# Find the linear approximation for f(x) = cube root of 1+3x at a=0 and use it to estimate cube root of 1.03

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### 1 Answer

To find the linear approximation to a function, use the Taylor series truncated after the first two terms. That is,

`f(x)=sum_{k=0}^infty1/{k!}f^{(k)}(a)(x-a)^k`

Since a=0, this means that the linear approximation is

`f(x)approx f(0)+f'(0)x`

Now we know that

`f(0)=(1+3(0))^{1/3}=1`

and since

`f'(x)=(1+3x)^{-2/3}`

then

`f'(0)=1`

**So the linear approximation is **

**`f(x)approx 1+x` **

**and the approximation to the cube root of 1.03 is with x=0.01 to get `1.03^{1/3}approx 1.01` .**