How to find tangential and normal components of acceleration at a given time (example problem)

An example problem involving the tangential and normal components of acceleration is solved algebraically, numerically (MATLAB) and graphically (Calcplot3d).

Problem statement:
An object in motion is described with the position function r(t)=〈t^3+t+1, t^2−2t+1〉. Compute the object's acceleration vector, tangential component of acceleration, and normal component of acceleration at time t=1/2.