Show your work and use as many words as possible to make sure it is clear what you do and, more

importantly, why. Please submit a single hw$02.$pdf for your work on paper.

$1.$ Convert the binary number $1010\mathrm{.}011$ to the decimal format.

$2.$ Find the standard binary form of $1010\mathrm{.}011.$

$3.$ Convert the decimal number $-13\mathrm{.}15625$ to the binary format.

$4.$ Find the standard binary form of $-13\mathrm{.}15625.$

$5.$ For the $64-$bit representation of $-13\mathrm{.}15625,$ give sign, mantissa, and exponent separately.

$6.$ What is the next $64-$bit number after the binary number $1010\mathrm{.}011?$

$7.$ What is next $64-$bit number before $-13\mathrm{.}15625?$

$8.$ Assuming f $($x$)$ and f $($x$)$ are given functions, use Taylor series

f $($x$0+$ h$)=$ f $($x$0)+$ hf $($x$0)+$ f $($x$0)$h$2/2!+$ f $($c$)$h$3/3!$

which is valid for a c such that x$0<=$ c $<=$ x$0+$h$,$ to derive an approximate formula for the

second order derivative f $$ at the point x$0.$ State explicitly what is the approximation

and what is the error of the formula. Hint: error is what is dropped in the answer.

$$ Now implement your formula in MATLAB for f $($x$)=$ x cos$($x$)$ and f $($x$)=$ cos$($x$)$

x sin$($x$)$ at x$0=\backslash$pi $/2.$ Report the smallest h such that the formula does not suffer from

round$-$offs and preserves its order of convergence $($by the way, what is it$?).$ Submit

hw$02$q$8.$m$.$ Hint: to evaluate the error, you will need to calculate f $($x$)$ by hand.

$2$