I have done problem one, and I don't know how to move on. Please help.
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
p1 (x) = x
p5 (x) = x ?
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
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
1 ? x2 dx.
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.
? 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.
This question was answered on: Sep 18, 2020
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