white (taw933) – HW14 – benzvi – (55600)
1
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printout
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have
19
questions.
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before answering.
001
10.0points
Find the derivative of
f
when
f
(
θ
) = ln (cos 3
θ
)
.
1.
f
′
(
θ
) =

1
sin 3
θ
2.
f
′
(
θ
) =

3 cot 3
θ
3.
f
′
(
θ
) =
3
cos 3
θ
4.
f
′
(
θ
) = cot 3
θ
5.
f
′
(
θ
) =

3 tan 3
θ
correct
6.
f
′
(
θ
) = 3 tan 3
θ
Explanation:
By the Chain Rule,
f
′
(
θ
) =
1
cos(3
θ
)
d
dθ
(cos 3
θ
) =

3 sin 3
θ
cos 3
θ
.
Consequently,
f
′
(
θ
) =

3 tan 3
θ
.
002
10.0points
Find the slope of the line tangent to the
graph of
ln(
xy
)

2
x
= 0
at the point where
x
= 1.
1.
slope =
1
2
e
−
2
2.
slope =
e
2
correct
3.
slope =

1
2
e
−
2
4.
slope =
e
−
2
5.
slope =

e
2
6.
slope =

1
2
e
2
Explanation:
Differentiating implicitly with respect to
x
we see that
1
xy
parenleftBig
y
+
x
dy
dx
parenrightBig

2 = 0
,
in which case
dy
dx
=

y
(1

2
x
)
x
=

e
2
x
(1

2
x
)
x
2
because, by exponentiation,
y
=
e
2
x
x
.
Consequently, at
x
= 1,
slope =
dy
dx
vextendsingle
vextendsingle
vextendsingle
x
=1
=
e
2
.
003
10.0points
Find the derivative of
f
(
t
) =
1 + ln
t
4

ln
t
.
1.
f
′
(
t
) =
5
t
(4

ln
t
)
2
correct
2.
f
′
(
t
) =
4
t
(1 + ln
t
)
2
3.
f
′
(
t
) =

4 ln
t
t
(1 + ln
t
)
2
4.
f
′
(
t
) =

5
t
(4

ln
t
)
2
5.
f
′
(
t
) =
4 ln
t
(1 + ln
t
)
2
6.
f
′
(
t
) =

5
(4

ln
t
)
2
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white (taw933) – HW14 – benzvi – (55600)
2
Explanation:
By the Quotient Rule,
f
′
(
t
) =
(4

ln
t
)(1
/t
) + (1 + ln
t
)(1
/t
)
(4

ln
t
)
2
=
(4

ln
t
) + (1 + ln
t
)
t
(4

ln
t
)
2
.
Consequently,
f
′
(
t
) =
5
t
(4

ln
t
)
2
.
004
10.0points
Find the derivative of
f
when
f
(
x
) = 2 ln(
x

radicalbig
x
2

3)
,
(
x >
3)
.
1.
f
′
(
x
) =

4
√
x
2

3
2.
f
′
(
x
) =
4
√
x
2

3
3.
f
′
(
x
) =
2
√
x
2

3
4.
f
′
(
x
) =

2
√
x
2

3
correct
5.
f
′
(
x
) =
1
√
x
2

3
6.
f
′
(
x
) =

1
√
x
2

3
Explanation:
By the Chain Rule
f
′
(
x
) =
2
x

√
x
2

3
parenleftBig
1

x
√
x
2

3
parenrightBig
=

2
√
x
2

3
.
005
10.0points
Determine
f
′
(
x
) when
f
(
x
) =
e
(3ln(
x
5
))
.
1.
f
′
(
x
) =
1
x
e
3ln(
x
5
)
2.
f
′
(
x
) = 15
x
14
correct
3.
f
′
(
x
) = 15(ln
x
)
e
3ln(
x
5
)
4.
f
′
(
x
) =
e
15
/x
5.
f
′
(
x
) = 14
x
15
6.
f
′
(
x
) =
3
x
2
e
3ln(
x
5
)
Explanation:
Since
r
ln
x
= ln
x
r
,
e
ln
x
=
x ,
we see that
f
(
x
) =
e
(ln
x
15
)
=
x
15
.
Consequently,
f
′
(
x
) = 15
x
14
.
006
10.0points
Find the derivative of
f
(
x
) = ln
radicalBigg
1 + 2
x
2
1

2
x
2
.
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 Spring '08
 schultz
 Derivative, Differential Calculus

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