# For <mstyle displaystyle="true"> f ( t )

For $f\left(t\right)=\left(\mathrm{ln}t-{e}^{t},\frac{{t}^{2}}{{e}^{t}}\right)$ what is the distance between f(2) and f(4)?
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Mihevcekd
Step 1
The distance between two points $P1\left(x1,y1\right)$ and $P2\left(x2,y2\right)$ is given as $\sqrt{{\left(x1-x2\right)}^{2}+{\left(y1-y2\right)}^{2}}$
Step 2
with $f\left(t\right)=\left(\mathrm{ln}\left(t\right)-{e}^{t},\frac{{t}^{2}}{{e}^{t}}\right)$ we get
$f\left(2\right)=\left(\mathrm{ln}\left(2\right)-{e}^{2},\frac{4}{{e}^{2}}\right)$
$f\left(4\right)=\left(\mathrm{ln}\left(4\right)-{e}^{4},\frac{16}{{e}^{4}}\right)$
so the distance is given by $\sqrt{{\left({e}^{4}-{e}^{2}-\mathrm{ln}\left(2\right)\right)}^{2}+{\left(\frac{4}{{e}^{2}}-\frac{16}{{e}^{4}}\right)}^{2}}$