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Inverse Functions Inverse functions are functions that undo one another In other words, if f1 (x) is the inverse of f(x), then f(f1 (x)) = f1 (f(x)) = xWe can use this definition to solve.

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Ng h J m j d ;. 12 Let f(x;y) be a di erentiable function, and let u= x yand v= x−yFinda constant such that (fx) 2 (f y) 2 = ((f u) 2 (f v) 2) Solution By the chain rule (fx)2 (f y)2 =(f u f v)2 (f u−f v)2 =2((f u)2 (f v)2)Thus =2 13 Find the directional derivative D ~ufat the given point in the direction indicated by the angle (a) f(x;y)= p 5x−4y;(2;1);. INTERPLANET i C ^ v l b g j ̌ E c ʔ̃T C g ł B V A C e 𐏎 X V B c T C g Ȃ ł͂̃ C i b v E i ł ͂ ܂ B ʃ C g ܂ B5400 ~ ȏ ő B ŒZ o ׁB ȃ^ C Z E N J ÁB.

The parent function is f(x) = cos(x) The following transformations have been applied a = 2 (Vertical stretch by a factor of 2) b = −1 (Reflection in the yaxis) h = 90° (Translation 90° to the right) k = 8 (Translation 8 units up) Practice Questions 1 The graph of f(x) = x3 was reflected in the yaxis, compressed vertically by a factor of. , ¨ON(j( 'K ¨(B 9( D E;) men orqG Cs t v w "Q ;OW(P( D E;) X*9ON)G ) 'E7' ,&'. H, I, J, K, L, M, N, O, P Q R S, T U V, DoubleU, X, Y and Z, A B C D E F G H I J K L M N O P Q R S T U V W X Y Z Now I know my ABC's Next time.

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Suppose there exists a surjection f∶A→ B Recall that SBS ≤SAS if and only if there exists an injection from Bto A We’ll construct one presently De ne a function g∶B→ s follows For each b∈B, we know there exists at least one a∈Asuch that f(a) =b Set g(b) equal to one such a. X ^ , X g J h, f C X g ^, f U C i N8 G ݃A C e ̔ I ܂ B1996 N ɃC g G ݁@Marge i } W j 𓡎} s ɊJ ƁA ȗ 24 N ԊF l Ɏx ĉc Ƃ Ă ܂ ɗL ܂ B. Kr »þ5# ‚F÷8 v µ±H , îo»Ø ÿOñ ÉÛ›ð ”À Å3©À䞀w M3B¹Ô ‹ul ßÔ(îi6 eo V°,6ñˆâ^Š;ûŸSR2A@ Š1@ fëZ°Ó!Uˆn¹“îç¢{Ÿè)¤ IÉä’NI=Iª Ó@ ¤ ›d͵ä¸ûˆ>™8©“Ö(} 1íÒ¬Bœ*r3š@nhwÐßD4ÍG Gü{ÈÝ ÝÏò¤Ð Ô7å‚ð תþ>”\f ֭ͬ‰ ¨= ¨4ÄCL € €/Eþ©>• "}Áõ4.

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JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS 92, (19) Fuzzy Numbers J G DUKMAN, H VAN HAERINGEN, AND S J DE LANGE Department of Mathematics and Informatics, Delft University of Technology, Delft, The Netherlands Submitted by L Zadeh Fuzzy numbers have been introduced by Zadeh in order to deal with imprecise numerical quantities in a practical way. P X g @ V R ΁E p X g ̔̔ N J j R E X s ` A c A @ I W i p X g n C ̐ ( N J j R) ŏ򉻂ƃp h ܂ B Ζ{ ̎ ͂𖳌 ɍ ߂ A A p X g ̔̔. Favorite Answer a) Integrate the i, j, k entries with respect to x, y, z, respectively f (x, y, z) = xyz g (y, z) f (x, y, z) = xyz h (x, z) f (x, y, z) = xyz 9z^2 j (x, y) Putting this all together, f (x, y, z) = xyz 9z^2 ( C).

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K ߌ U } p i T O ʃC f b N X. Let f G!G=Nwith kernel kerf= N Consider fj H H!H=N, then jImfj HjjjG=Njand jImfj HjjjHj(image divides both domain and codomain) But jG=N and jHjare relatively prime So Im fj H = 1 which implies that H N= kerf 3219 Prove that if N is a normal subgroup of the nite group Gand (jNj;jG Nj) = 1 then N is the unique subgroup of Gof order jNj 1. 1 4 , q;F (G tpa p y;) 3 0 kg vi l As ;s x U ng h Us;.

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G E T C G e B X g u g C g C g C I v o G ̕ 䗠 ł́A o ԑO ْ̋ Ă o i ɊF ܂ B ́u K ^ F Ă I v u ܂ I v Ƃ 悯 ̂ ߂̂ ܂ Ȃ B l Ɂu g C g C g C I v Ƃ C ߂Ă ̐ i J ܂ B ̓X ^ C X g ̊F ɃG 𑗂邽 ߥ B ̐ ܂őf Z ċz B ă_ W C Č N ɐ ܂ B R ̌b ݂ Ղ z @ \ i ł B. F i E ւ E E ̑ ̃T r X E E ʖ E җ{ u E X N E w Z j ̃ C Y E C t B j e B 戵. Exercise 2 Calculate the curl of the following vector fields F(x,y,z) (click on the green letters for the solutions) (a) F = xi−yj zk, (b) F = y3ixyj −zk, (c) F = xiyj zk p x2 y2 z2, (d) F = x2i2zj −yk Here is a review exercise before the final quiz Exercise 3 Let f be a scalar field and F(x,y,z) and G(x,y,z) be vector fields.

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Lf c is any real number and if f(x) = c for all x, then f ' (x) = 0 for all x That is, the derivative of a constant function is the zero function It is easy to see this geometrically Referring to Figure 1, we see that the graph of the constant function f(x) = c is a horizontal line. Is a factorisation of f(x) over the integers Suppose that f(x) = a nxn a n 1xn 1 a 0 g(x) = b dx d b d 1x 1 b 0 h(x) = c exe c e 1xe 1 c 0 for some n, dand e>1 As a 0 = b 0c 0 is not divisible by p2 either b 0 or c 0 is not divisible by p Possibly switching g(x) and h(x) we may assume that b. A) 5 kg B) 6 kg C) 8 kg D) k hw hJ 1 5 Fwp aPL f i sf;.

07 N x 0706 0704 0703 0702 0701 g C x g E Z ~ i ۂ̓ C x g E Z ~ i. Curl The second operation on a vector field that we examine is the curl, which measures the extent of rotation of the field about a point Suppose that F represents the velocity field of a fluid Then, the curl of F at point P is a vector that measures the tendency of particles near P to rotate about the axis that points in the direction of this vector The magnitude of the curl vector at P. 2 mOj ;j k h d p.

A, B, C, D, E, F, G;. 07 N x 0706 0704 0703 0702 0701 g C x g E Z ~ i ۂ̓ C x g E Z ~ i. Z − px 2 zp − y ie z2 = 0 − xq 2 zq − x z2 ie ( 2 zp − y ) − xq − z − 2px ( 2 zq − x ) = 0 2 z z ie px 2 − q ( xy − 2 z 2 ) = zx(3) Form the partial differential equation by eliminating the arbitrary function ‘f’from z = f ( 2 x y ) g ( 3x − y )Solution Given z = f ( 2 x y ) g ( 3x − y.

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