Use a linear approximation (or differentials) to estimate the given number?

NetherCraft 0

32.05^(4/5)

1 Answer

  • Put f(x) = x^(4/5) and do a linearization about the point x_0 = 32. Note that 32.05 = x_0 + Δx where Δx = 0.05.

    f(x_0 + Δx) ≈ f(x_0) + f ‘(x_0)Δx

    f ‘ (x) = (4/5) x^(-1/5) ==> f ‘ (x_0) = (4/5)(32)^(-1/5) = (4/5)/2 = 2/5.

    (32.05)^(4/5) ≈ (32)^(4/5) + (2/5)(0.05) = 16 + 1/50 = 16.02.

    A calculator generated decimal approximation to the value is 16.019997—very close!

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