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Example: a. Find the Taylor series
expansion of f(x) = √ x
for a = 4, n = 2
where
4 ≤ x
≤ 4.1 . b.
Estimate the accuracy to six decimal places. c.
Check result by
graphing | R2(x) | .
f(x)
= x 1/2
, f(4) =
2, f ' (x) =
1/2 x ˗ 1/2
, f ' (4) =
1/4
f '' (x)
= ˗ 1/4 x ˗
3/2 , f '' (4)
= 1 / 32 , f ''' (x) = 3/8 x ˗ 5/2 ,
f ''' (4) = 3/256
T2(x) =
2 + (1/4) (x ˗ 4) +
(˗ 1 / 32) ( x ˗ 4)2 / (2!) (result for part a)
| R2(x) | ≤
M / ( 2 + 1)! | x ˗ 4 | 2+ 1 =
M / (3!) | x ˗ 4 |3 =
Pick the largest value of M in 4 ≤ x
≤ 4.1, M = |
3/256 | and
| x ˗ 4 | = 0.1
| R2(x) | ≤
( 3/256) / ( 3!) | 0.1 | 3 =
0.00000195 = 0.000002
(result for part b)
Finally graph the function (part c) | R2(x) | =
| √ x ˗ T2(x)
| for 4
≤ x 4.1
From the above graph,
the maximum error to six decimal places for the second order
Taylor series expansion
of f(x) is R2(4.1) = 0.000001923. (result for part c)
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