Chadwick Ryan
10/29/2023 · High School
calcule \( \frac{d}{d r}\left(r(t) \cdot r^{\prime}(t)\right) \) 4. Encuentre la longitud de la curva \( \vec{r}(t)=e^{-t} \cos t i+e^{-t} \) sent j sobre el int ervalo \( \left[0, \frac{\pi}{2}\right] \) 5. Dado \( \vec{r}(t)=2 e^{t} i+e^{t} \cos t \mathrm{j}+e^{\prime} \) sent \( k \) determine el vector tangente unitario en \( t=0 \)
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1. La derivada de \( r(t) \cdot r^{\prime}(t) \) es \( r^{\prime}(t) \cdot r^{\prime}(t) + r(t) \cdot r^{\prime\prime}(t) \).
2. La longitud de la curva \( \vec{r}(t) \) en el intervalo \( \left[0, \frac{\pi}{2}\right] \) es \( \sqrt{2} \left(-e^{-\frac{\pi}{2}} + 1\right) \).
3. El vector tangente unitario en \( t=0 \) para \( \vec{r}(t) \) es \( \frac{2}{\sqrt{6}} \, \mathbf{i} + \frac{1}{\sqrt{6}} \, \mathbf{j} + \frac{1}{\sqrt{6}} \, \mathbf{k} \).
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