In-class work for 15 June 2012
1.) Given the position function $\vec{r}(t) = < \log(t), \sin(2t), \cos(2t) >$ find the velocity, acceleration, and speed.

2.) Find the velocity and position of particles with acceleration $\vec{a}(t) = < 2, 5t, 12t^2 >$ with initial velocity $\vec{v}(0) = < 0,0,1 >$ and initial position $\vec{r}(0) = < 1,0,0 >$.

3.) A projectile is fired with initial speed of $64$ meters per second with angle of elevation $45^o$. What is the range, maximum height, and speed of impact of the projectile?

4.) What are the tangential and normal components of the acceleration vector given $\vec{r}(t) = < \cos(\pi t), \sin(\pi t), t^3 >$?

5.) The position function of a particle is given by $\vec{r}(t) = < 1+t^2, t, 3t^2 - 17t >$. When is its speed a minimum?

6.) What force is required so that the a particle of mass $m$ has the position function $\vec{r}(t) = < 3t^3, 7t^2, 3t^3 >$?

7.) A cannon is fired at an angle of elevation of $60^o$. What is the muzzle speed if the maximum height of the shell is $600m$?

8.) Show that if a particle moves with constant speed, then the velocity and acceleration vectors are orthogonal.

9.) Find the domain of $f(x,y) = ...$
    a.) $xy$
    b.) $\frac{1}{x}y$
    c.) $\frac{1}{xy}$.

10.) Draw level curves of the function $f(x,y) = x^3-y$ for heights $k=-1,0,1$.