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

For $f\left(t\right)=\left(\frac{{e}^{t}}{t},4t+\frac{1}{t}\right)$ what is the distance between f(2) and f(5)?
You can still ask an expert for help

• Questions are typically answered in as fast as 30 minutes

Solve your problem for the price of one coffee

• Math expert for every subject
• Pay only if we can solve it

gutinyalk
Step 1
We will substitute first 2 and then 5:
$f\left(2\right)=\left(\frac{{e}^{2}}{2},4\cdot 2+\frac{1}{2}\right)=\left(\frac{{e}^{2}}{2},\frac{17}{2}\right)$
$f\left(5\right)=\left(\frac{{e}^{5}}{5},4\cdot 5+\frac{1}{5}\right)=\left(\frac{{e}^{5}}{5},\frac{101}{5}\right)$
Step 2
Then we will apply the distance formula:
$d=\sqrt{{\left({x}_{2}-{x}_{1}\right)}^{2}+{\left({y}_{2}-{y}_{1}\right)}^{2}}$
$d=\sqrt{{\left(\frac{{e}^{5}}{5}-\frac{{e}^{2}}{2}\right)}^{2}+{\left(\frac{101}{5}-\frac{17}{2}\right)}^{2}}$
$=\sqrt{{\left(\frac{2{e}^{5}-5{e}^{2}}{10}\right)}^{2}+{\left(\frac{101\cdot 2-17\cdot 5}{10}\right)}^{2}}$
$=\sqrt{\frac{4{e}^{10}-20{e}^{7}+25{e}^{4}}{100}+\frac{202-85}{100}}$
$=\frac{\sqrt{4{e}^{10}-20{e}^{7}+25{e}^{4}+117}}{10}$