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Theorem 5. If f x is continuous at x x0 and f x0 > 0 [or f x0 < 0], there exists an interval about x x0 in which f x > 0 [or f x < 0]. Theorem 6. If a function f x is continuous in an interval and either strictly increasing or strictly decreasing, the inverse function f 1 x is single-valued, continuous, and either strictly increasing or strictly decreasing.

1.0 0.5

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4.3 The standard normal curve used in probability and statistics is defined by the function f (x) = Sketch the graph for 3 x 3.

f[x_]= 1 Exp 1 x 2 ; 2 2 Plot[f[x], {x, 3, 3}]

Thus, the value of the integral is obtained without reference to the curve joining P1 and P2 . This notion of the independence of path of line integrals of certain vector elds, important to theory and application, is characterized by the following three theorems: Theorem 1. A necessary and su cient condition that A dr be independent of path is that there C exists a scalar function such that A r .

4.4 Plot the graphs y = sin x , y = 2 sin x , and y = 3 sin x from 2 to 2 on the same set of axes.

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, Plot[{Sin[x] 2 Sin[x], 3 Sin[x]}, {x, 2 o, 2 o}]

Along with the mapping of access control attributes and permissions, Samba also translates file and path names into their respective UNIX, DOS, and Windows forms (Table 11.2). In general, the translation entails applying an algorithm (Table 11.4) to convert long case-sensitive UNIX file names into case-insensitive DOS 8.3 format. The method is similar to, but different from, that employed by Windows 9x and Windows NT when transforming long file names (LFN) into a DOS 8.3 representation. The ability of a client to support LFN or DOS 8.3 format is negotiated during session setup. Table 11.2 File Name Characteristics

3 2 1

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2 1 2 3

4.5 The graphs of inverse functions are symmetric with respect to the line y = x. Plot the inverse functions f ( x ) = x 2, 0 x 2, and f 1 ( x ) = x , 0 x 4, as solid curves and the line y = x as a dotted line and observe the symmetry.

Theorem 2. A necessary and su cient condition that the line integral, is that r A 0. A dr be independent of path

g1 = Plot[ x 2 , {x, 0, 2}]; g2 = Plot[ x , {x, 0, 4}]; g3 = Plot[x, {x, 0, 4}, PlotStyle Dashing[{0, 0, .01}]]; Show[g1, g2, g3, AspectRatio Automatic, PlotRange {{0, 4}, Automatic}]

File name 127 chars All chars except \ / : * " < > | Win32, LFN, Universal Naming Convention (UNC) names

4.6 Sketch the graphs of y = x 2 , y = x 2, and y = x 2 sin 10 x , 2 x 2 , on a single set of axes enclosed by a frame.

Plot[{x2, x2, x2 Sin[10 x]}, {x, 2 o, 2 o}, Frame True]

Theorem 3. If r A 0, then the line integral of A over an allowable closed path is 0, i.e., A dr 0.

40 6 4 2 0 2 4 6

4.7 The family of Chebyshev polynomials is used in approximation theory and numerical analysis. Mathematica represents these polynomials as ChebyshevT[n, x]. On a single set of axes, using some device to distinguish the curves, plot a labeled graph showing the Chebyshev polynomials of degrees 2, 3, and 4.

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PlotLegends` Plot[{ChebyshevT[2, x], ChebyshevT[3, x], ChebyshevT[4, x]}, {x, 2, 2}, PlotStyle {GrayLevel[0], GrayLevel[.4], GrayLevel[.7]}, PlotLegend {"T2", "T3", "T4"}, LegendPosition {1,0}]

If C is a plane curve, then Theorem 3 follows immediately from Green s theorem, since in the plane case r A reduces to @A1 @A2 @y @x

PlotLegends` Plot[{ChebyshevT[2, x], ChebyshevT[3, x], ChebyshevT[4, x]}, {x, 2, 2}, PlotStyle {Red, Green, Blue}, PlotLegend {"T2", "T3", "T4"}, LegendPosition {1, 0}]

Sketch the graphs of y = 1 + sin x , 0 x 2 , y = 2 + sin x , 2 x 4 , and y = 3 + sin, 4 x 6 on one set of axes.

File name 255 chars All chars except \ / : * " < > | Win32, LFN, Universal Naming Convention (UNC) names

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