How do you use a linear approximation to approximate the value of #root(3)(27.5)#?

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Answer 1 · 2017-08-23T17:47:12

We let #f(x) = root(3)(a)#. Then #f(27) = root(3)(27) = 3#

We compute the derivative of #f(x)# as being

#f(a) = a^(1/3) -> f'(a) = 1/3a^(-2/3)#

Hence,

#f'(27) = 1/3(27)^(-2/3) =1/3(3)^(-2) = 1/3(1/9) = 1/27#

Recalll the equation of a line is given by

#y - y_1 = m(x - a)#

#y - 3 = 1/27(x - 27)#

#y = 1/27x - 1+ 3#

#y = 1/27x + 2#

So the approximation at #x = 27.5# would be

#y ~~ 1/27(27.5) + 2 ~~ 3.0185185...#

If we use a calculator to compute we get #3.0184054# apprxoaimtely, so this approximation is very good.

Hopefully this helps!