of calc 1.4 f(b)-f(a) Theorem 1.9(Extreme Value Theorem) Iff∈C[a,b].then llx∈[a,b In addition, if f is differentiable on(a, b), then the numbers ci and c2 occur either at the endpoints of [a, b] or where fis zero. (See Figure 1.5.) Figure 1.5 C2 Research work on the design of algorithms mathematics began in the 1960s. The first system to be operational, As mentioned in the preface, we will use the computer algebra system Maple wheneve in the 1970s, was a LISP-based appropriate. Computer algebra systems are particularly useful for symbolic differentiation system called MACSYMA and plotting graphs. Both techniques are illustrated in Example 1 Example 1 Use Maple to find the absolute minimum and absolute maximum values of f(x)=5cos 2x- 2rsin 2xf(x) on the intervals (a)[1, 2, and (b)[0.5, 11 Solution There is a choice of Text input or Math input under the Maple C 2D Math option. The Text input is used to document worksheets by standard text information in the document. The Math input option is used to execute Maple commands. Maple input 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. no be copied, scanned, or duplicated, in whole or in part Due to1.1 Review of Calculus 5 Figure 1.4 y a c b x Slope f(c) Parallel lines Slope b a f(b) f(a) y  f(x) Theorem 1.9 (Extreme Value Theorem) If f ∈ C[a, b], then c1, c2 ∈ [a, b] exist with f (c1) ≤ f (x) ≤ f (c2), for all x ∈ [a, b]. In addition, if f is differentiable on (a, b), then the numbers c1 and c2 occur either at the endpoints of [a, b] or where f  is zero. (See Figure 1.5.) Figure 1.5 y a c2 c1 b x y  f(x) Research work on the design of algorithms and systems for performing symbolic mathematics began in the 1960s. The first system to be operational, in the 1970s, was a LISP-based system called MACSYMA. As mentioned in the preface, we will use the computer algebra system Maple whenever appropriate. Computer algebra systems are particularly useful for symbolic differentiation and plotting graphs. Both techniques are illustrated in Example 1. Example 1 Use Maple to find the absolute minimum and absolute maximum values of f (x) = 5 cos 2x − 2x sin 2xf (x) on the intervals (a) [1, 2], and (b) [0.5, 1] Solution There is a choice of Text input or Math input under the Maple C 2D Math option. The Text input is used to document worksheets by adding standard text information in the document. The Math input option is used to execute Maple commands. Maple input 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.