How to use linear approximation to the square root function to estimate square roots 4.400 ?

Question

How to use linear approximation to the square root function to estimate square roots             4.400      ?

Attagswalmec6 · Accepted Answer

Using binomial expansion,                              a          2                +        b              =    a                  (        1        +                  b                      a            2                          )                              1          2                      =    a          (      1      +              1        2                    b                  a          2                    )      , nearly.Here, choose a = 66 and b = 44.Now,             4400        =                            66          2                +        44              =                            66          2                +        44              =    66                  (        1        +                  44                      66            2                          )                              1          2                          66          (      1      +              1        2                    (                  44                      66            2                          )            )        =    66          (      1      +              1        198            )        =    66    +          1      3        =    66.33  , nearly.

gatumisz3f6 · Accepted Answer

For function,       f  , the linear approximation at       a   is given by      L          (      x      )        =          f              (        a        )              +    f    ′          (      a      )              (      x      -      a      )      Note that the equation of the line tangent to the graph at             (      a      ,              f                  (          a          )                    )       has slope       m    =    f    ′          (      a      )       and it has point-slope equation      y    -          f              (        a        )              =    f    ′          (      a      )              (      x      -      a      )       Solving for       a   gives us the linear approximation to       f   at       a  .We are not told what to use for       a  , but we want to eventually use       x    =    4.4  .  We want       a   to be a number &quot;close to&quot; 4.4 for which it is relatively easy to calculate             f              (        a        )            .In this example, we want a number whose square root is easy to find and is near to       4.4  .The &#039;obvious&#039; (once you see it) choice is       a    =    4  So,            f              (        x        )              =          x       and             f              (        a        )              =          f              (        4        )              =    2        f    ′          (      x      )        =          1              2                  t                x             so       f    ′          (      a      )        =    f    ′          (      4      )        =          1      4            L          (      x      )        =    2    +          1      4              (      x      -      4      )      And we finish with       L          (      4.4      )        =    2    +          1      4              (      4.4      -      4      )        =    2.1  If there is a typo in the questionIf the question should ask us to approximate                     4        ,        400              we should find a different       a              60      2        =    3600   and             70      2        =    4900   so, for a rough approximation, use       a    =    70  A little more math will demonstrate that            66      2        =    4356   and             67      2        =    4489  Since       4356   is close to       4400  , use       a    =    4356  orTo get a rough estimate use             4400        =                  4        ×        100        ×        11              =    20          11       and approximate             11       using       a    =    9

How to use linear approximation to the square root function to estimate square roots sqrt 4.400?

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