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Multivariable Calculus

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Difficult

Reparametrize Curve with Respect to Arc Length

MVCALC-JAENDE

Reparameterize the curve:

$$\textbf{r}(t) =e^t\cos t\,\textbf{i}+e^t\sin t\,\textbf{j}+2e^t\,\textbf{k}$$

...with respect to arc length, given that $t$ starts at $t=0$ and continues in the positive direction.

A

$\textbf{a}(t) =e^{(s/2+1)}\cos (s/2+1)\,\textbf{i}+e^{(s/2+1)}\sin (s/2+1)\,\textbf{j}+2e^{(s/2+1)}\,\textbf{k}$

B

$\textbf{a}(t) =e^{(s/\sqrt{6}+1)}\cos (s/\sqrt{6}+1)\,\textbf{i}+e^{(s/\sqrt{6}+1)}\sin (s/\sqrt{6}+1)\,\textbf{j}+2e^{(s/\sqrt{6}+1)}\,\textbf{k}$

C

$\textbf{a}(t) =e^{(s/\sqrt{6}+1)}\cos \ln(s/\sqrt{6}+1)\,\textbf{i}+e^{(s/\sqrt{6}+1)}\sin \ln(s/\sqrt{6}+1)\,\textbf{j}+2e^{(s/\sqrt{6}+1)}\,\textbf{k}$

D

$\textbf{a}(t) =(s/\sqrt{6}+1)\cos \ln(s/\sqrt{6}+1)\,\textbf{i}+(s/\sqrt{6}+1)\sin \ln(s/\sqrt{6}+1)\,\textbf{j}+2(s/\sqrt{6}+1)\,\textbf{k}$

E

$\textbf{a}(t) =(s/2+1)\cos \ln(s/2+1)\,\textbf{i}+(s/\sqrt{6}+1)\sin \ln(s/2+1)\,\textbf{j}+2(s/2+1)\,\textbf{k}$