QUESTIONS

A common engineering problem involves determining the temperature at which a specific chemical reaction reaches equilibrium. Suppose the equilibrium temperature TT $($in degrees Celsius$)$ of a chemical reaction can be described by the equation:

$f(T)=T^3-2T^2+T-5=0$

You are required to find the temperature TT at which this reaction reaches equilibrium using various numerical methods. You will use the interval $1,3$ for all methods. Solve the problem using the following methods, and implement each in MATLAB. Ensure to include all necessary steps, calculations, and code.

a$)(20$ pts$)$ Graphical Method

Plot the function $f(T)$ over the interval $1,3.$

Identify and estimate the root visually.

b$)(20$ pts$)$ Bisection Method

Use the bisection method to find the root of the equation $f(T)=0$ with a tolerance of ${10}^{-6}.$

Detail each iteration step, showing the interval, midpoint, function values, and the updated interval.

c$)(20$ pts$)$ False Position Method

Use the false position method to find the root of the equation equation $f(T)=0$ with a tolerance of ${10}^{-6}.$

Detail each iteration step, showing the interval, function values, and the updated interval.

d$)$ pts$)$ Fixed Point Iteration

Reformulate the equation $f(T)=0$ into a fixed$-$point form $T=g(T).$

Choose an appropriate $g(T)$ and an initial guess $T_0.$

Use the fixed$-$point iteration method to find the root with a tolerance of ${10}^{-6}.$

Detail each iteration step, showing the current guess, next guess, and the difference.

e$)(20$ pts$)$ Comparison and Discussion

Compare the results obtained from the four methods.

Discuss the convergence and accuracy of each method.

Explain the advantages and disadvantages of each method in the context of this problem.

MATLAB Task: Implement all the methods in MATLAB and provide the code and final root estimate. Please avoid using several files instead one $.$m file for all solutions with