Homework # 7 - Math 250
The problems on this homework must be done alone. The honor code is in effect.
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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. \(\vec{r}(t) = \langle \cos(2t),\sin(2t)\rangle\)?
3. \(\vec{r}(t) = \langle -2\cos(t),-2\sin(t)\rangle\)?
4. \(\vec{r}(t) = \langle 2\cos(t),-2\sin(t)\rangle\)?
5. \(\vec{r}(t) = \langle 2\cos(t),\sin(t)\rangle\)?
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.
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.