Use the Linear Approximation to estimate Δ f = f ( 3.5 ) − f ( 3 ) for f ( x ) = 3 1 + x 2 a) Δ f ≈b) Δ f =

Question

Use the Linear Approximation to estimate   Δ  f  =  f  (  3.5  )  −  f  (  3  )     for     f  (  x  )  =      3          1      +              x        2            a)   Δ  f  ≈b)   Δ  f  =

Emmanuel Russo · Accepted Answer

Step 1Using linear approximation  Δ  f  =            d      f              d      x        Δ  x  =      d          d      x            (          3              1        +                  x          2                      )    Δ  x    ⇒  Δ  f  =            [                        −          3          ×          2          x                          (          1          +                      x            2                                )            2                                                |                          x          =          3                    ]              Δ      x          ∵  Δ  x  =  3.5  −  3  =  0.5    ⇒  Δ  f  =            −      18        100    ×  0.5  ⇒  Δ  f  =  −  0.09Actual chang   Δ      f    2    =  f  (  3.5  )  −  f  (  3  )    ⇒  Δ      f    2    =      3          1      +      (      3.5              )        2              −      3          1      +              3        2              =      3    13.25    −      3    10      ⇒  Δ      f    2    =  −  0.07358    Error  =  Δ  f  −  Δ      f    2      =  −  0.09  +  0.07358  =  −  0.01641

Use the Linear Approximation to estimate Delta f=f(3.5)-f(3) for f(x)=3/(1+x^2)

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