The Limit Definition of a Derivative by xiaoyounan

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									I.      The Limit Definition of a Derivative
     A. Explanation

      f(x) = x2




                    ( +       ( +    ))

      (x,f(x))
                      Secant line

                    Tangent line




                                                                         The Limit Definition of a
        1. Find the slope of the secant line:
                                                                               Derivative
        2. Make     smaller so msec is closer to mtan
                                                                               ( + )         ( )
                                                                          i

                                                                                       OR
                          ( + )        ( )
                             +                                                       ( + )   ( )
                                                                             i
        3. Find the limit as      of the secant line
     B. Examples
           a. f(x) 2x+3 find f ’(x)

             2( +     )+3      (2 + 3)                          2 +2    +3       2    3
         i                                              i



             2
         i                                                  i   2   2
b. f(x)=x2-2x Find f ‘ (x) and find f ‘ (1)

       ( +      )           2( +       )   (   2 )
 i

           +2           +          2       2   +2
 i

           (2 +              2)
 i

 i     2 +              2    2     2

     ( )   2        2

     (1)   2(1)         2

								
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