 jippie771h

2021-12-04

Working with vector-valued functions For each vector-valuedfunction r, carry out the following steps.
a. Evaluate , if each exists.
b. Find ${r}^{\prime }\left(t\right)$ and evaluate ${r}^{\prime }\left(0\right)$.
c. Find $rt\right)$.
d. Evaluate $\int r\left(t\right)dt.$
$r\left(t\right)=⟨t+1,{t}^{2}-3⟩$ Dona Hall

Tofind:
a) $\underset{t⇒0}{lim}r\left(\begin{array}{c}t\end{array}\right)and\underset{t⇒\mathrm{\infty }}{lim}r\left(\begin{array}{c}t\end{array}\right)$
b) r'(t) and evaluate r'(0).
c) r''(t)
d) $\int r\left(t\right)dt$
d) fr(t)dr
Given vector valued function is $r\left(t\right)=$
a) To find: $\underset{t⇒0}{lim}r\left(\begin{array}{c}t\end{array}\right)and\underset{t⇒\mathrm{\infty }}{lim}r\left(\begin{array}{c}t\end{array}\right)$
$\underset{t⇒0}{lim}r\left(\begin{array}{c}t\end{array}\right)=\underset{t⇒0}{lim}=<1,-3>$
and
$\underset{t⇒\mathrm{\infty }}{lim}r\left(\begin{array}{c}t\end{array}\right)=\underset{t⇒\mathrm{\infty }}{lim}=$ does not exist
b) To find r'(t) and evaluate r'(0):

Substitute t=0 in r'(t),
$⇒{r}^{\prime }\left(0\right)=<1,0>$
c) To find: r''(t)
$r{}^{″}\left(t\right)=\frac{d}{dt}\left({r}^{\prime }\left(t\right)\right)=<\frac{d}{dt}\left(1\right),\frac{d}{dt}\left(2t\right)\ge <0,2>$
d) To find $\int r\left(t\right)dt:$
$\int r\left(t\right)dt=\int dt\phantom{\rule{0ex}{0ex}}=\int \left(t+1\right)dt,\int \left({t}^{2}-3\right)dt$
$=<\frac{{t}^{2}}{2}+t+C,\frac{{t}^{3}}{3}-3t+D>$
Therefore,
a) $\underset{t⇒0}{lim}r\left(\begin{array}{c}t\end{array}\right)=<1,-3>$
b)
${r}^{\prime }\left(0\right)=<1,0>$
c)$r{}^{″}\left(t\right)=<0,2>$
d) $\int r\left(t\right)dt=<\frac{{t}^{2}}{2}+t+C,\frac{{t}^{3}}{3}-3t+D>$

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