Question

The local linear approximation to the function `f` at `x=1` is `y=2x+8` What is the value of `f(1)+f'(1)`? ​

Answer 1

`f(1)+f'(1) = 12`

Explanation:

If `y=f(x)` is a differentiable function, then the linear approximation around `x=1` is:

`f(x) ~= f(1) +f'(1)(x-1) = f'(1)x + f(1)-f'(1)`

Comparing with the linear approximation `y=2x+8` we get:

`f'(1)x + f(1)-f'(1) = 2x+8`

and equating the coefficient of the same degree in `x`:

`f'(1) = 2`

`f(1)-f'(1) = 8 => f(1) = 10`

So:

`f(1)+f'(1) = 12`