Assignment 6
Due Monday, April 14, 2008
1. Explain the relationship between limits of functions and the concept of the derivative. (Why do we
concern ourselves about limits in a calculus class? If we are taking limits of
functions
when we
compute a derivative, what function is involved in that limit?)
2. Let
f
(
x
) = sin
x
and
a
=
π
/
6
.
In class we agreed that the limit of
f
(
x
) as
x
approaches
a
is
L
=
1
2
.
That is, we agreed that
lim
x
→
π
6
sin
x
=
1
2
.
This problem explores the meaning of this statement; throughout this problem
f
(
x
) = sin
x
and
a
=
π
/
6
.
(a) For each
, below, compute the best
δ
, to four decimal places, such that if 0
<

x

a

<
δ
then

f
(
x
)

L

<
.
i.
= 0
.
1
ii.
= 0
.
05
iii.
= 0
.
01
(b) Suppose
is given and
very
small.
What will be the ”best” choice here for
δ
in the limit
definition? (Note that in this part of the problem, unlike part (a), you are not being given
.
Your answer should describe
δ
in terms of the unknown
.
)
3. For this problem, let
f
(
x
) =
x
3

27
x

3
and
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 Spring '08
 STAFF
 Calculus, Derivative, Limits, Continuous function, decimal places, limit deﬁnition, /6.

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