i$)$ A vertical force $P=a_{k}N$ acts on the end of a cantilever beam with a rectangular cross$-$section width of $a_{c}m$ and a height that changes linearly. Divide the beam length into $20$ equal parts and calculate the deflection and rotation at the midpoint of the beam using the finite difference method. $(20$ points$)L=4m,h_0=30cm,h_L=15cm,E=10$GPa

$h_{{?}_{?}}(0)$

ii$)$ When the loading and support conditions are as shown in the figure $($uniformly distributed load and simply supported at the right end$),$ find the values and locations of the maximum deflection, maximum positive rotation, maximum positive moment, and maximum positive shear force. $(30$ points$)$ Again, divide the system into $20$ equal parts for your calculations.

You may use electronic spreadsheets for your calculations. However, you must also write clearly in your assignment what you did, how you did it$,$ and why you did it$.$

a$=27$