8.3: Arc Length and Curvature
- Page ID
- 144338
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- 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.
- 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.
- 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.
- 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.
- 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.
- Find the arc length of \(\mathbf{r}(t) = (2t^2 + 3, t^2 + 1)\) for \(t \in [1, 4]\).
- Find the arc length of \(\mathbf{r}(t) = 2t\,\mathbf{i} + \sin t\,\mathbf{j} + \cos t\,\mathbf{k}\) for \(t \in [-\pi, \pi]\).
- 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.
- 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.
- 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 \(t = 1\).
- 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 \(t = 0\).
- 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 \(t = 0\).
- Find the curvature of \(\mathbf{r}(t) = (\sin t, 4\cos t)\).
- Find the curvature of \(\mathbf{r}(t) = t\,\mathbf{i} + t\,\mathbf{j} + t^2\,\mathbf{k}\).