In-class work for 12 June 2012
1.) Find the domain of $\vec{r}(t) = \left< \frac{t+1}{t-1}, \sqrt{t}, t \right>$.

2.) Find the limit $\displaystyle\lim_{t \rightarrow 0} < t^2, e^t+1, \frac{\sin t}{t} >.$

3.) Find a vector equation and a parametric equation joining $P=(0,0,2)$ and $Q=(-1,2,3)$.

4.) Find a vector equation of the curve of intersection of the cylinder $x^2+y^2=4$ and the plane $x+2z=5$.

5.) Compute the derivative of $\vec{r}(t) = < e^t \sin t, \cos t, t^5 >$.

6.) Find the unit tangent vector $\vec{T}(t)$ at $t=0$ of $\vec{r}(t) = < te^{3t}, \sin(t)+3, 7t^2 >$.

7.) Let $\vec{r}(t) = < \frac{1}{t}, \log(t), 5t >$. Compute $\displaystyle\int_2^3 \vec{r}(t) dt.$

8.) If $\vec{u}(t) = < t^2, 3t, \sin t >$ and $\vec{v}(t) = < 3t, e^t, \cos t >$, then what is $\frac{d}{dt} [ \vec{u}(t) \times \vec{v}(t) ]$?