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Unformatted text preview: 1 Study Guide for Prelim 1 Math 192  Spring 1997 Written by Don Allers; Revised by Sean Carver 1 Improper Integrals Idea: A proper integral is an integral R b a f ( x ) dx, where a and b are both finite and f ( x ) is continuous and finite on [ a,b ]. For proper integrals, the fundamental theorem of calculus tells us that if F is an antiderivative of f (i.e. if F(x) = f(x)) then R b a f ( x ) dx = F ( b ) F ( a ) . We will define an improper integral as a limit of proper integrals. Comment The integral R b a f ( x ) dx is improper if: (i) a or b is infinite, or (ii) f ( x ) is not finite on all of [ a,b ]. The fundamental theorem of calculus does not apply directly to this class of integrals. Indeed, the definition of a definite integral given in Chapter 4 explicitly assumes that the integral is proper. Without the definitions below, an improper integral would be an expression with no meaning. Solution Method: 1. If a is finite, and f ( x ) is continuous on [ a, ] we define R a f ( x ) dx = lim c R c a f ( x ) dx. 2. If b is finite, and f ( x ) is continuous on [ ,b ] we define R b f ( x ) dx = lim c  R b c f ( x ) dx. 3. If a,b are finite, and f ( x ) is continuous on ( a,b ] we define R b a f ( x ) dx = lim c a + R b c f ( x ) dx. 4. If a,b are finite, and f ( x ) is continuous on [ a,b ) we b define R b a f ( x ) dx = lim c b R c a f ( x ) dx. 5. If the integral is improper at multiple points on the domain [ a,b ], we divide this domain into subintervals [ a,c 1 ] , [ c 1 ,c 2 ] ,..., [ c n 1 ,c n ] , [ c n ,b ] so that each subinterval contains only one problem point at one of its endpoints, and no problem points in its interior. Then define R b a f ( x ) dx = R c 1 a f ( x ) dx + R c 2 c 1 f ( x ) dx + ... R c n c n 1 f ( x ) dx + R b c n f ( x ) dx, applying the above rules to each piece. Eg: (a) Z 1 x ln xdx, (b) Z dx x 2 + 1 , (c) Z  2 xdx ( x 2 + 1) 2 , (d) Z 1 1 dx x 2 / 3 . 2 Sequences Idea: Given a sequence of numbers { a n } = a 1 ,a 2 ,a 3 ,... , the question is whether or not these numbers are converging to some limit a . We say that { a n } converges to a , written a n a , as n , if the distance  a n a  between the sequence points and a goes to zero as n increases. Extend the Sequence to a Real Function Idea: Compute the limit of a sequence as the limit of a function f ( x ) as x . When? Use this method if the expression defining the sequence a n can be considered as a real function in the variable n whose limit as n can be computed easily. Method: To compute the limit of the sequence, take take the limit of its defining expression as you would take the limit of a real function. Remember LHopitals Rule. (The theorem behind this method is the following: If f ( n ) = a n and lim x f ( x ) = L then lim n a n = L. 2 2 SEQUENCES Continuous Function Theorem Idea: It would be nice if we could move a limit inside a function, that is, if we could say...
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This test prep was uploaded on 09/26/2007 for the course MATH 1920 taught by Professor Pantano during the Fall '06 term at Cornell University (Engineering School).
 Fall '06
 PANTANO
 Calculus, Improper Integrals, Infinite Series, Integrals, Mathematical Series

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