m148a1 - a,b if and only if f x and f x are integrable on...

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MATH 148 Assignment #1 Due: Monday, January 17 1) For each of the following limits of Riemann Sums identify the correspond- ing integral. In each case, sketch the graph of the function over the region of integration and find the limit (integral). a) lim n →∞ n i =1 ( i n ) 2 ( 1 n ) b) lim n →∞ n i =1 q 1 - ( - 1 + 2 i n ) ( 2 n ) c) lim n →∞ n i =1 sin( - π 2 + ( i ) π n ) ( π n ) 2) a) Show that if f ( x ) and g ( x ) are both integrable on [ a,b ], then h ( x ) = f ( x ) + g ( x ) is integrable on [ a,b ] and that Z b a h ( x ) dx = Z b a f ( x ) dx + Z b a g ( x ) dx. (Hint: Show that for any partition P we have that L ( f,P ) + L ( g,P ) L ( h,P ) U ( h,p ) U ( f,P ) + U ( g,P ) . ) b) Show that if f ( x ) is integrable on [ a,b ], then h ( x ) = | f ( x ) | is integrable on [ a,b ] (Hint: Show that for any partition P we have that U ( h,P ) - L ( h,P ) U ( f,P ) - L ( f,P ) . ) c) Give an example to show that it is possible for h ( x ) = | f ( x ) | to be integrable on [ a,b ] but f ( x ) is not. d) Let h ( x ) = max { f ( x ) ,g ( x ) } . Show that if f ( x ) and g ( x ) are both in- tegrable on [ a,b ], then h ( x ) is integrable on [ a,b ]. (Hint: First show that h ( x ) = f ( x ) + g ( x )+ | f ( x ) - g ( x ) | 2 . ) 1
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e) Let f + ( x ) = ± f ( x ) if f ( x ) 0 0 if f ( x ) < 0 . and let f - ( x ) = ± 0 if f ( x ) 0 - f ( x ) if f ( x ) < 0 . Show that f ( x ) is integrable on [
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Unformatted text preview: a,b ] if and only if f + ( x ) and f-( x ) are integrable on [ a,b ]. Moreover, in this case, show that Z b a f ( x ) dx = Z b a f + ( x ) dx-Z b a f-( x ) dx. 3) Show that if f ( x ) is contiunous on [ a,b ] with a < b , f ( x ) ≥ 0 for all x ∈ [ a,b ] and R b a f ( x ) dx = 0, then f ( x ) = 0 for all x ∈ [ a,b ]. 4) a) Assume that | f ( x ) | < M for all x ∈ [ a,b ]. Show that given any ± > with a < b , if n > M ( b-a ) 2 ± , then | S ( f,P n )-Z b a f ( x ) dx | < ± where S ( f,P n ) is any Riemann Sum associated with the regular-n par-tition on [ a,b ]. b) How large must n be so that | S ( f,P n )-Z . 2 e-x 2 dx | < 1 1000 where f ( x )-e-x 2 and S ( f,P n ) is any Riemann Sum associated with the regular-n partition on [0 , . 2]. 2...
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m148a1 - a,b if and only if f x and f x are integrable on...

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