CHAPTER 1 Mathematical Preliminaries and Error Analysis Figure 1.2 The tangent line has slope '(ro xo, f(ro)) Theorem 1.6 If the function f is differentiable at xo then f is continuous at xo. theorem attributed to michel The next theorems are of fundamental importance in deriving methods for error esti (1652-1719)appeared in mation. The proofs of these theorems and the other unreferenced results in this section can 1691 in a little-known treatise be found in any standard calculus text entitled Metode pour resound The set of all functions that have n continuous derivatives on X is denoted Cn(X), and es egalites. Rolle originally the set of functions that have derivatives of all orders on X is denoted C (X). Polynomia criticized the calculus that was developed by Isaac Newton and ational,trigonometric, exponential, and logarithmic functions are in C(X), where X Gottfried Leibniz. but later consists of all numbers for which the functions are defined. when X is an interval of the becam real line, we will again omit the parentheses in this notation. Theorem 1.7(Rolle's Theorem) upposef E Cla, b] and f is differentiable on(a, b). If f(a)=f(b), then a number c in (a, b)exists with f(c)=0.(See Figure 1.3.) fa)=f(b) Theorem 1.8(Mean Value Theorem) If f E Cla, b] and f is differentiable on(a, b), then a number c in(a, b) exists with(See f'(c) f(b-f(a) Copyright 2010 Cengage Learning. All Rights May no be copied, scanned, or duplicated, in whole or in part Due to maternally aftec the overall leaning expenence. Cengage Learning4 CHAPTER 1 Mathematical Preliminaries and Error Analysis Figure 1.2 x y y  f(x) (x0, f(x0)) f(x0) x0 The tangent line has slope f(x0) Theorem 1.6 If the function f is differentiable at x0, then f is continuous at x0. The next theorems are of fundamental importance in deriving methods for error esti￾mation. The proofs of these theorems and the other unreferenced results in this section can be found in any standard calculus text. The theorem attributed to Michel Rolle (1652–1719) appeared in 1691 in a little-known treatise entitled Méthode pour résoundre les égalites. Rolle originally criticized the calculus that was developed by Isaac Newton and Gottfried Leibniz, but later became one of its proponents. The set of all functions that have n continuous derivatives on X is denoted Cn(X), and the set of functions that have derivatives of all orders on X is denoted C∞(X). Polynomial, rational, trigonometric, exponential, and logarithmic functions are in C∞(X), where X consists of all numbers for which the functions are defined. When X is an interval of the real line, we will again omit the parentheses in this notation. Theorem 1.7 (Rolle’s Theorem) Suppose f ∈ C[a, b] and f is differentiable on (a, b). If f (a) = f (b), then a number c in (a, b) exists with f  (c) = 0. (See Figure 1.3.) Figure 1.3 x f(c)  0 a c b f(a)  f(b) y y  f(x) Theorem 1.8 (Mean Value Theorem) If f ∈ C[a, b] and f is differentiable on (a, b), then a number c in (a, b) exists with (See Figure 1.4.) f  (c) = f (b) − f (a) b − a . Copyright 2010 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it