Note that, the iso-parametric shape functions are obtained using Lagrange Multipliers. For an Nth degree polynomial approximation, there should be N + 1 shape functions. Note that, the shape functions should have a value unity (eg. 1) at the node that they belong to and zero (eg 0) at the other nodes. For example, a first-order approximation would be given as: u(x) =(x) u(x)+(x) (x-2) where u(x) and u(x) represents the nodal values of the function u at the first and second nodes Here, (x)= and 2(x)= - 3-31 a) If the element extends between coordinates-1 x+1, what are the shape functions in simplified forms (eg, x = -1 and x2 = +1)? b) Show that (-1) = 1 and (+1)=0. Also show that (-1) = 0 and 2(+1)=1 c) If the approximation is extended to second order (quadratic) polynomials as u(x) =(x) u(x)+(x) u(x)+(x) u(x3) Show that with above discussion selection of the base functions as: (x)=(x)=- and (x)= = 41-423-3 1-11-Jy Xy-X, Ty-Tz would be a proper choice (e.g. 1(x) = 1 and 1(x2)=(x)=0 and for the others, the similar rule applies). d) For the quadratic approximation given above and with node locations x =-1, x = 0 and x, = +1 find explicit functions for 1.2 and 3