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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
a
= 3
.
(a) Compute
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This note was uploaded on 04/02/2009 for the course MTH 142 taught by Professor Staff during the Spring '08 term at Sam Houston State University.
 Spring '08
 STAFF
 Calculus, Derivative, Limits

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