 osi4a2nxk

2021-12-03

1) Given the vector-valued function $r\left(t\right)=⟨{e}^{-t},-e{t}^{-t},t\mathrm{ln}t⟩$
a- Evaluate $\underset{t⇒0}{lim}r\left(t\right)$ and Evaluate $\underset{t⇒\mathrm{\infty }}{lim}r\left(t\right)$
b- Find r '(t) and evaluate r '(0)
c- Find r''(t)
d- Evaluate $\int r\left(t\right)dt$ oces3y

a- Evaluate the limits
$\underset{t⇒0}{lim}r\left(t\right)=\underset{t⇒0}{lim}⟨{e}^{-t},-t{e}^{-t},t\mathrm{ln}t⟩$
$=⟨\underset{t⇒0}{lim}{e}^{-t},\underset{t⇒0}{lim}-t{e}^{-t},\underset{t⇒0}{lim}t\mathrm{ln}t⟩$
$=⟨{e}^{0},\left(0\right){e}^{0},\left(0\right)\mathrm{ln}0⟩$
$=⟨1,0,0⟩$
$\underset{t⇒\mathrm{\infty }}{lim}r\left(t\right)=\underset{t⇒\mathrm{\infty }}{lim}⟨{e}^{-t},-t{e}^{-t},t\mathrm{ln}t⟩$
$=⟨\underset{t⇒\mathrm{\infty }}{lim}{e}^{-t},\underset{t⇒\mathrm{\infty }}{lim}-t{e}^{-t},\underset{t⇒\mathrm{\infty }}{lim}t\mathrm{ln}t⟩$
$=\mathrm{\infty }$
The limit does not exist because as tapproaches infinity, the limit of tIn(t) does not exist.
b- Find r'(t) and evaluate r'(0)
$r\left(t\right)=\frac{d}{dt}⟨{e}^{-t},-t{e}^{-t},t\mathrm{ln}t⟩$
$=⟨\frac{d}{dt}\left({e}^{-t}\right),\frac{d}{dt}\left(-t{e}^{-t}\right),\frac{d}{dt}\left(t\mathrm{ln}t\right)⟩$
$=⟨{e}^{-t},={e}^{-t}+t{e}^{-t},\mathrm{ln}\left(t\right)+1⟩$
${r}^{\prime }\left(t\right)=⟨-{e}^{-0},-{e}^{0}+{\left(0\right)}^{-0},\mathrm{ln}\left(0\right)+1⟩$
$=⟨-1,-1,\mathrm{\infty }⟩$
The value of r’(0) is undefined
c- Find r''(t)
We have ${r}^{\prime }\left(t\right)=⟨-{e}^{-t},{e}^{-t}+t{e}^{-t},\mathrm{ln}\left(t\right)+1⟩$
$rt\right)=\frac{d}{dt}⟨-{e}^{-t},{e}^{-t}+t{e}^{-t}$

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