Math 4181H, Honors
Calculus I Autumn 2015
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)
§1: 19(a)(b)(c)(d) )
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)
Due: Fri Sep. 4
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Due on Fri, Sep. 11:
§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...
§3: 24a, 26, 27c; §4: 9, 16,18
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Bonus
problem Extra
credit, you can submit it until Fri Sept 18. Worth a total of
100p.
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Due
on Fri, Sep. 11: §4 (Appendix 1) 2; (Appendix 2) 1; (Appendix 3) 5
§5 3 (i), (iv), (v); 7, 20, 25,26, 31, 33 (i) (iii); 34 ,39
(v) (vi) and: Bonus
points
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Recommended problems
§7: 3 (ii), 6, 7, 8, 10, 11
§8: 6, 7, 8(a)
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Due Fri. Oct 9:
§10: 15,25,27,28,29,31
§10: 15,25,27,28,29,31
§11: 4,15, 28, 42, 60
§11: 38,43,53
Very useful but optional (not graded): §11:
34,37,54,55,58
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Due Monday Oct 19
§12: 5b, 8, 14,21,24
§13: 13,14,16,20,31ab, 39;
Very useful but
optional (not graded): §12: 1
(i)-(iv), 4, 5(a), 7 (i),(ii), (iii)
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Due Monday Oct 26
§14:
9,10,18,23e.
§15:
3ab,9ab,13,16
Optional bonus problems The
area.
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Due Monday Nov 2
§18:
7,8,12,42(a),(b),47,49
Optional bonus problems Defining
powers from first principles.
Due Fri Nov 20
§20: 1 (i)(ii)(iii),7,8,11,13,14,23,24
§23: 1 (i)(ii)(ix)(xviii, 8,9,11(a), 15(c),28 (a,b,c,d,e),29
Due Mon Nov 29
§19:
36(a,b)
§24: 1(i,iv),2(v),11(a,b),12(a),24,25.
Due Mon Dec 7
§26:
10
§27: 4, 9(iii)(vi)(vii), 11c,g, 16a,b
Bonus
problems