Chapter 10 Functions - Nanyang Technological University
Y f:Xo Y X S f s (s) Image of a Set A function f is one-to-one (or injective), if and only if f(x) = f (y) implies x = y for all x and y in the domain of f In words: ^All elements in the domain of f have different images_ Mathematical Description: f:Ao B is one-to-one x 1, x 2 A (f(x 1)=f(x 2) Æ x 1 = x 2) or f:Ao B is one-to-one x 1, x 2
1 Systems of Linear Equations - BU
ax+by = f a0x+b0y = f0 (1 1) of two equations in two unknowns, x and y Numbers a,b,a0,b0 are the coefficients of the system and numbers f,f0 are the right-hand sides of the system We shall try to write x and y separately in terms of the given coefficients and right-hand sides If in system (1 1) it happens that b = 0 and a0 = 0, then the
Chapter Four Many-Place Predicates
Andria is taller than x T(ax) x is a friend of y F(xy) z sees Bella S(zb) Cary gave x to y G(cxy) z sent u to v S(zuv) We can use any capital letter for a many-place predicate (adding subscripts if desired) You can tell whether a predicate letter is being used as a one-place predicate or a many-place predicate by seeing
Real Analysis HW 2 Solutions - Brown
so that fA ngare disjoint,A n E nand S 1 k=0 A k= S 1 k=0 E k Using countable additivity and monotonicity m0(E) m 0 [1 k=0 E k = m0 [1 k=0 A k = X1 k=0 m(A k) 1 k=0 m0(E k): (ii) The proof is identical to the proof of in the case of Lebesgue, which relies on the properties of monotonicity, countable additivity, and the excision property 3
FONCTIONS - Généralités
x ax bx co f 2 x 7-1)Résumé : f x ax bx c 2 et az0 1° Dans le repére 0; ;ij la courbe C f c’est une parabole de sommet W;DE et d’axe de symétrie la droite x D 2° Les variations de f a 0 Si a 0 7-2) Exemple : 1° Soit f une fonction numérique tq : f x x x2 2 4 2 on a f est une fonction polynôme donc D f On a a 2 et b 4 et c 2 f x ax
Volumul corpurilor de rotatie - Matematic
y=f(x) Ca aplicaŃie a formulei (6 ) vom calcula volumul corpului generat de rotirea suprafeŃei limitată de Ox, y=sinx, 0#x# π 2 în jurul lui Oy Vom avea: 2 2 2 0 3 sin 0 sin0 2 sin 4 2 2 4 V x xdx π π π π π π = − − = = − ∫ căci x xdx x x xdx xsin cos cos sin 0 2 0 2 0 2 0 2 1 π π π π ∫ ∫=− + = =
CURS07-10
y +F z v z)dt r r []W =L2MT−2 iar unitatea de m ăsur a lucrului mecanic în SI este J (Joule) 1 J = 1kg m2/s2; 1 J este lucrul mecanic efectuat de o forţă de 1 N pe un drum de 1 m în direcţia forţei • Dacă vectorul forţă este constant, atunci: W F r F r F (r r ) F r Fd (F r) r r r r r r r r r r r
MANUAL DE FÍSICA 2n BATXILLERAT
Exemple: A =3cos t i−2t2 j 5t2 k; d A dt =−3sin t i−4t j 10t k Integrals indefinides La integral és l'operació inversa de la derivació: x2 ↔ 2x, la integral de 2x és x2 La notació que es fa servir és la següent: ∫2xdx= x2 on dx indica el diferencial respecte la variable que s'integra
Superf´ıcies d’`area m´ınima i bombolles de sab´o
L’exemple m´es senzill de superf´ıcie a l’espai s´on els gr`afics de les funcions z= f(x,y) Per exemple z= ax+by´es un pla, z= a2(x2+y2) ´es un paraboloide etc La nostra intu¨ıci´o diu que el pla te curvatura nul la pel paraboloide la curvatura augmenta amb a2 Dos tipus de curvatura: 1 Curvatura de Gauss (an`aloga a la cur
SOLUCIONARI Unitat 4 - XTEC
Fals Per exemple la funció f(x) = x3 Øs crei-xent en tots els reals i en canvi f ’(0) = 0 b) Si f(x) Øs decreixent en el punt x = x0, aleshores f™(x0) £ 0 Veritat, ja que si la funció Øs decreixent la funció derivada no Øs positiva c) Si f(x) Øs creixent a l™esquerra del punt x = a i decreixent a la dreta del mateix
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