1. Maclaurin and Taylor Polynomials Objective: Improve on the local linear approximation for higher order polynomials.

2. Local Quadratic Approximations Remember we defined the local linear approximation of a function f at x 0 as. In this formula, the approximating function is a first-degree polynomial. The local linear approximation of f at x 0 has the property that its value and the value of its first derivative match those of f at x 0.

3. Local Quadratic Approximations If the graph of a function f has a pronounced “bend” at x 0, then we can expect that the accuracy of the local linear approximation of f at x 0 will decrease rapidly as we progress away from x.

4. Local Quadratic Approximations One way to deal with this problem is to approximate the function f at x 0 by a polynomial p of degree 2 with the property that the value of p and the values of its first two derivatives match those of f at x 0. This ensures that the graphs of f and p not only have the same tangent line at x 0, but they also bend in the same direction at x 0. As a result, we can expect that the graph of p will remain close to the graph of f over a larger interval around x 0. The polynomial p is called the local quadratic approximation of f at x = x 0.

5. Local Quadratic Approximations We will try to find a formula for the local quadratic approximation for a function f at x = 0. This approximation has the form where c 0, c 1, and c 2 must be chosen so that the values of and its first two derivatives match those of f at 0. Thus, we want

6. Local Quadratic Approximations The values of p(0), p / (0), and p // (0) are as follows:

7. Local Quadratic Approximations Thus it follows that The local quadratic approximation becomes

8. Example 1 Find the local linear and local quadratic approximations of e x at x = 0, and graph e x and the two approximations together.

9. Example 1 Find the local linear and local quadratic approximations of e x at x = 0, and graph e x and the two approximations together. f(x)= e x, so The local linear approximation is The local quadratic approximation is

10. Example 1 Find the local linear and local quadratic approximations of e x at x = 0, and graph e x and the two approximations together. Here is the graph of the three functions.

11. Maclaurin Polynomials It is natural to ask whether one can improve the accuracy of a local quadratic approximation by using a polynomial of degree 3. Specifically, one might look for a polynomial of degree 3 with the property that its value and the values of its first three derivatives match those of f at a point; and if this provides an improvement in accuracy, why not go on to polynomials of higher degree?

12. Maclaurin Polynomials We are led to consider the following problem:

13. Maclaurin Polynomials We will begin by solving this problem in the case where x 0 = 0. Thus, we want a polynomial such that

14. Maclaurin Polynomials But we know that

15. Maclaurin Polynomials But we know that Thus we need

16. Maclaurin Polynomials This yields the following values for the coefficients of p(x)

17. Maclaurin Polynomials This leads us to the following definition.

18. Example 2 Find the Maclaurin Polynomials p 0, p 1, p 2, p 3, and p n for e x.

19. Example 2 Find the Maclaurin Polynomials p 0, p 1, p 2, p 3, and p n for e x. We know that so

20. Example 2 Find the Maclaurin Polynomials p 0, p 1, p 2, p 3, and p n for e x. The graphs of e x and all four approximations are shown.

21. Taylor Polynomials Up to now we have focused on approximating a function f in the vicinity of x = 0. Now we will consider the more general case of approximating f in the vicinity of an arbitrary domain value x 0.

22. Taylor Polynomials Up to now we have focused on approximating a function f in the vicinity of x = 0. Now we will consider the more general case of approximating f in the vicinity of an arbitrary domain value x 0. The basic idea is the same as before; we want to find an nth-degree polynomial p with the property that its value and the values of its first n derivatives match those of f at x 0. However, rather than expressing p(x) in powers of x, it will simplify the computation if we express it in powers of x – x 0 ; that is

23. Taylor Polynomial This leads to the following definition:

24. Example 3 Find the first four Taylor Polynomials for lnx about x = 2.

25. Example 3 Find the first four Taylor Polynomials for lnx about x = 2. Let f(x) = lnx. Thus

26. Example 3 Find the first four Taylor Polynomials for lnx about x = 2. This leads us to:

27. Example 3 Find the first four Taylor Polynomials for lnx about x = 2. The graphs of all five polynomials are shown.

28. Sigma Notation Frequently we will want to express a Taylor Polynomial in sigma notation. To do this, we use the notation f k (x 0 ) to denote the k th derivative of f at x = x o, and we make the connection that f 0 (x 0 ) denotes f(x 0 ). This enables us to write

29. Example 4 Find the n th Maclaurin polynomials for (a) (b) (c)

30. Example 4 Find the n th Maclaurin polynomials for (a) (b) (c) (a)

31. Example 4 Find the n th Maclaurin polynomials for (a) (b) (c) (a) This leads us to:

32. Example 4 Find the n th Maclaurin polynomials for (a) (b) (c) (a) This leads us to:

33. Example 4 Find the n th Maclaurin polynomials for (a) (b) (c) (a) The graphs are shown.

34. Example 4 Find the n th Maclaurin polynomials for (a) (b) (c) (b) We start with:

35. Example 4 Find the n th Maclaurin polynomials for (a) (b) (c) (b) This leads us to:

36. Example 4 Find the n th Maclaurin polynomials for (a) (b) (c) (b) This leads us to:

37. Example 4 Find the n th Maclaurin polynomials for (a) (b) (c) (b) The graphs are shown.

38. Example 4 Find the n th Maclaurin polynomials for (a) (b) (c) You need to memorize these!

39. Example 4 Find the n th Maclaurin polynomials for (a) (b) (c) You need to memorize these!

40. Example 4 Find the n th Maclaurin polynomials for (a) (b) (c) You need to memorize these!

41. Example 4 Find the n th Maclaurin polynomials for (a) (b) (c) (c) We start with:

42. Example 4 Find the n th Maclaurin polynomials for (a) (b) (c) (c) We start with:

43. Example 4 Find the n th Maclaurin polynomials for (a) (b) (c) (c) This leads us to You should memorize this as well.

44. The nth Remainder It will be convenient to have a notation for the error in the approximation. Accordingly, we will let denote the difference between f(x) and its nth Taylor polynomial: that is This can also be written as:

45. The nth Remainder The function is called the nth remainder for the Taylor series of f, and the formula below is called Taylor’s formula with remainder.

46. The nth Remainder Finding a bound for gives an indication of the accuracy of the approximation. The following theorem provides such a bound. The bound, M, is called the Lagrange error bound.

47. Example 6 Use an nth Maclaurin polynomial for to approximate e to five decimal places.

48. Example 6 Use an nth Maclaurin polynomial for to approximate e to five decimal places. The nth Maclaurin polynomial for is from which we have

49. Example 6 Use an nth Maclaurin polynomial for to approximate e to five decimal places. Our problem is to determine how many terms to include in a Maclaurin polynomial for to achieve five decimal-place accuracy; that is, we want to choose n so that the absolute value of the nth remainder at x = 1 satisfies

50. Example 6 Use an nth Maclaurin polynomial for to approximate e to five decimal places. To determine n we use the Remainder Estimation Theorem with, and I being the interval [0, 1]. In this case it follows that where M is an upper bound on the value of for x in the interval [0, 1].

51. Example 6 Use an nth Maclaurin polynomial for to approximate e to five decimal places. However, is an increasing function, so its maximum value on the interval [0, 1] occurs at x = 1; that is, on this interval. Thus, we can take M = e to obtain

52. Example 6 Use an nth Maclaurin polynomial for to approximate e to five decimal places. Unfortunately, this inequality is not very useful because it involves e, which is the very quantity we are trying to approximate. However, if we accept that e < 3, then we can use this value. Although less precise, it is more easily applied.

53. Example 6 Use an nth Maclaurin polynomial for to approximate e to five decimal places. Thus, we can achieve five decimal-place accuracy by choosing n so that or This happens when n = 9.

54. Homework Page 684 1, 3, 7, 9, 11, 15-21 odd 31, 32