8. What is the maximum error possible in using
the approximation
≈ − ! + !
when − . ≤ ≤ . ?
For what value of is this approximation
accurate to within . ?
What is the smallest degree of the Taylor
polynomial we can use to approximate if we
want the error in [ . , . ] to be less
than ?
9. What is the maximum error possible in using
the approximation
≈ − ! + !
when − . ≤ ≤ . ?
10. What is the maximum error possible in using
the approximation
≈ − ! + !
when − . ≤ ≤ . ?
Taylor’s Inequality:
| ( )| | |
!
11. What is the maximum error possible in using
the approximation
≈ − ! + !
when − . ≤ ≤ . ?
Taylor’s Inequality:
| ( )| | |
!
( )
| |=|− |≤ , =
12. What is the maximum error possible in using
the approximation
≈ − ! + !
when − . ≤ ≤ . ?
Taylor’s Inequality:
| ( )| | |
!
( )
| |=|− |≤ , =
when − . ≤ ≤ .
13. What is the maximum error possible in using
the approximation
≈ − ! + !
when − . ≤ ≤ . ?
Taylor’s Inequality:
| ( )| | |
!
( )
| |=|− |≤ , =
when − . ≤ ≤ .
.
.
!
14. For what value of is this approximation
accurate to within . ?
15. For what value of is this approximation
accurate to within . ?
We want
| |
< .
!
16. For what value of is this approximation
accurate to within . ?
We want
| |
< .
!
| | < . · ! .
17. For what value of is this approximation
accurate to within . ?
We want
| |
< .
!
| | < . · ! .
| |< .
18. What is the smallest degree of the Taylor
polynomial we can use to approximate if we want
the error in [ . , . ] to be less than ?
19. What is the smallest degree of the Taylor
polynomial we can use to approximate if we want
the error in [ . , . ] to be less than ?
.
| ( )| | | .
! !
20. What is the smallest degree of the Taylor
polynomial we can use to approximate if we want
the error in [ . , . ] to be less than ?
.
| ( )| | | .
! !
.
| ( )| | | .
! !
21. What is the smallest degree of the Taylor
polynomial we can use to approximate if we want
the error in [ . , . ] to be less than ?
.
| ( )| | | .
! !
.
| ( )| | | .
! !
.
| ( )| | | .
! !
22. What is the smallest degree of the Taylor
polynomial we can use to approximate if we want
the error in [ . , . ] to be less than ?
.
| ( )| | | .
! !
.
| ( )| | | .
! !
.
| ( )| | | .
! !
So we need the Taylor polynomial of degree 8.