Consider a function $f$:$[0,1]R$ defined as $f(x)=x^3+3x+d.$ To discretize

this function, uniformly sample the domain at $10$ locations. Now find the numerical first

and the second order differentiation of $f$ from its discretized version. Also represent both

quantities as Af and find A for both cases. Here, the vector $f$ represents discretized

version of $f.$ Compare your results with the analytical results. Here, $d$ represents the

number of characters in your first name.

The gradient of a function $f$:$R^{n}R$ is defined as $gra{d}_{x}f={\left[\begin{array}{cccc}\frac{\text{d}\text{e}\text{l}\text{f}}{del{x}_1}& \frac{\text{d}\text{e}\text{l}\text{f}}{del{x}_2}& \text{c}\text{d}\text{o}\text{t}\text{s}& \frac{\text{d}\text{e}\text{l}\text{f}}{del{x}_n}\end{array}\right]}^{TT}.$ For

example, if $f(x)=c^{TT}x,$ then $gra{d}_{x}f=c.$

The Rodriguez's formula for the rotation of a point $x$ is defined as $Rx=xcos(||v|{|}_2)+$

$(hat(v)x)sin(||v|{|}_2)+$hat$(v)hat(v{)}^{TT}x(1-cos(||v|{|}_2))$ using the axis$-$angle rotation representation,

where hat$(v)$ is the axis of rotation and $||v|{|}_2$ is the angle of rotation.

The general camera is modeled by the below equation that maps a $3$D m$]$point $P_w$ to

a $2$ D pixel $p.$

$\left[\begin{array}{c}n\end{array}\right]$

$m$

$1=\left[\begin{array}{cccccccccc}a& 0& c_x& 0& b& c_y& 0& 0& 1& 0\end{array}\right]$

$0$

$0\left[\begin{array}{ccc}R& -RT& 1\end{array}\right]$

$0^{TT}\left[\begin{array}{c}{P}_w\end{array}\right]$

$1$

$=\left[\begin{array}{ccccccc}l& 0& c_x& b& c_y& 0& 1\end{array}\right]$

$0$

$0\left[\begin{array}{ccccccc}{r}_1& r_2& r_3& r_5& r_6& r_8& r_9\end{array}\right]$

$r_7$

$r_4\left[\begin{array}{cccccccccc}1& 0& 0& -T_x& 1& 0& -T_y& 0& 1& -T_z\end{array}\right]$

$0$

$0\left[\begin{array}{c}{P}_w\end{array}\right]$

$1$

$\left[\begin{array}{c}p\end{array}\right]$

$1=K[R-RT]\left[\begin{array}{c}{P}_w\end{array}\right]$

$1$