The dataset provided tracks the evolution of temperature over the course of a summer

day. The goal is to calculate the trend of the temperature by using different numerical

schemes: forward and centered differences $($CD$)$ of second and fourth order.

Using the formulas from the numerical schemes discussed in class, fill in the dataset

with the calculated temperatures using the fourth$-$order centered difference scheme,

which is described by the following expression:

$\frac{\text{d}\text{e}\text{l}\text{T}}{\text{d}\text{e}\text{l}\text{t}}=\frac{4}{3}(\frac{T_{n+1}-T_{n-1}}{2t})-\frac{1}{3}(\frac{T_{n+2}-T_{n-2}}{4t})+O(t^4)$

Here, $T_n=T(t=nt).$

For $t=10h,$ detail your calculations for each scheme and interpret the results by

considering the sign of the trend.

Employing the Taylor series expansion for the functions $T(t+-t)$ and $T(t+-2t),$

derive the formula $(1)$ for the fourth$-$order centered difference scheme.$\backslash$table$[[,$Time $($t in hour$),$Temp. $($T in deg C$),$Forward $($unit$?),$CD of $2$nd order $($unit$?),$CD of $4$th order $($unit$?)],[0,t0=0,23\mathrm{.}2,$None,None,None$],[1,t1=2,19\mathrm{.}9,$None,None,None$],[2,t2=4,18\mathrm{.}4,$None,None,None$],[3,t3=6,19\mathrm{.}8,$None,None,None$],[4,t4=8,22\mathrm{.}2,$None,None,None$],[5,t5=10,24\mathrm{.}4,$None,None,None$],[6,t6=12,25\mathrm{.}8,$None,None,None$],[7,t7=14,26\mathrm{.}7,$None,None,$],[8,t8=16,27\mathrm{.}8,$None,None,None$],[9,t9=18,27\mathrm{.}9,$None,None,None$],[10,t10=20,27\mathrm{.}3,$None,None,None$]]$