CHAPTER 1- Mathematical Preliminaries and Error Analysis Maple gives the response f solve(-12 sin(2r)-4.x cos(2x),x,. 1) This indicates that Maple is unable to determine the solution. The reason is obvious once the graph in Figure 1.6 is considered. The function f is always decreasing on this interval so no solution exists. Be suspicious when Maple returns the same response it is given; it is as if it was questioning your request. In summary, on [0.5, 1] the absolute maximum value is f(0.5)=1.86004545 and the absolute minimum value is f(1)=-3899329036, accurate at least to the places The following theorem is not generally presented in a basic calculus course, but is derived by applying Rolle's Theorem successively to f, f,..., and, finally, to fn-l) This result is considered in Exercise 23 Theorem 1.10(Generalized Rolle's Theorem Suppose f E Cla, b] is n times differentiable on(a, b). If f(r)=0 at the n+ l distinct numbers a xo <x1 <.. xn< b, then a number c in(xo, xn), and hence in(a, b) exists with fm)(c)=0 We will also make frequent use of the Intermediate Value Theorem. Although its state ment seems reasonable, its proof is beyond the scope of the usual calculus course. It can however, be found in most analysis texts Theorem 1.11(Intermediate Value Theorem) If f E Cla, b] and K is any number between f(a) and f(b), then there exists a number c in(a, b) for which f(c)=K Value Theorem. In this example there are two other possibilitie teed by the Intermediate Figure 1.7 shows one choice for the number that is guara 1.7 (a, f(a)) f(b) (b, f(b)) Example 2 Show that x5-2x3+3x2-1=0 has a solution in the interval [0, 1 Solution Consider the function defined by f(x)=x5-2x3+3x2-1. The function f continuous on [0, 1. In addition Copyright 2010 Cengage Learning. All Rights t materially affect the overall leaming eaperience Cengage Learning reserves the right to remo rty commen may be suppressed from the eBook andor eChaptert'sh. May no be copied, scanned, or duplicated, in whole or in part Due to8 CHAPTER 1 Mathematical Preliminaries and Error Analysis Maple gives the response f solve(−12 sin(2x) − 4x cos(2x), x, .5 . . 1) This indicates that Maple is unable to determine the solution. The reason is obvious once the graph in Figure 1.6 is considered. The function f is always decreasing on this interval, so no solution exists. Be suspicious when Maple returns the same response it is given; it is as if it was questioning your request. In summary, on [0.5, 1] the absolute maximum value is f (0.5) = 1.86004545 and the absolute minimum value is f (1) = −3.899329036, accurate at least to the places listed. The following theorem is not generally presented in a basic calculus course, but is derived by applying Rolle’s Theorem successively to f , f  , ... , and, finally, to f (n−1) . This result is considered in Exercise 23. Theorem 1.10 (Generalized Rolle’s Theorem) Suppose f ∈ C[a, b] is n times differentiable on (a, b). If f (x) = 0 at the n + 1 distinct numbers a ≤ x0 < x1 < ... < xn ≤ b, then a number c in (x0, xn), and hence in (a, b), exists with f (n) (c) = 0. We will also make frequent use of the Intermediate Value Theorem. Although its state￾ment seems reasonable, its proof is beyond the scope of the usual calculus course. It can, however, be found in most analysis texts. Theorem 1.11 (Intermediate Value Theorem) If f ∈ C[a, b] and K is any number between f (a) and f (b), then there exists a number c in (a, b) for which f (c) = K. Figure 1.7 shows one choice for the number that is guaranteed by the Intermediate Value Theorem. In this example there are two other possibilities. Figure 1.7 x y f(a) f(b) y  f(x) K (a, f(a)) (b, f(b)) a c b Example 2 Show that x5 − 2x3 + 3x2 − 1 = 0 has a solution in the interval [0, 1]. Solution Consider the function defined by f (x) = x5 − 2x3 + 3x2 − 1. The function f is continuous on [0, 1]. In addition, 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