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Title School District of Palm Beach County Mach Lakes Bulletin #30 Week of April 23 Author Created Date PMZ b a f(x)dt This is a little tricky since the integral is in terms of t, but f is an expression in x, it will be constant with respect to t The answer f(x)(b−a) 3 Evaluate, where possible If not, state why (a) Z 4 1 x2 −x1Problem 12 (333) A function f on a;b is called a step function if there exists a partition P= fa= u 0
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Theorem 17 If f A!Band g B!Care bijections, then g f A!C is a bijection Proof Composition of surjections is a surjection, and compositions of injections are injections De nition 112 (Inverse Function) If f A !B is a bijection, then its inverse, f 1 B !AThe force on the wire is given by F = IL × B The direction of L × B is the negative ydirection Since L and B are perpendicular to each other, the magnitude F = ILB Details of the calculation F = ILB = (24 A)(075 m)(16 T) = 2 N The force on the section of wire is F =_ e h I h _ l _ e ^ b j H PDLO GRPXFVUHGVXU#PWV UV YDVSLWDFLVXU#JPDLO FRP b g l _ j g _ l Z ^ j _ k Z ZZZ GRPVXUGXOLFD UV > h f m q _ g b d Z k j _ ^ b o r d h e Z ± ;



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0 ∈ Zx and suppose that x−r divides f(x) for some r ∈ Q Then we must have f(r) = 0 We will use this fact to prove that in fact r ∈ Z If r = 0 then there is nothing to prove So assume r 6= 0 and write r = a/b with a,b ∈ Z, b 6= 0 and ( a,b) = 1 Then 0 = f(r) = f(a/b) = a b n a n−1 a b n−1 ···a 1 a b a 0 MultiplyingIf f(x) has a zero in Q then it has a zero m2Z and mdivides a 0 Proof If is a zero of f(x) then (x ) is a linear factor of Qx By Gauss f(x) must have a linear factor in Z, f(x) = (ax b)g(x) Looking at the leading coe cients, we must have that adivides 1 So a= 1 Possibly replacing g(x) by g(x) we may assume that a= 1D H G D M J K ;



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