Differentiability: Difference between revisions

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<math>f_1: D \to {\mathbb R}</math> that is continuous in <math>x_0</math> such that
<math>f_1: D \to {\mathbb R}</math> that is continuous in <math>x_0</math> such that


:<math> f(x) = f(x_0) + (x-x_0) f_1(x) </math>
:<math> f(x) = f(x_0) + (x-x_0) f_1(x) \,.</math>




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         { size: 1, name: 'f_1', color: 'black', fixed: true, trace: true});
         { size: 1, name: 'f_1', color: 'black', fixed: true, trace: true});
      
      
var txt = board.create('text', [2, 7, function() {  
var txt = board.create('text', [0.5, 7, function() {  
         return '( ' +  
         return '( ' +  
               fx.Y().toFixed(2) + ' - (' + fx0.Y().toFixed(2) +  
               fx.Y().toFixed(2) + ' - (' + fx0.Y().toFixed(2) +  
               ') ) / (' +  
               ') ) / ( ' +  
               fx.X().toFixed(2) + ' - (' + fx0.X().toFixed(2) +
               fx.X().toFixed(2) + ' - (' + fx0.X().toFixed(2) +
               ') ) = ' + ((fx.Y()-fx0.Y())/(fx.X()-fx0.X())).toFixed(3);
               ') ) = ' + ((fx.Y()-fx0.Y())/(fx.X()-fx0.X())).toFixed(3);

Revision as of 19:35, 22 January 2019

If the function [math]\displaystyle{ f: D \to {\mathbb R} }[/math] is differentiable in [math]\displaystyle{ x_0\in D }[/math] then there is a function [math]\displaystyle{ f_1: D \to {\mathbb R} }[/math] that is continuous in [math]\displaystyle{ x_0 }[/math] such that

[math]\displaystyle{ f(x) = f(x_0) + (x-x_0) f_1(x) \,. }[/math]


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