Question

Find the linear approximation of f(x)=ln x at x=1 and use it to estimate ln(1.19 )?

1)L(x)=
2)ln1.19

Answer 1

See below.

Explanation:

We start by finding the equation of the tangent line at `x =1` to the graph of `y = lnx`.

`f(1) = ln(1) = 0`

We know the derivative of `f(x)` will be `f'(x) = 1/x`.

`f'(1) = 1/1 = 1`

Therefore, the equation of the tangent is:

`y - y_1 = m(x - x_1)`
`y - 0 = 1(x - 1)`
`y = x -1`

Therefore, an approximation for `ln(1.19)` would be

`y = 1.19 - 1 = 0.19`

The actual value, obtained using a calculator, would be `0.174`. This means the percent error is

`(0.19 - 0.174)/0.174 * 100 = 9.1 %`

Which seems quite large, yet this is an approximation, so the estimation isn't too terrible.

Hopefully this helps!