James Stewart-Calculus_ Early Transcendentals-Cengage Learning (2015)

LIES MY CALCULATOR AND COMPUTER TOLD ME ■ 9
11. Proof that lim
=0:From1 1 it follows that 1 , and integration gives
1
1
+1 =
lim
0
1
+1 = lim
1
0
1
1
0
=
+1 ,thatis, 1
+1
, and since
+1
=0, it follows from the Squeeze Theorem that lim
+1 =0. Of course, the expression
1( +1)on the left side could have been replaced by 0 and the proof would still be correct.
1
Calculations: Using the formula = 1
1! + 2 2! + ···+ 1
! with an 11-digit calculator:
!
0 17182818284 7 01404151360 14 507636992
1 07182818284 8 01233210880 15 771455488
2 04365636568 9 01098897920 16 12353287808
3 03096909704 10 00988979200 17 21001589274
4 02387638816 11 00878771200 18 37802960693
5 01938194080 12 00545254400 19 71825635317
6 01629164480 13 02911692800 20 14365127163
It is clear that the values calculated from the direct reduction formula will diverge to . If we instead calculate
using the reduction formula in Maple (with Digits:=16), we get some odd results: 20 = 1000, 28 =10 14 ,
29 =0 and 30 =10 17 , for examples. But for larger , the results are at least small and positive (for example,
1000 0001.) For 32,175, we get the delightful object too large error message. If, instead of using the
reduction formula, we integrate directly with Maple, the results are much better.
13. We can start by expressing and in terms of () =( 1) ( 6= 0), where (0) = 1 to make
continuous at 0 (by l’Hospital’s Rule). Namely, =1+(), =1 () and
sinh =
1+() [1 ()]
2
() and (), thus presenting no loss of accuracy due to subtraction.
= 1 [()+()], where the addition involves only positive numbers
2
Stewart: Calculus, Sixth Edition. ISBN: 0495011606. © 2008 Brooks/Cole. All rights reserved.
Another form, which calls the function only once:
We write sinh = ( ) 2 1
= [1 + ()]2 1
2 2[1 + ()]
that sinh
is an even function, so replacing by || does not change its value.
= 1+ 1 || (||) 2
(||)
, taking advantage of the fact
1+|| (||)
Stewart: Calculus: Early Transcendentals, Seventh Edition. ISBN: 0538497904. (c) 2012 Brooks/Cole. All rights reserved.