$=.!()(+)()$

$.$

a$.[2$ points$]$ Setting x $=0\mathrm{.}25,$ and using the cos function for f$,$ write down an expression for

the AD$.[!$ is the exact derivative of f$,$ and you know that well.$]$

b$.[3$ points$]$ Now write a MATLAB function that can compute the AD for any value of x or h$.$

c$.[3$ points$]$ Using a suitable graph, show the output of your function with x fixed at $0\mathrm{.}25,$ and

with values of h ranging from $10-1$ to $10-18.$

d$.[3$ points$]$ An expression for an upper limit of the AD can be found using a Taylor series.

This is

$="$#$ $!!($&$)$#

$(.$

Where $$ is the exact second derivative of f$.$ Calculate and plot the expression EA on top of the

expression for AD at x $=0\mathrm{.}25,$ using the same range of h values.

e$.[2$ points$]$ Is there a portion of the graph where the EA agrees with or is larger than the AD$?$

Over what range does this happen?

f$.[2$ points$]$ Is there a portion of the graph where the EA is smaller than the AD$?$ Over what

range does this happen?

g$.[5$ points$]$ Why would EA and AD agree in some places and not in others? If there is an h

value where a switch between agreeing and not agreeing happens, is that number important?

h$.[$BONUS $3$ points$]$ Plot the EA and AD comparisons for x $=0\mathrm{.}1$ and $/3.$ Compare and

contrast the three graph sets. How is each pair of EA and AD graphs different, and why?