Question Details

[answered] MATH125 FALL 2016 COMPUTER ASSIGNMENT DUE: In particular, y


I need help writing these questions in code according to Matlab (mainly questions 3 and 4).?


MATH125

 

FALL 2016

 

COMPUTER ASSIGNMENT

 

DUE: NOV 17 In this assignment we use MATLAB to perform numerical computations and generate graphics. In order to complete this assignment, you need to know the basic

 

syntax of MATLAB (to be covered in the discussion class on Nov 3). In particular,

 

you need to know how to define and plot functions and to write simple loops. Your

 

submitted assignment should contain all codes and outputs of the programs, as well

 

as necessary explanation. While you are encouraged to discuss with fellow students,

 

your must write your codes yourself and write up your own work.

 

1. (Warm up) Consider the function f (x) = sin x. Also consider the following

 

polynomials:

 

p1 (x) = x

 

x3

 

6

 

x5

 

x3

 

+

 

p5 (x) = x ?

 

6

 

120

 

p3 (x) = x ? These polynomials are approximations of the sine curve. Plot f , p1 , p3 and

 

p5 on the same graph over the interval [??, ?].

 

2. (Simple loop, complicated behaviors) The computer allows us to simulate

 

the dynamics of many systems (e.g. weather). As an example, write a loop

 

that performs the following computation:

 

(i) Start with a number x0 in (0, 1).

 

(ii) Given xn , define xn+1 by xn+1 = 3.6xn (1 ? xn ). For example, x1 =

 

3.6x0 (1 ? x0 ), x2 = 3.6x1 (1 ? x1 ), and so on.

 

(iii) Repeat (ii) until we get x100 .

 

Plot a single output sequence (xn versus n for n = 0, 1, 2, . . . , 100) for your

 

choice of x0 in (0, 1). You will see that the output changes a lot even if

 

you change the condition x0 just a little bit. This is a classic example of

 

chaotic behavior.

 

3. (Method of bisection) The method of bisection is a method to find a root

 

of the equation f (x) = 0, where f is continuous. It is based on the intermediate value theorem. We will use the method as described here:

 

https://en.wikipedia.org/wiki/Bisection method#The method

 

The input of the method is the following: a continuous function f , and an

 

interval [a, b] for each f (a) and f (b) have opposite signs.

 

(a) Consider the cubic polynomial f (x) = x3 ? x ? 2 and [a, b] = [1, 2]

 

(this is the example in Wikipedia). Try to reproduce the table there,

 

and find an approximate root x? of f (x) = 0 in [1, 2] which is accurate

 

up to 2?10 .

 

Date: October 25, 2016.

 

1 2 MATH125 (b) Consider the function f (x) = sin x?x?0.5, and [a, b] = [?2, ?1]. Run

 

the method of bisection with 20 iterations. For each iteration, record

 

the value of the midpoint as well as the value of f at the midpoint.

 

4. (Numerical integration) In this problem we compute the Riemann sums

 

corresponding to the definite integral

 

Z 1p

 

1 ? x2 dx.

 

0 ?

 

The exact value of this integral is ?4 . Write f (x) = 1 ? x2 .

 

(a) Write a script that performs the following:

 

? Input: a positive integer n, such as n = 100.

 

Pn

 

? Output: the value of the Riemann sum i=1 f (xi ) n1 , where xi

 

is chosen according to the midpoint rule.

 

(b) Compute the Riemann sum for n = 5, 10, 25, 50, 100, 200, 500, 1000.

 

Arrange your results in a table form.

 

(c) Recall that the exact value of the integral is ?4 . For each n, let In be

 

the Riemann sum with n terms and ? En = ? In 4

 

be the error of the approximation. Plot En against n, and log10 En

 

against log10 n. Briefly comment on the results.

 


Solution details:
STATUS
Answered
QUALITY
Approved
ANSWER RATING

This question was answered on: Sep 18, 2020

PRICE: $15

Solution~0001013685.zip (25.37 KB)

Buy this answer for only: $15

This attachment is locked

We have a ready expert answer for this paper which you can use for in-depth understanding, research editing or paraphrasing. You can buy it or order for a fresh, original and plagiarism-free copy from our tutoring website www.aceyourhomework.com (Deadline assured. Flexible pricing. TurnItIn Report provided)

Pay using PayPal (No PayPal account Required) or your credit card . All your purchases are securely protected by .
SiteLock

About this Question

STATUS

Answered

QUALITY

Approved

DATE ANSWERED

Sep 18, 2020

EXPERT

Tutor

ANSWER RATING

GET INSTANT HELP/h4>

We have top-notch tutors who can do your essay/homework for you at a reasonable cost and then you can simply use that essay as a template to build your own arguments.

You can also use these solutions:

  • As a reference for in-depth understanding of the subject.
  • As a source of ideas / reasoning for your own research (if properly referenced)
  • For editing and paraphrasing (check your institution's definition of plagiarism and recommended paraphrase).
This we believe is a better way of understanding a problem and makes use of the efficiency of time of the student.

NEW ASSIGNMENT HELP?

Order New Solution. Quick Turnaround

Click on the button below in order to Order for a New, Original and High-Quality Essay Solutions. New orders are original solutions and precise to your writing instruction requirements. Place a New Order using the button below.

WE GUARANTEE, THAT YOUR PAPER WILL BE WRITTEN FROM SCRATCH AND WITHIN YOUR SET DEADLINE.

Order Now