1.
pi/2.
2.
Q(t) = tan(2/3 t^{3/2} + pi/4).
3.
(a) y(t) = 2/(1 + 3 exp(2t)).
(b) There is a stable equilibrium at y(t) = 0
and an unstable equilibrium at y(t) = 2.
4.
(a) The level curves are the circles centered at the origin.
The level curve for f(x, y) = c is the circle with radius e^{c/2}.
(b) 0.
5.
(a) L(x, y) = 2/5 x + 4/5 y - 2 + ln 5.
(b) The value decreases by approximately 0.008.
(c) 2x + 4y - 5z = 10 - 5 ln 5.
6.
(a) pi/3.
(b) 2x + y + z = 3.
7.
(a) -4/25.
(b) The direction pointing straight at the origin,
i.e., the direction of the vector
| -1 |
| -2 |
| 0 |.
(c) The direction of the vector
| -2 |
| 1 |
| 9 |
is one such. Any non-zero vector perpendicular to the gradient
vector will give a suitable direction.
Reminder: I will hold a "last chance" question-and-problem session on Thursday, May 26, 5:30 pm, in Geology 3656. And questions can always be posted on our virtual office hours board.