1. MAT 1236 Calculus III Section 11.7 Taylor Series

2. HW… WebAssign 11.7 Quiz :11.6, 11.7 Exam is coming up!
Bring the tutoring record on Tuesday

3. Preview
Define the Taylor Series (Foundation of many kinds of pointwise approximations)
Maclaurin Series is a special case of the Taylor Series with 0 as the center
Show that

4. Theorem
That is, if 𝑓 can be represented by a power series, we know how to find all the coefficients 𝑐𝑛

5. Theorem
That is, if 𝑓 can be represented by a power series, we know how to find all the coefficients 𝑐𝑛
WHY?

6. Why?

7. Why?

8. Definition: Taylor Series
It is a Maclaurin Series if 𝑎=0, that is

9. Definition: Taylor Series
It is a Maclaurin Series if 𝑎=0, that is

10. Example 1
Find the Maclaurin Series of

11. Example 1
Find the Maclaurin Series of

12. Example 2
Find the Maclaurin Series of

13. Example 2
Find the Maclaurin Series of
Cheap!

14. Example 3
Find the Maclaurin Series of
Cheap!

15. Example 4
(a) Find the Maclaurin Series of

16. Example 4
(b) Evaluate as an power series 

17. Example 4 (b) Evaluate as an power series 
Hints (i) 𝑓 𝑥 = 𝑒 𝑥 , 𝑒  =𝑓(?)

18. Example 4 (b) Evaluate as an power series 
Hints (i) 𝑓 𝑥 = 𝑒 𝑥 , 𝑒  =𝑓(?)
(ii) 𝑓 𝑥 = 𝑒 𝑥 , 𝑒 −𝑥 2 =𝑓(?)

19. Example 5
Find the Taylor Series of
We need the formula for for all 𝑛

20. Example 5

21. Expectations
In quizzes and exams, you are expected to give solutions with all the details as shown in the last example.

22. Hints
To see/justify the pattern, do not multiply/combine the constants in the derivatives
Check that the pattern actually works for all 𝑛, especially 𝑛=0