运筹学 12

may not exist.
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Example 1
2 2 1. Show that lim x 2 x 2 2(2) 0 x 2 1 2. Show that lim does not exist. x0 x
Solution
1. To verify this result, evaluate x2-2x for values of x close to, but not equal to, 2. 2. To verify this result, observe that as x gets near 0, becomes either a very large positive number or a very 1 1 large x negative number.. Thus, as x approaches 0, x will not approach any single number.
F ( x ) f ( x ) dx
The definite integral of f(x) from x=a to x=b is b written f (damental Theorem of Calculus states that if f(x) is continuous fro all x satisfying a ≤ x ≤ b, then
f ( x, y) dx g ( y) y
h( y )
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12.3 Basic Rules of Probability
Definition: Any situation where the outcome is uncertain is called an experiment. Definition: For any experiment, the sample space S of the experiment consists of all possible outcomes for the experiment. Definition: An event E consists of any collection of points (set of outcomes) in the sample space. Definition: A collection of events E1, E2,…,En is said to be a mutually exclusive collection of events if for i ≠ j (i=1,2,…,n and j=1,2,…n), Ei and Ej have no points in common.
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Example 2
Bakeco orders sugar from Sugarco. The perpound purchase price of the sugar depends on the size of the order (see Table 1). Let x = number of pounds of sugar purchased by Bakeco f(x) = cost of ordering x pounds of sugar Then f(x) =25x for 0 ≤ x < 100 f(x) =20x for 100 ≤ x ≤ 200 f(x) =15x for x > 200 For all values of x, determine if x is continuous or discontinuous.
Taylor Series Expansion
In our study of queuing theory (Chapter 8), we will need the Taylor series expansion of a function f(x).
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Given that fn-1(x) exists for every point on the interval [a,b], Taylor’s theorem allows us to write for any h satisfying 0 ≤ h ≤ b – a,
Description
A review of some basic topics in calculus and probability, which will be useful in later chapters.
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12.1 Review of Differential Calculus
The limit is one of the most basic ideas in calculus. Definition: The equation
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Higher Derivatives
We define f(2)(a)=f ′′(a) to be the derivative of the function f ′(x) at x=a. Similarly, we can define (if it exists) f(n)(a) to be the derivative of f(n-1)(x) at x=a. Thus, for Example 3, f′′ (p) = 3000e-p(-1) – 3000e-p(1-p)
Chapter 12
Review of Calculus and Probability
to accompany Operations Research: Applications and Algorithms 4th edition
by Wayne L. Winston
Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc.
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Example 2 – cont’d
Solution It is clear that
x100
lim f ( x )
and
x 200
lim f ( x )
do not exist. Thus, f(x) is discontinuous at x=100 and x=200 and is continuous for all other values of x satisfying x ≥ 0.
(1)
f (a h) f (a)
i 1
i n
where (1) will hold for some number p between a and a + h. Equation (1) is the nth-order Taylor series expansion of the f(x) about a.
f (i ) (a) i f ( n1) ( p) n1 h h i! (n 1)!
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Example 4
Find the first-order Taylor series expansion of e-x about x = 0. Solution Since f′(x) = -e-x and f′′(x) = -e-x, we know that (1) will hold on any interval [0,b]. Also, f(0) = 1, f′(0) = -1, and f′′(x) = e-x. Then (1) yields the following firstorder Taylor series expansion for e-x about x=0:
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Definition: A function f(x) is continuous at a point a if
lim f ( x ) f ( a )
xa
If f(x) is not continuous at x=a, we say that f(x) is discontinuous (or has a discontinuity) at a.

b
a
f ( x)dx F (b) F (a)
where F(x) is any indefinite integral of f(x).
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12.2 Differentiation of Integrals
In order to differentiate a function whose value depends on an integral you should let f(x, y) be a function of variables x and y, and let g(y) and h(y) be functions of y. Then
F ( y)
h( y ) g ( y)
f ( x, y)dx
is a function only of y. Leibniz’s rule for differentiating an integral states that
F ( y)
h( y )
g ( y)
f ( x, y )dx, then F ( y ) h( y) f (h( y ), y) g ( y ) f ( g ( y), y )
2 xi x j
to denote a second-order partial derivative.
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Review of Integral Calculus
A knowledge of the basics of integral calculus is valuable when studying random variables. Consider two functions: f(x) and F(x). If F′(x), we say that F(x) is the indefinite integral of f(x). The fact that F(x) is the indefinite integral of f(x) is written