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8.3: Arc Length and Curvature

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    144338
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    1. Determine any values of \(t\) at which the vector-valued function \(\mathbf{r}(t) = (t, 0)\) is not smooth. Graph the function and describe its behavior at the points where it is not smooth.
       
    2. Determine any values of \(t\) at which the vector-valued function \(\mathbf{r}(t) = t^3\,\mathbf{i}\) is not smooth. Graph the function and describe its behavior at the points where it is not smooth.
       
    3. Determine any values of \(t\) at which the vector-valued function \(\mathbf{r}(t) = \left(\dfrac{1}{3}t^3 - \dfrac{1}{2}t^2, 0\right)\) is not smooth. Graph the function and describe its behavior at the points where it is not smooth.
       
    4. Determine any values of \(t\) at which the vector-valued function \(\mathbf{r}(t) = (t^3, 5t^2)\) is not smooth. Graph the function and describe its behavior at the points where it is not smooth.
       
    5. Determine any values of \(t\) at which the vector-valued function \(\mathbf{r}(t) = \sqrt[3]{t}\,\mathbf{i} + t\,\mathbf{j}\) is not smooth. Graph the function and describe its behavior at the points where it is not smooth.
       
    6. Find the arc length of \(\mathbf{r}(t) = (2t^2 + 3, t^2 + 1)\) for \(t \in [1, 4]\).
       
    7. Find the arc length of \(\mathbf{r}(t) = 2t\,\mathbf{i} + \sin t\,\mathbf{j} + \cos t\,\mathbf{k}\) for \(t \in [-\pi, \pi]\).
       
    8. Find an arc length parameterization from \(t = 0\) for the curve \(\mathbf{r}(t) = \dfrac{1}{2}t^2\mathbf{i} - \dfrac{1}{3}t^3\mathbf{j}\) with the same orientation as the original curve.
       
    9. Find an arc length parameterization from \(t = 0\) for the curve \(\mathbf{r}(t) = (\cos t, t, \sin t)\) with the same orientation as the original curve.
       
    10. Given \(\mathbf{r}(t) = \dfrac{1}{3}t^3\,\mathbf{i} + \dfrac{1}{2}t^2\,\mathbf{j}\), find the values of \(\mathbf{T}(t)\) and \(\mathbf{N}(t)\) at \(t = 1\).
       
    11. Given \(\mathbf{r}(t) = (2\cos t, 2\sin t, t)\), find the values of \(\mathbf{T}(t)\), \(\mathbf{N}(t)\), and \(\mathbf{B}(t)\) at \(t = 0\).
       
    12. Given \(\mathbf{r}(t) = t\,\mathbf{i} + e^t\sin t\,\mathbf{j} + e^t\cos t\,\mathbf{k}\), find the values of \(\mathbf{T}(t)\), \(\mathbf{N}(t)\), and \(\mathbf{B}(t)\) at \(t = 0\).
       
    13. Find the curvature of \(\mathbf{r}(t) = (\sin t, 4\cos t)\).
       
    14. Find the curvature of \(\mathbf{r}(t) = t\,\mathbf{i} + t\,\mathbf{j} + t^2\,\mathbf{k}\).

    8.3: Arc Length and Curvature is shared under a not declared license and was authored, remixed, and/or curated by LibreTexts.

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