(2) (a) Sketch and give explicit mathematical expressions for all the shape functions \(\hat{\phi}_i(\xi)\) on
the master element \(\hat{\Omega} = [-1, 1]\) for the case in which \(\hat{\Omega}\) is a fourth order (polynomial)
element.
(b) Consider two elements in some given FE mesh: a quadratic element \(\Omega_2 = [4mm, 5mm]\)
and a fourth order element \(\Omega_7 = [10mm, 12.5mm]\). Derive the mathematical expres-
sions for the third shape function \(\phi_3^e(x)\) of each of these two elements (i.e. e = 2 and
e = 7). For element \(\Omega_2\), you can use the expressions for the quadratic shape functions
\(\hat{\phi}_i(\xi)\), i = 1, 2, 3, of the quadratic master element \(\hat{\Omega}\) that were introduced in class as
an example. Use MATLAB to print the graphs of these two shape functions. Make
sure your graph is one worthy of an engineer, i.e. proper labels, titles, etc..