Lecture 31 – Approximating Functions

1. Lecture 31 – Approximating Functions Consider the following: Now, use the reciprocal function and tangent line to get an approximation. 3 1 2

2. 2.01 2

3. First derivative gave us more information about the function (in particular, the direction). For values of x near a the linear approximation given by the tangent line should be better than the constant approximation. Second derivative will give us more information (curvature). For values of x near a the quadratic approximation should be better than the linear approximation.

4. What quadratic is used as the approximation? Key idea: Need to have quadratic match up with the function and its first and second derivatives at x = a.

5. Use p2(x) to get a better approximation. 2.01 2

6. GraphicalExample at x = 0 3 1 2

7. What higher degree polynomial is appropriate? Key idea: Need to have nth degree polynomial match up with the function and all of its derivatives at x = a.

8. The coefficients, ck, for the nth degree Taylor polynomial approximating the function f(x) at x = a have the form:

9. Lecture 32 – Taylor Polynomials Def: The Taylor polynomial of order n for function f at x = a: The remainder term for using this polynomial: for some c betweenx and a. whereM provides a bound on how big the n+1st derivative could possibly be.

10. Estimatethe maximum error in approximating the reciprocal function at x = 2 with an 8th order Taylor polynomial on the interval [2, 3].

12. What is the actual maximum error in approximating the reciprocal function at x = 2 with an 8th order Taylor polynomial on the interval [2, 3]?

13. What nth degree polynomial would you need in order to keep the error below .0001?

14. To keep error below .0001, need to keep Rn below .0001.

15. Lecture 33 – Taylor Series The Taylor series centered at x = a: is a power series with The Taylor series centered at x = 0 is called a Maclaurin series:

16. Example 1 Find the Maclaurin series for f (x) = sin x.

17. Example 2 Find the Maclaurin series for f (x) = ex.

18. For what values of x will the last two series converge? Ratio Test: Series converges for Series converges for

19. Consider the graphs:

20. Example 3 Find the Maclaurin series for f (x) = ln(1 + x).

21. For what values of x will the series converge?

22. Example 4 Creating new series for:

23. Lecture 34 – More Taylor Series Create and use other Taylor series like was done with power series.

24. Example 1

25. Example 2

26. Example 3

27. Example 4

28. Example 5