Problem $1$: Answer all questions below for a system with a $4-$bit mantissa and a$3-$bit signed exponent.

Machine number

Q$1(20$ points$)$: Convert the decimal number, $0\mathrm{.}150$ to floating$-$point binary.

Q$2(20$ points$)$: Convert the desimal number, $0\mathrm{.}1667$ to floating$-$point binary.

Round$-$off errors

Q$3(30$ points$)$: Estimate the true relative error $($fraction$)$ for the following addition.

$0\mathrm{.}150+0\mathrm{.}1667$

Problem $2(30$ points$)$: Use Taylor's expansion with a second$-$order approximation to evaluate the first derivate of a function, $f^{\text{'}}(x).$

Determine the forward finite difference for $f^{\text{'}}(x)$ and the order of the truncation error.