Final Review
Answers
p. 1
Disclaimer: This answer sheet may contain errors, hopefully there are not many.
1.
a.
1
x
+2 = pos # which means that
)
2
(
2
+
=
+
ξ
ξ
and you cancel
b.
1
x
+2 = neg # which means that
)
2
(
2
+

=
+
ξ
ξ
and you cancel
c.
1
divide each term by the highest power in the denominator which is
x
(Note:
2
x
x
=
)
d.
0
divide each term by the highest power in the denominator which is
x
2
e.
6
1

you get
0
0
, so use L’Hopital’s Rule repeated until you no longer get
0
0
f.
DNE
x
y
sin
=
does not “settle down” to a single value and continually oscillates
2.
a.
3
1
see problem 1c
b.
DNE
Compute
4
5
lim
4

+
→
ξ
ξ
and
4
5
lim
4


→
ξ
ξ
and see that they are not the same value
c.
DNE
(
1
3
lim
)
(
lim
2
2
2
=

=
+
+
→
→
η
η
θ
η
η
3
1
lim
)
(
lim
2
2
=
+
=
+
+
→
→
η
η
θ
η
η
)
d.
5
Compute the left and righthand limits
e.
4
1

Simplify by first using a common denominator
3.
a.
b.
4.
Write the equation as
0
cos
=

ξ
ξ
, then show that for some value of
x
x
x

χοσ
= neg # and for a
different value of
x
x
x

χοσ
= pos #.
(Hint: find an
x
value that gives these two results).
Then explain
how you know that
x
x

χοσ
is continuous and use the Intermediate Value Theorem to claim that for some
of
x
,
0
cos
=

ξ
ξ
. (Or in terms of the original problem there is some
x
value where
x
x
=
χοσ
.)
5.
a.
2
2
)
(

=
′
ξ
ξ
φ
b.
2
1
)
(
x
x
f

=
′
6.
0
2
)
2
sin
(
cos
×

+
=
′
ξ
ξ
ψ
and
0
0
2

=
÷
′
π
ψ
.
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Final Review
Answers
p. 2
Disclaimer: This answer sheet may contain errors, hopefully there are not many.
7.
π
cm
2
/min
?
=
δτ
δΑ
when
50
=
ρ
01
.
0
=
δτ
δρ
Area of a circle:
2
r
A
π
=
Be sure to take the derivative of the Area equation BEFORE plugging in any
numbers.
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 Fall '08
 STAFF
 Calculus, Derivative, Cos, lim θ

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