Linear Approximation

Definition (Linear Approximation) If and its partial derivatives and are defined on an open region containing the point and and are continuous at then



so that

.

Example (Linear Approximation) Using linear approximation and differentials to find an approximate value for

    Solution. Consider the function and observe that we can easily calculate Therefore, we take and



Since



we have

          
          
  
  
  
  
  
  
  
  
  

This approximation is accurate to two decimal places.

Example (Linear Approximation) Using linear approximation and differentials to find an approximate value for Use differentials to find an approximate value for  



    Solution. Consider the function



and observe that we can easily calculate Therefore, we take and in
    

    
Since




and


    
we have

  
  
  
  
  
  
  
  
  
  
  
    
This approximation is accurate to two decimal places.  

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