wtm369 – Review 3 – Helleloid – (58250)
1
This
printout
should
have
19
questions.
Multiplechoice questions may continue on
the next column or page – find all choices
before answering.
001
10.0 points
Find the value of
x
when
x
= 8 log
4
parenleftBig
1
16
parenrightBig
−
7 log
3
27
without using a calculator.
1.
x
=
−
34
2.
x
= 37
3.
x
= 33
4.
x
=
−
33
5.
x
=
−
37
correct
Explanation:
By properties of logs,
log
4
parenleftBig
1
16
parenrightBig
= log
4
parenleftBig
1
(4)
2
parenrightBig
=
−
2
,
while
log
3
27 = log
3
(3)
3
= 3
.
Consequently,
x
=
−
21
−
16 =
−
37
.
002
10.0 points
Simplify the expression
f
(
x
) = 8
2(log
8
e
)ln
x
as much as possible.
1.
f
(
x
) = 2
x
2.
f
(
x
) =
x
8
3.
f
(
x
) =
e
17
4.
f
(
x
) =
x
16
5.
f
(
x
) =
x
2
correct
Explanation:
By the property of inverse functions,
8
log
8
y
=
y,
e
ln
y
=
y .
Consequently,
f
(
x
) = 8
2(log
8
e
)ln
x
= (8
log
8
e
)
2ln
x
=
e
ln
x
2
=
x
2
.
003
10.0 points
Five functions are defined by the table
x
f
(
x
)
g
(
x
)
h
(
x
)
F
(
x
)
G
(
x
)
1
2
3
4
5
6
9
6
2
7
8
1
2
8
4
6
9
7
7
2
8
6
4
3
3
8
1
6
2
9
2
4
9
8
2
7
Which of these functions is not 11?
1.
h
(
x
)
2.
G
(
x
)
correct
3.
g
(
x
)
4.
F
(
x
)
5.
f
(
x
)
Explanation:
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wtm369 – Review 3 – Helleloid – (58250)
2
The function
G
(
x
) is not 11 since
G
(1) =
G
(5)
.
keywords:
inverse function, function values
11 function
004
10.0 points
If
f
is the function defined by
f
(
x
) = 3
x
2
(
x
≤
0)
,
determine the inverse function,
f
−
1
, of
f
.
1.
f
−
1
(
x
) =
−
radicalbigg
x
3
,
(
x
≥
0)
correct
2.
f
−
1
(
x
) =
radicalbigg
−
x
3
,
(
x
≤
0)
3.
f
−
1
(
x
) =
−
radicalbigg
−
x
3
,
(
x
≤
0)
4.
f
−
1
(
x
) =
radicalbigg
x
3
,
(
x
≥
0)
5.
f
−
1
(
x
) =
radicalbigg
−
x
3
,
(
x
≥
0)
Explanation:
The function defined by
f
(
x
) = 3
x
2
is one
toone so long as
x
is restricted to
x
≤
0;
hence the inverse function,
f
−
1
, exists. Now
3
x
2
=
y,
(
y >
0)
=
⇒
x
=
±
radicalbigg
y
3
.
To determine
f
−
1
we interchange the roles of
x
and
y
. But the original function
f
is defined
only for
x
≤
0. Consequently,
f
−
1
(
x
) =
−
radicalbigg
x
3
,
(
x
≥
0)
.
005
10.0 points
On (
−
1
,
1) the function
f
(
x
) = 4 +
x
2
+ tan
parenleftBig
πx
2
parenrightBig
has an inverse
g
. Find the value of
g
′
(4).
(Hint:
find the value of
f
(0)).
1.
g
′
(4) =
2
π
correct
2.
g
′
(4) =
π
2
3.
g
′
(4) =
4
π
4.
g
′
(4) = 1
5.
g
′
(4) =
π
4
Explanation:
By definition of inverse function:
f
(
g
(
x
)) =
x ,
g
(
f
(
x
)) =
x .
Applying the Chain Rule to the first of these
equations, we see that
f
′
(
g
(
x
))
g
′
(
x
) = 1
,
in which case
g
′
(
x
) =
1
f
′
(
g
(
x
))
.
Thus to find the value of
g
′
(4) we need to
know the value of
g
(4) as well as
f
′
(
x
). Now
f
′
(
x
) = 2
x
+
π
2
sec
2
parenleftBig
πx
2
parenrightBig
.
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 Fall '08
 schultz
 Calculus, Inverse function, $100, Logarithm, $290, $3200

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