What square root best approximates the point on the graph?
1
2
3
4
5
6
7
1
N15
128
153

Using binomial expansion,

√

a

2

+

b

=

a

(

1

+

b

a

2

)

1

2

=

a

(

1

+

1

2

b

a

2

)

, nearly.

Here, choose a = 66 and b = 44.

Now,

√

4400

=

√

66

2

+

44

=

√

66

2

+

44

=

66

(

1

+

44

66

2

)

1

2

66

(

1

+

1

2

(

44

66

2

)

)

=

66

(

1

+

1

198

)

=

66

+

1

3

=

66.33

, nearly.

Note that the error is of order 66(1/198)^2=O(.002), nearly.

Explanation:

pepin the short[a] (german: pippin der kurze, french: pépin le bref, c. 714 – 24 september 768) was the king of the franks from 751 until his death. he was the first of the carolingians to become king.[b][2]

the younger son of the frankish prince charles martel and his wife rotrude, pepin's upbringing was distinguished by the ecclesiastical education he had received from the monks of st. denis. succeeding his father as the mayor of the palace in 741, pepin reigned over francia jointly with his elder brother carloman. pepin ruled in neustria, burgundy, and provence, while his brother carloman established himself in austrasia, alemannia and thuringia. the brothers were active in suppressing revolts led by the bavarians, aquitanians, saxons, and the alemanni in the early years of their reign. in 743, they ended the frankish interregnum by choosing childeric iii, who was to be the last merovingian monarch, as figurehead king of the franks.

lee and grant, both of whom held the highest rank in their respective armies, had known each other slightly during the mexican-american war (1846-48) and began their dialogue by exchanging awkward personal inquiries. characteristically, grant had arrived in his mud-splattered field uniform while lee had turned out in full dress attire, complete with sash and sword.

explanation: