1.1 Review of calculus Definition 1.2 Let f be a function defined on a set X of real numbers and xo E X. Then f is continuous The basic concepts of calculus at xo if and its applications were developed in the late 17th and lim f(r)=f(xo) arly 18th centuries. but the mathematically precise concepts The function f is continuous on the set X if it is continuous at each number in X of limits and continuity were not described until the time of The set of all functions that are continuous on the set X is denoted C(X). When X is an interval of the real line, the parentheses in this notation are omitted. For example, the (789-1857) Heinrich Eduard set of all functions continuous on the closed interval [a, b] is denoted CI Heine(1821-1881), and Karl R denotes the set of all real numbers, which also has the interval notation(-oo, oo). So Weierstrass(1815-1897) in the the set of all functions that are continuous at every real number is denoted by C(R)or by atter portion of the 19th century.C(-∞,∞) The limit of a sequence of real or complex numbers is defined in a similar manner Definition 1.3 Let (rnIng_, be an infinite sequence of real numbers. This sequence has the limit x(converges to x)if, for any e>0 there exists a positive integer N(E)such that lrn -xl E, whenever () lim x=x,orxn→xasn→∞, means that the sequence InnIng converges tox. Theorem 1.4 If f is a function defined on a set X of real numbers and xo E X, then the following a.f is continuous at b. If (n ne, is any sequence in X converging to xo, then limn-oo f(rn)=f(ro).B The functions we will consider when discussing numerical methods will be assumed to be continuous because this is a minimal requirement for predictable behavior. Functions that are not continuous can skip over points of interest, which can cause difficulties when attempting to approximate a solution to a problem. Differentiability More sophisticated assumptions about a function generally lead to better approximation results. For example, a function with a smooth graph will normally behave more predictably than one with numerous jagged features. The smoothness condition relies on the conce of the derivative Definition 1.5 Let f be a function defined in an open interval containing xo. The function f is differentiable if f(xo)= lim f(x)-f(x0) exists. The number f(xo)is called the derivative of f at xo. A function that has a derivative at each number in a set x is differentiable on x The derivative of f at xo is the slope of the tangent line to the graph of f at (xo, f(ro)) as shown in Figure 1.2 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 Learning1.1 Review of Calculus 3 Definition 1.2 Let f be a function defined on a set X of real numbers and x0 ∈ X. Then f is continuous at x0 if lim x→x0 f (x) = f (x0). The function f is continuous on the set X if it is continuous at each number in X. The set of all functions that are continuous on the set X is denoted C(X). When X is an interval of the real line, the parentheses in this notation are omitted. For example, the set of all functions continuous on the closed interval [a, b] is denoted C[a, b]. The symbol R denotes the set of all real numbers, which also has the interval notation (−∞,∞). So the set of all functions that are continuous at every real number is denoted by C(R) or by C(−∞,∞). The basic concepts of calculus and its applications were developed in the late 17th and early 18th centuries, but the mathematically precise concepts of limits and continuity were not described until the time of Augustin Louis Cauchy (1789–1857), Heinrich Eduard Heine (1821–1881), and Karl Weierstrass (1815 –1897) in the latter portion of the 19th century. The limit of a sequence of real or complex numbers is defined in a similar manner. Definition 1.3 Let{xn}∞ n=1 be an infinite sequence of real numbers. This sequence has the limit x (converges to x) if, for any ε > 0 there exists a positive integer N(ε) such that |xn − x| < ε, whenever n > N(ε). The notation lim n→∞ xn = x, or xn → x as n → ∞, means that the sequence {xn}∞ n=1 converges to x. Theorem 1.4 If f is a function defined on a set X of real numbers and x0 ∈ X, then the following statements are equivalent: a. f is continuous at x0; b. If {xn}∞ n=1 is any sequence in X converging to x0, then limn→∞ f (xn) = f (x0). The functions we will consider when discussing numerical methods will be assumed to be continuous because this is a minimal requirement for predictable behavior. Functions that are not continuous can skip over points of interest, which can cause difficulties when attempting to approximate a solution to a problem. Differentiability More sophisticated assumptions about a function generally lead to better approximation results. For example, a function with a smooth graph will normally behave more predictably than one with numerous jagged features. The smoothness condition relies on the concept of the derivative. Definition 1.5 Let f be a function defined in an open interval containing x0. The function f is differentiable at x0 if f  (x0) = lim x→x0 f (x) − f (x0) x − x0 exists. The number f  (x0) is called the derivative of f at x0. A function that has a derivative at each number in a set X is differentiable on X. The derivative of f at x0 is the slope of the tangent line to the graph of f at (x0, f (x0)), as shown in Figure 1.2. 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