Hi How are you? This homework you have to use the matlab.And here the questions

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ENGR 328 Homework 3

Due: 02/07/2019 @ 2:00 pm

Please upload a PDF of your homework including generated plots and any.m files you created.

1. The following infinite series can be used to approximate ex:

é? =1+x+

3

+ +..

3!

+

n!

(a) Prove that this Maclaurin series expansion is a special case of the Taylor series expansion

with xi -0 and h=x.

(b) Use the Taylor series to estimate f(x)= eat Xiti = 1 for x = 0.25. Employ the zero-, first-,

second-, and third-order versions and compute the lel for each case,

2. Use zero-through third-order Taylor series expansions to predict f(3) for

f(x) = 25×2 – 6x + 7x – 88

using a base point at x = 1. Compute the true percent relative error for each approximation.

3. Use a centered difference approximation of O(h?) to estimate the second derivative of the

function examined in problem 2. Perform the evaluation at x = 2 using step sizes of h = 0.2 and

0.1. Compare your estimates with the true value of the second derivative. Interpret your results

on the basis of the remainder term of the Taylor series expansion

4. Consider the function f(x)= x3 – 2x + 4 on the interval [-2, with h=0.25. Use the forward,

backward, and centered finite difference approximations for the first and second derivatives so as

to graphically illustrate which approximation is most accurate. Graph all three first-derivative

finite difference approximations along with the theoretical, and do the same for the second

derivative as well.

5. Develop a well-structured MATLAB function to compute the Maclaurin series expansion for

the sine function as described below.

sin x=x-

+..

+

3!

5!

Pattern your function after the one for the exponential function below.

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5. Develop a well-structured MATLAB function to compute the Maclaurin series expansion for
the sine function as described below.
sin x= x -
5!
Pattern your function after the one for the expo
function below.
function [fx,ea, iter] = Iter Meth(x, es,maxit)
% Maclaurin series of exponential function
% [fx,ea, iter] - Iter Meth(x,es,maxit)
% input:
% x= value at which series evaluated
% es = stopping criterion (default = 0.0001)
% maxit=maximum iterations (default = 50)
% output:
% fx = estimated value
% ea = approximate relative error (%)
% iter = number of iterations
% defaults:
if nargin < 2 isempty(es), es = 0.0001;end
if nargin
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