Exercise

$2$

$.$

$$A function and its approximations

$($

$3$

$0$

$$points

$)$

Consider

$$

$$

$=$

$0$

$\backslash$infty

$$

$3$

$$

$($

$3$

$$

$)$

$!$

$,$

$$that is

$,$

$$the power series obtained by keeping every third term in the

Taylor expansion of exp

$($

$$

$)$

$.$

$$This power series corresponds to the true function

$$

t

r

u

e

$($

$$

$)$

$=$

$1$

$3$

$($

$$

$$

$$

$($

$$

$)$

$+$

$2$

exp

$($

$-$

$$

$2$

$)$

$$

$$

$$

$($

$3$

$$

$2$

$)$

$)$

For any possible integer

$$

$,$

$$define the

$$

$$term power series approximation

$$

a

p

p

r

o

x

$($

$$

$,$

$$

$)$

:

$=$

$$

$$

$=$

$0$

$$

$-$

$1$

$$

$3$

$$

$($

$3$

$$

$)$

$!$

$.$

Submit a MATLAB code

$($

$.$

m file

$)$

$$named YourlastnameYourfirstnameHW

$2$

p

$2$

$.$

m that plots

$2$

$$

$$line plots for the functions

$$

t

r

u

e

$($

in black solid line

$)$

$$versus xin

$[$

$-$

$5$

$,$

$5$

$]$

$($

in the horizontal

axis

$)$

$.$

$$In the same figure window, plot

$$

a

p

p

r

o

x

$($

$$

$,$

$$

$)$

$$for

$$

$=$

$2$

$($

in red dashed line

$)$

$,$

$$

$=$

$3$

$($

in

green dashed line

$)$

$,$

$$

$=$

$4$

$($

in blue dashed line

$)$

$.$

We shared a starter code YourlastnameYourfirstnameHW

$2$

p

$2$

$$

$$inside the the CANVAS File

section Files

$/$

MATLAB Files

$/$

Homework

$2$

$.$

$$You only need to complete lines

$1$

$1$

$$and

$2$

$0$

$$in that

starter code, then rename the file appropriately with your first and last names.

Hint: Look up sqrt

$,$

$$exp$,$ cos

$,$

$$sum$,$ factorial and power

$($

$$

$$

$$

$($

$.$

$)$

$)$

$$in MATLAB documen

$-$

tation. Also, intuition suggests that as

$$

$$increases$,$

$$

a

p

p

r

o

x

$$should get closer to

$$

t

r

u

e

$.$