please complete these assignments the direction are fairly straightforward1 Newton’s method Newton’s...

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please complete these assignments the direction are fairly straightforward1 Newton’s method Newton’s method is a root-finding algorithm. The way it works is pretty simple. Let f be a differentiable function for all x ∈ R. Pick a starting point, x0. Then we compute x1 = x0 − f(x0) f ′(x0) . If f(x1) is not close enough to 0, the process is repeated as xn+1 = xn − f(xn) f ′(xn) until sufficiently precise value is reached. Read more about this here 2 Final Project part 1 Part 1 of final project is to create a script that uses Newton’s method to find a root of a polynomial. You can use either MATLAB or Python and you can choose to do it numerically or symbolically. However, if you choose to use one software, you will need to use the other one for part 2. Similarly, if you choose to do part 1 numerically, then you will need to do part 2 symbolically. My recommendation is to use MATLAB and to do numerically. Here’s what I expect from your script. 1. Start by asking the order, say n, of the polynomial we want to work with. If the input is not a positive integer, then print ‘n must be a positive integer’ and ask for the input again. This must be repeated until an acceptable value is provided. 2. Ask for the coefficients of the polynomial. For i  10 10 or f ′(xm)  10 −14 and f ′(xm) = 0, then stop the procedure and print something like ‘The method failed. Pick another starting point.’. 8. If you found xm as desired, then print ‘The solution to the given polynomial is xm.’. (xm needs to be the actual value) Extra credit: Create a plot that illustrates what you did to find the solution. 1 https://en.wikipedia.org/wiki/Newton%27s_method 	Newton's method 	Final Project part 1   1 Taylor Series A Taylor series is a representation of a function as an infinite sum of polynomials. Given a differentiable function f(x), the Taylor series centered at c is ∞∑ n=0 f (n)(c) n! (x− c)n. As we learned before, a Taylor series representation is a great way to estimate a function. Of course, since we only want to estimate, we will deal with finite Taylor polynomial rather than a Taylor series. m-th degree Taylor polynomial of a function f(x) centered c is m∑ n=0 f (n)(c) n! (x− c)n. 2 Final Project part 2 In this project, you will write a script that will plot both a function and m-th degree Taylor polynomial of the function. If you used MATLAB(or Python) for part 1, then you must use Python(or MATLAB). Similarly, if you did part 1 numerically(or symbolically), then you must do part 2 symbolically(or numerically) 1. The script must take a function(f), degree of a Taylor polynomial(m) and the center(c). If not specified take m = 10 and c = 0. 2. If m is not a positive integer, then print ‘n needs to be a positive integer.’ 3. If c is not a real number, then print ‘c needs to be a real number.’ 4. Create a figure that prints both the function f and the Taylor polynomial. Make sure x-axis displays from c− 10 to c + 10. Also, two lines must be different color and include a legend to show which one is which. 5. If you are doing this numerically, estimate the derivatives using ∆x = 10−6. If you are doing this symbolically, calculate the exact values using diff command. 6. Make sure graphs look smooth. Here are some tips! 1. If you are using Python, then you can import math package and use math.factorial(n) to calculate n!. 2. If you are doing symbolically, make sure you feed a symbol instead of a function. That is, you need to provide sp.exp(x) as an argument rather than f(x) which is a function. (You could use a function as an argument but then you will need to change that to symbol.) 1 	Taylor Series 	Final Project part 2   1 Final project part 3 Write a script that defines two new classes, circle and rectangle. Class circle should satisfy the followings. 1. Take (x, y) coordinate of the center as arguments. (0, 0) should be the default value. 2. Take the radius of the circle as an argument. r = 1 should be the default value. 3. Create a method that calculates the area of the circle. 4. Create a method that calculates the circumference of the circle. 5. Create a method that determines whether or not the given point (x, y) is inside the circle. Print ‘Given point is in the circle.’ or ‘Given point is not in the circle.’ as result. 6. Extra Credit(5 pts): Create a method that graphs the given circle. Class rectangle should satisfy the followings. 1. Take (x, y) coordinate of the center as arguments. (0, 0) should be the default value. 2. Take the width(x direction) and height(y direction) as arguments. 1 should be the default value for both. 3. Create a method that determines whether or not the rectangle is a square. Print ‘It is a square.’ or ‘It is not a square.’ as result. 4. Create a method that calculates the area of the rectangle. 5. Create a method that calculates the perimeter of the rectangle. 6. Create a method that determines whether or not the given point (x, y) is inside the rectangle. Print ‘Given point is in the rectnagle.’ or ‘Given point is no in the rectangle.’ as result. 7. Extra Credit(5 pts): Create a method that graphs the given rectangle. 1 	Final project part 3   Project 2 September 13, 2021 In mathematics, there are many simple hard problems; problems that look simple but are amazingly hard. Collatz conjecture is one of these problems. So what is Collatz conjecture? Let’s take any positive integer m. If m is even, then we divide it by 2, m1 = m 2 . If m is odd, then we multiply it by 3 and add 1, m1 = 3m + 1. We will repeat this process with m1 to create m2, m3, · · · . The conjecture claims that if we will eventually end up with 1. Let’s look at an example. 1. Starting value: m = 3. 2. m1 = 3 ∗ 3 + 1 = 10. 3. m2 = 10 2 = 5. 4. m3 = 5 ∗ 3 + 1 = 16. 5. m4 = 16 2 = 8. 6. m5 = 8 2 = 4. 7. m6 = 4 2 = 2. 8. m7 = 2 2 = 1. So it takes 7 steps to get 1 if you start from 3. So far, we tested the conjecture for starting values up to 268 and all starting values eventually end up with 1. Now, mathematicians have little to no clue how to show this conjecture is true for all numbers (or show that the conjecture is false). What’s even more interesting is the size of the starting value does not directly tell us the number of total steps needed. For example, if you start with 32, it only takes 5 steps to reach 1. However, if you start with 27, it takes 111 steps to reach 1. In this project, we will investigate Collatz conjecture using MATLAB. 1. Write a script that will take the starting value as an input and test whether it reaches 1. (a) If it takes more than 100,000 steps, the script should stop and print It takes more than 100,000 steps to test the conjecture. (b) If there exists 1

Mathews · Accepted Answer

greynodes/circle.m
classdef circle % classs name is defined as circle
    properties % properties are saved in the file
        xcenter=0;
        ycenter=0;
        radius=1;
    end
    
    methods
        function Area = getArea(newcircle)% function used to create the area of the circle
            %RECTANGLE Construct an instance of this class
            %   Detailed explanation goes here
            Area=(newcircle.radius*3.14*newcircle.radius);
        end
        
        function perimeter = getperimeter(newcircle)% function used to create the perimeter of the circle
            perimeter= 2*3.14*newcircle.radius;
            
            function newcircle =circle(Radius,Xcenter,Ycenter)
            newcircle.

1 Newton’s method Newton’s method is a root-finding algorithm. The way it works is pretty simple. Let f be a differentiable function for all x ∈ R. Pick a starting point, x0. Then we compute x1 = x0 −...

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