Use a linear approximation to estimate the number

Question

Use a linear approximation to estimate the number sqrt(9.05)

Answer 1

You can use the linear approximation formula L(x) = f(a) + f'(a)(x-a), where f(x) = √x, a = 9 ( value for which we know the square root), and x = 9.05.

 

f(x) = √x   so, f'(x) = 1/(2√x)

 

Therefore, √9.05 ≈ L(9.05) = f(9) + f'(9)(9.05 - 9)

 

                                         = 3 + (1/6)(0.05) = 3.008333333 (accurate to 4 decimal places) 

Answer 2

You must first establish a new coordinate system of known values. 

 

We know that sqrt(9) = 3 and the sqrt(16) = 4

 

This gives us two inputs and two outputs of a function. 

 

Using these two points we find the slope between them. Because 16 and 9 are the inputs they become the x values and 4 and 3 become the respective y values. 

 

So we have the slope between (9,3) and (16,4) is (4-3)/(16-9) = 1/7

 

This gives the equation starting at (9,3) as a hypothetical origin as 

 

y = x/7 + 3

 

In this equation assume x = (given - 9)

 

So x would be (9.05-9) or 0.05

 

y = (0.05/7) + 3 = 3.00714