Help with linear approximations. Can anyone help me out?

Using the linear approximation (1+x)^k= 1+kx(1+x)k=1+kx , find an approximation of f(x) = (4+3x)^(1/3)f(x)=(4+3x)13.

I tried doing (1+(3+3x))^(1/3)(1+(3+3x))13. and then I solved it and got 2+x, but that doesn't appear to be the answer! A walk through would be super nice!

1 Answer
Mar 12, 2018

(4+3x)^(1/3) ~= 4^(1/3)+ x/4^(2/3)(4+3x)13413+x423

Explanation:

(4+3x)^(1/3) = (4(1+3/4x))^(1/3) = 4^(1/3)(1+3/4x)^(1/3)(4+3x)13=(4(1+34x))13=413(1+34x)13

we can now use the linear approximation:

(1+x)^k ~= 1+kx(1+x)k1+kx

(4+3x)^(1/3) ~= 4^(1/3)(1+1/3 3/4x)(4+3x)13413(1+1334x)

and simplifying:

(4+3x)^(1/3) ~= 4^(1/3)+ x/4^(2/3)(4+3x)13413+x423

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