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G A T X E G A _ T X y V E p c F t @ C X g _ Ԏ _ G A T X_ L b g F W v E i d d o_ O _ i j_ r R _ G A T X y R C C g F v q O q S P V Q b z ԍ t g A b v ␳ p w p G A T X Ƃ āA ́A _ 炩 o l ̂ ߁A ו ςނƃR i O ̃ 傫 Ȃ ܂ B ̃ } T g G A T X Ƃ Ă ߂ ܂ B H Ŏ t ł u { g I d l v B A b v/ _ E p r C o u A z X i ` u j t. When I think of y=f(x), i Think of y = f(x)= 1, x = 1, x =2, then y =f(x) =2, x =3, then y= f(x)=3, and so on Hi John, I find it helps sometimes to think of a function as a machine, one where you give a number as input to the machine and receive a number as the output. And and and Pause here for a class discussion Point has the same coordinates as point , except its coordinate has the opposite sign.

Type Notes Uploaded By sammm_y;. Y n 2fx f(x) = 0gfor all nand lim n!1y n = y Since f is continuous, by Theorem 402 we have f(y) = lim n!1f(y n) = lim n!10 = 0 Hence y2fx f(x) = 0g, so fx f(x) = 0gcontains all of its limit points and is a closed subset of R 3 Let Xand Y be closed subsets of R Prove that X Y is a closed subset of R2 State and prove a generalization. U J t F I C ^ v ́A Ƃ t C ^ ̓ BL сA l b g ̋C ɂȂ l ^ Ȃǂ̘b 𒆐S ɁA Ă œ ɕK v ̂Ȃ Љ l T C g ł B l ^ ̃^ R ~ ҂ Ă ܂ B y14 N4 21 z y m z2 ڂ̃g N C x g u J t F I E p e B ` ` v 5 25 ɊJ  ܂ I @ Ƌ X y V Q X g o \ ł I.

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ʎВc @ l \ ی𗬋@ \ Ấu j N t F X e B o v A5 9 ( ) ɃJ l M z ŊJ Â ꂽ B t F X ́A ʂ Č𗬂 ߁A ɓ Ă̐e P Ȃ 邱 Ƃ ړI Ƃ A6 ڂ } ͓ ĂV c Q B ̎Q ҂ f 炵 n j \ B. R Ɏg 鍂 ׂ̃t ʐ^ f ޏW ł B 10,000 ̐ E ő勉 ̖ 摜 A ʐ^ T C g B u O HP A C X g ɂ p B p p A H ܂߂ăt i j ł BNo 1973 v C x g r ` ̗. S = Y C G Or S = (Y T C) (T G) These equations are the same, because the two Ts in the second equation cancel each other, but each reveals a different way of thinking about national saving In particular, the second equation separates national saving into two pieces private saving (Y T C) and public saving (T G).

} S ݓX ̏ڂ C x g ͌ T C g ł B f 1st f i1000 ~ ł ɂȂ ܂ j. G(x) = sup{x y T − g ∗ (y) y ∈ R n} where ∗g (y) is the conjugate of g(x) We provide this theorem without proof, and omit further discussion of lower semicontinuity, but we can safely assume that the functions that we will deal with satisfy this Note thatthis condition implies g ∗∗ = , is, the conjugate. 116 = H > B R G B D g Z F b g g h _ h e h ` d b y m g b \ _ j k b l _ l “ K \ B \ Z g J b e k d b”, L h f 53, K \I 1 1, F _ o Z g b a Z p b y, _ e _ d l j b n.

S t \ E R y \ ̓` F S t B Ŋy C x g 񖞍ځb ` F S t F y R X F { F y s ԉ ` F S t F y R X ӂ. 2 (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. = y = ex First, for m = 1, it is true Next, assume that it is true for k, then d k1 dxk1 ex = d dx d dxk ex = d dx (ex) = ex By the axiom of induction, it is true for all positive integer m 3 13 Another Definition of the Exp Function.

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The Fokker CX was a Dutch biplane scout and light bomber designed in 1933 It had a crew of two (a pilot and an observer). The slope of the line is the value of , and the yintercept is the value of Slope yintercept Slope yintercept Any line can be graphed using two points Select two values, and plug them into the equation to find the corresponding values. The slope of the line is the value of , and the yintercept is the value of Slope yintercept Slope yintercept Any line can be graphed using two points Select two values, and plug them into the equation to find the corresponding values.

Title PhysRevB Author Aziz Abdellahi Created Date 9/8/15 PM. N u c Y @ c A C x g u z K ΐV ԉΑ ƐM B ӍՁI v y z @ @ @ @ 쌧 z K S x m x m Ԓn. T P B Y J P Q O Y W V P J Q Y W M S Y A P J B Y M V I X L M B D Q 2 4 L R N M A Q Q D F Z R T L Y N Y A E O R L Y P 4 T E 4 S Y 4 P T R S Y S N R Y A E N R 7 4 my whole family figured it out but i can't?.

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How far away are they from the xaxis?. Now, let ðE,≤Þ be an ordered set and T be a given operator on ðE, ≤Þ reversing the order such that xTxor for all x∈E We take a part A of E and a monotone maps f A→A;. 选择一项 a g(x) = 2e5x15 b g(x) = 3e5x10 c g(x) = €5x–6 d g(x) = 2e5x dy dt = = 7y, and y = 3 when t = 0 Solve the equation 选择一项 a y = e71 b y = 3 e71 C y = 2 e71 d y = 3e7t.

I è & s v j v ~ ¯ R t y Q v } Q V F Ö J ê i ¼ L ¯ ¯ è y ¸ ~ y ´ q è Ó ê ­ µ å ñ Ï Ó § ë Á Õ â ¦ _ L ¤ F Ö ¥ T y ² C y l s O Q 8 y T N Í J ê ó y 8 y Ì ß ± ¤ < ¹ u ¥ ~ è y i t s ë 0 S y » < V ² C d }. > = < ;. Example 122 Solve y y 0 with given initial values y 0 y 0 Now ex and e x are solutions of this differential equation, so the general solution is a linear combination of these But we won’t have as easy a time finding a solution like (123), since these functions do not have the initial values 1 0;.

DB h B ² Q D٬8. ê ¦ · Ü Æ Ó § ñ 3& u t y A ¢ = = b q O À b q O h y , y # ê y Ð ï s u è ® ß b Ï J î ½ ­ O j b d } î ½ v ­ v J u O Ö û z ² 0 s O j b d } ) _ Z k ` O } ê 8 ð ¬ d v M j Ü · ­ y ¥ = ¨ ­ ¡ ¦ ¿ ¯ Á Æ y q u t y ô. How far away are they from the yaxis?.

G H I J K L M N O P Q R S T U V W X Y Z \ ^ _ ` a b c d P Q e f g h i W I J j k l m n a U d R. Solve y 00 4 y sin2 t y 0 10 y 0 0 331 solve y 00 7 y School University of Waterloo;. Course Title MATH 24;.

W E Y s T ̎Γ x Ɓw B ` x ̈̑ Ȑl K I ƃ I i h ̓s s _ r ` K ₵ ܂ m v C x g c A n g X J i I v V i c A. Y * Ê I ­ ß X ÿ “ Ï ù » s H ˜ B Ï C B m = § ë Ò b Æ D 0 í _ ) e N ¿ ù R % › ï Ò ’ ‡ ¶ ° ) > < T N O ¥ Y ï F 8 Î € ô È c Y z ¬ W 7 K ) G , @ þ æ þ < ö § ± ú o Ä q r L Ç Ã ü ¸ ò ³ à Õ É K Û ( y !. There's nothing more I can do with this, and I can't find a fully numerical value because I don't have a number to plug in for the t So my answer is g(3) = 4 – t Given that f (x) = 3x 2 2x, find f (h) Everywhere that my formula has an "x", I now plug in an "h" I start with the formula they gave me.

Ƃ C X g t f Ƃ t f ށE t C X g t C X g. 2 (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. G b v y W C X g X V i j n C I L ł I i u O j n ߂܂ i j Illustration no18" ǂ ".

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