91Ó°ÊÓ

Sketch the graph of the plane curve given by the vector-valued function and, at the point on the curve determined by \(\mathbf{r}\left(t_{0}\right),\) sketch the vectors \(\mathbf{T}\) and \(\mathbf{N}\). Note that \(\mathbf{N}\) points toward the concave side of the curve. $$ \mathbf{r}(t)=3 \cos t \mathbf{i}+2 \sin t \mathbf{j} \quad t_{0}=\pi $$

Use a graphing utility to graph the function. In the same viewing window, graph the circle of curvature to the graph at the given value of \(x\). $$ y=\ln x, \quad x=1 $$

Find the indefinite integral. $$ \int\left(\frac{1}{t} \mathbf{i}+\mathbf{j}-t^{3 / 2} \mathbf{k}\right) d t $$

Sketch the space curve represented by the intersection of the surfaces. Then represent the curve by a vector-valued function using the given parameter. $$\text { Surfaces } \quad \text { Parameter }$$ $$ z=x^{2}+y^{2}, \quad x+y=0 \quad x=t $$

Find \(\mathbf{T}(t), \mathbf{N}(t), a_{\mathrm{T}},\) and \(a_{\mathrm{N}}\) at the given time \(t\) for the space curve \(\mathbf{r}(t) .\) $$ \mathbf{r}(t)=t \mathbf{i}+2 t \mathbf{j}-3 t \mathbf{k} \quad t=1 $$

Use a graphing utility to graph the function. In the same viewing window, graph the circle of curvature to the graph at the given value of \(x\). $$ y=e^{x}, \quad x=0 $$

Find \(\mathbf{T}(t), \mathbf{N}(t), a_{\mathrm{T}},\) and \(a_{\mathrm{N}}\) at the given time \(t\) for the space curve \(\mathbf{r}(t) .\) $$ \mathbf{r}(t)=4 t \mathbf{i}-4 t \mathbf{j}+2 t \mathbf{k} \quad t=2 $$

Find the indefinite integral. $$ \int\left(\ln t \mathbf{i}+\frac{1}{t} \mathbf{j}+\mathbf{k}\right) d t $$

Use a graphing utility to graph the function. In the same viewing window, graph the circle of curvature to the graph at the given value of \(x\). $$ y=\frac{1}{3} x^{3}, \quad x=1 $$

Sketch the space curve represented by the intersection of the surfaces. Then represent the curve by a vector-valued function using the given parameter. $$\text { Surfaces } \quad \text { Parameter }$$ $$ z=x^{2}+y^{2}, \quad z=4 \quad x=2 \cos t $$