cmake plots for comparing the crank-nicolson scheme with the forward and backward euler schemes in the same test problem as in exercise filenamegrowth_cn py exercise understand finite differences via taylor series the taylor series around point can for function xbe written xd ac ax / ax ac dx dx ax / dx di ax / is dx for function of timeas addressed in our ode problemswe would use instead of instead of xand time point tn instead of au td tn tn tn / tn tn dt dt tn tn / dt di tn tn / is dt aa forward finite difference approximation to the derivative areads tn tn tn we can justify this formula mathematically through taylor series write up the taylor series for tn (around tn as given above)and then solve the expression with respect to tn identifyon the right-hand sidethe finite difference approximation and an infinite series this series is then the error in the finite difference approximation if is assumed small ( < ) will be much larger than which will be much larger than and so on the leading order term in the series for the errori the error with the least power of is good approximation of the error identify this term brepeat afor backward differenceu tn tn tn tt this timewrite up the taylor series for tn taround tn solve with respect to tn /and identify the leading order term in the error how is the error compared to the forward difference