Linear Approximation

Question

Use differentials or Linear Approximation to approximate the value shown (note each method will give the same result for your chosen x = a)

Cube root of 26.1

(Round to four decimal places)

Answer 1

if y = f(x) = x^(1/3) Then dy = (1/3)x^(-2/3) dx here let x = 27 and then delta x = 26.1 - 27 = -.9

or stated another way f(x₂) - f(x₁) = f'(x₁)(x₂ - x₁) or f(26.1) - f(27) = (1/3)(27)^(-2/3)(26.1-27)

f(26.1) = -.0333 + 3 = 2.9666

The taylor's method says f(x) = f(x_o) + f'(x_o)(x-x_o) which is basically the same as above. x = 26.1, x_o = 27

or f(26.1) = f(27) + (1/3)(27)^(-2/3)(26.1-27) = 3 - .0333 = 2.9666