Randolph College
Department of Mathematics and Computer Science

 

Homework # 7 - Math 250


The problems on this homework must be done alone.  The honor code is in effect.

 First name:   Last name:


 

In questions 1 through 5 what motion is described by the function?

 

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

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.

 

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

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.

 

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

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.

 

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

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.

 

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

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.

 

6.  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.

 

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

 

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

 

9.  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.

 

10.  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.