The force-deflection characteristic of a spring is described by \(F=500 x+2 x^{3}\), where the force \((F)\) is in newton and the deflection \((x)\) is in millimeters. Find

(a) the linearized spring constant at \(x=10 \mathrm{~mm}\) and

(b) the spring forces at \(x=9 \mathrm{~mm}\) and \(x=11 \mathrm{~mm}\) using the linearized spring constant. Also find the error in the spring forces found in (b).