1. Taylor Series John Weiss

2. Approximating Functions f(0)= 4 What is f(1)? f(x) = 4? f(1) = 4?

3. Approximating Functions f(0)= 4, f’(0)= -1 What is f(1)? f(x) = 4 - x? f(1) = 3?

4. Approximating Functions f(0)= 4, f’(0)= -1, f’’(0)= 2 What is f(1)? f(x) = 4 – x + x2? (same concavity) f(1) = 4?

5. Approximating Functions f(x) = sin(x) What is f(1)? f(0) = 0, f’(0) = 1 f(x) = 0 + x? f(1) = 1?

6. Approximating Functions f(x) = sin(x) f(0) = 1, f’(0) = 1, f’’(0) = 0, f’’’(0) = -1,… What is f(1)? i.e . What is sin(1)?

7. Famous Mathematicians James Gregory (1671) Brook Taylor (1712) Colin Maclaurin (1698-1746) Joseph-Louis Lagrange (1736-1813) Augustin-Louis Cauchy (1789-1857)

8. Approximations Linear Approximation Quadratic Approximation

9. Taylor’s Theorem Let k≥1 be an integer and be k times differentiable at . Then there exists a function such that Note: Taylor Polynomial of degree k is:

10. Works for Linear Approximations

11. Works for Quadratic Approximations

12. f(x) = sin(x)Degree 1

13. f(x) = sin(x)Degree 3

14. f(x) = sin(x)Degree 5

15. f(x) = sin(x)Degree 7

16. f(x) = sin(x)Degree 11

17. Implications If fand g have the same value and all of the same derivatives at a point, then they must be the same function!

18. Proof: If f and g are smooth functions that agree over some interval, they MUST be the same function Let f and g be two smooth functions that agree for some open interval (a,b), but not over all of R Define h as the difference, f – g, and note that h is smooth, being the difference of two smooth functions. Also h=0 on (a,b), but h≠0 at other points in R Without loss of generality, we will form S, the set of all x>a, such that f(x)≠0 Note that a is a lower bound for this set, S, and being a subset of R, S is complete so S has a real greatest lower bound, call it c. c, being a greatest lower bound of S, is also an element of S, since S is closed Now we see that h=0 on (a,c), but h≠0 at c. So, h is discontinuous at c, so then h cannot be smooth Thus we have reached a contradiction, and so f and g must agree everywhere!

19. Suppose f(x) can be rewritten as a power series…

20. Entirety (Analytic Functions) A function f(x) is said to be entire if it is equal to its Taylor Series everywhere Entire sin(x) Not Entire log(1+x)

21. Proof: sin(x) is entire Maclaurin Series sin(0)=0 sin’(0)=1 sin’’(0)=0 sin’’’(0)=-1 sin’’’’(0)=0 sin’’’’’(0)=1 sin’’’’’’(0)=0 … etc.

22. Proof: sin(x) is entire Lagrange formula for the remainder: Let be k+1 times differentiable on (a,x) and continuous on [a,x]. Then for some z in (a,x)

23. Proof: sin(x) is entire First, sin(x) is continuous and infinitely differentiable over all of R If we look at the Taylor Polynomial of degree k Note though for all z in R

24. Proof: sin(x) is entire However, as k goes to infinity, we see Applying the Squeeze Theorem to our original equation, we obtain that as k goes to infinity and thus sin(x) is entire since it is equal to its Taylor series

25. Maclaurin Series Examples Note:

26. Applications Physics Special Relativity Equation Fermat’s Principle (Optics) Resistivity of Wires Electric Dipoles Periods of Pendulums Surveying (Curvature of the Earth)

27. Special Relativity Let . If v ≤ 100 m/s Then according to Taylor’s Inequality (Lagrange)

28. Lagrange Remainder Lagrange formula for the remainder: Let be k+1 times differentiable on (a,x) and continuous on [a,x]. Then for some z in (a,x)

29. Special Relativity Let . If v ≤ 100 m/s Then according to Taylor’s Inequality (Lagrange)

30. The End