(x,2))-Math.sin(x);};
var n=35,d=.03;
var cxy=Array(n).fill(d).map((r,x)=>[x*r,controlf(x*r)]);
    sxy3=Array(n).fill(d).map((r,x)=>[x*r,sumf(3,x*r)]);
    sxy5=Array(n).fill(d).map((r,x)=>[x*r,sumf(5,x*r)]);
Highcharts.chart('container3',{
    chart:{type:'line'},credits:{enabled:false},
    xAxis:{title:{text:'x'}},yAxis:{title:{text:'y'}},
    title:{text:'Sum Function and Control Function'},
    series:[{name:'sum function n=3',color:'#3636ff',
             marker:{radius:4,symbol:'triangle-down'},data:sxy3},
            {name:'sum function n=5',color:'#36ff36',
             marker:{radius:4,symbol:'square'},data:sxy5},
            {name:'control function',color:'#ff3636',
             marker:{radius:2,symbol:'circle',opacity:.5},data:cxy}]});

Thebe Python: SymPy Integration

Status:No Kernel Attached

✒️ SymPy Integration

!pip install -q sympy
import sympy as sp
sp.init_printing(use_unicode=True)
x,a,b=sp.symbols('x,a,b')
f21=x*(a*x**2+b)**(1/3)
intf21=sp.integrate(f21,x).simplify()
display([f21,intf21])
sp.pprint(75*'=')
p=sp.plot(f21.subs({a:1,b:2}),(x,-10,10),
          line_color='#3636ff',ylim=(-50,50),show=False)
ip=sp.plot(intf21.subs({a:1,b:2}),(x,-10,10),
           line_color='#36ff36',ylim=(-50,50),show=False)
p.extend(ip); p.show()

Tuesday, August 10, 2021

Function and Their Graphs 2

Functions and Their Graphs

JS & Functions

<script src='https://code.highcharts.com/highcharts.js'></script> was added in the page head.

Thebe Python: SymPy Integration

✒️ SymPy Integration