Use linear approximation to estimate the number sqrt (9.05)

Linear approximation is used when the value needed is close to a known value.

Here, we know sqrt (9) = 3.

Let

f (x) = sqrt (x)

f' (x) = 1 / (2sqrt (x))

Apply linear approximation

f (9.05) = f (9) + f' (x) dx

=sqrt (9) + 0.05 / (2sqrt (9))

=3+.05 / (6)

=3.008333

exact value is 3.008322, estimate is off by. 0004%