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1.   Find the unit tangent vector to the curve \({\bf r}(t) = \langle 3te^t, \frac{4}{\ln 2} 2^t \rangle\) at \(t = 0\)

a.  \({\bf T}(0) = \langle 0, 1 \rangle \)

b.  \({\bf T}(0) = \langle 3, 4\rangle \)

c.  \({\bf T}(0) = \langle 0, 1\rangle \)

d.  \({\bf T}(0) = \langle \frac{3}{5}, \frac{4}{5} \rangle \)

e.  \({\bf T}(0) = \langle \frac{1}{2}, \frac{\sqrt{3}}{2}\rangle \))

2.   Find the principal unit normal vector to the curve \({\bf r}(t) = \cos(t^2), \sin(t^2) \rangle\) where \(t > 0\)?

a.  \({\bf N}(t) = \langle -\cos(t^2), -\sin(t^2)\rangle \)

b.  \({\bf N}(t) = \langle t^2\cos(t^2), t^2\sin(t^2)\rangle \)

c.  \({\bf N}(t) = \langle -t^2\sin(t^2), t^2\cos(t^2)\rangle \)

d.  \({\bf N}(t) = \langle \sin(t^2), \cos(t^2)\rangle \)

e.  \({\bf N}(t) = \langle -t^2\sin(t^2), -t^2\cos(t^2)\rangle \)

3.  Evaluate the indefinite integral. \(\int \langle t e^t, e^{-t} \rangle dt\)

a.  \(\langle \frac{1}{2}t^2 e^t + c_1, -e^{-t} + c_2 \rangle \)

b.  \(\langle \frac{1}{2}t^2 e^t + c_1, e^{-t} + c_2 \rangle \)

c.  \(\langle te^t - e^t + c_1, -e^{t} + c_2 \rangle \)

d.  \(\langle te^t + e^t + c_1, e^{-t} + c_2 \rangle \)

e.  \(\langle te^t - e^t + c_1, -e^{-t} + c_2 \rangle \)

4.  What does it mean if an object has a constant acceleration of \(\langle 0, 0\rangle \)?

a.  The object is stopped.

b.  The object is moving in a circle at constant speed.

c.  The object is moving in a straight line at constant speed or is stopped.

d.  The object is moving in a circle at constant speed or is stopped.

e.  The object is moving in a circle at constant speed or a straight line at constant speed.

5.  What's does the following equation represent \(r = z\)? (You're meant to infer the coordinate system we're using here.)

a.  Two straight lines in the rz-plane.

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

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

d.  A cone centered around the z-axis.

e.  A cone centered around the r-axis.

6.  What's does the following equation represent \(r^2 + z^2 = 1\)?

a.  A sphere of radius 1.

b.  Just the point at the origin.

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

d.  A hyperboloid of two sheets opening on the z-axis.

e.  A parabolic hyerboloid.

7.  What's does the following set of equations represent \(\rho \leq 1, 0 \leq \theta \leq \frac{\pi}{2}, 0\leq \phi \leq \frac{\pi}{2})?

a.  A solid sphere of radius 1.

b.  Half of a solid sphere of radius 1.

c.  One third of a solid sphere of radius 1.

d.  One quarter of a solid sphere of radius 1.

e.  One eighth of a solid sphere of radius 1.

8.   Find the curvature, \(K\), for the curve \({\bf r}(t) = \langle t, t^2, t^3 \rangle\) at \(t = 1\).

a.  \(\frac{1}{2} \)

b.  \(\frac{1}{\sqrt{2}}\)

c.  \(\frac{\sqrt{19}}{7\sqrt{14}}\)

d.  \(\frac{\sqrt{19}}{\sqrt{14}} \)

e.  \(\frac{1}{\sqrt{14}} \)

9.   Find the length of the curve \(C:\vec{r}(t) = \langle \cos t, \sin t, \frac{3}{4}t \rangle \) where \(0 \leq t \leq 4\pi\).

a.  \(\pi\)

b.  \(2\pi\)

c.  \(3\pi\)

d.  \(4\pi\)

e.  \(5\pi\)