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MA 123 |
Problems for Week
3
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September 22, 2005
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Your work on these problems will be collected in section on Tuesday, September 27.
Problems for September 20. Sec. 2.3: Probs. 6; 8; 9; 12; 13; 17; 19; 23; 25; 26; 27; 40..
Sec. 2.4: Probs 1; 3; 4; 8; 11; 12; 23; 27; 33; 35; 44.
Problems for September
22. Sec.
2.4: Probs 23; 27; 33; 35;
44.
Sec. 2.5: Probs. 4; 5; 6; 10a,b; 16; 19; 20; 22.
EXTRA
CREDIT: Some food for
thought
1. The definition of limit on page101 of the text is
incomplete. Explain why and
salvage the definition.
2. The popular parlor game Take it to the
Limit is played with two players. The first player is called Epsilon and
the second player is called Delta. The game is played with a deck of cards
called center>
1. Delta begins by turning over the top
Challenge card. Both players
make a mental note the information on the
card:
f(x),
a, and L. 2. Epsilon calls out a positive real
number and declares that to be the value of e. 3. Delta calls out another positive
real
number and declares that to be the value of d. 4. Epsilon calls out a real number and
declares that to be x.
Who
wins? Once the
above four 0in 5.4pt 0in 5.4pt'>
a.
x is not equal
to
a; b. | x – a | < d; c. |
f(x)
– L | >= e. Otherwise,
Delta is the winner. A
Sample Round.
Suppose the first Challenge card shows the function f(x) =
x2, and the numbers a =
2, and L = 5. Epsilon calls out e
=
.5, then Delta
calls out d = .02. Finally, Epsilon calls out x = 2.01. Who wins? To
answer this we consider the conditions (a), (b), (c) above. Since x = 2.01 and a = 2 we have x is not equal to
a, so the condition (a) is
true. Moreover, since | x - a | = | 2.01 –
2 | = .01 and since d = .02 we have | x – a | <
d, so condition (b) is also true. Finally, we consider condition (c). We
calculate f(x) = (2.01)2 = 4.0401, so since L = 5
we have | f(x) – L | = .9599 which is greater than
.5, the value of e. Hence
condition (c) is also satisfied.
Thus all three conditions are satisfied and Epsilon is declared the
winner of the round. Questions. Analyze the game of Take It To The
Limit to decide who, if anyone, has a
winning strategy. For example, in
the above sample round, does Epsilon have a winning strategy? What if we change L from L=5 to L=4? Now who has the winning strategy? For any given Challenge card, how can you decide who has a
winning strategy? (Hint: Read carefully the
formal definition
of limit (correctly!) given in Appendix D starting on page A33 of the
book.) Note: The above section numbers refer to our
textbook: Calculus:
Concepts and Contexts, (Third
Edition), by James Stewart, Brooks/Cole Publishing Company;
2005.