Let x=square root 6 + square root 7. What two consecutive integers does x lie between?

Question

Can you pls explain in words, how you got the answer. I know that the answer is 5 and 6 by using the calculator, but I don't know how to do square root 6 plus square root 7. Pls, help. 

Answer 1

You can use linear approximation.

 

L(x) = f(x₀) + f'(x₀)(x - x₀)

 

Let     L(x₁) = √6

         L(x₂) = √7

 

f(x) = √x

 

First, we find the derivative of √x.

 

f'(x) = 1 / (2√x)

 

 

For L(x₁) , x₀= 4

For L(x₂) , x₀ = 9

 

 

L(x₁) = 2 + (1/4)(x - 4)

        = (1/4)x + 1

 

 

L(x₂) = 3 + (1/6)(x - 9)

        = (1/6)x - (3/2) + 3

        = (1/6)x + (3/2)

 

 

Then,

 

x = L(x₁) + L(x₂) = [(1/4)x + 1] + [(1/6)x + (3/2)]

 

x = [(1/4)(6) + 1] + [(1/6)(7) + (3/2)]

 

x = (3/2) + 1 + (7/6) + (3/2)

 

x = (9 + 6 + 7 + 9) / 6

 

x = (18 + 13) / 6

 

x = 31 / 6

 

x = 5 1/6

 

 

This number is between 5 and 6.