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1.  What's does the following equation represent \(36x^2 + 9 y^2 + 4z^2 = 36\)?

a.  A sphere of radius 6 centered at the origin.

b.  An elipsoid centered at the origin, crossing the x-axis at 1 and -1.

c.  An elipsoid centered at the origin, crossing the x-axis at 6 and -6.

d.  A hyperboloid of one sheet.

e.  A hyperboloid of two sheets.

2.  What's does the following equation represent \(36x^2 - 9 y^2 + 4z^2 = 36\)?

a.  A sphere of radius 6 centered at the origin.

b.  An elipsoid centered at the origin, crossing the x-axis at 1 and -1.

c.  An elipsoid centered at the origin, crossing the x-axis at 6 and -6.

d.  A hyperboloid of one sheet.

e.  A hyperboloid of two sheets.

3.  What's does the following equation represent \(36x^2 - 9 y^2 + 4z^2 = -36\)?

a.  A sphere of radius 6 centered at the origin.

b.  An elipsoid centered at the origin, crossing the x-axis at 1 and -1.

c.  An elipsoid centered at the origin, crossing the x-axis at 6 and -6.

d.  A hyperboloid of one sheet.

e.  A hyperboloid of two sheets.

4.  What's does the following equation represent \(x + y^2 + z^2 = 0\)?

a.  A paraboloid opening on the positive z-axis.

b.  A paraboloid opening on the negative z-axis.

c.  A paraboloid opening on the positive x-axis.

d.  A paraboloid opening on the negative x-axis.

e.  A paraboloid opening on the positive y-axis.

In questions 5 through 9 what motion is described by the function?

5.  \(\vec{r}(t) = \langle 2\cos(t),2\sin(t)\rangle\)?

a.  Counterclockwise motion around a circle of radius 1 at constant speed 2.

b.  Counterclockwise motion around a circle of radius 2 at constant speed 2.

c.  Clockwise motion around a circle of radius 1 at constant speed 2.

d.  Clockwise motion around a circle of radius 2 at constant speed 2.

e.  Counterclockwise motion around an elipse.

6.  \(\vec{r}(t) = \langle \cos(2t),\sin(2t)\rangle\)?

a.  Counterclockwise motion around a circle of radius 1 at constant speed 2.

b.  Counterclockwise motion around a circle of radius 2 at constant speed 2.

c.  Clockwise motion around a circle of radius 1 at constant speed 2.

d.  Clockwise motion around a circle of radius 2 at constant speed 2.

e.  Counterclockwise motion around an elipse.

7.  \(\vec{r}(t) = \langle -2\cos(t),-2\sin(t)\rangle\)?

a.  Counterclockwise motion around a circle of radius 1 at constant speed 2.

b.  Counterclockwise motion around a circle of radius 2 at constant speed 2.

c.  Clockwise motion around a circle of radius 1 at constant speed 2.

d.  Clockwise motion around a circle of radius 2 at constant speed 2.

e.  Counterclockwise motion around an elipse.

8.  \(\vec{r}(t) = \langle 2\cos(t),-2\sin(t)\rangle\)?

a.  Counterclockwise motion around a circle of radius 1 at constant speed 2.

b.  Counterclockwise motion around a circle of radius 2 at constant speed 2.

c.  Clockwise motion around a circle of radius 1 at constant speed 2.

d.  Clockwise motion around a circle of radius 2 at constant speed 2.

e.  Counterclockwise motion around an elipse.

9.  \(\vec{r}(t) = \langle 2\cos(t),\sin(t)\rangle\)?

a.  Counterclockwise motion around a circle of radius 1 at constant speed 2.

b.  Counterclockwise motion around a circle of radius 2 at constant speed 2.

c.  Clockwise motion around a circle of radius 1 at constant speed 2.

d.  Clockwise motion around a circle of radius 2 at constant speed 2.

e.  Counterclockwise motion around an elipse.

10.  Evaluate the limit.   $$ \ lim_{t \rightarrow \infty} \langle te^{(-t)}, t^{(1/t)} \rangle.$$

a.  \(\langle e, 0\rangle\)

b.  \(\langle 1/e, 0\rangle\)

c.  \(\langle 1, 0\rangle\)

d.  \(\langle 0, 1\rangle\)

e.  The limit does not exist.

11.   Compute the derivative. $$\frac{d}{dt} \langle \sin^2(t), t^t\rangle $$

a.  \(\langle \sin 2t, t^t \rangle\)

b.  \(\langle 2\sin t\cos t, t^{(t-1)} \rangle\)

c.  \(\langle \sin t\cos t, (1+\ln t) t^t \rangle\)

d.  \(\langle \sin t\cos t, t^t \rangle\)

e.  \(\langle 2\sin t\cos t, (1+\ln t) t^t \rangle\)

12.  Find the largest open intervals upon which the curve \(C: \vec{r}(t) = \langle t^2, t^3 \rangle\)is smooth.

a.  \((-\infty,\infty)\)

b.  \((-\infty,0), (0,\infty)\)

c.  \((-\infty,-1), (-1, 0), (0, 1), (1,\infty)\)

d.  \((-\infty,-1), (-1, 1), (1,\infty)\)

e.  \((-\infty, 0), (0, 1), (1,\infty)\)

13.  Say \(\vec{r}(t) = \langle t^2, t^2 \rangle \) describes the position of a particle at time t. Describe the motion of this particle.

a.  The particle travels along a line and switches direction at time t = 0.

b.  The particle travels along a line, stops at t = 0 but does not switch direction.

c.  The particle travels along a parabola and switches direction at t = 0

d.  The particle travels along a parabola, stops at t = 0 but does not switch direction.

e.  The particle travels along a parabola but never stops.

14.  Say \(\vec{r}(t) = \langle t^3, t^3 \rangle \) describes the position of a particle at time t. Describe the motion of this particle.

a.  The particle travels along a line and switches direction at time t = 0.

b.  The particle travels along a line, stops at t = 0 but does not switch direction.

c.  The particle travels along a cubic curve and switches direction at t = 0

d.  The particle travels along a cubic curve, stops at t = 0 but does not switch direction.

e.  The particle travels along a cubic curve but never stops.