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If f (c) > 0, then f (x) > 0 for all x in some open interval containing c, and the graph would be concave upward in that interval, contradicting the assumption that there is an in ection point at x = c We get a similar contradiction if f (c) < 0, for in that case, the graph would be concave downward in an open interval containing c EXAMPLES.

Example 2.9 Calculate the time in days, hours, minutes, and seconds for the epoch day 324.95616765.

LeftTrim(ExprS) RightTrim(ExprS)

This represents the 324th day of the year plus 0.95616765 mean solar day. The decimal fraction in hours is 24 0.95616765 22.9480236; the decimal fraction of this expressed in minutes is 0.9480236 60 56.881416; the decimal fraction of this expressed in seconds is 0.881416 60 52.88496. Thus, the epoch is day 324, at 22 h, 58 m, 52.88 s.

(a) Consider the graph of y = x 3 [see Fig. 23-4(a)]. Here y = 3x 2 and y = 6x. Since y > 0 when x > 0, and y < 0 when x < 0, the curve is concave upward when x > 0, and concave downward when x < 0. There is an in ection point at the origin, where the concavity changes. This is the only possible in ection point, for if y = 6x = 0, then x must be 0.

NoSpaces(Exprs) Returns ExprS without any spaces, even within the string not just leading and trailing spaces. Substring(ExprS,ExprN1,ExprN2) Returns a string consisting of the characters from ExprS starting at character ExprN1 up to and including the ExprN2 character. If ExprN1<1 then rst character is assumed. If ExprN> the length of the string then the last character is assumed. If ExprN<1 or ExprN2> the length of the string then the last character is assumed. Left(ExprS,ExprN) Returns a string containing ExprN characters from ExprS starting from rst character. This is equivalent to Substring(ExprS,1,ExprN). If ExprN< 1 then 1 is assumed. If ExprN > the length of the string then the length of the string is assumed.

Universal time coordinated is equivalent to Greenwich mean time (GMT), as well as Zulu (Z) time. There are a number of other universal time systems, all interrelated ( Wertz, 1984) and all with the mean solar day as the fundamental unit. For present purposes, the distinction between these systems is not critical, and the term universal time (UT), will be used from now on. For computations, UT will be required in two forms: as a fraction of a day and in degrees. Given UT in the normal form of hours, minutes, and seconds, it is converted to fractional days as UTday 1 ahours 24 minutes 60 seconds b 3600 (2.18)

2.9.3 Julian dates*

Right(ExprS,ExprN) Returns a string containing ExprN characters from ExprS ending with the last character. This is equivalent to Substring(ExprS, Length(ExprS) + 1 ExprN, ExprN). If ExprN < 1 then 1 is assumed. If ExprN > the length of the string then the length of the string is assumed. Extract(ExprS1,ExprS2,ExprN) This will return a string that is the ExprNth part of ExprS1 separated by the separator ExprS2. ExprS1 is a string with data separated by characters speci ed in ExprS2. The Extract() function will return the data part that is the ExprNth part. If ExprN is greater than the number of parts Extract() will return the last part. If ExprS1 does not contain the separator characters speci ed in ExprS2 then ExprS1 will be returned. If ExprN is not a number an error will be given. If it is not an integer it will be made into an integer. If it is less than 1 then it is converted to 1.

360

(b) If f (c) = 0, the graph of f need not have an in ection point at x = c. For instance, the graph of f (x) = x 4 [see Fig. 23-4(b)] has a relative minimum, not an in ection point, at x = 0, where f (x) = 12x 2 = 0.

a = test,5,2.4 b = , print Extract(a,b,2) print Extract(a,b,3) print Extract(a,b,6) b = ; print Extract(a,b,1)

(2.19)