Use a linear approximation (or differentials) to estimate the given number.

Question

99.4

Answer 1

To approximate √99.4, we have to use formula for linear approximation as instructed, and it is:

f(x)≈ f(a)+f′(a)(x−a)

Since you are dealing with square root,

 

f(x)=√x

Therefore using derivative,

f'(x) = 1/(2√x)

Here, x=99.4 and I’m going to choose a =100 because that’s the closest perfect square number to 99.4. Therefore:

f(a) = f(100) = √100 = 10

f'(a) = f'(100) =1/(2√100) = 1/20

f(x)=f(99.4)≈ f(100)+f′(100)(99.4−100)

≈ 10+(1/20)(-0.6)

≈ 10+(1/20)(-0.6)

≈ 10 + (-0.03)

≈ 9.97