Exercises
UF Calculus Set 21
1. Examine the graph of
f
(
x
)
shown. Around which points would the linearizations seem to be
the least accurate for small changes in
x
? Where would it seem to be most accurate? What
feature of
f
seems to determine this? (Note: we will soon see that this feature can be studied
with the second derivative, which is why it is prominent in our error formula.)
2. Write the linearization of the function at the points indicated. Sketch the function and the two
linearizations on the same axes, and answer the question provided.
(a)
f
(
x
) = ln(
x
)
at
(1
,
0)
and
(
e,
1)
; which gives the better approximation for
ln(2)
?
(b)
f
(
x
) =
e
x
at
(0
,
1)
and
(1
,e
)
; which gives the better approximation for
√
e
?
(c)
f
(
x
) =
√
1 +
x
at
(0
,
1)
and
(3
,
2)
; which gives the better approximation for
√
3
?
3. For each of the functions below, write the differential, and calculate
dy
for the given values
of
x
and
dx
. How accurate is
dy
in approximating
Δ
y
(what is their difference) to the nearest
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 Spring '11
 T
 Calculus, Approximation, Derivative, dx, better approximation

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