calc

find linear approximations for each of the following and put a bound on the error of the estimate.

1.

(7.985)^1/3 using f(x)=x^1/3 and a=8

2.
(.9997)^100 using f(x)=x^100 and a=1

please help

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asked by C
  1. Taylor's series expansion for f(a+h) gives:
    f(a+h)=f(a)+hf'(a)+h²f"(a)/2+...
    It can be shown that the sum of the third and subsequent terms does not exceed ε where
    ε=h²f"(a+ξh)
    where ξ takes on a value to maximize the absolute value of &epsilon. Normally, ξ is either at 0 or 1.

    So the estimation becomes:
    f(a+h)=f(a)+hf'(a)±ε

    1.
    (7.985)^1/3 using f(x)=x^1/3 and a=8

    f(x)=x^(1/3)
    f'(x)=(1/3)x^(-2/3)
    f"(x)=-(2/9)x^(-5/3)
    a=8
    h=-0.015
    f(8-0.015)=f(7.985)
    =f(8)-0.015f'(8)
    =2-0.015*(1/3)/4
    =2-0.00125
    =1.99875
    Error bound,
    ε
    =(1/2)(-0.015)²f"(8) approx.
    =0.0001125*0.00694
    =0.000000781

    Correct value
    = 1.998749217935179
    Actual error
    = 1.998749217935179-1.99875
    = -0.000000782

    I will leave #2 as an exercise for you.

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    posted by MathMate
  2. Sorry, the error term:
    ε=h²f"(a+ξh)

    should have read:
    ε=h²f"(a+ξh)/2!

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