For a simply supported beam under a point load in the mid$-$span, the maximum deflection occurs at the mid$-$point of the beam. It can be calculated through the equation below:

Where $P=$ a point load $(N),L=$ length $(m),E=$ the modulus of elasticity $(\frac{N}{m^2}),$ and $I=$ the moment of inertia $(m^4).$

Assume that in the field you used some sensors$/$equipment to take the measurements listed in the table below. Measurements for each parameter are obviously subject to some errors, as mentioned in the last column.

a$)$ Use the Taylor series expansion to find the error propagation in maximum deflection.

b$)$ Based on these measurements, determine the possible range of maximum deflection.

$\backslash$table$[[$Parameter$,$Value,Error in measurement,Value$],[tilde(P),800,$widetilde$(P),50],[tilde(L),5,$widetilde$(L),0\mathrm{.}05],[tilde(E),{7\mathrm{.}5}^{**}{10}^9,$widetilde$(E),5^{**}{10}^2$