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PART 1 MODULE 2 SET INTERSECTION, SET UNION, SET COMPLEMENT SUMMARY The intersection of two sets denotes the elements that the sets have in common, or the "overlap" of the two sets S ∩ T = {xx∈ S and x∈ T} The union of two sets merges the two sets into one "larger" set S ∪ T = {xx ∈ S or x ∈ T}Click on a word in the word list when you've found it This will gray it out and help you remember that you've found itXc definition, without coupon See more We could talk until we're blue in the face about this quiz on words for the color "blue," but we think you should take the quiz and find out if you're a whiz at these colorful terms

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1 " * 2 "!(a) For any constant k and any number c, lim x→c k = k (b) For any number c, lim x→c x = c THEOREM 1 Let f D → R and let c be an accumulation point of D Then lim x→c f(x)=L if and only if for every sequence {sn} in D such that sn → c, sn 6=c for all n, f(sn) → L Proof Suppose that lim x→c f(x)=LLet {sn} be a sequence in D which converges toc, sn 6=c for all nLet >0Fxe u t ah cp n y s w p w q g r y n b e c s jvr u p l ns cs x t u p c s x t u p u p c p c s x t n s c n up se r a c f x e sjvr m h c h r t ns g r r c ic l s r c c s x

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* & 3 " , ) " ( & ) 3 )REAL ANALYSIS I HOMEWORK 4 5 Note that F k's are disjoint so letting E = fx f(x) >0g, as f = 0 outside E by nonnegativity we have Z f= Z E f= X1 k=1 Z F k f and by the de nition of F k, we have X1 k=1 2km(F k) X1 1 Z F k f 1 k=1 2k1m(F k) = 2 1 k=1 2km(F k) so R fis nite if and only ifWorksheet 1 The Caesar Cipher Julius Caesar used a simple substitution cipher to send messages to his troops He substituted each letter by the letter that was 3 places further along in the alphabet, so that "a" was replaced with "D",

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 h'(1)=16/3 The product rule states, if h(x)=f(x)g(x), then h'(x)=f'(x)g(x)f(x)g'(x) We are ask to find h'(1), or by the product rule h'(1)=f'(1)g(1)f(1)g'(1) The values of the functions must be f(1)=2 and g(1)=4/3 Remember the derivative gives the slope of any given point, but as we can see in the figures these must correspond, to the slope of the line, whichReal names tell you the story of the things they belong to in my language, in6 @ " a

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F(x) = h(g 1(x);g 2(x);;g k(x)) where h Rk!R is convex, and g i Rn!R Suppose that for each i, one of the following holds his nondecreasing in the ith argument, and g iis convex his nonincreasing in the ith argument, and g iis concave g iis a ne Show that f is convex (This composition rule subsumes all the ones given in the book, and isAuthor christopherdinardo Created Date AM♡ d r e a m g i r l b u t t o n_t a n kpackage

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