The document discusses using Taylor series approximations to evaluate expressions. It finds the maximum error of approximating e-0.5 as 1 - 0.5 + 0.5^2, determines the value of x for which this approximation is accurate to within 0.01, and establishes that an 8th degree Taylor polynomial is needed to approximate e^x in [-1,1] with an error less than 0.001.

1. 11.11 Applications of
Taylor Series
−
Evaluate correct to within an error
of 0.01.

2. 11.11 Applications of
Taylor Series
−
Evaluate correct to within an error
of 0.01.
− ∞ (− ) ∞
= = ! = = (− ) !
= − ! + ! − ! + ···

3. 11.11 Applications of
Taylor Series
−
Evaluate correct to within an error
of 0.01.
− ∞ (− ) ∞
= = ! = = (− ) !
= − ! + ! − ! + ···
−
= − · ! + · ! − · ! + ··· +

4. − ∞ (− ) ∞
= = ! = = (− ) !
= − ! + ! − ! + ···
−
= − · ! + · ! − · ! + ··· +

5. − ∞ (− ) ∞
= = ! = = (− ) !
= − ! + ! − ! + ···
−
= − · ! + · ! − · ! + ··· +
−
= − · ! + · ! − · ! + ···

6. − ∞ (− ) ∞
= = ! = = (− ) !
= − ! + ! − ! + ···
−
= − · ! + · ! − · ! + ··· +
−
= − · ! + · ! − · ! + ···
= − + − + − ···

7. − ∞ (− ) ∞
= = ! = = (− ) !
= − ! + ! − ! + ···
−
= − · ! + · ! − · ! + ··· +
−
= − · ! + · ! − · ! + ···
= − + − + − ···
≈ .

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.

23. Taylor approximation of .
+
= ( ) = + + ···
=
( + )! ! ! !

24. Taylor approximation of .
+
= ( ) = + + ···
=
( + )! ! ! !

25. Taylor approximation of .
+
= ( ) = + + ···
=
( + )! ! ! !

26. Taylor approximation of .
+
= ( ) = + + ···
=
( + )! ! ! !

27. Taylor approximation of .
+
= ( ) = + + ···
=
( + )! ! ! !

28. Taylor approximation of .
+
= ( ) = + + ···
=
( + )! ! ! !

29. Taylor approximation of .
+
= ( ) = + + ···
=
( + )! ! ! !

30. Taylor approximation of .
+
= ( ) = + + ···
=
( + )! ! ! !

31. Taylor approximation of .
+
= ( ) = + + ···
=
( + )! ! ! !

32. Taylor approximation of .
+
= ( ) = + + ···
=
( + )! ! ! !

33. Taylor approximation of .
+
= ( ) = + + ···
=
( + )! ! ! !

34. Taylor approximation of .
+
= ( ) = + + ···
=
( + )! ! ! !

35. Taylor approximation of .
+
= ( ) = + + ···
=
( + )! ! ! !