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12 Review of Calculus and Probability

# 12 Review of Calculus and Probability - Review of Calculus...

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Review of Calculus and Probability We review in this chapter some basic topics in calculus and probability, which will be useful in later chapters. 12.1 Review of Integral Calculus In our study of random variables, we often require a knowledge of the basics of integral calculus, which will be briefly reviewed in this section. Consider two functions: f ( x ) and F ( x ). If F ( x ) f ( x ), we say that F ( x ) is the indefi- nite integral of f ( x ). The fact that F ( x ) is the indefinite integral of f ( x ) is written F ( x ) f ( x ) dx The following rules may be used to find the indefinite integrals of many functions ( C is an arbitrary constant): (1) dx x C af ( x ) dx a f ( x ) dx ( a is any constant) [ f ( x ) g ( x )] dx f ( x ) dx g ( x ) dx x n dx n x n 1 1 C ( n 1) x 1 dx ln x C e x dx e x C a x dx ln a x a C ( a 0, a 1) [ f ( x )] n f ( x ) dx [ f n ( x )] n 1 1 C ( n 1) f ( x ) 1 f ( x ) dx ln f ( x ) C

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For two functions u ( x ) and v ( x ), u ( x ) v ( x ) dx u ( x ) v ( x ) v ( x ) u ( x ) dx (Integration by parts) e f ( x ) f ( x ) dx e f ( x ) C a f ( x ) f ( x ) dx a ln f ( a x ) C ( a 0, a 1) The concept of an integral is important for the following reasons. Consider a function f ( x ) that is continuous for all points satisfying a x b . Let x 0 a , x 1 x 0 , x 2 x 1 , . . . , x i x i 1 , x n x n 1 b , where b n a . From Figure 1, we see that as approaches zero (or equivalently, as n grows large), i n i 1 f ( x i ) will closely approximate the area under the curve y f ( x ) between x a and x b . If f ( x ) is continuous for all x satisfying a x b , it can be shown that the area under the curve y f ( x ) between x a and x b is given by lim 0 i n i 1 f ( x i ) which is written as b a f ( x ) dx or the definite integral of f ( x ) from x a to x b . The Fundamental Theorem of Cal- culus states that if f ( x ) is continuous for all x satisfying a x b , then b a f ( x ) dx F ( b ) F ( a ) where F ( x ) is any indefinite integral of f ( x ). F ( b ) F ( a ) is often written as [ F ( x )] b a . Ex- ample 1 illustrates the use of the definite integral. E X A M P L E 1 Suppose that at time t (measured in hours, and the present t 0), the rate a ( t ) at which customers enter a bank is a ( t ) 100 t . During the next 2 hours, how many customers will enter the bank? Customer Arrivals at a Bank 708 C H A P T E R 1 2 Review of Calculus and Probability y x y = f ( x ) x 1 x 0 = a b = x n x n – 1 x 2 x 3 F I G U R E 1 Relation of Area and Definite Integral
1 2 . 1 Review of Integral Calculus 709 Solution Let t 0 0, t 1 t 0 , t 2 t 1 , . . . , t n t n 1 2 (of course, 2 n ). Be- tween time t i 1 and time t i , approximately 100 t i customers will arrive. Therefore, the to- tal number of customers to arrive during the next 2 hours will equal lim 0 i n i 1 100 t i (see Figure 2). From the Fundamental Theorem of Calculus, lim 0 i n i 1 100 t i 2 0 (100 t ) dt [50 t 2 ] 2 0 200 0 200 Thus, 200 customers will arrive during the next 2 hours. P R O B L E M S Group A a ( t ) a ( t ) = 100 t = 0.1 t t 1 t 2 t 3 t 4 t 5 t 6 t 7 t 8 t 9 t 10 t 11 t 12 t 13 t 14 t 15 t 16 t 17 t 18 t 19 0 = t 0 t 2 0 = 2 200 150 100 50 F I G U R E 2 Relation of Total Arrivals in Next 2 Hours to Area under a ( t ) Curve 1 The present is t 0. At a time t years from now, I earn income at a rate e 2 t . How much money do I earn during the next 5 years?

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12 Review of Calculus and Probability - Review of Calculus...

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