Using Linear Approximation, estimate Δf for a change in x from x=a to x=b. Use the estimate to approximate f(b), and find the error using the calculator. f(x)=x1/3, a=8, b=10.

Answer 1

f(x) = x^1/3; Δf ≈ df = f'(x)dx = f'(a)(b - a); f'(x) = 1/3 x^-2/3; a = 8, b = 10; Δx = 10 - 8 = 2; f'(8) = 1/3(8^{- 2/3}) = 1/12;

So, Δf·≈ 1/12·2 = 1/6; f(10) ≈ f(8) + df = 8^1/3 + 1/6 = 2 + 1/6 = 2.1667

Error = I10^1/3 - 2.1667I = I2.1544 - 2.1667I = 0.0123