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Unformatted text preview: Example
Find the linear approximation L(x) to the function
f (x) = tan π
2 x
3π at the point x = 1/2 and use it to approximate tan 16 . Solution: The linear approximation to
π
x
2 f (x) = tan
at the point x = 1/2 is given by L(x) = f (1/2) + f (1/2)(x − 1/2),
where, using the Chain Rule,
f (x) = sec2
so
f (1/2) = sec2 π
π
x·
,
2
2 π1
π
·
·
22
2 and
f (1/2) = tan = √ 2 2 π
2 = π, π1
·
= 1.
22 It then follows that
L(x) = 1 + π (x − 1/2). To approximate tan 3π
16 using L(x), we need to ﬁrst determine which value of x gives
f (x) = tan 3π
.
16 Clearly x must be such that
π
3π
x=
,
2
16
or
x= 2 3π
3
·
=.
π 16
8 Then
tan 3π
16 3
3
≈L
,
8
8 =f where 3
31
π
−
=1+π
=1− .
8
82
8
π
3π
is 1 − .
So the desired approximation to tan
16
8
L .
(Note that the approximation satisﬁes L(3/8) = .61 while the exact value is given by
.
tan(3π/16) = .67.) 2 ...
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This note was uploaded on 04/02/2012 for the course MTH 132 taught by Professor Kihyunhyun during the Fall '10 term at Michigan State University.
 Fall '10
 KIHYUNHYUN
 Calculus, Approximation, Linear Approximation

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