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x2 + y 2 4x}, {x, 2}, {y, 2}]

is granted to the resource (Figure 3.8).

1.53711, y 2.3627}

x2 + y 2 4x}, {x, 3}, {y, 6}]

V (c)

2.4552, y 6.02802}

spiral = PolarPlot[p, {p, 0, 6 o}]; ellipse = ContourPlot[4 x2 + 9 y2 400, {x, 10, 10}, {y, 20, 20}, ContourStyle Dashing[.02]]; Show[spiral, ellipse]

SAM information can be replicated on one or more additional domain controllers called Backup Domain Controllers (BDC). The presence of a BDC assists in ensuring fault tolerance for the SAM directory. A BDC will also respond to a validation request at times when the PDC is busy or unavailable due to a system or network failure. The existence of a BDC is not required, but it should be carefully considered for large, geographically distributed domains.

k 2 b a2 : 2

5 5

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The graph shows three points of intersection that appear to be near (4, 6), ( 8, 4), and ( 9, 2). To convert the polar equation to rectangular, we use the transformations r = x 2 + y 2 and = tan 1 ( y / x) . However, Newton s method is more stable if we write this as tan( x 2 + y 2 ) = y / x . FindRoot[{Tan[ x2 + y 2 ] y/x, 4x2 + 9y 2 400}, {x, 4}, {y, 6}] FindRoot[{Tan[ x2 + y 2 ] y/x, 4x2 + 9y 2 400}, {x, 8}, {y, 4}] FindRoot[{Tan[ x2 + y 2 ] y /x, 4x2 + 9y 2 400}, {x, 9}, {y, 2}]

3.93476, y 6.1289} 8.04703, y 3.95785} 9.38786, y 2.29668}

(Shell Method) Find the volume obtained by orbiting the region of part (b) about the y-axis. Compare this volume with that obtained in part (b). b V 2 x kx dx 2 kb3 =3

near the point (1, 2, 3).

In instances when an organization's business rules, administrative structure, or geographic layout doesn't neatly fit the single domain model, multiple domains can be set up and logically connected through trust relationships. Users in a trusted domain are allowed to access resources and participate in group membership in the local domain without having to duplicate SAM directory entries in both domains. This feature provides a means for distributing namespace administration functions in large enterprises. Trust relationships can be broken down into three models, one-way trust, two-way trust, and complete trust. In a one-way trust model, domain A permits access to local resources by domain B, but domain B does not return the favor by permitting access to its resources. In a two-way trust environment, access to resources in domains A and B are permitted in both directions. In the complete-trust model, all domains have access to all resources, no matter where they are located. Complete trust or combinations with the other models are very difficult to manage.

1 =5 z

FindRoot[{x + y + z 6, Sin[x] + Cos[y] + Tan[z] 1, Exp[x] + Sqrt[y] + 1/z 5}, {x, 1}, {y, 2}, {z, 3}]

MISCELLANEOUS PROBLEMS 5.28 If f x and g x are continuous in a; b , prove Schwarz s inequality for integrals: b 2 b b f x g x dx @ f f x g2 dx fg x g2 dx

1.23382, y 1.5696, z 3.19658}

Because they are so prevalent in algebra, Mathematica offers commands that are devoted exclusively to polynomials.

- 46 -

PolynomialQ[expression, variable] yields True if expression is a polynomial in variable, and False otherwise. Variables[polynomial] gives a list of all independent variables in polynomial. Coefficient[polynomial, form] gives the coefficient of form in polynomial. Coefficient[polynomial, form, n] gives the coefficient of form to the nth power in polynomial. CoefficientList[polynomial, variable] gives a list of the coefficients of powers of variable in polynomial, starting with the 0th power.

Hence by Problem 1.13 of 1, using (1) with b b b B2 f f x g2 dx; C f x g x dx A2 g x g2 dx;

PolynomialQ[x2 + 3 x + 2, x] True PolynomialQ[x2 + 3 x + 2/x, x] False PolynomialQ[x2 + 3 x + 2/y, x] True PolynomialQ[x2 + 3 x + 2/y, y] False

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