91Ó°ÊÓ

Find a vector function that describes the following curves. Intersection of the cylinder \(x^{2}+y^{2}=4\) with the plane \(x+z=6\)

Find a vector function that describes the following curves. Intersection of the cone \(z=\sqrt{x^{2}+y^{2}}\) and plane \(z=y-4\)

Find the derivatives of \(\mathbf{u}(t), \quad \mathbf{u}^{\prime}(t), \quad \mathbf{u}^{\prime}(t) \times \mathbf{u}(t)\) \(\mathbf{u}(t) \times \mathbf{u}^{\prime}(t), \quad\) and \(\mathbf{u}(t) \cdot \mathbf{u}^{\prime}(t) .\) Find the unit tangent vector. $$ \mathbf{u}(t)=\left\langle e^{t}, e^{-t}\right\rangle $$

Find the derivatives of \(\mathbf{u}(t), \quad \mathbf{u}^{\prime}(t), \quad \mathbf{u}^{\prime}(t) \times \mathbf{u}(t)\) \(\mathbf{u}(t) \times \mathbf{u}^{\prime}(t), \quad\) and \(\mathbf{u}(t) \cdot \mathbf{u}^{\prime}(t) .\) Find the unit tangent vector. $$ \mathbf{u}(t)=\left\langle t^{2}, 2 t+6,4 t^{5}-12\right\rangle $$

Evaluate the following integrals. $$ \int\left(\tan (t) \sec (t) \mathbf{i}-t e^{3 t} \mathbf{j}\right) d t $$

Evaluate the following integrals. $$ \int_{1}^{4} \mathbf{u}(t) d t, \text { with } \mathbf{u}(t)=\left\langle\frac{\ln (t)}{t}, \frac{1}{\sqrt{t}}, \sin \left(\frac{t \pi}{4}\right)\right\rangle $$

Find the length for the following curves. $$ \mathbf{r}(t)=\langle 3(t), 4 \cos (t), 4 \sin (t)\rangle \text { for } 1 \leq t \leq 4 $$

Find the length for the following curves. $$ \mathbf{r}(t)=2 \mathbf{i}+t \mathbf{j}+3 t^{2} \mathbf{k} \text { for } 0 \leq t \leq 1 $$

Reparameterize the following functions with respect to their arc length measured from \(t=0\) in direction of increasing \(t .\) $$ \mathbf{r}(t)=2 t \mathbf{i}+(4 t-5) \mathbf{j}+(1-3 t) \mathbf{k} $$

Reparameterize the following functions with respect to their arc length measured from \(t=0\) in direction of increasing \(t .\) $$ \mathbf{r}(t)=\cos (2 t) \mathbf{i}+8 t \mathbf{j}-\sin (2 t) \mathbf{k} $$