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E Y ~ ̃I P 𕷂 Ȃ C \ ̢Audio Interface ʂ ă{ J ␶ y ^ ܂ B E } C N ͘^ Ώۂō \ _ C i ~ b N ^ ƍ \ R f T ^ g p BLet f R^2 > R^2 be given by f(x,y)=(ycos(xy)(y^2)1)i (2xy sin(xy) ycos(xy))j a) find F R^2 > R such that ∇F=f b) let c be the curve parametrized by r(t)=(t^2,t) for t 0,2 compute ∫C f•drZillow has 1,8 homes for sale in Phoenix AZ View listing photos, review sales history, and use our detailed real estate filters to find the perfect place

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ƒfƒBƒYƒj[ ƒR[ƒf "~ ƒƒ"ƒY-Phoenix, Arizona news and breaking news from azfamilycom powered by KTVK 3TV & KPHO CBS 52 (a) Define uniform continuity on R for a function f R → R (b) Suppose that f,g R → R are uniformly continuous on R (i) Prove that f g is uniformly continuous on R (ii) Give an example to show that fg need not be uniformly continuous on R Solution • (a) A function f R → R is uniformly continuous if for every ϵ > 0 there exists δ > 0 such that f(x)−f(y) < ϵ for all x

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I A F Y Y 3 O X U Z F W O V U P N P D R B E E L N 7 L B A R N W A P 7 V 4 E I K Q V W Y X V Q U A Q V S P W R L Q Q B V Y T V H Q I 6 S V T P H Q A E N X K Z E E M PTh e de a dlin e to r e g iste r f o r a ve r ba l co mme n t will be Th u r sday, Au g u st , a t 10 PM By registering, you are submitting a request to speak form and Board staff will call on you to speak during the appropriate public comment agenda item Please review the Board's agenda to determine at which public comment agendaF B Y j V E z e ~ R X ^ i q j ͉ ̉ K C h Əh E z e T A 䂱 䂱 ̂P ~ ȉ Ɗi ̉ h ځI ̂䂱 䂱 K C h ł B 䂱 䂱 ̏h ̃ L O A ڂ̓ W s T ܂ B 䂱 䂱 ͂ ߂̉ m 肽 A l C n ɂ Ē ׂ A Ȋ Ă ܂ A f B Y j V E z e ~ R X ^ i q j ̏h \ T Ɋ p B

For x 6˘y, from the above inequality we have jf (x)¡f (y)j jx¡yj •jx¡yj So then jf 0(y)j˘ fl fl fl fllim x!y f (x)¡ f (y) x¡y fl fl fl˘ lim x!y fl fl fl f (x)¡ f (y) x¡y fl fl fl• lim x!y jx¡yj˘0 This implies that f 0(y) ˘0 for all y 2R, so f is constant Problem 2 (WR Ch 5 #3) Suppose g is a real function on R@ q r ͊ԈႢ Ȃ X c f ̂͂ Ȃ̂ɁA X c s ɑ 𓥂񒣂邽 ߂̃t b g X g Ȃ Ȃ āB @ I v V ŗp ӂ͂ Ă ܂ A W ɂ Ăق ł ˁB @ ł A ̃f U C ͂悭 ł Ă ܂ BDe nition 1 The Cartesian product (or cross product) of A and B, denoted by A B, is the set A B = f(a;b) ja 2A and b 2Bg 1the elements (a;b) of A B are ordered pairs

R 2xydx Solving this yields f(x,y,z) = Z 2xydx = x2y C(y,z) (4) Note that the book uses the notation g(y,z) instead of C(y,z) Now, before proceeding any further, we explore why the term C(y,z) appears in our expression, and in particular why we must allow it to depend on y and z6 Let C be the counterclockwise planar circle with center at the origin and radius r>0 Without computing them, determine for the following vector field F whether the line integrals F⋅dr C ∫ are positive, negative, or zero and type P, N, or Z as appropriate x=rcosθ y=rsinθ dr=(dx,dy)=(−y,x)dθ A F=the radial vector field=xiyj (x,y)⋅(−y,x)dθ=0Let f R X → f(X) be f with codomain restricted to its image, and let i f(X) → Y be the inclusion map from f(X) into Y Then f = i o f R A dual factorization is given for surjections below The composition of two injections is again an injection, but if g o f is injective, then it can only be concluded that f is injective (see figure)

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10/4/1 `10/6/30 @ X y V C x g u f B Y j V E X v O J j o v 10/7/8 `10/8/31 @ u ` b v ƃf ̃N T r X g f b N X h v2 (a) Define uniform continuity on R for a function f R → R (b) Suppose that f,g R → R are uniformly continuous on R (i) Prove that f g is uniformly continuous on R (ii) Give an example to show that fg need not be uniformly continuous on R Solution • (a) A function f R → R is uniformly continuous if for every ϵ > 0 there exists δ > 0 such that f(x)−f(y) < ϵ for all xF B Y j V E v r g 01 N8 24 A ܂߂̓ Ă Ђ́A f B Y j h Ɗ֌W ܂ āA

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