Lecture 20: Taylor’s Theorem and the Definition of Riemann Sums

18.100A: Complete Lecture Notes Lecture 20: Taylor’s Theorem and the Definition of Riemann Sums Taylor’s Theorem Remark 1. Taylor’s theorem is essentially the Mean Value Theorem for higher order derivatives. Definition 2 (n-times Differentiable) We say f : I → R is n-times differentiable on J ⊂ I if f 0 , f 00 , . . . , f (n) exist at every point in J. Notation 3 We denote the n-th derivative of f as f (n) (as used above). Theorem 4 (Taylor) Suppose f : [a, b] → R is continuous and has n continuous derivatives on [a, b] such that f (n+1) exists on (a, b). Given x0 , x ∈ [a, b], there exists a c ∈ (x0 , x) such that f (x) = n X 1 (k) f (n+1) (c) f (x0 )(x − x0 )k + (x − x0 )n+1 . k! (n + 1)! k=0 Denote the large sum as Pn (x) and the last term with Rn (x). Definition 5 Pn (x) is the n-th order Taylor polynomial for f at x0 . Rn (x) is the n-th order remainder term. We will essentially apply the Mean Value Theorem n + 1 times to prove Taylor’s theorem. Proof : Let x, x0 ∈ [a, b]. If x = x0 then any c will satisfy the theorem. So, suppose x 6= x0 . Let Mx,x0 = f (x)−Pn (x) (x−x0 )n+1 . Hence, f (x) = Pn (x) + Mx,x0 (x − x0 )n+1 . Now, for 0 ≤ k ≤ n, f k (x0 ) = Pn(k) (x0 ). Let g(s) = f (s) − Pn (s) − Mx,x0 (s − x0 )n+1 (notably, n + 1-times differentiable. Then, g(x0 ) = f (x0 ) − Pn (x0 ) − Mx,x0 (x0 − x0 )n+1 = 0 g 0 (x0 ) = f 0 (x0 ) − Pn0 (x0 ) − Mx,x0 (n + 1)(x0 − x0 )n = 0 .. . g (n) (x0 ) = f (n) (x0 ) − Pn(n) (x0 ) − Mx,x0 (n + 1)!(x0 − x0 ) = 0. 1 Now, notice that g(x) = 0 and g(x0 ) = 0. By the MVT, there exists an x1 ∈ (x0 , x) such that g 0 (x1 ) = 0. Thus, g 0 (x0 ) = 0 and g 0 (x1 ) = 0. Therefore, ∃x2 ∈ (x0 , x1 ) such that g 00 (x2 ) = 0. Continuing, we analogously find xn between x0 and xn−1 such that g (n) (xn ) = 0. Then, finally, g (n) (x0 ) = 0 and g (n) (xn ) = 0 implies ∃c ∈ (x0 , xn ) (and thus between x0 and x) such that g (n+1) (c) = 0. We may compute dn+1 Mx,x0 (s − x0 )n+1 = Mx,x0 (n + 1)!. ds(n+1) (n+1) Furthermore, Pn (c) = 0 since Pn is a polynomial of degree n. Thereforee, 0 = g (n+1) (c) = f (n+1) (c) − Mx,x0 (n + 1)!, which implies Mx,x0 = f (n+1)! (c) (n+1)! and thus f (x) = Pn (x) + f (n+1) (c) (x − x0 )n+1 . (n + 1)! Theorem 6 (Second Derivative Test) Suppose f : (a, b) → R has two continuous derivatives. If x0 ∈ (a, b) such that f 0 (x0 ) = 0 and f 00 (x0 ) > 0, then f has a strict relative minimum at x0 . Proof : Since f 00 is continuous at x0 and lim f 00 (c) = f 00 (x0 ) > 0, c→x0 we have that ∃δ > 0 such that for all c ∈ (x0 − δ, x0 + δ), f 00 (c) > 0. Let x ∈ (x0 − δ, x0 + δ) (as you will show in your homework). Then, by Taylor’s theorem, ∃c between x and x0 (hence c ∈ (x0 − δ, x0 + δ)) such that f (x) = f (x0 ) + f 00 (c) (x − x0 )2 ≥ f (x0 ), 2 with f (x) > f (x0 ) if x 6= x0 . The Riemann Integral Remark 7. Riemann integration is the first rigorous theory of ‘area’ that agrees with experience (areas of rectangles, triangles, circles), and it is the inverse of differentiation. However, it is not a complete theory of area (see Lebesgue integration). The Riemann Integral Definition 8 We define the set C([a, b]) := {f : [a, b] → R | f is continuous}. 2 Definition 9 (Partition) A partition x of [a, b] is a finite set x = {a = x0 < x1 < · · · < xn = b}. The norm of x, denoted kxk, is the number kxk := max{x1 − x0 , x2 − x1 , . . . , xn − xn−1 }. Definition 10 (Tag) If x is a partition, a tag of x is a finite set ξ = {ξ1 , . . . , ξn } such that a = x0 ≤ ξ1 ≤ x1 ≤ ξ2 ≤ x2 ≤ · · · ≤ xn−1 ≤ ξn ≤ xn = b. The pair (x, ξ) is referred to as a tagged partition. Example 11 Consider the tagged partition (x, ξ) = ({1, 3/2, 2, 3}, {5/4, 7/4, 5/2}. Then, kxk = max{3/2 − 1, 2 − 3/2, 3 − 2} = 1. Definition 12 (Riemann sum) The Riemann sum of f corresponding to (x, ξ) is the number Sf (x, ξ) := n X f (ξk )(xk − xk−1 ). k=1 Let’s try to understand how to interpret this using a picture. For f ∈ C([a, b]) positive, Sf (x, ξ) is an approximate area under the graph of f . As kxk → 0, we should expect these approximate areas to converge to a number A, which we interpret as the area under the curve f on the interval [a, b]. Question 13. Do these approximate sums actually converge? We will answer this question and more during the next few lectures. 3