Honors AP Calculus / Mr. Hansen

Name: _______KEY__________

5/20/2010 [rev. 6/7/2010]

 

 

Quiz on Lagrange Error Term

 

1.

Prove: If f (x) = sinh x, then  for all  and all .

 

 

 

Note: For all real numbers a, b, we know (or should know) that  This result, known as the triangle inequality, is related to the Pythagorean Theorem in the sense that the shortest distance between two points is always the hypotenuse. Adding lengths individually will always give a result that is at best equal, and often greater. The triangle inequality is used in the proof that follows.

 

 

 

Since f (n) (x) is always either sinh x or cosh x, and since e is known to be less than 3, we have the following for all real x:

 

 

 

 

 

 

 

 

 

2.

Use the result of #1 to find the maximum possible error when the Maclaurin series for sinh x is used to estimate sinh 4 with 6 nonzero terms. Warning:

 

 

 

By Taylor’s Theorem,

 

sinh x = f (x) =

 

 

 

Note that all the even-numbered terms (0, 2, 4, etc.) are 0. Thus the Lagrange error bound after 6 nonzero terms is

 

 for some .

 

 

 

Since sinh x is positive and increasing on (0, 4) [considered known, but easily proved by observing that sinh x has a derivative, cosh x, that is positive everywhere], |sinh x| is bounded on (0, 4) by sinh 4, and by #1, we know that a bound for sinh 4 is 34.

 

 

 

Therefore, 2.837.

 

 

 

Several students made use of the fact that all even-numbered terms in the Maclaurin expansion are zero. Therefore, we might as well say that we have used 12, not 11, terms in achieving the first 6 nonzero terms. The advantage of doing that is that then the “n + 1” in the Lagrange error term can be 13 instead of 12. This gives a bound of

 

 

 

0.873, which is a tighter bound. Either method is fine.