use linear approx. to estimate the given number

Question

use linear approx. to estimate the given number

 

3√1001 (cubed root)

 

 

Answer 1

Let f(x) = ³ √x.  Then f'(x) = (1/3)x^-2/3

 

Let a = 1000   f(a) = ³ √a = 10

 

f(x) ≈ L(x) = f(a) + f'(a)(x-a)

 

So, ³ √1001 = f(1001) ≈  L(1001) = f(1000) + f'(1000)(1001-1000)

 

                   ≈ 10 + (1/3)(1000)^-2/3(1)

 

                   = 10 + 1/300

 

                   = 10.003333333 

 

 

Answer 2

10³ = 1000

11³ = 1331

 

I would definitely estimate 10