This homework shows effects $of$ errors caused because computer representations $of$ numbers have finite precision. $[It$ uses $an$ example from numerical differentiation, which $is$ covered after the midterm.$]In$ numerical differentiation, the approximate first derivative $of$ a continuous function $f$ can $be$ calculated from values $at$ two points using the formula $f^{\text{'}}$~~$f^{\text{'}}=\frac{f(x+h)-f(x)}{h}$ PART $1$ Where the notation $\frac{?}{?}$ means the first derivative and $his$ the distance between the two points. The $absolute$ difference between $f^{\text{'}}$ and $f^{\text{'}}isAD=|f^{\text{'}}(x)-f^{\text{'}}|=|f^{\text{'}}(x)-\frac{f(x+h)-f(x)}{h}|$ and this gives $us$ some idea $of$ the error $in$ approximating the derivative using $f^{\text{'}}$ instead $of{f}^{\text{'}}.A.[2$ points$]$ Set $x=\frac{}{3}$ and say that $fis$ the $sin$ function. Write down $an$ appropriate expression for the $ADin$ this case. $B.[3$ points$]$ Now, write a python function that can compute the $AD$ for any value $ofxorh.C.[3$ points$]$ Using a suitable graph, show the output $of$ the function with $x$ fixed $at\frac{}{3}$ and with values $ofh$ ranging from ${10}^{-1}to{10}^{-18}.$ PART $2An$ expression for $an$ upper $limitof$ the $AD$ can $be$ found from a Taylor series approximation. This $isIJ=h|f^{\text{'}\text{'}}(x){|}_{?}$

$A.[2$ points$]$ Set $x=\frac{}{3}$ and say that $fis$ the $sin$ function. Write down $an$ appropriate expression for the

$ADin$ this case.

$B.[3$ points$]$ Now, write a python function that can compute the $AD$ for any value $ofxorh.$

$C.[3$ points$]$ Using a suitable graph, show the output $of$ the function with $x$ fixed $at\frac{}{3}$ and with values $ofh$

ranging from ${10}^{-1}to{10}^{-18}.$

PART $2$

$An$ expression for $an$ upper $limitof$ the $AD$ can $be$ found from a Taylor series approximation. This $is$

$UL=\frac{h|f^{\text{'}\text{'}}(x)|}{2}$

where $"$ denotes the exact second derivative $off.$

$A.[3$ points$]$ Calculate and plot the expression $ULon$ top $of$ your estimate for $ADatx=\frac{}{3},$ using the

same range $ofh$ values.

$B.[2$ points total$]$ Describe where $on$ the graph $UL$ and $AD$ plots are different. $If$ they don't look different

make sure you are plotting enough points.

I. $[1$ points$]Is$ there a part $of$ the graph where $ULis$ larger than $AD?Ifso,$ over which range?

$II.[1$ points$]Is$ there a part $of$ the graph where $ULis$ smaller than $AD?Ifso,$ over which range?

$C.[2$ points$]Is$ there $an$