Math 4181H, Honors
Calculus I Autumn 2012
Homework
problems
§1: 5(ii),(iv)
(use only P1–P12, and justify
each step)
§1: 5(vi)
(use 5(iv))
§1: 8 (Here the idea is to derive P10–P12
from P1–P9 and P'10–P'13; your first step should be to define
the positive numbers P using only these new axioms.)
For the following problems,
please provide complete proofs. However, any of the results in
Chapter 1, and any of the facts in problems 3, 5, 12
(i),(ii),(iii), can be used without special mention or further
proof. Also, you may assume the following fact, which we will
prove later: Any positive number a has a positive square root.
§1: 7 (5(ix) and (x) are useful);
§1: 12(iv), (v), (vi); 14(a)(b)
§2: 1(ii), 2(ii), 3 a, d, e(i), e(ii)
Recommended problems:
(You don’t have to
turn these in, but they are very useful.
§1: 2; 3(iii); 4(iv), (xiii);
5 (i), (ix), (x); 10(ii); 11(ii), (iv), 14(c)
§2: 1(i), 2(i)
Required problems:
§1: 19(a)(b)(c)(d) )
§2: 14(a)
§2: 20
§3: 5(ii)(iv)(vi) (see
Problem
4
for
notation), 7, 9, 13 plus the following problem
related to Chap. 2
"Consider
the
set
ℕ2 of pairs (m,n) of natural numbers with
the following order relation: (M,N)>(m,n) if M>m or if M=m and
N>n. (For example, (2,1)>(1,100), (3,10)>(3,9); this
is called the lexicographic order.) Show that ℕ2
with this order is well ordered". This can shown in a
number of ways: a carefully organized induction, or an
inductive construction of an "infinite descending sequence" with
contradictory properties...
Assigned on Fri, Aug.
31: §3: 24a, 26, 27c; §4: 9, 16,18
Assigned on Tue, Sep. 4 §4
(Appendix 1) 2; (Appendix
2) 1; (Appendix 3) 5
Assigned on Mon, Sep. 10 §5 3 (i), (iv), (v); 7, and these:
More problems
Assigned on Tue, Sep. 11 §5: 20, 25,26,34. (due Fri Sep. 14)
Assigned on Thu, Sep. 13 §5: 31, 33 (i) (iii),39 (v) (vi); §6: 6,7 (due Fri Sep. 21)
Assigned on Mon, Sep. 17
§7: 3 (ii), 6, 7, 8,
10, 11(due Fri Sep. 21) Extra credit
problems
Assigned on Wed, Sep. 19
§8: 6, 7, 8(a)
Assigned on Tue, Sep. 25 §22: 1(iv), 2(ii),(v) 5abc,6ab + §9: 15,19,22a,23,24 (Due Fri, Sep.28)
Assigned on Thu, Sep. 27 §10:
15,25,27,28,29,31 (Due
Fri, Oct. 5)
Assigned on Mon, Oct 1 §11:
4,15, 28, 42, 60 (Due
Fri, Oct. 5)
Very useful but optional (not graded): §11: 34,37,54,55,58
Assigned on Wed, Oct 3 §11:
38,43,53+Appendix: 3,8
Assigned on Fri, Oct 5 §12:
5b,8,14, 21, 24
Very useful but optional (not graded): §12: 1 (i)-(iv), 4, 5(a), 7 (i),(ii), (iii)
Assigned on Mon, Oct 8 Optional bonus problems The
area.
Assigned on Thu Oct 11 (due Fri Oct 19) §13: 13,14,16,20,31ab, 39;
§14:9,10,18,23e.
Optional bonus problems Open sets, zero Lebesgue measure.
Assigned on Wed Oct 17 (due Fri, Oct 19) §15: 3 (a)(b), 9 (a)(b), 13, 16
Optional bonus problems Defining powers from first principles.
Assigned on Mon Oct 22 (due Fri, Oct 26) §18: 7,8,12,42(a),(b),47,49
Assigned on Thu, Oct 25 (due Fri, Nov. 2) §13: 25 (length of a curve); §19: 3(iv)(vii) (viii),
4(ii)(viii),5(iii)(v)(ix),8(ii)(iii)(vi)(vii),22(a)(b)
plus this
Extra credit problems
Assigned on Wed, Oct 31 (due Fri, Nov. 2) §20: 1 (i)(ii)(iii),7,8,11,13,14,23,24
Assigned on Fri, Nov 9 (due Fri, Nov. 16) §23: 1 (i)(ii)(ix)(xviii, 8,9,11(a),
15(c),28 (a,b,c,d,e),29
Assigned on Thu, Nov 15 (due Fri, Nov. 30) §19: 36(a,b),§24: 1(i,iv),2(v),11(a,b),12(a),24,25.
Assigned on Wed, Nov 21 As extra
credit problems: §25 7,10; §27: 1(i,v),2(v),3(ii),5
Assigned on Sat, Nov 24 Review
problems, graded as regular hw: Review Hw