Use the linear approximation to estimate Δ𝑓=𝑓(3.3)−𝑓(3) for 𝑓(𝑥)=6/(1+𝑥^2). Estimate actual change (give answer to six decimal places). Compute the linear approximation error and error %.

Answer 1

f(3) = 6/10 = 0.6; f'(x) = -12x/(1 + x²)²; f'(3) = -36/100 = - 0.36;

L(x) = 0.6 - 0.36(x - 3) = - 0.36x + 1.68; Δf ≈ L(3.3) - L(3) = - 0.36·3.3 + 1.68 - 0.6 = - 0.108

Exact value of Δf = f(3.3) - f(3) = 0.504626 - 0.6 = - 0.095374.

Error: Δ = I- 0.108 + 0.095374I = 0.012626;

% error: δ = Δ/IΔfI·100% = (0.012626/0.095374)·100% = 13.238409%