Question $1.$ Use MATLAB built$-$in functions to calculate sum, average, maximum, and minimum, all in one

function, for an array of numbers. The array of numbers should be input by user:

$-$ Instructions:

$1.$ Create a function named $$calculateStats$$ that takes a vector as input and returns the sum, average,

maximum, and minimum.

$2.$ Write a script that prompts the user to input a list of numbers, stores the numbers in a vector, and

then passes the vector to the $$calculateStat$$ function to display the results.

$3.$ The script should handle errors for invalid inputs $($e$.$g$.,$ non$-$numeric entries$).$

$-$ Guidelines for Report: Explain the purpose of each function, provide a flowchart of the algorithm, and

discuss any issues encountered.

Question $2.$ Create a MATLAB Script for Plotting Functions using $$fplot$,$ ask the user to enter a function $($e$.$g$.,$

sin$($x$)$ and x$^2).$ Specify the range for input x$,$ put title, and label x and y axes; finally, for better

illustration the plot should be in grid form.

Steps:

$-$ Prompt the user to input a function in the form of a MATLAB expression $($e$.$g$.,$sin$($x$),$exp$($x$),$

etc.$).$

$-$ Ask the user to specify the range of $$x$$ values.

$-$ Plot the function over the specified range using the $$fplot$$ function.

$-$ Guidelines for Report: Describe how to input functions in MATLAB, discuss limitations $($e$.$g$.,$

unsupported functions$),$ and include plots for multiple examples.

Question $3.$ Implement Fibonacci Sequence once with $$for loop$$ and once with $$while loop$,$ first ask the user to

enter the number of Fibonacci numbers and display the result.

$-$ Instructions:

$1.$ Write a script that generates the first $$n$$ Fibonacci numbers, where $$n$$ is specified by the user.

$2.$ Include options for using a $`$for$`$ loop and a $`$while$`$ loop. Compare the implementations.

$-$ Guidelines for Report: Explain how different loop structures work, include the scripts for both

approaches, and discuss the differences in performance or readability.

Question $4.$ Use a $$for loop$$ and conditional statement $$if$$ to create a script to find numbers divisible by both $3$

and $5$ in a given range $($e$.$g$.$:$100$ to $300)$; display the result using $$disp$(\text{'}$Numbers divisible by both $3$ and $5$

are:$\text{'})$;$$

$-$ Instructions:

$1.$ Prompt the user to specify a range $($e$.$g$.,$ from $$a$$ to $$b$).$

$2.$ Find and display all numbers within this range that meet the divisibility criteria.

$-$ Guidelines for Report: Describe the use of conditional statements for divisibility checks, include

results for various ranges, and discuss how different ranges affect the performance.

Question $5.$ a$)$ Write the scripts for the Bisection, Newton$-$Raphson and Secant Methods; start with the following

functions

bisectionRoot $=$ bisectionMethod$($f$,2,3,1$e$-6,100)$;

newtonRoot $=$ newtonRaphson$($f$,$ df$,2\mathrm{.}5,1$e$-6,100)$;

secantroot$=...$

for example f can be the following function

f $=$ @$($x$)$ x$^3-4*$x $-9$;

df $=$ @$($x$)3*$x$^2-4$;

$-$ Instructions: $1.$ Write a function for the Bisection method that accepts an initial interval, tolerance, and maximum

iterations.

$2.$ Implement a function for the Newton$-$Raphson method, requiring an initial guess and the derivative

of the function.

$3.$ Test these functions with the equation $$f$($x$)=$ x$^3-4$x $-9.$

$-$ Guidelines for Report: Provide the theoretical background of each method, include the MATLAB

code, and discuss the convergence behavior of these methods.

b$)$ For performance comparison, create a script that compares the results of these methods for different

functions.

$-$ Instructions:

$1.$ Create a script that calls these functions for various test cases, including functions with different

characteristics $($e$.$g$.,$ multiple roots, steep gradients$).$

$2.$ Measure the number of iterations and computational time.

$-$ Guidelines for Report: Present the results in tables and plots, explain why one method may outperform

the other for certain functions, and discuss the limitations$.$

Question $6.$ Develop an M$-$file function based on $$Gauss elimination with partial pivoting$$ to implement Gauss

elimination with partial pivoting. Modify the function so that it computes and returns the determinant

$($with the correct sign$)$ and detects whether the system is singular based on a near$-$zero determinant. For

the latter, define $$near$-$zero$$ as being when the absolute value of the determinant is below a tolerance

$($e$.$g$.$: $1105.)$ When this occurs, design the function so that an error message is displayed, and the

function terminates.

Here is the functions first line:

function$[$x$,$D$]=$GaussPivotNew$($A$,$b$,$tol$)$

where D$=$the determinant and tol$=$the tolerance, tol$=1105.$

Guidelines for Report: Explain the purpose of $$partial pivoting$$ method, and discuss any issues

encountered.

Conclusion

For each of the problems, write short conclusion.