Local linear approximations.?

Question

The side of a square is measure to be 10 ft., with a possible error of +/- .1 ft. (a) Use differentials to estimate the error in the calculated area. (b) Estimate the percentage errors in the side and the area. Please show work.

Answer 1

A = S^3

∆A = ∂A/∂S * ∆S

∆A = 3*S^2 * ∆S

∆S = ±0.1

∆A = ±3*10^2*0.1 = ±30 ft^2

For percentage area use

%E(side) = 100*∆S / S = ±100*0.1 / 10 = ±1%

%E(area) = 100*∆A / A = S = 100 * 3*S^2 * ∆S / S^3 = 100 * 3 * ∆S/S

so %E(area) = 3 * %E(side)

%E(area) = ±3%