Question

How do you estimate the quantity using the Linear Approximation of `(3.9)^(1/2)`?

Answer 1

Function `f(x) = x^(1/2) = sqrtx`

Find an `a` near `3.9` with `f(a)` easy to find.

Use `a=4`, so `f(a) = f(4) = sqrt4 = 2`

The linear approximation of `f` at `(a,f(a))` is a form of the equation for the tangent line at `(a,f(a))`

`L(x) = f(a) + f'(a)(x-a)`

For `f(x) = sqrtx`, we have `f'(x) = 1/(2sqrtx)`.

So, `f'(a) = f'(4) = 1/(2sqrt4) = 1/4`

`L(x) = 2+1/4(x-4)`

`L(3.9) = 2+0.25(-0.1) = 2-0.025 = 1.975`

(Note that: `1.975^2 = 3.900625` an

`sqrt3.9 ~~ 1.974842`.)