Use linear approximation and differential to approximate : 14^(1/4)

Question

Use linear approximation and differential toto approximate : 14^(1/4)

Answer 1

Consider the function f(x) = x^1/4. Then you are being asked for f(14). Since 2⁴ = 16, know that f(16) = 2. Also 16 is close to 14. Thus we will use a linear approximation near x=16.

 

Take the derivative to get f'(x) = (1/4)x^-3/4 so f'(16) = (1/4) 16^-3/4 = (1/4) 2^-3 = (1/4)(1/8) = 1/32

 

Then recall that if we know that f is differentiable in a neighborhood of a and we know f(a) and f'(a) then for any b close enough to a, we can approximate f(b) linearly as f(a)+(b-a)f'(a)

 

Here we have f(14) ≈ f(16)+(14-16)f'(16) = 2 - (2)(1/32) = 2 - 1/16 = 1.9375

 

Note how this is within 0.2% of the true value f(14) = 14^1/4 = 1.934336420267669...