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a uniform mesh for the elements are arbitrary Based on the equal division, the rst element is located on the normalized time subdomain [0, 1/16], the second element on the subdomain [1/16, 2/16], and so on through the sixteenth element on the subdomain [15/16, 1] The four Gaussian points x = 033992104 and 086113631 of the fourth Legendre polynomial de ned on [-1, 1] are scaled to t = ((b - a)x + b + a)/2 on the subdomain [a, b] for each element (Stroud and Secrest, 1986) For example, the rst four Gaussian points located on the subdomain, [0, 1/16], of the rst element are 000433949, 002062747, 004187253, and 005816051 For Eq (531), there are 66 constraints the two end conditions and the sixty-four values for the right sides of the differential equations to be satis ed at the Gaussian points Sixty-six B-splines of order six are used to formulate the system of linear equations in Eq (56) The knot sequence used to evaluate the splines for the collocation process is established by repeating four knots at each mesh point, t = 1/16, 2/16, 3/16, , and 15/16 At the end points, t = 0 and t = 1, the knots are repeated six times as necessary because of the order of the spliones (k = 6) After Eq (56) is solved for the unknown coef cients, the solution of Eq (531) is readily found for any point in the interval 0 t 1 enabling the dynamic behavior of the follower system to be determined

As the output shows, the value is scaled in terms of thousands In addition to the placeholders, a custom format specifier can contain other characters Any other characters are simply passed through, appearing in the formatted string exactly as they appear in the format specifier For example, this WriteLine( ) statement:

Using this information, we can sketch the graph of f (x) = loga x If a > 1 then ln a > 0 so that f (x) > 0 and f (x) < 0 The graph of f is exhibited in Fig 66

ConsoleWriteLine("Fuel efficiency is {0:### mpg}", 213);

142 10

produces this output:

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You can also use the escape sequences, such as \t or \n, if necessary The E and e placeholders cause a value to be displayed in scientific notation At least one 0, but possibly more, must follow the E or e The 0s indicate the number of decimal digits that will be displayed The decimal component will be rounded to fit the format Using an uppercase E causes an uppercase E to be displayed; using a lowercase e causes a lowercase e to be displayed To ensure that a sign character precedes the exponent, use the E+ or e+ forms To display a sign character for negative values only, use E, e, E-, or e- The ; is a separator that enables you to specify different formats for positive, negative, and zero values Here is the general form of a custom format specifier that uses the ; : positive-fmt;negative-fmt;zero-fmt

If 0 < a < 1 then ln a = ln(1/a) < 0 so that f (x) < 0 and f (x) > 0 The graph of f is sketched in Fig 67 Since g(x) = a x is the inverse function to f (x) = loga x, the graph of g is the re ection in the line y = x of the graph of f (Figs 66 and 67) See Figs 68, 69

FIGURE 526 Normalized acceleration of output motions for spline (k = 10) and optimized polydyne in Example 8

Part II:

Here is an example:

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