$1-(30\%)$ A steel beam $(E=200$GPa;$I=500105mm4)$ with a length of $(10000+x^{**})mm$ is subjected to

a uniformly distributed load of $w=5\mathrm{.}0k\frac{N}{m}($see figure below$).$ You want to find the point where the

deflection $(y)$ of the beam is maximum $($the point where $y^{\text{'}}=d\frac{y}{d}x=0)$ as well as the maximum value of

deflection. The elastic deflection function is given below:

$y=\frac{w}{nr}(L^{3}x-3L{x}^3+2x^4)$

$1$GPa$=1k\frac{N}{m}m2$

a$.$ Differentiate the function y to obtain the derivative $(y^{\text{'}})$ and substitute the given values. Ensure your

units are consistent.

b$.$ Use the graphical method and plot the functions $y$ and $y^{\text{'}}$ between $x=0$ and $x=L.($use the step size of

$100mm).$

c$.$ Based on the plotted graphs, approximate the value of $x$ that satisfies $y^{\text{'}}=0.$ Explain why this value of $x$

is related to maximum deflection.

d$.$ Use the Bisection Method to obtain a three $(3)$ significant digit approximation of the position of the

maximum deflection. Choose $x0=3500mm$ and $x1=5500mm.$ Use Excel and demonstrate a sample

calculation for the first iteration.

e$.$ Use the Regula Falsi method to get a three $(3)$ significant digit approximation of the position of the

maximum deflection. Choose $x0=3500mm$ and $x1=5500mm.$ Use Excel and demonstrate a sample

calculation for the first iteration.

f$.$ Compare the methods in $d$ and $e$ to see which one can find the root with more significant figures more

quickly.