Question

Help with linear approximations. Can anyone help me out?

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

I tried doing `(1+(3+3x))^(1/3)`. 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!

Answer 1

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

Explanation:

`(4+3x)^(1/3) = (4(1+3/4x))^(1/3) = 4^(1/3)(1+3/4x)^(1/3)`

we can now use the linear approximation:

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

`(4+3x)^(1/3) ~= 4^(1/3)(1+1/3 3/4x)`

and simplifying:

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