How to use a linear approximation or differentials to estimate 1 1002 ?

Question

How to use a linear approximation or differentials to estimate             1      1002      ?

esbocinamlgn · Accepted Answer

1      1002        =          1              1000        +        2                  =          1      1000        ⋅          1              1        +                  2          1000                    Now we can work with that last term if we write it as:      =          1      1000        ⋅          1              1        +        x              ,          |      x      |                     &amp;lt;&amp;lt;               1  The Taylor Expansion, a type of Binomial Expansion, is the simplest option:                     (        1        +        x        )            α        =    1    +    α    x    +    O          (              x        2            )       and here       α    =    -    1   and       x    =          2      1000            ⇒          1      1000              (      1      -      x      +      O              (                  x          2                )            )            =          1      1000              (      1      -              2        1000            )        =          1      1000        ⋅          998      1000            ≈    9.98    ×          10              -        4            If you enter the original value into your calculator, you&#039;ll receive almost that.

cerakank2n · Accepted Answer

Let             f              (        x        )              =          1      x        ,    x    ≠    0    ,                 so that,               f    ′          (      x      )        =    -          1              x        2              .  We know that,             f              (        x        +        δ        x        )              ≈          f              (        x        )              +    f    ′          (      x      )        ⋅    δ    x    .  Taking,       x    =    1000    ,                and              ,    δ    x    =    2    ,   we have,            f              (        1000        +        2        )              ≈          f              (        1000        )              +    2    f    ′          (      1000      )        ,    i    .    e    .    ,              1      1002        ≈          1      1000        +    2          (      -              1                  1000          2                    )        =    0.001    -    0.000002  Therefore final answer:       ∴          1      1002        ≈    0.000998

How to use a linear approximation or differentials to estimate 1/1002?

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