Write a script to compute machine $($about $1)$ using a while loop. Your code should

output the value of $$ along with a message explaining what $$ means. For extra credit use the

IEEE standard for double precision given in class to show by hand that your computed value is

correct.

$2.(25$ pts$)$ The following algorithm calculates a value similar to machine $$ called machine delta.

$1.$ Set $=1\mathrm{.}0$

$2.$ While $>0$ repeat

a$.=$

b$.=/2$

$3.$ Output $$

$4.$ Stop

Write a script to find and print machine delta. Explain what $$ is and why it is so different than

machine $.$ For extra credit use the IEEE standard for double precision given in class to show by

hand that your computed value is correct.

$3.(50$ pts$)$ For this problem you want to estimate the derivative of the function $=()$ at $$

$=.$ See Lab$01$ for the definition of $().$ Use both the forward difference,

$(0)=(0+)(0)$

$+()$

and the centered difference

$(0)=(0+)(0)$

$2+(2)$

For the step sizes, let $=1/21$

$,$ with $=1,2,...,.$ For $=10,$ find and save the absolute

true errors

$|,|=|()()|$

for each difference approximation and step size.

a$.$ Find the true value by hand using the definition of $()$ given in Lab$01.$

b$.$ Plot the log of the absolute error versus the log of $$ for both the forward and the

centered approximation on the same set of axes. Do not use Matlab$$s loglog command.

c$.$ Add a title, axes labels, and a legend.

d$.$ Use Matlab$$s $$Basic Fitting Tool$$ to find the slope, for each difference approximation.

e$.$ Report the slopes you find and explain their values mathematically. Hint: If an error is

$()$ then it is proportional to $,$ while if it is $(2),$ it is proportional to $2$

$.$

f$.$ Finally, experiment with $$ to determine the optimum step size, $,$ for each difference

approximation, that best balances the truncation and the round$-$off errors.