For the function $y=x^3-2x^2+x,$ calculate the value of $y$ for the following values of $x$ using element$-$by$-$element operations: $-2,-1,0,1,2,3,4.$

For the function $y=\frac{x^2-2}{x+4},$ calculate the value of $y$ for the following values of $x$ using element$-$by$-$element operations: $-3,-2,-1,0,1,2,3.$

The length $|u|($magnitude$)$ of a vector $u=+yj+zk$ is given by $|u|=\sqrt[\phantom{2}]{x^2+y^2+z^2}.$ Given the vector $u=23\mathrm{.}5i-17j+6k,$ determine its length two ways:

$($a$)$ Define the vector in MATLAB, and then write a mathematical expression that uses the components of the vector.

$($b$)$ Define the vector in MATLAB, then use element$-$by element operations to create a new vector with elements that are the squares of the elements of the original vector. Then use MATLAB built$-$in functions sum and sqret to calculate the length. All of these steps can be written in one command.

The unit vector $u_n$ in the direction of the vector $u=+yj+zk$ is given by $u_n=\frac{+yj+zk}{\sqrt[\phantom{2}]{x^2+y^2+z^2}}.$ Determine the unit vector of the vector $u=-8i-14j+25k$ by writing one MATLAB command.

Define $x$ and $y$ as the vectors $x=[1,3,5,7,9]$ and ; Then use them in the following expressions to calculate $z$ using element$-$by$-$element calculations.

$($a$)z=\frac{x{y}^2}{x+y}$

$($b$)z=x(x^2-y)-(x-y{)}^2$

Define $p$ and $w$ as scalars, $p=2\mathrm{.}3$ and define $w=5\mathrm{.}67,$ and, $t,x,$ and $y$ as the vectors $t=[1\mathrm{.}2,3\mathrm{.}4,5],x=[2\mathrm{.}8,2\mathrm{.}5,2\mathrm{.}2,1\mathrm{.}9,1\mathrm{.}6],$ and $y=[4\mathrm{.}7,10\mathrm{.}13,17].$ Then use these variables to calculate the following expressions using element$-$by$-$element calculations for the vectors.

$($b$)S=\frac{p(x+y{)}^2}{yw}+\frac{wxt}{py}$