ucsc
econ/ams
11
b
Review Questions 4
Solutions
1.
Compute the differentials of the functions below.
a.
y
=
x
2

3
x
+ 1
,
dy
= (2
x

3)
dx
b.
u
=
e
x
2

3
x
+1
,
du
=
e
x
2

3
x
+1
(2
x

3)
du
2.
Use differentials to estimate
3
√
28. Express your answer as a simple fraction,
a/b
,
not
in decimal form.
We use the approximation formula
f
(
x
0
+
dx
)
≈
f
(
x
0
) +
dy
. First, we identify the
function, which is straightforward:
f
(
x
) =
x
1
/
3
. Next, we identify
x
0
. We want to
set 28 =
x
0
+
dx
, and we want
dx
to be relatively small, and we also want
x
0
to
be a point for which it is easy to evaluate
f
(
x
). In other words we are looking for
a point,
x
0
, that is close to 28 and for which the cube root is known.
x
0
= 27 fills
the bill.
So, we have
f
(
x
) =
x
1
/
3
,
x
0
= 27 and
dx
= 28

27 = 1. Next we compute
dy
:
dy
=
f
0
(
x
0
)
dx
=
1
3
x

2
/
3
0
dx
=
1
3
27

2
/
3
·
1 =
1
27
.
Finally, we plug everything back into the approximation formula, above.
28
1
/
3
≈
27
1
/
3
+
dy
= 3 +
1
27
=
82
27
.
Note:
estimate is within 0
.
00045 of the true value of
3
√
28.
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 Spring '08
 BINICI
 Constant of integration, initial value, dx, dy

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