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Math 408D, Fall 2009 (Daniel Allcock)
M408D: Calculus II
TTh 12:30-2:00, CPE 2.214
Daniel Allcock, RLM 9.112, phone 1-1120
office hours: M 11-12, Tue 10-11 and by appointment.
(my last name)@math.utexas.edu.
TA: Nick Zufelt, RLM 11.152, phone 471-1684.
TA office hours: M 10-11, Tue 11-12, W 12-1.
This is where I will post homework assignments and other announcements.
For general information, see the first-day handout.
HW 1
12.1:
6,
8,
14,
22,
24,
28,
29,
34,
36,
54,
55,
57,
62,
68,
81
7.8:
10,
8,
14,
18,
26,
30,
31,
32,
40,
44,
46 (hint: a clean way is to change variables),
52,
54 (hint: at one point using continuity greatly simplifies things),
58 (scary but clean),
81
HW 2
Problem A: Compute the first 6 terms of the sequence xn=1-1/n. Compute the first 6 partial sums of the series Σn=1∞(1-1/n). How are the different? Does the sequence converge? Does the sequence of partial sums converge? Does the series converge?
12.2:
19,
22,
25,
26,
32,
34,
38,
60,
64,
76 (hint: use similar triangles to find the ratio of each triangle's radius to that of its smaller neighbor or neighbors)
Problem B: Give examples of sequences xn and yn with the property that Σ xn and Σ yn diverge but Σ (xn+yn) converges. How is this different from the sum rule for convergent series? (n ranges from 1 to ∞ in the sums.)
8.8:
2,
6,
18,
20,
28,
31,
32,
36 (hint: after finding the indefinite integral, use continuity to find the limit),
42
Problem C: integrate tan(x ) over the interval [-π/2,π/2].
HW 3
12.3:
2,
10,
15,
17,
18,
24,
28,
30,
32,
36
12.4:
4,
6,
8,
14,
16,
27,
32 (hint: bound n1/n from above),
40,
46
12.5:
6,
10,
12,
23,
26,
31,
32
HW 4
12.6:
1,
8,
11,
13,
21,
22,
28,
29,
31
38 (you don't have to do 38, just look at it. Ramanujan was an Indian mathematician with an uncanny knack for infinite series. He found many amazing series like these that you can't imagine a human discovering.)
12.7:
Nothing to turn in here; instead, do all the problems quickly,
simply noting what test you would use (ratio if you see it will work,
integral if you see how to do the integral, AST if you see the
conditions are satisfied, etc.) If you can see conv/div then note
that too. The point is to discover that the answer CAN often be just seen,
although calculculation may be required to verify your impulse. Here are my own terse notes on this that you can compare with.
12.8:
6,
8,
10,
13,
19,
28,
30,
37 (hint: this is easier after 12.9)
12.9:
1,
2,
8,
13,
18,
24,
34
HW 5
12.10:
2, 6, 8, 19, 25, 26, 31, 36, 43, 54, 55, 56, 58, 65, 66
12.11:
31, 35ab, 36a, 37
HW 6
13.1:
1,
6,
10,
12,
16,
23-32 (it is enough to say "closed half-space" or "open slab" or something similar; the point is to see qualitative information in the formulas. "closed" means the region's boundary is included, while "open" means that none of its boundary is included.),
39
13.2:
3,
5,
14,
16,
17,
19,
22,
25,
27,
28,
31,
34,
45
HW 7
13.3:
1, 7, 9, 11, 15, 20, 22, 24, 25, 36, 38, 40, 46, 47, 50, 51, 52, 56, 60
13.4:
1, 4, 5, 7, 8, 10, 12, 13, 16, 22, 32, 34, 36, 39, 43, 49
EXAM 1 grading scale
82+ A+
70+ A
68+ B+
62+ B
60+ B-
56+ C+
45+ C
40+ C-
38+ D+
32+ D
30+ D-
HW 8
13.5:
1, 3, 4, 5, 7, 10, 14, 16, 23, 26, 28, 31, 33, 35, 37, 38, 40, 41, 44, 48,
55, 61, 69, 70, 76 (on 7 and 10 you don't need to give the "symmetric" form)
13.6:
1, 6, 12, 16, 20, 21-28, 33, 36, 41, 42, 43, 46, 48
HW 9
11.1:
5, 8, 22 (be sure to describe the motion in words as well as formulas), 24,
25, 27, 28, 38, 41
AND: sketch the curve defined by x=-2-3t and y=3-2t, and compare with #5.
11.2:
1, 3, 6, 17, 18, 25, 42, 44, 48 (hint: first you have to find the values of t where the curve meets itself).
11.3:
7-12, 14, 25, 26, 27, 28, 32, 36 (Hint: for a good picture,
find the direction of the curve at the origin), 37, 57, 62
11.4:
1, 5, 10, 13, 37, 44, 46, 48
HW 10
14.1:
9, 12, 19-24, 26, 28, 41 (also say whether the paths cross)
14.2:
4, 8, 12, 23, 32, 35, 39, 51
15.1:
31, 36, 38, 39, 42, 43, 47, 65ac, 66ac
15.2:
2, 30, 33, 36, 38
HW 11
15.3:
5-8, 9, 10, 16, 26, 46, 48, 49, 50, 57, 60, 72, 75, 84
15.4:
1, 4, 17, 18, 32, 34, 38
EXAM 2 grading scale
100+ A+
88+ A
83+ B+
73+ B
68+ B-
65+ C+
60+ C
57+ C-
54+ D+
49+ D
46+ D-
HW 12
15.5:
1, 12, 13, 19, 36, 39, 42, 44, 52
15.6:
5, 9, 14 (caution: "directional derivative in the direction of v" means a unit vector), 22, 24, 34, 38, 44, 47, 52
15.7:
4, 12, 18 (but don't restrict x and y like the problem asks you to), 19, 36, 46, 50, 54
HW 13
due Tursday Dec 3 (the last day of
class).
15.8: 3 (Caution finding "the maximum"), 4, 5, 21, 26, 27. (On 21, they are asking which assumption in the formal statement of the method of Lagrange multipliers doesn't hold. They can't all hold or else the answer to (b) would be different.)
(Note: problem 40 was listed at first, but is not on the homework.)
16.1: 12, 13, 14
16.2: 2, 6, 9, 15 (and sketch the region of integration), 22 (hint: choose the right order to integrate), 24, 30 ("first octant" means that x, y and z are all nonnegative; also, if you understand the shape then the problem is easy even without calculus), 31, 36
16.3: (In every problem, sketch the region of integration even if the book doesn't explicitly ask you to.)
7, 14, 15, 21, 28, 33, 39, 40, 45, 46
Remember: I insist on the "x=..." or "y=..." or similar in the bounds on integrals when there are several variables floating around.