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Question
Let f(x) = `root(3)(x)`. Find the linear approximation at x = 27. Use the linear approximation to approximate `root(3)(27.2)`
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Solution
x = 27
f(x) = `root(3)(27)` = 3
We need to find the value of `root(3)(27.2)`
We know that
f(x0 + Δx) = f(x0) + f'(x0) Δx
f(27.2) = `3 + 1/(3x^(2/3)) xx 0.2`
= `3 + 1/(3(27)^(2/3)) xx 0.2`
= `3 + 1/(3 xx 9) x 0.2`
= `3 + 0.2/27`
= `3 + 2/270`
= 3 + 0.0074
= 3.0074
∴ Approximate value of `root(3)(27.2)` = 3.0074
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