Question

How do you use the tangent line approximation to approximate the value of `ln(1004)` ?

Answer 1

A formula for a tangent line approximation of a function f, also called linear approximation , is given by

`f(x)~~f(a)+f'(a)(x-a),`
which is a good approximation for `x` when it is close enough to `a`.

I'm not sure, but I think the question is about approximate the value `ln(1.004)`. Could you verify it please? Otherwise we will need to know an approximation to `ln(10).`

In this case, we have `f(x)=ln(x)`, `x=1.004` and `a=1`.

Since,
`f'(x)=1/x=>f'(1)=1` and `ln(1)=0`, we get

`ln(1.004)~~ln(1)+1*(1.004-1)=0.004.`