Question

Find the linear approximation of the function f(x) = 4 − x at a = 0. L(x) = $$ Incorrect: Your answer is incorrect. Use L(x) to approximate the numbers 3.9 and 3.99 . (Rou

Answer 1

`L(x)=4-x`

`L(3.9)=0.1\\L(3.99)=0.01`

Explanation:

The linear approximating polynomial is:
`L(x) = f(a) + f'(a)(x - a)`

Given:
`f(x) = 4 - x` at a=0

f(0)=4-0=4

f'(x)=-1, Therefore: f'(a)=-1

Therefore, the linear approximation of f(x) at a=0 is:

`L(x) = f(0) + f'(a)(x - 0)\\L(x)=4-x`

We then use our result to approximate 3.9 and 3.99.

`L(3.9)=4-3.9=0.1\\L(3.99)=4-3.99=0.01`