8.1: Vector-Valued Functions and Space Curves
- Page ID
- 144336
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- Given the vector-valued function \(\mathbf{r}(t) = (2t^2 + 3, \sqrt{3t - 2})\), find \(\mathbf{r}(2)\).
- Given the vector-valued function \(\mathbf{r}(t) = \sec t\mathbf{i} + \tan t\mathbf{j}\), find \(\mathbf{r}\left(\frac{\pi}{4}\right)\).
- Sketch the curve of the vector-valued function \(\mathbf{r}(t) = t\mathbf{i} + (1 - t^2)\mathbf{j}\).
- Sketch the curve of the vector-valued function \(\mathbf{r}(t) = (t, t^2, t)\).
- Sketch the curve of the vector-valued function \(\mathbf{r}(t) = (\cos t, \sin t)\). Be sure to indicate the orientation of the curve.
- Sketch the curve of the vector-valued function \(\mathbf{r}(t) = \sin t\mathbf{i} + \cos t\mathbf{j} + t\mathbf{k}\). Be sure to indicate the orientation of the curve.
- Find the domain of the vector-valued function \(\mathbf{r}(t) = \left(\dfrac{1}{t-3}, \sqrt{t^2 - 1}\right)\).
- Find the domain of the vector-valued function \(\mathbf{r}(t) = 2e^t\mathbf{i} + \ln t\mathbf{j} + \dfrac{1}{t^2 + 1}\mathbf{k}\).
- Evaluate \(\displaystyle \lim_{t \to 0} (2t + 1, \cos t)\), if possible.
- Evaluate \(\displaystyle \lim_{t \to 2} \left(\frac{t - 2}{t^2 - 4}, e^t\right)\), if possible.
- Evaluate \(\displaystyle \lim_{t \to \infty} \left(e^{-t}\mathbf{i} + \frac{3t^2 + 2t - 4}{2 - 5t^2}\mathbf{j} + \arctan(t)\mathbf{k}\right)\), if possible.
- Evaluate \(\displaystyle \lim_{t \to 0^+} \left( \frac{\sin t}{t}, \ln t, 4\right)\), if possible.
- Express the vector-valued function \(\mathbf{r}(t) = 2t\mathbf{i} - (3 + t)\mathbf{j}\) in Cartesian coordinates by eliminating the parameter \(t\), then sketch the graph of the vector-valued function. Be sure to indicate the orientation of the curve.
- Express the vector-valued function \(\mathbf{r}(t) = (t^3, t^2)\) in Cartesian coordinates by eliminating the parameter \(t\), then sketch the graph of the vector-valued function. Be sure to indicate the orientation of the curve.
- Express the vector-valued function \(\mathbf{r}(t) = \dfrac{1}{t}(\mathbf{i} + \mathbf{j})\) in Cartesian coordinates by eliminating the parameter \(t\), then sketch the graph of the vector-valued function. Be sure to indicate the orientation of the curve.
- Express the vector-valued function \(\mathbf{r}(t) = (2\sin t, -3\cos t)\) in Cartesian coordinates by eliminating the parameter \(t\), then sketch the graph of the vector-valued function. Be sure to indicate the orientation of the curve.
- Express the vector-valued function \(\mathbf{r}(t) = 3\sec^2t\mathbf{i} + \tan t\mathbf{j}\) in Cartesian coordinates by eliminating the parameter \(t\), then sketch the graph of the vector-valued function. Be sure to indicate the orientation of the curve.
- Find a vector-valued function that traces the line from point \(P(1, 0, 1)\) to point \(Q(0, 1, 0)\).
- Find a vector-valued function that traces the parabola \(y = x^2\) from left-to-right.
- Find a vector-valued function that traces the ellipse \(4x^2 + 9y^2 = 36\) in the counterclockwise direction.
- Find a vector-valued function that traces the hyperbola \(\dfrac{x^2}{16} - y^2 = 1\), where the right branch is oriented top-to-bottom.