2. In Section 4.1, we used the linearization L(x) to approximate a function f (x) near a point x = a:Section 4.1 We refer to L(x) as a “first-order” approximation to f (x) at x = a because f (x) and L(x) have the same value and the same first derivative at x = a:derivative A first-order approximation is useful only in a small interval around x = a. In this section we learn how to achieve greater accuracy over larger intervals using the higher-order approximations.

3. In what follows, assume that f (x) is defined on an open interval I and that all higher derivatives f (k) (x) exist on I. Let aopen interval We say that two functions f (x) and g (x) agree to order n at x = a if their derivatives up to order n at x = a are equal: g(x) “approximates f (x) to order n” at x = a. We define the nth Taylor polynomial centered at x = a as follows: Before proceeding to the examples, we write T n (x) in summation notation:

4. The first few Taylor polynomials are: Note that T 1 (x) is the linearization of f (x) at a. Note also that T n (x) is obtained from T n−1 (x) by adding on a term of degree n:

5. THEOREM 1 The polynomial T n (x) centered at a agrees with f (x) to order n at x = a, and it is the only polynomial of degree at most n with this property. This shows that the value and the derivatives of order up to n = 2 at x = a are equal. By convention, we regard f (x) as the zeroeth derivative, and thus f (0) (x) is f (x) itself. When a = 0, T n (x) is also called the nth Maclaurin polynomial.

6. Maclaurin Polynomials for e x Plot the third and fourth Maclaurin polynomials for f (x) = e x. Compare with the linear approximation.

7. Computing Taylor Polynomials Compute the Taylor polynomial T 4 (x) centered at a = 3 for

9. Find the Taylor polynomials T n (x) of f (x) = ln x centered at a = 1.

10. The derivatives form a repeating pattern of period 4: Cosine Find the Maclaurin polynomials of f (x) = cos x. f (j) (x)= f (j+4) (x). The derivatives at x = 0 also form a pattern:

11. Scottish mathematician Colin Maclaurin (1698–1746) was a professor in Edinburgh. Newton was so impressed by his work that he once offered to pay part of Maclaurin’s salary. Horizon

12. How far is the horizon? Valerie is at the beach, looking out over the ocean. How far can she see? Use Maclaurin polynomials to estimate the distance d, assuming that Valerie’s eye level is h = 1.7 m above ground. What if she looks out from a window where her eye level is 20 m?distance Valerie can see a distance d = Rθ, the length of the circular arc AH. Our key observation is that θ is close to zero (both θ and h are much smaller than shown in the figure), so we lose very little accuracy if we replace cos θ by its second Maclaurin polynonomial Maclaurin Polynomials

13. How far is the horizon? Valerie is at the beach, looking out over the ocean. How far can she see? Use Maclaurin polynomials to estimate the distance d, assuming that Valerie’s eye level is h = 1.7 m above ground. What if she looks out from a window where her eye level is 20 m?distance Valerie can see a distance d = Rθ, the length of the circular arc AH. Furthermore, h is very small relative to R, so we may replace R + h by R to obtain

14. Valerie can see a distance d = Rθ, the length of the circular arc AH. Furthermore, h is very small relative to R, so we may replace R + h by R to obtain The earth’s radius is approximately R ≈ 6.37 × 10 6 m, so In particular, we see that d is proportional to Valerie’s eye level is h = 1.7 m

15. THEOREM 2 Error Bound Assume that f (n+1) (x) exists and is continuous. Let K be a number such that |f (n+1) (u)| ≤ K for all u between a and x. Then where T n (x) is the nth Taylor polynomial centered at x = a. Using the Error Bound Apply the error bound to |ln 1.2 − T 3 (1.2)| where T 3 (x) is the third Taylor polynomial for f (x) = lnx at a = 1. Check your result with a calculator.

16. Using the Error Bound Apply the error bound to |ln 1.2 − T 3 (1.2)| where T 3 (x) is the third Taylor polynomial for f (x) = ln x at a = 1. Step 1. Find a value of K. Therefore, we find a value of K such that |f (4) (u)| ≤ K for all u between a = 1 and x = 1.2. As we computed earlier, f (4) (x) = −6x −4 Step 2. Apply the error bound. Step 3. Check the result.

17. Observe that ln x and T 3 (x) are indistinguishable near x = 1.2.

18. Approximating with a Given Accuracy Let T n (x) be the nth Maclaurin polynomial for f (x) = cos x. Find a value of n such that | cos 0.2 − T n (0.2)| < 10 −5 Step 1. Find a value of K. Step 2. Find a value of n. It’s not possible to solve this inequality for n, but we can find a suitable n by checking several values:

19. The error in T n (x) is the absolute value |R n (x)|. R n (x) can be represented as an integral. Taylor’s Theorem: Version I Assume that f (n+1) (x) exists and is continuous. Then Taylor’s Theorem: Version II (Lagrange) Assume f (n+1) (x) exists and is continuous. Then

20. For reference, we will include a table of standard Maclaurin and Taylor polynomials.