Το σχολικό βιβλίο μαθηματικών θετικών σπουδών Γ Λυκείου λύσεις των ασκήσεων

["2.10 x \u2212\u221e \u20131 1 3 +\u221e f \u2032 (x) \u2013\u2013 0+ f \u2033(x) +0 \u20130 + + f ( x) \u2013 16 0 +\u221e \u2212\u221e T.M. \u03a3.\u039a. \u201316 T.E.ii) \u25cf \u0397 f \u03bf\u03c1\u03af\u03b6\u03b5\u03c4\u03b1\u03b9 \u03c3\u03c4\u03bf \u0391 = (\u2212\u221e,1) \u222a (1, +\u221e)\u25cf \u0397 f \u03b5\u03af\u03bd\u03b1\u03b9 \u03c3\u03c5\u03bd\u03b5\u03c7\u03ae\u03c2 \u03c3\u03c4\u03bf \u0391, \u03c9\u03c2 \u03c1\u03b7\u03c4\u03ae.\u25cf \u0393\u03b9\u03b1 \u03ba\u03ac\u03b8\u03b5 x \u2208 \u0391 \u03b9\u03c3\u03c7\u03cd\u03b5\u03b9 f \u2032( x) = \u22122 , \u03bf\u03c0\u03cc\u03c4\u03b5 (x \u22121)2 f \u2032(x) \u2260 0 \u03b3\u03b9\u03b1 \u03ba\u03ac\u03b8\u03b5 x \u2208 \u0391.\u03a4\u03bf \u03c0\u03c1\u03cc\u03c3\u03b7\u03bc\u03bf \u03c4\u03b7\u03c2 f \u2032 \u03c6\u03b1\u03af\u03bd\u03b5\u03c4\u03b1\u03b9 \u03c3\u03c4\u03bf\u03bd \u03c0\u03b1\u03c1\u03b1\u03ba\u03ac\u03c4\u03c9 \u03c0\u03af\u03bd\u03b1\u03ba\u03b1, \u03b1\u03c0\u03cc \u03c4\u03bf\u03bd \u03bf\u03c0\u03bf\u03af\u03bf \u03c0\u03c1\u03bf\u03c3-\u03b4\u03b9\u03bf\u03c1\u03af\u03b6\u03bf\u03c5\u03bc\u03b5 \u03c4\u03b1 \u03b4\u03b9\u03b1\u03c3\u03c4\u03ae\u03bc\u03b1\u03c4\u03b1 \u03bc\u03bf\u03bd\u03bf\u03c4\u03bf\u03bd\u03af\u03b1\u03c2 \u03c4\u03b7\u03c2 f . x \u2212\u221e 1 +\u221e f \u2032 (x) \u2013 \u2013 f ( x)\u0393\u03b9\u03b1 \u03ba\u03ac\u03b8\u03b5 x\u2208\u0391 \u03b9\u03c3\u03c7\u03cd\u03b5\u03b9 f \u2032\u2032( x) = \u22122 \u22122(x \u22121) = (x 4 . (x \u22121)4 \u2212 1)3\u03a4\u03bf \u03c0\u03c1\u03cc\u03c3\u03b7\u03bc\u03bf \u03c4\u03b7\u03c2 f \u2033 \u03c6\u03b1\u03af\u03bd\u03b5\u03c4\u03b1\u03b9 \u03c3\u03c4\u03bf\u03bd \u03c0\u03b1\u03c1\u03b1\u03ba\u03ac\u03c4\u03c9 \u03c0\u03af\u03bd\u03b1\u03ba\u03b1, \u03b1\u03c0\u03cc \u03c4\u03bf\u03bd \u03bf\u03c0\u03bf\u03af\u03bf \u03c0\u03c1\u03bf\u03c3-\u03b4\u03b9\u03bf\u03c1\u03af\u03b6\u03bf\u03c5\u03bc\u03b5 \u03c4\u03b1 \u03b4\u03b9\u03b1\u03c3\u03c4\u03ae\u03bc\u03b1\u03c4\u03b1 \u03c3\u03c4\u03b1 \u03bf\u03c0\u03bf\u03af\u03b1 \u03b7 f \u03c3\u03c4\u03c1\u03ad\u03c6\u03b5\u03b9 \u03c4\u03b1 \u03ba\u03bf\u03af\u03bb\u03b1 \u03c0\u03c1\u03bf\u03c2 \u03c4\u03b1 \u03ac\u03bd\u03c9 \u03ae\u03c0\u03c1\u03bf\u03c2 \u03c4\u03b1 \u03ba\u03ac\u03c4\u03c9 \u03ba\u03b1\u03b9 \u03c4\u03b1 \u03c3\u03b7\u03bc\u03b5\u03af\u03b1 \u03ba\u03b1\u03bc\u03c0\u03ae\u03c2. x \u2212\u221e 1 +\u221e + f \u2033 (x) \u2013 f ( x) 150","2.10\u25cf \u0395\u03af\u03bd\u03b1\u03b9 lim f (x) = lim x +1 = 1, \u03bf\u03c0\u03cc\u03c4\u03b5 \u03b7 \u03b5\u03c5\u03b8\u03b5\u03af\u03b1 y = 1 \u03b5\u03af\u03bd\u03b1\u03b9 \u03bf\u03c1\u03b9\u03b6\u03cc\u03bd\u03c4\u03b9\u03b1 x \u2192 \u2212\u221e x\u2192\u2212\u221e x \u22121\u03b1\u03c3\u03cd\u03bc\u03c0\u03c4\u03c9\u03c4\u03b7 \u03c4\u03b7\u03c2 Cf \u03c3\u03c4\u03bf \u2212\u221e.\u039f\u03bc\u03bf\u03af\u03c9\u03c2 lim f (x) = 1, \u03bf\u03c0\u03cc\u03c4\u03b5 \u03b7 y = 1 \u03b5\u03af\u03bd\u03b1\u03b9 \u03bf\u03c1\u03b9\u03b6\u03cc\u03bd\u03c4\u03b9\u03b1 \u03b1\u03c3\u03cd\u03bc\u03c0\u03c4\u03c9\u03c4\u03b7 \u03ba\u03b1\u03b9 \u03c3\u03c4\u03bf +\u221e. x \u2192 +\u221e\u0395\u03c0\u03af\u03c3\u03b7\u03c2 lim f (x) = lim x +1 = \u2212\u221e, lim f (x) = +\u221e, \u03bf\u03c0\u03cc\u03c4\u03b5 \u03b7 \u03b5\u03c5\u03b8\u03b5\u03af\u03b1 x = 1 \u03b5\u03af\u03bd\u03b1\u03b9 x \u22121 x \u21921\u2212 x \u21921\u2212 x \u21921+\u03ba\u03b1\u03c4\u03b1\u03ba\u03cc\u03c1\u03c5\u03c6\u03b7 \u03b1\u03c3\u03cd\u03bc\u03c0\u03c4\u03c9\u03c4\u03b7 \u03c4\u03b7\u03c2 Cf .\u25cf \u03a3\u03c7\u03b7\u03bc\u03b1\u03c4\u03af\u03b6\u03bf\u03c5\u03bc\u03b5 \u03c4\u03bf\u03bd \u03c0\u03af\u03bd\u03b1\u03ba\u03b1 \u03bc\u03b5\u03c4\u03b1\u03b2\u03bf\u03bb\u03ce\u03bd \u03c4\u03b7\u03c2 f \u03ba\u03b1\u03b9 \u03c7\u03b1\u03c1\u03ac\u03c3\u03c3\u03bf\u03c5\u03bc\u03b5 \u03c4\u03b7 \u03b3\u03c1\u03b1\u03c6\u03b9\u03ba\u03ae\u03c4\u03b7\u03c2 \u03c0\u03b1\u03c1\u03ac\u03c3\u03c4\u03b1\u03c3\u03b7. x \u2212\u221e 1 +\u221ef \u2032 (x)f \u2033(x) \u2013 \u2013 \u2013f ( x) 1 + \u2212\u221e +\u221e 1iii) \u25cf \u0397 f \u03bf\u03c1\u03af\u03b6\u03b5\u03c4\u03b1\u03b9 \u03c3\u03c4\u03bf \u0391 = R. \u25cf \u0397 f \u03b5\u03af\u03bd\u03b1\u03b9 \u03c3\u03c5\u03bd\u03b5\u03c7\u03ae\u03c2 \u03c3\u03c4\u03bf R \u03c9\u03c2 \u03c0\u03bf\u03bb\u03c5\u03c9\u03bd\u03c5\u03bc\u03b9\u03ba\u03ae. \u25cf \u0393\u03b9\u03b1 \u03ba\u03ac\u03b8\u03b5 x \u2208R \u03b9\u03c3\u03c7\u03cd\u03b5\u03b9 f \u2032(x) = 4x3 \u2212 4x = 4x(x2 \u22121), \u03bf\u03c0\u03cc\u03c4\u03b5 f \u2032(x) = 0 \u21d4 x = 0 \u03ae x = \u2013 1 \u03ae x = 1. \u03a4\u03bf \u03c0\u03c1\u03cc\u03c3\u03b7\u03bc\u03bf \u03c4\u03b7\u03c2 f \u2032 \u03c6\u03b1\u03af\u03bd\u03b5\u03c4\u03b1\u03b9 \u03c3\u03c4\u03bf\u03bd \u03c0\u03b1\u03c1\u03b1\u03ba\u03ac\u03c4\u03c9 \u03c0\u03af\u03bd\u03b1\u03ba\u03b1, \u03b1\u03c0\u03cc \u03c4\u03bf\u03bd \u03bf\u03c0\u03bf\u03af\u03bf \u03c0\u03c1\u03bf\u03c3\u03b4\u03b9\u03bf\u03c1\u03af\u03b6\u03bf\u03c5\u03bc\u03b5 \u03c4\u03b1 \u03b4\u03b9\u03b1\u03c3\u03c4\u03ae\u03bc\u03b1\u03c4\u03b1 \u03bc\u03bf\u03bd\u03bf\u03c4\u03bf\u03bd\u03af\u03b1\u03c2 \u03ba\u03b1\u03b9 \u03c4\u03b1 \u03c4\u03bf\u03c0\u03b9\u03ba\u03ac \u03b1\u03ba\u03c1\u03cc\u03c4\u03b1\u03c4\u03b1 \u03c4\u03b7\u03c2 f. x \u2212\u221e \u20131 0 1 +\u221ef \u2032 (x) \u2013 0 + 0 \u2013 0 + 0 T.\u039c.f(x) \u20131 \u20131 T.E. T.E.\u2014 \u0393\u03b9\u03b1 \u03ba\u03ac\u03b8\u03b5 x \u2208R \u03b9\u03c3\u03c7\u03cd\u03b5\u03b9 f \u2032\u2032(x) = 12x2 \u2212 4 = 4(3x2 \u22121), \u03bf\u03c0\u03cc\u03c4\u03b5 f \u2032\u2032(x) = 0 \u21d4 x = 3 \u03ae x = \u2212 3 . 33 151","2.10 \u03a4\u03bf \u03c0\u03c1\u03cc\u03c3\u03b7\u03bc\u03bf \u03c4\u03b7\u03c2 f \u2033 \u03c6\u03b1\u03af\u03bd\u03b5\u03c4\u03b1\u03b9 \u03c3\u03c4\u03bf\u03bd \u03c0\u03b1\u03c1\u03b1\u03ba\u03ac\u03c4\u03c9 \u03c0\u03af\u03bd\u03b1\u03ba\u03b1, \u03b1\u03c0\u03cc \u03c4\u03bf\u03bd \u03bf\u03c0\u03bf\u03af\u03bf \u03c0\u03c1\u03bf\u03c3\u03b4\u03b9\u03bf\u03c1\u03af\u03b6\u03bf\u03c5\u03bc\u03b5 \u03c4\u03b1 \u03b4\u03b9\u03b1\u03c3\u03c4\u03ae\u03bc\u03b1\u03c4\u03b1 \u03c3\u03c4\u03b1 \u03bf\u03c0\u03bf\u03af\u03b1 \u03b7 f \u03c3\u03c4\u03c1\u03ad\u03c6\u03b5\u03b9 \u03c4\u03b1 \u03ba\u03bf\u03af\u03bb\u03b1 \u03c0\u03c1\u03bf\u03c2 \u03c4\u03b1 \u03ac\u03bd\u03c9 \u03ae \u03c0\u03c1\u03bf\u03c2 \u03c4\u03b1 \u03ba\u03ac\u03c4\u03c9 \u03ba\u03b1\u03b9 \u03c4\u03b1 \u03c3\u03b7\u03bc\u03b5\u03af\u03b1 \u03ba\u03b1\u03bc\u03c0\u03ae\u03c2.x \u2212\u221e \u22123 3 +\u221e 3 3f \u2033(x) + 0 \u2013 0 +f ( x) \u22125 \u22125 9 9 \u03a3.\u039a. \u03a3.\u039a.\u25cf \u0395\u03af\u03bd\u03b1\u03b9 lim f (x) = lim (x4 \u2212 2x2 ) = lim x4 = +\u221e.x \u2192 \u2212\u221e x \u2192 \u2212\u221e x \u2192 \u2212\u221elim f (x) = lim x4 = +\u221e.x \u2192 +\u221e x \u2192 +\u221e\u0397 Cf \u03b4\u03b5\u03bd \u03ad\u03c7\u03b5\u03b9 \u03b1\u03c3\u03cd\u03bc\u03c0\u03c4\u03c9\u03c4\u03b5\u03c2 \u03c3\u03c4\u03bf \u2212\u221e, +\u221e, \u03b1\u03c6\u03bf\u03cd \u03b5\u03af\u03bd\u03b1\u03b9 \u03c0\u03bf\u03bb\u03c5\u03c9\u03bd\u03c5\u03bc\u03b9\u03ba\u03ae \u03c4\u03b5\u03c4\u03ac\u03c1\u03c4\u03bf\u03c5\u03b2\u03b1\u03b8\u03bc\u03bf\u03cd.\u25cf \u03a3\u03c7\u03b7\u03bc\u03b1\u03c4\u03af\u03b6\u03bf\u03c5\u03bc\u03b5 \u03c4\u03bf\u03bd \u03c0\u03af\u03bd\u03b1\u03ba\u03b1 \u03bc\u03b5\u03c4\u03b1\u03b2\u03bf\u03bb\u03ce\u03bd \u03c4\u03b7\u03c2 f \u03ba\u03b1\u03b9 \u03c7\u03b1\u03c1\u03ac\u03c3\u03c3\u03bf\u03c5\u03bc\u03b5 \u03c4\u03b7 \u03b3\u03c1\u03b1\u03c6\u03b9\u03ba\u03ae\u03c4\u03b7\u03c2 \u03c0\u03b1\u03c1\u03ac\u03c3\u03c4\u03b1\u03c3\u03b7.x \u2212\u221e \u20131 \u22123 0 3 1 +\u221e 3 0 3f \u2032 (x) \u2013 0 + + \u2013 0+ \u2013 +f \u2033(x) + + 0\u2013 \u2013 0+ +\u221e+\u221e \u22125 0 \u22125 \u20131 9 T.M. 9 T.E.f(x) \u20131 T.E. \u03a3.\u039a. \u03a3.\u039a.\u03a3\u03c7\u03cc\u03bb\u03b9\u03bf\u0395\u03c0\u03b5\u03b9\u03b4\u03ae \u03b3\u03b9\u03b1 \u03ba\u03ac\u03b8\u03b5 x \u2208R \u03b9\u03c3\u03c7\u03cd\u03b5\u03b9 f (\u2212x) = (\u2212x)4 \u2212 2(\u2212x)2 = x4 \u2212 2x2 = f (x), 152","2.10\u03b7 f \u03b5\u03af\u03bd\u03b1\u03b9 \u03ac\u03c1\u03c4\u03b9\u03b1, \u03bf\u03c0\u03cc\u03c4\u03b5 \u03b7 \u03b3\u03c1\u03b1\u03c6\u03b9\u03ba\u03ae \u03c4\u03b7\u03c2 \u03c0\u03b1\u03c1\u03ac\u03c3\u03c4\u03b1\u03c3\u03b7 \u03b5\u03af\u03bd\u03b1\u03b9 \u03c3\u03c5\u03bc\u03bc\u03b5\u03c4\u03c1\u03b9\u03ba\u03ae \u03c9\u03c2 \u03c0\u03c1\u03bf\u03c2\u03c4\u03bf\u03bd \u03ac\u03be\u03bf\u03bd\u03b1 \u03c4\u03c9\u03bd y.2. i) \u25cf \u0397 f \u03bf\u03c1\u03af\u03b6\u03b5\u03c4\u03b1\u03b9 \u03c3\u03c4\u03bf A = R*.\u25cf \u0397 f \u03b5\u03af\u03bd\u03b1\u03b9 \u03c3\u03c5\u03bd\u03b5\u03c7\u03ae\u03c2 \u03c3\u03c4\u03bf R*, \u03c9\u03c2 \u03c1\u03b7\u03c4\u03ae. x 2\u2212 1 1 x2\u25cf \u0393\u03b9\u03b1 \u03ba\u03ac\u03b8\u03b5 x \u2208 R * \u03b9\u03c3\u03c7\u03cd\u03b5\u03b9 f \u2032( x) = 1\u2212 x2 = , \u03bf\u03c0\u03cc\u03c4\u03b5 f \u2032(x) = 0 \u21d4 x = 1 \u03ae x = \u2013 1.\u03a4\u03bf \u03c0\u03c1\u03cc\u03c3\u03b7\u03bc\u03bf \u03c4\u03b7\u03c2 f \u2032 \u03c6\u03b1\u03af\u03bd\u03b5\u03c4\u03b1\u03b9 \u03c3\u03c4\u03bf\u03bd \u03c0\u03b1\u03c1\u03b1\u03ba\u03ac\u03c4\u03c9 \u03c0\u03af\u03bd\u03b1\u03ba\u03b1, \u03b1\u03c0\u03cc \u03c4\u03bf\u03bd \u03bf\u03c0\u03bf\u03af\u03bf\u03c0\u03c1\u03bf\u03c3\u03b4\u03b9\u03bf\u03c1\u03af\u03b6\u03bf\u03c5\u03bc\u03b5 \u03c4\u03b1 \u03b4\u03b9\u03b1\u03c3\u03c4\u03ae\u03bc\u03b1\u03c4\u03b1 \u03bc\u03bf\u03bd\u03bf\u03c4\u03bf\u03bd\u03af\u03b1\u03c2 \u03ba\u03b1\u03b9 \u03c4\u03b1 \u03b1\u03ba\u03c1\u03cc\u03c4\u03b1\u03c4\u03b1 \u03c4\u03b7\u03c2 f. x \u2212\u221e \u20131 0 1 +\u221ef \u2032 (x) + 0 \u2013 \u2013 0+ \u20132 2f(x) T.\u039c. T.\u0395.\u2014 \u0393\u03b9\u03b1 \u03ba\u03ac\u03b8\u03b5 x \u2208R* \u03b9\u03c3\u03c7\u03cd\u03b5\u03b9 f \u2032\u2032( x) = 2x3 \u2212 2x(x2 \u22121) = 2 , \u03bf\u03c0\u03cc\u03c4\u03b5 x4 x3 f \u2033(x) \u2260 0 \u03b3\u03b9\u03b1 \u03ba\u03ac\u03b8\u03b5 x \u2208R*.\u03a4\u03bf \u03c0\u03c1\u03cc\u03c3\u03b7\u03bc\u03bf \u03c4\u03b7\u03c2 f \u2033 \u03c6\u03b1\u03af\u03bd\u03b5\u03c4\u03b1\u03b9 \u03c3\u03c4\u03bf\u03bd \u03c0\u03b1\u03c1\u03b1\u03ba\u03ac\u03c4\u03c9 \u03c0\u03af\u03bd\u03b1\u03ba\u03b1, \u03b1\u03c0\u03cc \u03c4\u03bf\u03bd \u03bf\u03c0\u03bf\u03af\u03bf \u03c0\u03c1\u03bf\u03c3-\u03b4\u03b9\u03bf\u03c1\u03af\u03b6\u03bf\u03c5\u03bc\u03b5 \u03c4\u03b1 \u03b4\u03b9\u03b1\u03c3\u03c4\u03ae\u03bc\u03b1\u03c4\u03b1 \u03c3\u03c4\u03b1 \u03bf\u03c0\u03bf\u03af\u03b1 Cf \u03c3\u03c4\u03c1\u03ad\u03c6\u03b5\u03b9 \u03c4\u03b1 \u03ba\u03bf\u03af\u03bb\u03b1 \u03c0\u03c1\u03bf\u03c2 \u03c4\u03b1 \u03ac\u03bd\u03c9 \u03ae\u03c0\u03c1\u03bf\u03c2 \u03c4\u03b1 \u03ba\u03ac\u03c4\u03c9. x \u2212\u221e 0 +\u221e f \u2033(x) \u2013 + f ( x)\u25cf \u03a0\u03bb\u03ac\u03b3\u03b9\u03b5\u03c2 - \u03bf\u03c1\u03b9\u03b6\u03cc\u03bd\u03c4\u03b9\u03b5\u03c2 \u03b1\u03c3\u03cd\u03bc\u03c0\u03c4\u03c9\u03c4\u03b5\u03c2\u0397 \u03b1\u03c3\u03cd\u03bc\u03c0\u03c4\u03c9\u03c4\u03b7 \u03c4\u03b7\u03c2 Cf \u03c3\u03c4\u03bf \u2212\u221e \u03b5\u03af\u03bd\u03b1\u03b9 \u03c4\u03b7\u03c2 \u03bc\u03bf\u03c1\u03c6\u03ae\u03c2 y = \u03bbx + \u03b2, \u03cc\u03c0\u03bf\u03c5 \u03bb = lim f (x) = lim \uf8eb\uf8ed\uf8ec1 + 1 \uf8f6 =1 \u03ba\u03b1\u03b9 x x2 \uf8f8\uf8f7 x \u2192 \u2212\u221e x \u2192 \u2212\u221e \u03b2 = lim ( f (x) \u2212 \u03bbx) = lim \uf8eb x+ 1 \u2212 x \uf8f6 = lim 1 = 0. \uf8ec\uf8ed x \uf8f7\uf8f8 x x \u2192 \u2212\u221e x \u2192 \u2212\u221e x \u2192 \u2212\u221e\u0394\u03b7\u03bb\u03b1\u03b4\u03ae, \u03b5\u03af\u03bd\u03b1\u03b9 \u03b7 \u03b5\u03c5\u03b8\u03b5\u03af\u03b1 y = x.\u039f\u03bc\u03bf\u03af\u03c9\u03c2, \u03b1\u03c0\u03bf\u03b4\u03b5\u03b9\u03ba\u03bd\u03cd\u03b5\u03c4\u03b1\u03b9 \u03cc\u03c4\u03b9 \u03b7 \u03b5\u03c5\u03b8\u03b5\u03af\u03b1 y = x \u03b5\u03af\u03bd\u03b1\u03b9 \u03b1\u03c3\u03cd\u03bc\u03c0\u03c4\u03c9\u03c4\u03b7 \u03c4\u03b7\u03c2 Cf \u03c3\u03c4\u03bf +\u221e. 153","2.10\u25cf \u039a\u03b1\u03c4\u03b1\u03ba\u03cc\u03c1\u03c5\u03c6\u03b5\u03c2 \u03b1\u03c3\u03cd\u03bc\u03c0\u03c4\u03c9\u03c4\u03b5\u03c2\u0395\u03af\u03bd\u03b1\u03b9 lim f (x) = lim \uf8eb x + 1 \uf8f6 = \u2212\u221e \u03ba\u03b1\u03b9 \uf8ed\uf8ec x \uf8f8\uf8f7 x \u2192 0\u2212 x \u2192 0\u2212 lim f (x) = lim \uf8eb x + 1\uf8f6 = +\u221e. \uf8ec\uf8ed x \uf8f8\uf8f7 x \u2192 0+ x \u2192 0+\u0386\u03c1\u03b1 \u03b7 \u03b5\u03c5\u03b8\u03b5\u03af\u03b1 x = 0 \u03b5\u03af\u03bd\u03b1\u03b9 \u03ba\u03b1\u03c4\u03b1\u03ba\u03cc\u03c1\u03c5\u03c6\u03b7 \u03b1\u03c3\u03cd\u03bc\u03c0\u03c4\u03c9\u03c4\u03b7 \u03c4\u03b7\u03c2 Cf . \u0395\u03c0\u03af\u03c3\u03b7\u03c2 \u03ad\u03c7\u03bf\u03c5\u03bc\u03b5: \uf8eb x 2 + 1 \uf8f6 lim \uf8ec x \uf8f7 lim f (x) = \uf8edx \u2192 \u2212\u221e \uf8f8 = lim x = \u2212\u221e \u03ba\u03b1\u03b9 x \u2192 \u2212\u221e x \u2192 \u2212\u221e lim f (x) = lim x2 +1 = lim x = +\u221e. x \u2192 +\u221e xx \u2192 +\u221e x \u2192 +\u221e\u25cf \u03a3\u03c7\u03b7\u03bc\u03b1\u03c4\u03af\u03b6\u03bf\u03c5\u03bc\u03b5 \u03c4\u03bf\u03bd \u03c0\u03af\u03bd\u03b1\u03ba\u03b1 \u03bc\u03b5\u03c4\u03b1\u03b2\u03bf\u03bb\u03ce\u03bd \u03c4\u03b7\u03c2 f \u03ba\u03b1\u03b9 \u03c7\u03b1\u03c1\u03ac\u03c3\u03c3\u03bf\u03c5\u03bc\u03b5 \u03c4\u03b7 \u03b3\u03c1\u03b1\u03c6\u03b9\u03ba\u03ae\u03c4\u03b7\u03c2 \u03c0\u03b1\u03c1\u03ac\u03c3\u03c4\u03b1\u03c3\u03b7. x \u2212\u221e \u20131 0 1 +\u221e 0 f \u2032 (x) + 0 \u2013 \u2013 + 2 + f \u2033(x) \u2013 \u2013 + T.\u0395. +\u221e +\u221e \u20132 f ( x) T.\u039c. \u2212\u221e \u2212\u221e\u03a3\u03c7\u03cc\u03bb\u03b9\u03bf\u0395\u03c0\u03b5\u03b9\u03b4\u03ae \u03b3\u03b9\u03b1 \u03ba\u03ac\u03b8\u03b5 x \u2208R* \u03b9\u03c3\u03c7\u03cd\u03b5\u03b9 f (\u2212x) = (\u2212x)2 +1 = \u2212 x2 +1 = \u2212 f (x), \u2212x x 154","2.10\u03b7 f \u03b5\u03af\u03bd\u03b1\u03b9 \u03c0\u03b5\u03c1\u03b9\u03c4\u03c4\u03ae, \u03bf\u03c0\u03cc\u03c4\u03b5 \u03b7 \u03b3\u03c1\u03b1\u03c6\u03b9\u03ba\u03ae \u03c4\u03b7\u03c2 \u03c0\u03b1\u03c1\u03ac\u03c3\u03c4\u03b1\u03c3\u03b7 \u03b5\u03af\u03bd\u03b1\u03b9 \u03c3\u03c5\u03bc\u03bc\u03b5\u03c4\u03c1\u03b9\u03ba\u03ae \u03c9\u03c2 \u03c0\u03c1\u03bf\u03c2\u03c4\u03b7\u03bd \u03b1\u03c1\u03c7\u03ae \u039f.ii) \u25cf \u0397 f \u03bf\u03c1\u03af\u03b6\u03b5\u03c4\u03b1\u03b9 \u03c3\u03c4\u03bf \u0391 = (\u2212\u221e,1) \u222a (1, +\u221e) \u25cf \u0397 f \u03b5\u03af\u03bd\u03b1\u03b9 \u03c3\u03c5\u03bd\u03b5\u03c7\u03ae\u03c2 \u03c3\u03c4\u03bf \u0391, \u03c9\u03c2 \u03c1\u03b7\u03c4\u03ae. \u25cf \u0393\u03b9\u03b1 \u03ba\u03ac\u03b8\u03b5 x \u2208 \u0391 \u03b9\u03c3\u03c7\u03cd\u03b5\u03b9 f \u2032( x) = (2x \u22121)(x \u22121) \u2212 (x2 \u2212 x \u2212 2) = x2 \u2212 2x + 3 , (x \u22121)2 ( x \u22121)2\u03bf\u03c0\u03cc\u03c4\u03b5 f \u2032 (x) > 0 \u03b3\u03b9\u03b1 \u03ba\u03ac\u03b8\u03b5 x \u2208 \u0391.\u03a4\u03bf \u03c0\u03c1\u03cc\u03c3\u03b7\u03bc\u03bf \u03c4\u03b7\u03c2 f \u2032 \u03c6\u03b1\u03af\u03bd\u03b5\u03c4\u03b1\u03b9 \u03c3\u03c4\u03bf\u03bd \u03c0\u03b1\u03c1\u03b1\u03ba\u03ac\u03c4\u03c9 \u03c0\u03af\u03bd\u03b1\u03ba\u03b1 \u03b1\u03c0\u03cc \u03c4\u03bf\u03bd \u03bf\u03c0\u03bf\u03af\u03bf \u03c0\u03c1\u03bf\u03c3-\u03b4\u03b9\u03bf\u03c1\u03af\u03b6\u03bf\u03c5\u03bc\u03b5 \u03c4\u03b1 \u03b4\u03b9\u03b1\u03c3\u03c4\u03ae\u03bc\u03b1\u03c4\u03b1 \u03bc\u03bf\u03bd\u03bf\u03c4\u03bf\u03bd\u03af\u03b1\u03c2 \u03ba\u03b1\u03b9 \u03c4\u03b1 \u03b1\u03ba\u03c1\u03cc\u03c4\u03b1\u03c4\u03b1 \u03c4\u03b7\u03c2 f. x \u2212\u221e 1 +\u221e f \u2032 (x) + + f ( x)\u0393\u03b9\u03b1 \u03ba\u03ac\u03b8\u03b5 x \u2208 \u0391 \u03b9\u03c3\u03c7\u03cd\u03b5\u03b9: f \u2032\u2032( x) = (2x \u2212 2)( x \u22121)2 \u2212 2(x \u22121)(x2 \u2212 2x + 3) = \u22124 , (x \u22121)4 (x \u22121)3\u03bf\u03c0\u03cc\u03c4\u03b5 \u03c4\u03bf \u03c0\u03c1\u03cc\u03c3\u03b7\u03bc\u03bf \u03c4\u03b7\u03c2 f \u2033 \u03c6\u03b1\u03af\u03bd\u03b5\u03c4\u03b1\u03b9 \u03c3\u03c4\u03bf\u03bd \u03c0\u03b1\u03c1\u03b1\u03ba\u03ac\u03c4\u03c9 \u03c0\u03af\u03bd\u03b1\u03ba\u03b1, \u03b1\u03c0\u03cc \u03c4\u03bf\u03bd \u03bf\u03c0\u03bf\u03af\u03bf\u03c0\u03c1\u03bf\u03c3\u03b4\u03b9\u03bf\u03c1\u03af\u03b6\u03bf\u03c5\u03bc\u03b5 \u03c4\u03b1 \u03b4\u03b9\u03b1\u03c3\u03c4\u03ae\u03bc\u03b1\u03c4\u03b1 \u03c3\u03c4\u03b1 \u03bf\u03c0\u03bf\u03af\u03b1 \u03b7 Cf \u03c3\u03c4\u03c1\u03ad\u03c6\u03b5\u03b9 \u03c4\u03b1 \u03ba\u03bf\u03af\u03bb\u03b1 \u03c0\u03c1\u03bf\u03c2 \u03c4\u03b1 \u03ac\u03bd\u03c9\u03ae \u03c0\u03c1\u03bf\u03c2 \u03c4\u03b1 \u03ba\u03ac\u03c4\u03c9. x \u2212\u221e 1 +\u221e \u2013 f \u2033 (x) + f ( x)\u25cf \u03a0\u03bb\u03ac\u03b3\u03b9\u03b5\u03c2 - \u03bf\u03c1\u03b9\u03b6\u03cc\u03bd\u03c4\u03b9\u03b5\u03c2 \u03b1\u03c3\u03cd\u03bc\u03c0\u03c4\u03c9\u03c4\u03b5\u03c2\u0397 \u03b1\u03c3\u03cd\u03bc\u03c0\u03c4\u03c9\u03c4\u03b7 \u03c4\u03b7\u03c2 Cf \u03c3\u03c4\u03bf \u2212\u221e \u03b5\u03af\u03bd\u03b1\u03b9 \u03c4\u03b7\u03c2 \u03bc\u03bf\u03c1\u03c6\u03ae\u03c2 y = \u03bbx + \u03b2, \u03cc\u03c0\u03bf\u03c5 \u03bb = lim f (x) = lim x2 \u2212 x \u2212 2 =1 \u03ba\u03b1\u03b9 x x2 \u2212 x x \u2192 \u2212\u221e x \u2192 \u2212\u221e \u03b2 = lim ( f (x) \u2212 \u03bbx) = \uf8eb x2 \u2212x\u22122 \u2212 x \uf8f6 = lim \u22122 = 0. lim \uf8ec x \u22121 \uf8f7 x \u22121 x \u2192 \u2212\u221e \uf8edx \u2192 \u2212\u221e \uf8f8 x \u2192 \u2212\u221e 155","2.10\u0394\u03b7\u03bb\u03b1\u03b4\u03ae, \u03b5\u03af\u03bd\u03b1\u03b9 \u03b7 \u03b5\u03c5\u03b8\u03b5\u03af\u03b1 y = x.\u039f\u03bc\u03bf\u03af\u03c9\u03c2, \u03b7 y = x \u03b5\u03af\u03bd\u03b1\u03b9 \u03b1\u03c3\u03cd\u03bc\u03c0\u03c4\u03c9\u03c4\u03b7 \u03c4\u03b7\u03c2 Cf \u03c3\u03c4\u03bf +\u221e.\u25cf \u039a\u03b1\u03c4\u03b1\u03ba\u03cc\u03c1\u03c5\u03c6\u03b5\u03c2 \u03b1\u03c3\u03cd\u03bc\u03c0\u03c4\u03c9\u03c4\u03b5\u03c2\u0395\u03af\u03bd\u03b1\u03b9 lim f (x) = lim x2 \u2212 x \u2212 2 = +\u221e \u03ba\u03b1\u03b9 x \u21921\u2212 x\u21921\u2212 x \u22121 lim f (x) = lim x2 \u2212 x \u2212 2 = \u2212\u221e. x \u21921+ x\u21921+ x \u22121\u0386\u03c1\u03b1, \u03b7 \u03b5\u03c5\u03b8\u03b5\u03af\u03b1 x = 1 \u03b5\u03af\u03bd\u03b1\u03b9 \u03ba\u03b1\u03c4\u03b1\u03ba\u03cc\u03c1\u03c5\u03c6\u03b7 \u03b1\u03c3\u03cd\u03bc\u03c0\u03c4\u03c9\u03c4\u03b7 \u03c4\u03b7\u03c2 Cf.\u0395\u03c0\u03af\u03c3\u03b7\u03c2 \u03ad\u03c7\u03bf\u03c5\u03bc\u03b5: lim f (x) = lim x2 \u2212 x \u2212 2 = \u2212\u221e \u03ba\u03b1\u03b9 lim f (x) = +\u221e. x \u2192 \u2212\u221e x\u2192\u2212\u221e x \u22121 x \u2192 +\u221e\u25cf \u03a3\u03c7\u03b7\u03bc\u03b1\u03c4\u03af\u03b6\u03bf\u03c5\u03bc\u03b5 \u03c4\u03bf\u03bd \u03c0\u03af\u03bd\u03b1\u03ba\u03b1 \u03bc\u03b5\u03c4\u03b1\u03b2\u03bf\u03bb\u03ce\u03bd \u03c4\u03b7\u03c2 f \u03ba\u03b1\u03b9 \u03c7\u03b1\u03c1\u03ac\u03c3\u03c3\u03bf\u03c5\u03bc\u03b5 \u03c4\u03b7 \u03b3\u03c1\u03b1\u03c6\u03b9\u03ba\u03ae\u03c4\u03b7\u03c2 \u03c0\u03b1\u03c1\u03ac\u03c3\u03c4\u03b1\u03c3\u03b7. x \u2212\u221e 1 +\u221e f \u2032 (x) + + f \u2033(x) + \u2013 f ( x) +\u221e +\u221e \u2212\u221e \u2212\u221e3. \u25cf \u0395\u03af\u03bd\u03b1\u03b9 \u0391 = [ \u2013 \u03c0,\u03c0] \u25cf \u0397 f \u03b5\u03af\u03bd\u03b1\u03b9 \u03c3\u03c5\u03bd\u03b5\u03c7\u03ae\u03c2 \u03c3\u03c4\u03bf \u0391 \u03c9\u03c2 \u03ac\u03b8\u03c1\u03bf\u03b9\u03c3\u03bc\u03b1 \u03c3\u03c5\u03bd\u03b5\u03c7\u03ce\u03bd \u03c3\u03c5\u03bd\u03b1\u03c1\u03c4\u03ae\u03c3\u03b5\u03c9\u03bd. \u25cf \u0393\u03b9\u03b1 \u03ba\u03ac\u03b8\u03b5 x \u2208 \u0391 \u03b9\u03c3\u03c7\u03cd\u03b5\u03b9 f \u2032 (x) = 1+\u03c3\u03c5\u03bdx, \u03bf\u03c0\u03cc\u03c4\u03b5 156","\u0393\u0395\u039d\u0399\u039a\u0395\u03a3 \u0391\u03a3\u039a\u0397\u03a3\u0395\u0399\u03a3 2\u03bf\u03c5 \u039a\u0395\u03a6\u0391\u039b\u0391\u0399\u039f\u03a5 \u0393\u2019 \u039f\u039c\u0391\u0394\u0391\u03a3 f \u2032(x) = 0 \u21d4 x = \u2212\u03c0 \u03ae x = \u03c0.\u03a4\u03bf \u03c0\u03c1\u03cc\u03c3\u03b7\u03bc\u03bf \u03c4\u03b7\u03c2 f \u2032 \u03c6\u03b1\u03af\u03bd\u03b5\u03c4\u03b1\u03b9 \u03c3\u03c4\u03bf\u03bd \u03c0\u03b1\u03c1\u03b1\u03ba\u03ac\u03c4\u03c9 \u03c0\u03af\u03bd\u03b1\u03ba\u03b1, \u03b1\u03c0\u03cc \u03c4\u03bf\u03bd \u03bf\u03c0\u03bf\u03af\u03bf \u03c0\u03c1\u03bf\u03c3\u03b4\u03b9\u03bf\u03c1\u03af\u03b6\u03bf\u03c5\u03bc\u03b5\u03c4\u03b1 \u03b4\u03b9\u03b1\u03c3\u03c4\u03ae\u03bc\u03b1\u03c4\u03b1 \u03bc\u03bf\u03bd\u03bf\u03c4\u03bf\u03bd\u03af\u03b1\u03c2 \u03c4\u03b7\u03c2 f \u03ba\u03b1\u03b9 \u03c4\u03b1 \u03b1\u03ba\u03c1\u03cc\u03c4\u03b1\u03c4\u03b1 \u03b1\u03c5\u03c4\u03ae\u03c2. x \u2013\u03c0 +\u03c0 f \u2032 (x) 0 +0 f ( x)\u0393\u03b9\u03b1 \u03ba\u03ac\u03b8\u03b5 x \u2208 \u0391 \u03b9\u03c3\u03c7\u03cd\u03b5\u03b9 f \u2033(x) = \u2013 \u03b7\u03bcx, \u03bf\u03c0\u03cc\u03c4\u03b5 f \u2032\u2032(x) = 0 \u21d4 x = \u2212\u03c0 \u03ae x = 0 \u03ae x = \u03c0.\u03a4\u03bf \u03c0\u03c1\u03cc\u03c3\u03b7\u03bc\u03bf \u03c4\u03b7\u03c2 f \u2033 \u03c6\u03b1\u03af\u03bd\u03b5\u03c4\u03b1\u03b9 \u03c3\u03c4\u03bf\u03bd \u03c0\u03b1\u03c1\u03b1\u03ba\u03ac\u03c4\u03c9 \u03c0\u03af\u03bd\u03b1\u03ba\u03b1, \u03b1\u03c0\u03cc \u03c4\u03bf\u03bd \u03bf\u03c0\u03bf\u03af\u03bf\u03c0\u03c1\u03bf\u03c3\u03b4\u03b9\u03bf\u03c1\u03af\u03b6\u03bf\u03c5\u03bc\u03b5 \u03c4\u03b1 \u03b4\u03b9\u03b1\u03c3\u03c4\u03ae\u03bc\u03b1\u03c4\u03b1 \u03c3\u03c4\u03b1 \u03bf\u03c0\u03bf\u03af\u03b1 \u03b7 f \u03c3\u03c4\u03c1\u03ad\u03c6\u03b5\u03b9 \u03c4\u03b1 \u03ba\u03bf\u03af\u03bb\u03b1 \u03c0\u03c1\u03bf\u03c2 \u03c4\u03b1 \u03ac\u03bd\u03c9\u03ae \u03c0\u03c1\u03bf\u03c2 \u03c4\u03b1 \u03ba\u03ac\u03c4\u03c9 \u03ba\u03b1\u03b9 \u03c4\u03b1 \u03c3\u03b7\u03bc\u03b5\u03af\u03b1 \u03ba\u03b1\u03bc\u03c0\u03ae\u03c2. x \u2013\u03c0 0 \u03c0 f\u2033(x) 0 + 0 \u2013 0 f ( x) 0 \u03a3.\u039a.\u25cf \u0395\u03af\u03bd\u03b1\u03b9 f( \u2013 \u03c0) = \u2013 \u03c0 \u03ba\u03b1\u03b9 f(\u03c0) = \u03c0\u25cf \u03a3\u03c7\u03b7\u03bc\u03b1\u03c4\u03af\u03b6\u03bf\u03c5\u03bc\u03b5 \u03c4\u03bf\u03bd \u03c0\u03af\u03bd\u03b1\u03ba\u03b1 \u03bc\u03b5\u03c4\u03b1\u03b2\u03bf\u03bb\u03ce\u03bd \u03c4\u03b7\u03c2 f \u03ba\u03b1\u03b9 \u03c7\u03b1\u03c1\u03ac\u03c3\u03c3\u03bf\u03c5\u03bc\u03b5 \u03c4\u03b7 \u03b3\u03c1\u03b1\u03c6\u03b9\u03ba\u03ae\u03c4\u03b7\u03c2 \u03c0\u03b1\u03c1\u03ac\u03c3\u03c4\u03b1\u03c3\u03b7.x \u2013\u03c0 0 \u03c0f \u2032 (x) 0 + +0f \u2033(x) 0 + 0 \u2013 0f ( x) \u2013\u03c0 0 \u03c0 min \u03a3.\u039a. max\u0393\u0395\u039d\u0399\u039a\u0395\u03a3 \u0391\u03a3\u039a\u0397\u03a3\u0395\u0399\u03a3 2\u03bf\u03c5 \u039a\u0395\u03a6\u0391\u039b\u0391\u0399\u039f\u03a5 \u0393\u0384 \u039f\u039c\u0391\u0394\u0391\u03a31. i) \u0393\u03b9\u03b1 \u03ba\u03ac\u03b8\u03b5 x \u2208 (0, +\u221e) \u03b9\u03c3\u03c7\u03cd\u03b5\u03b9: f \u2032( x) = \u2212 1 \u03ba\u03b1\u03b9 g\u2032(x) = 2x \u2212 3, x2 157","\u0393\u0395\u039d\u0399\u039a\u0395\u03a3 \u0391\u03a3\u039a\u0397\u03a3\u0395\u0399\u03a3 2\u03bf\u03c5 \u039a\u0395\u03a6\u0391\u039b\u0391\u0399\u039f\u03a5 \u0393\u2019 \u039f\u039c\u0391\u0394\u0391\u03a3 \u03bf\u03c0\u03cc\u03c4\u03b5 f \u2032 (1) = \u20131 \u03ba\u03b1\u03b9 g\u2032(l) = \u20131. \u03a4\u03bf \u03c3\u03b7\u03bc\u03b5\u03af\u03bf \u0391(1,1) \u03b5\u03af\u03bd\u03b1\u03b9 \u03ba\u03bf\u03b9\u03bd\u03cc \u03c3\u03b7\u03bc\u03b5\u03af\u03bf \u03c4\u03c9\u03bd Cf \u03ba\u03b1\u03b9 Cg, \u03b1\u03c6\u03bf\u03cd f(1) = 1 \u03ba\u03b1\u03b9 g(l) = l. \u0395\u03c0\u03b5\u03b9\u03b4\u03ae \u03b9\u03c3\u03c7\u03cd\u03b5\u03b9 f \u2032 (1) = g\u2032(l), \u03bf\u03b9 \u03b5\u03c6\u03b1\u03c0\u03c4\u03cc\u03bc\u03b5\u03bd\u03b5\u03c2 \u03c4\u03c9\u03bd Cf , Cg \u03c3\u03c4\u03bf (1,1) \u03c4\u03b1\u03c5\u03c4\u03af\u03b6\u03bf\u03bd\u03c4\u03b1\u03b9. ii) \u0393\u0007 \u03b9\u03b1 \u03bd\u03b1 \u03b2\u03c1\u03bf\u03cd\u03bc\u03b5 \u03c4\u03b7 \u03c3\u03c7\u03b5\u03c4\u03b9\u03ba\u03ae \u03b8\u03ad\u03c3\u03b7 \u03c4\u03c9\u03bd Cf \u03ba\u03b1\u03b9 Cg \u03b2\u03c1\u03af\u03c3\u03ba\u03bf\u03c5\u03bc\u03b5 \u03c4\u03bf \u03c0\u03c1\u03cc\u03c3\u03b7\u03bc\u03bf \u03c4\u03b7\u03c2 \u03b4\u03b9\u03b1\u03c6\u03bf\u03c1\u03ac\u03c2 \u03d5(x) = g(x) \u2212 f (x) = x2 \u2212 3x + 3 \u2212 1 = (x \u22121)3 . xx \u0388\u03c7\u03bf\u03c5\u03bc\u03b5: \u03c6(x) < 0, \u03b3\u03b9\u03b1 \u03ba\u03ac\u03b8\u03b5 x \u2208 (0,1) \u03ba\u03b1\u03b9 \u03c6(x) > 0 \u03b3\u03b9\u03b1 \u03ba\u03ac\u03b8\u03b5 x \u2208 (1, +\u221e). \u0395\u03c0\u03bf\u03bc\u03ad\u03bd\u03c9\u03c2: \u2014 \u03b7 Cf \u03b5\u03af\u03bd\u03b1\u03b9 \u03c0\u03ac\u03bd\u03c9 \u03b1\u03c0\u03cc \u03c4\u03b7\u03bd Cg, \u03cc\u03c4\u03b1\u03bd x \u2208 (0,1) \u03ba\u03b1\u03b9 \u2014 \u03b7 Cg \u03b5\u03af\u03bd\u03b1\u03b9 \u03c0\u03ac\u03bd\u03c9 \u03b1\u03c0\u03cc \u03c4\u03b7\u03bd Cf , \u03cc\u03c4\u03b1\u03bd x \u2208 (1, +\u221e) (\u03c3\u03c7\u03ae\u03bc\u03b1).2. \u0007\u0398\u03b5\u03c9\u03c1\u03bf\u03cd\u03bc\u03b5 \u03c4\u03b7 \u03c3\u03c5\u03bd\u03ac\u03c1\u03c4\u03b7\u03c3\u03b7 \u03d5(x) = f (x) \u2212 g(x). \u0393\u03b9\u03b1 \u03ba\u03ac\u03b8\u03b5 x \u2208 R \u03b9\u03c3\u03c7\u03cd\u03b5\u03b9 \u03d5\u2032(x) = f \u2032(x) \u2212 g\u2032(x) > 0, \u03bf\u03c0\u03cc\u03c4\u03b5 \u03b7 \u03c6 \u03b5\u03af\u03bd\u03b1\u03b9 \u03b3\u03bd\u03b7\u03c3\u03af\u03c9\u03c2 \u03b1\u03cd\u03be\u03bf\u03c5\u03c3\u03b1 \u03c3\u03c4\u03bf R. \u0395\u03c0\u03bf\u03bc\u03ad\u03bd\u03c9\u03c2: \u2014 \u0393\u03b9\u03b1 x > 0 \u03b8\u03b1 \u03b5\u03af\u03bd\u03b1\u03b9: \u03d5(x) > \u03d5(0) \u21d4 f (x) \u2212 g(x) > f (0) \u2212 g(0) \u21d4 f (x) > g(x), \u03b1\u03c6\u03bf\u03cd f(0) = g(0) \u03ba\u03b1\u03b9 \u2014 \u0393\u03b9\u03b1 x < 0 \u03b8\u03b1 \u03b5\u03af\u03bd\u03b1\u03b9 \u03d5(x) < \u03d5(0) \u21d4 f (x) \u2212 g(x) < f (0) \u2212 g(0) \u21d4 f (x) < g(x), \u03b1\u03c6\u03bf\u03cd f(0) = g(0). 158","\u0393\u0395\u039d\u0399\u039a\u0395\u03a3 \u0391\u03a3\u039a\u0397\u03a3\u0395\u0399\u03a3 2\u03bf\u03c5 \u039a\u0395\u03a6\u0391\u039b\u0391\u0399\u039f\u03a5 \u0393\u2019 \u039f\u039c\u0391\u0394\u0391\u03a33. \u0391\u03c0\u03cc \u03c4\u03bf \u03bf\u03c1\u03b8\u03bf\u03b3\u03ce\u03bd\u03b9\u03bf \u03c4\u03c1\u03af\u03b3\u03c9\u03bd\u03bf \u039f\u0392\u039c \u03ad\u03c7\u03bf\u03c5\u03bc\u03b5: (\u0392\u039c) = 1\u2219 \u03b7\u03bc\u03b8 \u03ba\u03b1\u03b9 (\u039f\u039c) = 1\u2219 \u03c3\u03c5\u03bd\u03b8.\u0395\u03af\u03bd\u03b1\u03b9 \u03cc\u03bc\u03c9\u03c2 (\u0392\u0393) = 2(\u0392\u039c) = 2\u03b7\u03bc\u03b8 \u03ba\u03b1\u03b9 (AM) = (OA) + (\u039f\u039c) = 1+\u03c3\u03c5\u03bd\u03b8\u03bf\u03c0\u03cc\u03c4\u03b5 \u0395 = \u0395 (\u03b8 ) = 1 2\u03b7\u00b5\u03b8 (1+ \u03c3\u03c5\u03bd\u03b8 ) = \u03b7\u00b5\u03b8 (1+ \u03c3\u03c5\u03bd\u03b8 ). 2\u0393\u03b9\u03b1 \u03ba\u03ac\u03b8\u03b5 \u03b8 \u2208 (0,\u03c0 ) \u03b9\u03c3\u03c7\u03cd\u03b5\u03b9: \u0395\u2032(\u03b8) = \u03c3\u03c5\u03bd\u03b8(1 + \u03c3\u03c5\u03bd\u03b8) \u2013 \u03b7\u03bc2\u03b8 = \u03c3\u03c5\u03bd2\u03b8 \u2013 \u03b7\u03bc2\u03b8 + \u03c3\u03c5\u03bd\u03b8 = \u03c3\u03c5\u03bd2\u03b8 + \u03c3\u03c5\u03bd\u03b8,\u03bf\u03c0\u03cc\u03c4\u03b5 \u0395 \u2032(\u03b8 ) = 0 \u21d4 \u03c3\u03c5\u03bd2\u03b8 = \u2212\u03c3\u03c5\u03bd\u03b8 \u21d4 \u03c3\u03c5\u03bd2\u03b8 = \u03c3\u03c5\u03bd(\u03c0 \u2212\u03b8 ) \u21d4 2\u03b8 = \u03c0 \u2212\u03b8 , \u03b1\u03c6\u03bf\u03cd \u03b8 \u2208 (0,\u03c0 ) \u21d4\u03b8 =\u03c0 . 3\u03a4\u03bf \u03c0\u03c1\u03cc\u03c3\u03b7\u03bc\u03bf \u03c4\u03b7\u03c2 \u0395\u2032 \u03c6\u03b1\u03af\u03bd\u03b5\u03c4\u03b1\u03b9 \u03c3\u03c4\u03bf\u03bd \u03c0\u03b1\u03c1\u03b1\u03ba\u03ac\u03c4\u03c9 \u03c0\u03af\u03bd\u03b1\u03ba\u03b1 ( \u0395 \u2032 \uf8eb \u03c0 \uf8f6 > 0 \u03ba\u03b1\u03b9 \u0395 \u2032 \uf8eb \u03c0 \uf8f6 < 0), \uf8ed\uf8ec 6 \uf8f8\uf8f7 \uf8ec\uf8ed 2 \uf8f7\uf8f8\u03b1\u03c0\u03cc \u03c4\u03bf\u03bd \u03bf\u03c0\u03bf\u03af\u03bf \u03c0\u03c1\u03bf\u03c3\u03b4\u03b9\u03bf\u03c1\u03af\u03b6\u03bf\u03c5\u03bc\u03b5 \u03c4\u03b1 \u03b4\u03b9\u03b1\u03c3\u03c4\u03ae\u03bc\u03b1\u03c4\u03b1 \u03bc\u03bf\u03bd\u03bf\u03c4\u03bf\u03bd\u03af\u03b1\u03c2 \u03c4\u03b7\u03c2 \u0395 \u03ba\u03b1\u03b9 \u03c4\u03b1 \u03b1\u03ba\u03c1\u03cc\u03c4\u03b1\u03c4\u03b1\u03b1\u03c5\u03c4\u03ae\u03c2. \u03b80 \u03c0\/3 \u03c0 \u0395\u2032(\u03b8) + 0\u2013 33 \u0395(\u03b8) 4 max\u0386\u03c1\u03b1, \u03b7 \u03bc\u03ad\u03b3\u03b9\u03c3\u03c4\u03b7 \u03c4\u03b9\u03bc\u03ae \u03c4\u03bf\u03c5 \u03b5\u03bc\u03b2\u03b1\u03b4\u03bf\u03cd \u03b5\u03af\u03bd\u03b1\u03b9 3 3 \u03ba\u03b1\u03b9 \u03c0\u03b1\u03c1\u03bf\u03c5\u03c3\u03b9\u03ac\u03b6\u03b5\u03c4\u03b1\u03b9 \u03cc\u03c4\u03b1\u03bd \u03b8 = \u03c0 . 43 159","\u0393\u0395\u039d\u0399\u039a\u0395\u03a3 \u0391\u03a3\u039a\u0397\u03a3\u0395\u0399\u03a3 2\u03bf\u03c5 \u039a\u0395\u03a6\u0391\u039b\u0391\u0399\u039f\u03a5 \u0393\u2019 \u039f\u039c\u0391\u0394\u0391\u03a34. \u0393\u0007 \u03bd\u03c9\u03c1\u03af\u03b6\u03bf\u03c5\u03bc\u03b5 \u03cc\u03c4\u03b9 \u03c4\u03bf \u03bc\u03ae\u03ba\u03bf\u03c2 \u03c4\u03cc\u03be\u03bf\u03c5 \u03b8 rad \u03b5\u03af\u03bd\u03b1\u03b9 L = r \u22c5\u03b8 \u03b5\u03bd\u03ce \u03c4\u03bf \u03b5\u03bc\u03b2\u03b1\u03b4\u03cc\u03bd \u03ba\u03c5\u03ba\u03bb\u03b9\u03ba\u03bf\u03cd \u03c4\u03bf\u03bc\u03ad\u03b1 \u03b8 rad \u03b5\u03af\u03bd\u03b1\u03b9 \u0395 = 1 r2\u03b8. 2 \u0395\u03c0\u03bf\u03bc\u03ad\u03bd\u03c9\u03c2, \u03b7 \u03c0\u03b5\u03c1\u03af\u03bc\u03b5\u03c4\u03c1\u03bf\u03c2 \u03c4\u03bf\u03c5 \u03ba\u03c5\u03ba\u03bb\u03b9\u03ba\u03bf\u03cd \u03c4\u03bf\u03bc\u03ad\u03b1 \u03b5\u03af\u03bd\u03b1\u03b9: 2r + r\u03b8 = 20 \u21d4 \u03b8 = 20 \u2212 2r , 0 < r < 10 r \u03ba\u03b1\u03b9 \u03c4\u03bf \u03b5\u03bc\u03b2\u03b1\u03b4\u03cc\u03bd \u03c4\u03bf\u03c5 \u03b5\u03af\u03bd\u03b1\u03b9: \u0395 (r) = 1 r2 20 \u2212 2r = 10r \u2212 r2 , r \u2208 (0,10). 2r \u0393\u03b9\u03b1 \u03ba\u03ac\u03b8\u03b5 r \u2208 (0,10) \u03b9\u03c3\u03c7\u03cd\u03b5\u03b9 E\u2032(r) = 10 \u2013 2r, \u03bf\u03c0\u03cc\u03c4\u03b5 \u0395 \u2032(r) = 0 \u21d4 r = 5. To \u03c0\u03c1\u03cc\u03c3\u03b7\u03bc\u03bf \u03c4\u03b7\u03c2 E \u2032 (r), \u03b7 \u03bc\u03bf\u03bd\u03bf\u03c4\u03bf\u03bd\u03af\u03b1 \u03ba\u03b1\u03b9 \u03c4\u03b1 \u03b1\u03ba\u03c1\u03cc\u03c4\u03b1\u03c4\u03b1 \u03c4\u03b7\u03c2 \u0395 \u03c6\u03b1\u03af\u03bd\u03bf\u03bd\u03c4\u03b1\u03b9 \u03c3\u03c4\u03bf\u03bd \u03c0\u03b1\u03c1\u03b1\u03ba\u03ac\u03c4\u03c9 \u03c0\u03af\u03bd\u03b1\u03ba\u03b1. r0 5 10 \u0395\u2032(r) + 0\u2013 \u0395(r) 25 max\u0394\u03b7\u03bb\u03b1\u03b4\u03ae \u03b7 \u0395 \u03c0\u03b1\u03c1\u03bf\u03c5\u03c3\u03b9\u03ac\u03b6\u03b5\u03b9 \u03c3\u03c4\u03bf r = 5 \u03bc\u03ad\u03b3\u03b9\u03c3\u03c4\u03bf \u03c4\u03bf E(r) = 25. \u0395\u03c0\u03bf\u03bc\u03ad\u03bd\u03c9\u03c2 \u03bf \u03b1\u03bd\u03b8\u03cc\u03ba\u03b7\u03c0\u03bf\u03c2\u03ad\u03c7\u03b5\u03b9 \u03c4\u03b7 \u03bc\u03b5\u03b3\u03b1\u03bb\u03cd\u03c4\u03b5\u03c1\u03b7 \u03b4\u03c5\u03bd\u03b1\u03c4\u03ae \u03b5\u03c0\u03b9\u03c6\u03ac\u03bd\u03b5\u03b9\u03b1, \u03cc\u03c4\u03b1\u03bd \u03b7 \u03b1\u03ba\u03c4\u03af\u03bd\u03b1 \u03c4\u03bf\u03c5 \u03ba\u03cd\u03ba\u03bb\u03bf\u03c5 \u03b5\u03af\u03bd\u03b1\u03b9 r = 5 m.5. i) \u0391\u03c0\u03cc \u03c4\u03b1 \u03bf\u03c1\u03b8\u03bf\u03b3\u03ce\u03bd\u03b9\u03b1 \u03c4\u03c1\u03af\u03b3\u03c9\u03bd\u03b1 \u039f\u0391\u0393 \u03ba\u03b1\u03b9 \u039f\u0394\u0392 \u03ad\u03c7\u03bf\u03c5\u03bc\u03b5: \u03c3\u03c5\u03bd\u03b8 = (\u039f\u0393 ) = 1 \u03ba\u03b1\u03b9 \u03b7\u00b5\u03b8 = (\u039f\u2206) = 1 (\u039f\u0391) (\u039f\u0391) (\u039f\u0392 ) (\u039f\u0392 )\u03bf\u03c0\u03cc\u03c4\u03b5 (\u039f\u0391) = 1 \u03ba\u03b1\u03b9 (\u039f\u0392 ) = 1 , 0 < \u03b8 < \u03c0 \u03c3\u03c5\u03bd\u03b8 \u03b7\u00b5\u03b8 2ii) (\u0391\u0392 ) = (\u039f\u0391) + (\u039f\u0392 ) = 1 + 1 \u03b7\u00b5\u03b8 \u03c3\u03c5\u03bd\u03b8iii) \u0398\u0007 \u03b5\u03c9\u03c1\u03bf\u03cd\u03bc\u03b5 \u03c4\u03b7 \u03c3\u03c5\u03bd\u03ac\u03c1\u03c4\u03b7\u03c3\u03b7 f (\u03b8 ) = 1 + 1 \u03b7 \u03bf\u03c0\u03bf\u03af\u03b1 \u03b5\u03af\u03bd\u03b1\u03b9 \u03bf\u03c1\u03b9\u03c3\u03bc\u03ad\u03bd\u03b7 \u03c3\u03c4\u03bf \u03b7\u00b5\u03b8 \u03c3\u03c5\u03bd\u03b8 \u03c0 \u03c0\uf8eb 0, 2 \uf8f6 \u03ba\u03b1\u03b9 \u03c3\u03c5\u03bd\u03b5\u03c7\u03ae\u03c2 \u03c3\u03c4\u03bf \u03b4\u03b9\u03ac\u03c3\u03c4\u03b7\u03bc\u03b1. \u0395\u03c0\u03b9\u03c0\u03bb\u03ad\u03bf\u03bd, \u03b3\u03b9\u03b1 \u03ba\u03ac\u03b8\u03b5 x \u2208 \uf8eb 0, 2 \uf8f6 \u03b9\u03c3\u03c7\u03cd\u03b5\u03b9:\uf8ed\uf8ec \uf8f8\uf8f7 \uf8ec\uf8ed \uf8f8\uf8f7 160","\u0393\u0395\u039d\u0399\u039a\u0395\u03a3 \u0391\u03a3\u039a\u0397\u03a3\u0395\u0399\u03a3 2\u03bf\u03c5 \u039a\u0395\u03a6\u0391\u039b\u0391\u0399\u039f\u03a5 \u0393\u2019 \u039f\u039c\u0391\u0394\u0391\u03a3 f \u2032(\u03b8 ) = \u2212\u03c3\u03c5\u03bd\u03b8 + \u03b7\u00b5\u03b8 = \u03b7\u00b53\u03b8 \u2212 \u03c3\u03c5\u03bd3\u03b8 , \u03b7\u00b52\u03b8 \u03c3\u03c5\u03bd2\u03b8 \u03b7\u00b52\u03b8 \u22c5 \u03c3\u03c5\u03bd2\u03b8\u03bf\u03c0\u03cc\u03c4\u03b5 f \u2032(\u03b8 ) = 0 \u21d4 \u03b7\u00b53\u03b8 \u2212 \u03c3\u03c5\u03bd3\u03b8 = 0 \u21d4 \u03b7\u00b5\u03b8 = \u03c3\u03c5\u03bd\u03b8 \u21d4 \u03b8 = \u03c0 , 4\u03b1\u03c6\u03bf\u03cd \u03b8 \u2208 \uf8eb 0, \u03c0 \uf8f6 . \uf8ed\uf8ec 2 \uf8f8\uf8f7\u03a4\u03bf \u03c0\u03c1\u03cc\u03c3\u03b7\u03bc\u03bf \u03c4\u03b7\u03c2 f \u2032 \u03c6\u03b1\u03af\u03bd\u03b5\u03c4\u03b1\u03b9 \u03c3\u03c4\u03bf\u03bd \u03c0\u03b1\u03c1\u03b1\u03ba\u03ac\u03c4\u03c9 \u03c0\u03af\u03bd\u03b1\u03ba\u03b1, \u03b1\u03c0\u03cc \u03c4\u03bf\u03bd \u03bf\u03c0\u03bf\u03af\u03bf\u03c0\u03c1\u03bf\u03c3\u03b4\u03b9\u03bf\u03c1\u03af\u03b6\u03bf\u03c5\u03bc\u03b5 \u03c4\u03b1 \u03b4\u03b9\u03b1\u03c3\u03c4\u03ae\u03bc\u03b1\u03c4\u03b1 \u03bc\u03bf\u03bd\u03bf\u03c4\u03bf\u03bd\u03af\u03b1\u03c2 \u03c4\u03b7\u03c2 f \u03ba\u03b1\u03b9 \u03c4\u03b1 \u03b1\u03ba\u03c1\u03cc\u03c4\u03b1\u03c4\u03b1 \u03b1\u03c5\u03c4\u03ae\u03c2. \u03b80 \u03c0 \u03c0 4 2 f \u2032 (\u03b8) \u2013 0 + f(\u03b8) 2 2 min\u0394\u03b7\u03bb\u03b1\u03b4\u03ae, \u03b7 f \u03c3\u03c4\u03bf \u03b8 = \u03c0 \u03c0\u03b1\u03c1\u03bf\u03c5\u03c3\u03b9\u03ac\u03b6\u03b5\u03b9 \u03b5\u03bb\u03ac\u03c7\u03b9\u03c3\u03c4\u03bf f \uf8eb\u03c0 \uf8f6 = 2 2. 4 \uf8ec\uf8ed 4 \uf8f7\uf8f8\u0395\u03c0\u03bf\u03bc\u03ad\u03bd\u03c9\u03c2, \u03c4\u03bf \u03bc\u03b5\u03b3\u03b1\u03bb\u03cd\u03c4\u03b5\u03c1\u03bf \u03b4\u03c5\u03bd\u03b1\u03c4\u03cc \u03bc\u03ae\u03ba\u03bf\u03c2 \u03c4\u03b7\u03c2 \u03c3\u03ba\u03ac\u03bb\u03b1\u03c2, \u03c0\u03bf\u03c5 \u03bc\u03c0\u03bf\u03c1\u03b5\u03af, \u03b1\u03bd \u03bc\u03b5\u03c4\u03b1\u03c6\u03b5\u03c1\u03b8\u03b5\u03af\u03bf\u03c1\u03b9\u03b6\u03cc\u03bd\u03c4\u03b9\u03b1 \u03bd\u03b1 \u03c3\u03c4\u03c1\u03af\u03c8\u03b5\u03b9 \u03c3\u03c4\u03b7 \u03b3\u03c9\u03bd\u03af\u03b1, \u03b5\u03af\u03bd\u03b1\u03b9 2 2m \u2245 2,8m.6. i) \u25cf \u0397 \u03c3\u03c5\u03bd\u03ac\u03c1\u03c4\u03b7\u03c3\u03b7 f \u03ad\u03c7\u03b5\u03b9 \u03c0\u03b5\u03b4\u03af\u03bf \u03bf\u03c1\u03b9\u03c3\u03bc\u03bf\u03cd \u03c4\u03bf \u0391 = (0, +\u221e) \u25cf \u0397 f \u03b5\u03af\u03bd\u03b1\u03b9 \u03c3\u03c5\u03bd\u03b5\u03c7\u03ae\u03c2 \u03c3\u03c4\u03bf \u0391.\u25cf \u0393\u03b9\u03b1 \u03ba\u03ac\u03b8\u03b5 x > 0 \u03b9\u03c3\u03c7\u03cd\u03b5\u03b9 f \u2032( x) = 1 \u2212 ln x , \u03bf\u03c0\u03cc\u03c4\u03b5 x2 f \u2032(x) = 0 \u21d4 ln x = 1 \u21d4 x = e.\u03a4\u03bf \u03c0\u03c1\u03cc\u03c3\u03b7\u03bc\u03bf \u03c4\u03b7\u03c2 f \u2032 \u03c6\u03b1\u03af\u03bd\u03b5\u03c4\u03b1\u03b9 \u03c3\u03c4\u03bf\u03bd \u03c0\u03b1\u03c1\u03b1\u03ba\u03ac\u03c4\u03c9 \u03c0\u03af\u03bd\u03b1\u03ba\u03b1, \u03b1\u03c0\u03cc \u03c4\u03bf\u03bd \u03bf\u03c0\u03bf\u03af\u03bf\u03c0\u03c1\u03bf\u03c3\u03b4\u03b9\u03bf\u03c1\u03af\u03b6\u03bf\u03c5\u03bc\u03b5 \u03c4\u03b1 \u03b4\u03b9\u03b1\u03c3\u03c4\u03ae\u03bc\u03b1\u03c4\u03b1 \u03bc\u03bf\u03bd\u03bf\u03c4\u03bf\u03bd\u03af\u03b1\u03c2 \u03ba\u03b1\u03b9 \u03c4\u03b1 \u03c4\u03bf\u03c0\u03b9\u03ba\u03ac \u03b1\u03ba\u03c1\u03cc\u03c4\u03b1\u03c4\u03b1 \u03c4\u03b7\u03c2 f. x 0 e +\u221e f \u2032 (x) + 0 \u2013 1 f(x) e max 161","\u0393\u0395\u039d\u0399\u039a\u0395\u03a3 \u0391\u03a3\u039a\u0397\u03a3\u0395\u0399\u03a3 2\u03bf\u03c5 \u039a\u0395\u03a6\u0391\u039b\u0391\u0399\u039f\u03a5 \u0393\u2019 \u039f\u039c\u0391\u0394\u0391\u03a3\u25cf \u0393\u03b9\u03b1 \u03ba\u03ac\u03b8\u03b5 x > 0 \u03b9\u03c3\u03c7\u03cd\u03b5\u03b9 f \u2032\u2032( x) = \u2212x \u2212 2x(1\u2212 ln x) = 2 ln x \u2212 3 , \u03bf\u03c0\u03cc\u03c4\u03b5 x4 x3 f \u2032\u2032(x) = 0 \u21d4 ln x = 3 \u21d4 x = e3\/2. 2\u03a4\u03bf \u03c0\u03c1\u03cc\u03c3\u03b7\u03bc\u03bf \u03c4\u03b7\u03c2 f \u2033 \u03c6\u03b1\u03af\u03bd\u03b5\u03c4\u03b1\u03b9 \u03c3\u03c4\u03bf\u03bd \u03c0\u03b1\u03c1\u03b1\u03ba\u03ac\u03c4\u03c9 \u03c0\u03af\u03bd\u03b1\u03ba\u03b1, \u03b1\u03c0\u03cc \u03c4\u03bf\u03bd \u03bf\u03c0\u03bf\u03af\u03bf \u03c0\u03c1\u03bf\u03c3\u03b4\u03b9\u03bf\u03c1\u03af-\u03b6\u03bf\u03c5\u03bc\u03b5 \u03c4\u03b1 \u03b4\u03b9\u03b1\u03c3\u03c4\u03ae\u03bc\u03b1\u03c4\u03b1 \u03c3\u03c4\u03b1 \u03bf\u03c0\u03bf\u03af\u03b1 \u03b7 Cf \u03b5\u03af\u03bd\u03b1\u03b9 \u03ba\u03c5\u03c1\u03c4\u03ae \u03ae \u03ba\u03bf\u03af\u03bb\u03b7 \u03ba\u03b1\u03b9 \u03c4\u03b1 \u03c3\u03b7\u03bc\u03b5\u03af\u03b1 \u03ba\u03b1\u03bc\u03c0\u03ae\u03c2\u03c4\u03b7\u03c2. x0 e3\/2 +\u221e f \u2033(x) \u20130+ f ( x) 3 2e3\/ 2 \u03a3.\u039a.\u25cf \u03a0\u03bb\u03ac\u03b3\u03b9\u03b5\u03c2 - \u03bf\u03c1\u03b9\u03b6\u03cc\u03bd\u03c4\u03b9\u03b5\u03c2 \u03b1\u03c3\u03cd\u03bc\u03c0\u03c4\u03c9\u03c4\u03b5\u03c2\u0397 \u03b1\u03c3\u03cd\u03bc\u03c0\u03c4\u03c9\u03c4\u03b7 \u03c4\u03b7\u03c2 Cf \u03c3\u03c4\u03bf +\u221e \u03b5\u03af\u03bd\u03b1\u03b9 \u03c4\u03b7\u03c2 \u03bc\u03bf\u03c1\u03c6\u03ae\u03c2 y = \u03bbx + \u03b2, \u03cc\u03c0\u03bf\u03c5 \uf8eb0\uf8f6 1 \uf8ed\uf8ec 0 \uf8f8\uf8f7\u03bb = lim ln x lim x = lim 1 =0 \u03ba\u03b1\u03b9 x2 = 2x 2x2 x \u2192 +\u221e x \u2192 +\u221e x \u2192 +\u221e 1\u03b2 = lim ( f (x) \u2212 \u03bb x) = lim ln x = lim x = 0. x \u2192 +\u221e xx \u2192 +\u221e 1x\u2192+\u221e\u0386\u03c1\u03b1, \u03b7 \u03b5\u03c5\u03b8\u03b5\u03af\u03b1 y = 0 \u03b5\u03af\u03bd\u03b1\u03b9 \u03bf\u03c1\u03b9\u03b6\u03cc\u03bd\u03c4\u03b9\u03b1 \u03b1\u03c3\u03cd\u03bc\u03c0\u03c4\u03c9\u03c4\u03b7 \u03c4\u03b7\u03c2 Cf \u03c3\u03c4\u03bf +\u221e. lim f (x) = lim ln x 1\u0395\u03c0\u03b5\u03b9\u03b4\u03ae, \u03b5\u03c0\u03b9\u03c0\u03bb\u03ad\u03bf\u03bd, xx\u21920 = lim \uf8eb x \u22c5 ln x \uf8f6 = \u2212\u221e, \u03b7 \u03b5\u03c5\u03b8\u03b5\u03af\u03b1 x = 0 x \u2192 0+ \uf8ec\uf8ed \uf8f7\uf8f8 x \u2192 0+\u03b5\u03af\u03bd\u03b1\u03b9 \u03ba\u03b1\u03c4\u03b1\u03ba\u03cc\u03c1\u03c5\u03c6\u03b7 \u03b1\u03c3\u03cd\u03bc\u03c0\u03c4\u03c9\u03c4\u03b7 \u03c4\u03b7\u03c2 Cf .\u25cf \u03a3\u03c7\u03b7\u03bc\u03b1\u03c4\u03af\u03b6\u03bf\u03c5\u03bc\u03b5 \u03c4\u03bf\u03bd \u03c0\u03af\u03bd\u03b1\u03ba\u03b1 \u03bc\u03b5\u03c4\u03b1\u03b2\u03bf\u03bb\u03ce\u03bd \u03c4\u03b7\u03c2 f \u03ba\u03b1\u03b9 \u03c7\u03b1\u03c1\u03ac\u03c3\u03c3\u03bf\u03c5\u03bc\u03b5 \u03c4\u03b7 \u03b3\u03c1\u03b1\u03c6\u03b9\u03ba\u03ae \u03c4\u03b7\u03c2\u03c0\u03b1\u03c1\u03ac\u03c3\u03c4\u03b1\u03c3\u03b7.x0 e e3\/2 +\u221ef \u2032 (x) + 0 \u2013 \u2013f \u2033(x) \u2013 \u20130 + 0f ( x) 1 3 \u2212\u221e e 2e3\/ 2 \u03a3.\u039a. T.M. 162","\u0393\u0395\u039d\u0399\u039a\u0395\u03a3 \u0391\u03a3\u039a\u0397\u03a3\u0395\u0399\u03a3 2\u03bf\u03c5 \u039a\u0395\u03a6\u0391\u039b\u0391\u0399\u039f\u03a5 \u0393\u2019 \u039f\u039c\u0391\u0394\u0391\u03a3ii) \u0395\u03af\u03bd\u03b1\u03b9 \u03b1 \u03b1 +1 > (\u03b1 +1)\u03b1 \u21d4 ln \u03b1 \u03b1 +1 > ln(\u03b1 +1)\u03b1 \u21d4 (\u03b1 +1) ln\u03b1 > \u03b1 ln(\u03b1 +1) \u21d4 ln\u03b1 > ln(\u03b1 +1) \u03b1 \u03b1 +1 \u21d4 f (\u03b1 ) > f (\u03b1 +1). \u0397 \u03c4\u03b5\u03bb\u03b5\u03c5\u03c4\u03b1\u03af\u03b1 \u03b1\u03bd\u03b9\u03c3\u03cc\u03c4\u03b7\u03c4\u03b1 (\u03ac\u03c1\u03b1 \u03ba\u03b1\u03b9 \u03b7 \u03c0\u03c1\u03ce\u03c4\u03b7) \u03b5\u03af\u03bd\u03b1\u03b9 \u03b1\u03bb\u03b7\u03b8\u03ae\u03c2, \u03b1\u03c6\u03bf\u03cd e < \u03b1 < \u03b1 + 1 \u03ba\u03b1\u03b9 \u03b7 f \u03b5\u03af\u03bd\u03b1\u03b9 \u03b3\u03bd\u03b7\u03c3\u03af\u03c9\u03c2 \u03c6\u03b8\u03af\u03bd\u03bf\u03c5\u03c3\u03b1 \u03c3\u03c4\u03bf [e, +\u221e).iii) \u0393\u03b9\u03b1 \u03ba\u03ac\u03b8\u03b5 x > 0 \u03ad\u03c7\u03bf\u03c5\u03bc\u03b5: 2x = x2 \u21d4 ln 2x = ln x2 \u21d4 x ln 2 = 2 ln x \u21d4 ln 2 = ln x \u21d4 f (2) = f (x). 2x \u0394\u03b7\u03bb\u03b1\u03b4\u03ae \u03b7 \u03b5\u03be\u03af\u03c3\u03c9\u03c3\u03b7 2x = x2 \u03ad\u03c7\u03b5\u03b9 \u03c4\u03cc\u03c3\u03b5\u03c2 \u03bb\u03cd\u03c3\u03b5\u03b9\u03c2 \u03c3\u03c4\u03bf (0, +\u221e), \u03cc\u03c3\u03b5\u03c2 \u03b5\u03af\u03bd\u03b1\u03b9 \u03bf\u03b9 \u03c4\u03b9\u03bc\u03ad\u03c2 \u03c4\u03bf\u03c5 x > 0 \u03b3\u03b9\u03b1 \u03c4\u03b9\u03c2 \u03bf\u03c0\u03bf\u03af\u03b5\u03c2 \u03b7 \u03c3\u03c5\u03bd\u03ac\u03c1\u03c4\u03b7\u03c3\u03b7 f \u03c0\u03b1\u03af\u03c1\u03bd\u03b5\u03b9 \u03c4\u03b7\u03bd \u03c4\u03b9\u03bc\u03ae f (2) = ln 2 . 2 \u0395\u03c0\u03b5\u03b9\u03b4\u03ae 22 = 22 \u03ba\u03b1\u03b9 24 = 42, \u03b7 \u03b5\u03be\u03af\u03c3\u03c9\u03c3\u03b7 2x = x2 \u03ad\u03c7\u03b5\u03b9 \u03c3\u03c4\u03bf (0, +\u221e) \u03bb\u03cd\u03c3\u03b5\u03b9\u03c2 \u03c4\u03b9\u03c2 x = 2 \u03ba\u03b1\u03b9 x = 4. \u0398\u03b1 \u03b1\u03c0\u03bf\u03b4\u03b5\u03af\u03be\u03bf\u03c5\u03bc\u03b5 \u03c4\u03ce\u03c1\u03b1 \u03cc\u03c4\u03b9 \u03b1\u03c5\u03c4\u03ad\u03c2 \u03b5\u03af\u03bd\u03b1\u03b9 \u03bc\u03bf\u03bd\u03b1\u03b4\u03b9\u03ba\u03ad\u03c2. \u03a0\u03c1\u03ac\u03b3\u03bc\u03b1\u03c4\u03b9 \u03c3\u03cd\u03bc\u03c6\u03c9\u03bd\u03b1 \u03bc\u03b5 \u03c4\u03bf \u03b5\u03c1\u03ce\u03c4\u03b7\u03bc\u03b1 i): \u2014 \u03b7 f \u03c3\u03c4\u03bf (0,e] \u03b5\u03af\u03bd\u03b1\u03b9 \u03b3\u03bd\u03b7\u03c3\u03af\u03c9\u03c2 \u03b1\u03cd\u03be\u03bf\u03c5\u03c3\u03b1. \u0386\u03c1\u03b1 \u03c4\u03b7\u03bd \u03c4\u03b9\u03bc\u03ae f(2) \u03b8\u03b1 \u03c4\u03b7\u03bd \u03c0\u03ac\u03c1\u03b5\u03b9 \u03bc\u03b9\u03b1 \u03c6\u03bf\u03c1\u03ac, \u03b3\u03b9\u03b1 x = 2. \u2014 \u03b7 f \u03c3\u03c4\u03bf [e, +\u221e) \u03b5\u03af\u03bd\u03b1\u03b9 \u03b3\u03bd\u03b7\u03c3\u03af\u03c9\u03c2 \u03c6\u03b8\u03af\u03bd\u03bf\u03c5\u03c3\u03b1. \u0386\u03c1\u03b1 \u03c4\u03b7\u03bd \u03c4\u03b9\u03bc\u03ae f(4) \u03b8\u03b1 \u03c4\u03b7\u03bd \u03c0\u03ac\u03c1\u03b5\u03b9 \u03bc\u03cc\u03bd\u03bf \u03bc\u03b9\u03b1 \u03c6\u03bf\u03c1\u03ac. 163","\u0393\u0395\u039d\u0399\u039a\u0395\u03a3 \u0391\u03a3\u039a\u0397\u03a3\u0395\u0399\u03a3 2\u03bf\u03c5 \u039a\u0395\u03a6\u0391\u039b\u0391\u0399\u039f\u03a5 \u0393\u2019 \u039f\u039c\u0391\u0394\u0391\u03a3 \u0395\u03c0\u03bf\u03bc\u03ad\u03bd\u03c9\u03c2, \u03bf\u03b9 \u03bb\u03cd\u03c3\u03b5\u03b9\u03c2 \u03c4\u03b7\u03c2 2x = x2 \u03b5\u03af\u03bd\u03b1\u03b9 \u03b1\u03ba\u03c1\u03b9\u03b2\u03ce\u03c2 \u03b4\u03cd\u03bf, \u03bf\u03b9 x1 = 2 \u03ba\u03b1\u03b9 x2 = 4.7. i) \u0398\u0007 \u03b5\u03c9\u03c1\u03bf\u03cd\u03bc\u03b5 \u03c4\u03b7 \u03c3\u03c5\u03bd\u03ac\u03c1\u03c4\u03b7\u03c3\u03b7 f (x) = \u03b1 x + \u03b2 x , \u03b7 \u03bf\u03c0\u03bf\u03af\u03b1 \u03bf\u03c1\u03af\u03b6\u03b5\u03c4\u03b1\u03b9 \u03c3\u03c4\u03bf R \u03ba\u03b1\u03b9 \u03b5\u03af\u03bd\u03b1\u03b9 \u03c0\u03b1\u03c1\u03b1\u03b3\u03c9\u03b3\u03af\u03c3\u03b9\u03bc\u03b7 \u03c3\u2019 \u03b1\u03c5\u03c4\u03cc. \u0395\u03c0\u03b5\u03b9\u03b4\u03ae f(0) = 2 \u03ad\u03c7\u03bf\u03c5\u03bc\u03b5: f (x) \u2265 f (0) \u03b3\u03b9\u03b1 \u03ba\u03ac\u03b8\u03b5 x \u2208R , \u03c0\u03bf\u03c5 \u03c3\u03b7\u03bc\u03b1\u03af\u03bd\u03b5\u03b9 \u03cc\u03c4\u03b9 \u03b7 f \u03c3\u03c4\u03bf x0 = 0 \u03c0\u03b1\u03c1\u03bf\u03c5\u03c3\u03b9\u03ac\u03b6\u03b5\u03b9 \u03b5\u03bb\u03ac\u03c7\u03b9\u03c3\u03c4\u03bf, \u03bf\u03c0\u03cc\u03c4\u03b5 \u03c3\u03cd\u03bc\u03c6\u03c9\u03bd\u03b1 \u03bc\u03b5 \u03c4\u03bf \u03b8\u03b5\u03ce\u03c1\u03b7\u03bc\u03b1 \u03c4\u03bf\u03c5 Fermat \u03b9\u03c3\u03c7\u03cd\u03b5\u03b9 f \u2032 (0) = 0. \u0395\u03af\u03bd\u03b1\u03b9 \u03cc\u03bc\u03c9\u03c2 f \u2032(x) = \u03b1 x ln\u03b1 + \u03b2 x ln \u03b2 , \u03bf\u03c0\u03cc\u03c4\u03b5 f \u2032(0) = 0 \u21d4 ln\u03b1 + ln \u03b2 = 0 \u21d4 ln(\u03b1\u03b2 ) = 0 \u21d4 \u03b1\u03b2 = 1. ii) \u0393\u0007 \u03b9\u03b1 \u03ba\u03ac\u03b8\u03b5 x \u2208 R \u03b9\u03c3\u03c7\u03cd\u03b5\u03b9. \u0398\u03b5\u03c9\u03c1\u03bf\u03cd\u03bc\u03b5, \u03c4\u03ce\u03c1\u03b1, \u03c4\u03b7 \u03c3\u03c5\u03bd\u03ac\u03c1\u03c4\u03b7\u03c3\u03b7 f (x) = \u03b1 x \u2212 x \u22121, \u03b7 \u03bf\u03c0\u03bf\u03af\u03b1 \u03bf\u03c1\u03af\u03b6\u03b5\u03c4\u03b1\u03b9 \u03c3\u03c4\u03bf R \u03ba\u03b1\u03b9 \u03b5\u03af\u03bd\u03b1\u03b9 \u03c0\u03b1\u03c1\u03b1\u03b3\u03c9\u03b3\u03af\u03c3\u03b9\u03bc\u03b7 \u03c3\u2019 \u03b1\u03c5\u03c4\u03cc. \u0395\u03c0\u03b5\u03b9\u03b4\u03ae f (0) = \u03b1 0 \u2212 0 \u22121 = 0, \u03ad\u03c7\u03bf\u03c5\u03bc\u03b5 f(x) \u2265 f(0) \u03b3\u03b9\u03b1 \u03ba\u03ac\u03b8\u03b5 x \u2208 R. \u0386\u03c1\u03b1 \u03b7 f \u03c3\u03c4\u03bf x0 = 0 \u03c0\u03b1\u03c1\u03bf\u03c5\u03c3\u03b9\u03ac\u03b6\u03b5\u03b9 \u03b5\u03bb\u03ac\u03c7\u03b9\u03c3\u03c4\u03bf, \u03bf\u03c0\u03cc\u03c4\u03b5 \u03c3\u03cd\u03bc\u03c6\u03c9\u03bd\u03b1 \u03bc\u03b5 \u03c4\u03bf \u03b8\u03b5\u03ce\u03c1\u03b7\u03bc\u03b1 \u03c4\u03bf\u03c5 Fermat, \u03b9\u03c3\u03c7\u03cd\u03b5\u03b9 f \u2032 (0) = 0. \u0395\u03af\u03bd\u03b1\u03b9 \u03cc\u03bc\u03c9\u03c2: f \u2032(x) = \u03b1 x ln\u03b1 \u22121 \u03bf\u03c0\u03cc\u03c4\u03b5 f \u2032(0) = 0 \u21d4 \u03b1 0 ln\u03b1 \u22121 = 0 \u21d4 ln\u03b1 = 1 \u21d4 \u03b1 = e.8. i) \u2014\u0393\u03b9\u03b1 \u03ba\u03ac\u03b8\u03b5 x \u2208 R \u03b9\u03c3\u03c7\u03cd\u03b5\u03b9: f \u2032 (x) = ex \u03ba\u03b1\u03b9 f \u2033(x) = ex > 0 \u03b3\u03b9\u03b1 \u03ba\u03ac\u03b8\u03b5 x \u2208 R.\u0386\u03c1\u03b1 \u03b7 f \u03b5\u03af\u03bd\u03b1\u03b9 \u03ba\u03c5\u03c1\u03c4\u03ae \u03c3\u03c4\u03bf R.\u2014 \u0393\u03b9\u03b1 \u03ba\u03ac\u03b8\u03b5 x \u2208 (0, +\u221e) \u03b9\u03c3\u03c7\u03cd\u03b5\u03b9:g\u2032(x) = 1 \u03ba\u03b1\u03b9 g \u2032\u2032( x) = \u2212 1 < 0. x x2\u0386\u03c1\u03b1 \u03b7 f \u03b5\u03af\u03bd\u03b1\u03b9 \u03ba\u03bf\u03af\u03bb\u03b7 \u03c3\u03c4\u03bf (0, +\u221e).ii) \u0397 \u03b5\u03be\u03af\u03c3\u03c9\u03c3\u03b7 \u03c4\u03b7\u03c2 \u03b5\u03c6\u03b1\u03c0\u03c4\u03bf\u03bc\u03ad\u03bd\u03b7\u03c2 \u03c4\u03b7\u03c2 Cf \u03c3\u03c4\u03bf \u03c3\u03b7\u03bc\u03b5\u03af\u03bf \u0391(0,1) \u03b5\u03af\u03bd\u03b1\u03b9: y \u22121 = f \u2032(0)(x \u2212 0) \u21d4 y \u22121 = x \u21d4 y = x +1, 164","\u0393\u0395\u039d\u0399\u039a\u0395\u03a3 \u0391\u03a3\u039a\u0397\u03a3\u0395\u0399\u03a3 2\u03bf\u03c5 \u039a\u0395\u03a6\u0391\u039b\u0391\u0399\u039f\u03a5 \u0393\u2019 \u039f\u039c\u0391\u0394\u0391\u03a3 \u03b5\u03bd\u03ce \u03c4\u03b7\u03c2 Cg \u03c3\u03c4\u03bf \u03c3\u03b7\u03bc\u03b5\u03af\u03bf (1,0) \u03b5\u03af\u03bd\u03b1\u03b9: y \u2212 0 = g\u2032(1)(x \u22121) \u21d4 y = x \u22121.iii) \u0007\u03b1) \u0007\u0395\u03c0\u03b5\u03b9\u03b4\u03ae \u03b7 f \u03b5\u03af\u03bd\u03b1\u03b9 \u03ba\u03c5\u03c1\u03c4\u03ae \u03c3\u03c4\u03bf R \u03b7 \u03b5\u03c6\u03b1\u03c0\u03c4\u03bf\u03bc\u03ad\u03bd\u03b7 \u03c4\u03b7\u03c2 Cf \u03c3\u03c4\u03bf \u03c3\u03b7\u03bc\u03b5\u03af\u03bf (0,1) \u03b2\u03c1\u03af\u03c3\u03ba\u03b5\u03c4\u03b1\u03b9 \u03ba\u03ac\u03c4\u03c9 \u03b1\u03c0\u03cc \u03c4\u03b7\u03bd Cf . \u0386\u03c1\u03b1 \u03b9\u03c3\u03c7\u03cd\u03b5\u03b9: ex \u2265 1+ x \u03b3\u03b9\u03b1 \u03ba\u03ac\u03b8\u03b5 x \u2208 R. \u0397 \u03b9\u03c3\u03cc\u03c4\u03b7\u03c4\u03b1 \u03b9\u03c3\u03c7\u03cd\u03b5\u03b9 \u03bc\u03cc\u03bd\u03bf \u03cc\u03c4\u03b1\u03bd x = 0.\u03b2) \u0007\u0395\u03c0\u03b5\u03b9\u03b4\u03ae \u03b7 g \u03b5\u03af\u03bd\u03b1\u03b9 \u03ba\u03bf\u03af\u03bb\u03b7 \u03c3\u03c4\u03bf (0, +\u221e) \u03b7 \u03b5\u03c6\u03b1\u03c0\u03c4\u03bf\u03bc\u03ad\u03bd\u03b7 \u03c4\u03b7\u03c2 Cg \u03c3\u03c4\u03bf \u03c3\u03b7\u03bc\u03b5\u03af\u03bf \u0392(1,0) \u03b2\u03c1\u03af\u03c3\u03ba\u03b5\u03c4\u03b1\u03b9 \u03c0\u03ac\u03bd\u03c9 \u03b1\u03c0\u03cc \u03c4\u03b7\u03bd Cg. \u0386\u03c1\u03b1, \u03b9\u03c3\u03c7\u03cd\u03b5\u03b9: x \u22121 \u2265 ln x \u03b3\u03b9\u03b1 \u03ba\u03ac\u03b8\u03b5 x \u2208 (0, +\u221e).\u0397 \u03b9\u03c3\u03cc\u03c4\u03b7\u03c4\u03b1 \u03b9\u03c3\u03c7\u03cd\u03b5\u03b9 \u03cc\u03c4\u03b1\u03bd x = 1.iv) \u0393\u03b9\u03b1 \u03ba\u03ac\u03b8\u03b5 x \u2208 R \u03b9\u03c3\u03c7\u03cd\u03b5\u03b9 x \u22121 \u2264 x +1,\u03bf\u03c0\u03cc\u03c4\u03b5, \u03bb\u03cc\u03b3\u03c9 \u03c4\u03bf\u03c5 \u03b5\u03c1\u03c9\u03c4\u03ae\u03bc\u03b1\u03c4\u03bf\u03c2 iii), \u03ad\u03c7\u03bf\u03c5\u03bc\u03b5ln x \u2264 x \u22121 < x +1 < ex , x > 0.\u0386\u03c1\u03b1 ln x < ex, \u03b3\u03b9\u03b1 \u03ba\u03ac\u03b8\u03b5 x > 0.9. i) H \u03c3\u03c5\u03bd\u03ac\u03c1\u03c4\u03b7\u03c3\u03b7 f(x) = ex \u2013 \u03bbx \u03b5\u03af\u03bd\u03b1\u03b9 \u03c0\u03b1\u03c1\u03b1\u03b3\u03c9\u03b3\u03af\u03c3\u03b9\u03bc\u03b7 \u03c3\u03c4\u03bf R \u03bc\u03b5 f \u2032(x) = ex \u2212 \u03bb. \u0395\u03af\u03bd\u03b1\u03b9 f \u2032(x) = 0 \u21d4 ex \u2212 \u03bb = 0 \u21d4 x = ln \u03bb. \u03a4\u03bf \u03c0\u03c1\u03cc\u03c3\u03b7\u03bc\u03bf \u03c4\u03b7\u03c2 f \u2032 , \u03b7 \u03bc\u03bf\u03bd\u03bf\u03c4\u03bf\u03bd\u03af\u03b1 \u03ba\u03b1\u03b9 \u03c4\u03b1 \u03b1\u03ba\u03c1\u03cc\u03c4\u03b1\u03c4\u03b1 \u03c4\u03b7\u03c2 f \u03c6\u03b1\u03af\u03bd\u03bf\u03bd\u03c4\u03b1\u03b9 \u03c3\u03c4\u03bf\u03bd \u03c0\u03af\u03bd\u03b1\u03ba\u03b1. x \u2212\u221e ln\u03bb +\u221ef \u2032 (x) \u2013 0+ f(x) min\u0395\u03c0\u03bf\u03bc\u03ad\u03bd\u03c9\u03c2, \u03b7 f \u03c0\u03b1\u03c1\u03bf\u03c5\u03c3\u03b9\u03ac\u03b6\u03b5\u03b9 \u03b5\u03bb\u03ac\u03c7\u03b9\u03c3\u03c4\u03b7 \u03c4\u03b9\u03bc\u03ae \u03b3\u03b9\u03b1 x = ln \u03bb \u03c4\u03b7\u03bd 165","\u0393\u0395\u039d\u0399\u039a\u0395\u03a3 \u0391\u03a3\u039a\u0397\u03a3\u0395\u0399\u03a3 2\u03bf\u03c5 \u039a\u0395\u03a6\u0391\u039b\u0391\u0399\u039f\u03a5 \u0393\u2019 \u039f\u039c\u0391\u0394\u0391\u03a3 f (ln \u03bb) = eln\u03bb \u2212 \u03bb ln \u03bb = \u03bb \u2212 \u03bb ln \u03bb = \u03bb(1\u2212 ln \u03bb).ii) \u0399\u03c3\u03c7\u03cd\u03bf\u03c5\u03bd \u03bf\u03b9 \u03b9\u03c3\u03bf\u03b4\u03c5\u03bd\u03b1\u03bc\u03af\u03b5\u03c2 R R R \u21d4 min f (x) \u2265 0 \u21d4 \u03bb(1\u2212 ln \u03bb) \u2265 0 \u21d4 1\u2212 ln \u03bb \u2265 0 \u21d4 ln \u03bb \u2264 1 \u21d4 \u03bb \u2264 e.\u0386\u03c1\u03b1, \u03b7 \u03bc\u03b5\u03b3\u03b1\u03bb\u03cd\u03c4\u03b5\u03c1\u03b7 \u03c4\u03b9\u03bc\u03ae \u03c4\u03bf\u03c5 \u03bb, \u03b3\u03b9\u03b1 \u03c4\u03b7\u03bd \u03bf\u03c0\u03bf\u03af\u03b1 \u03b9\u03c3\u03c7\u03cd\u03b5\u03b9 ex \u2265 \u03bb x \u03b3\u03b9\u03b1 \u03ba\u03ac\u03b8\u03b5 x \u2208 R,\u03b5\u03af\u03bd\u03b1\u03b9 \u03b7 \u03bb = e.iii) \u0393\u0007 \u03b9\u03b1 \u03bd\u03b1 \u03b5\u03c6\u03ac\u03c0\u03c4\u03b5\u03c4\u03b1\u03b9 \u03b7 \u03b5\u03c5\u03b8\u03b5\u03af\u03b1 y = ex \u03c4\u03b7\u03c2 \u03b3\u03c1\u03b1\u03c6\u03b9\u03ba\u03ae\u03c2 \u03c0\u03b1\u03c1\u03ac\u03c3\u03c4\u03b1\u03c3\u03b7\u03c2 \u03c4\u03b7\u03c2 g(x) = ex, \u03b1\u03c1\u03ba\u03b5\u03af \u03bd\u03b1 \u03c5\u03c0\u03ac\u03c1\u03c7\u03b5\u03b9 \u03c3\u03b7\u03bc\u03b5\u03af\u03bf x0 \u03c4\u03ad\u03c4\u03bf\u03b9\u03bf, \u03ce\u03c3\u03c4\u03b5 \u03b7 \u03b5\u03c6\u03b1\u03c0\u03c4\u03bf\u03bc\u03ad\u03bd\u03b7 \u03c4\u03b7\u03c2 Cg \u03c3\u03c4\u03bf \u0391(x0,g(x0)) \u03bd\u03b1 \u03c4\u03b1\u03c5\u03c4\u03af\u03b6\u03b5\u03c4\u03b1\u03b9 \u03bc\u03b5 \u03c4\u03b7\u03bd y = ex. \u0393\u03b9\u03b1 \u03bd\u03b1 \u03b9\u03c3\u03c7\u03cd\u03b5\u03b9 \u03b1\u03c5\u03c4\u03cc, \u03b1\u03c1\u03ba\u03b5\u03af \uf8f1 g (x0 ) = e \u22c5 x0 \u21d4 \uf8f1\uf8f4e x0 = ex0 \u21d4 x0 = 1. \uf8f2 g \u2032(x0 ) = e \uf8f2 =e \uf8f3 \uf8f4\uf8f3e x0\u0395\u03c0\u03bf\u03bc\u03ad\u03bd\u03c9\u03c2 \u03b7 y = ex \u03b5\u03c6\u03ac\u03c0\u03c4\u03b5\u03c4\u03b1\u03b9 \u03c4\u03b7\u03c2 Cg \u03c3\u03c4\u03bf \u03c3\u03b7\u03bc\u03b5\u03af\u03bf \u0391(l,e). \u03b51 : y \u2212 ex0 = ex0 (x \u2212 x0 ). (1)10. i) \u0393\u03b9\u03b1 x \u2260 0 \u03b5\u03af\u03bd\u03b1\u03b9 f (x) \u2212 f (0) = x2\u03b7\u00b5 1 = x\u03b7\u00b5 1 . x x\u22120 x x\u0395\u03c0\u03b5\u03b9\u03b4\u03ae x\u03b7\u00b5 1 \u2264 x \u03ad\u03c7\u03bf\u03c5\u03bc\u03b5 x \u2212 x \u2264 x\u03b7\u00b5 1 \u2264 x . x\u038c\u03bc\u03c9\u03c2 lim x = lim (\u2212 x ) = 0. x\u21920 x\u21920 166","\u0393\u0395\u039d\u0399\u039a\u0395\u03a3 \u0391\u03a3\u039a\u0397\u03a3\u0395\u0399\u03a3 2\u03bf\u03c5 \u039a\u0395\u03a6\u0391\u039b\u0391\u0399\u039f\u03a5 \u0393\u2019 \u039f\u039c\u0391\u0394\u0391\u03a3\u0386\u03c1\u03b1 lim \uf8eb x\u03b7\u00b5 1 \uf8f6 = 0. \uf8ed\uf8ec x \uf8f8\uf8f7 x\u21920\u0395\u03c0\u03bf\u03bc\u03ad\u03bd\u03c9\u03c2 f \u2032 (0) = 0.\u0391\u03c6\u03bf\u03cd f(0) = 0 \u03ba\u03b1\u03b9 f \u2032 (0) = 0, \u03b7 \u03b5\u03c5\u03b8\u03b5\u03af\u03b1 y = 0 \u03b5\u03af\u03bd\u03b1\u03b9 \u03b5\u03c6\u03b1\u03c0\u03c4\u03cc\u03bc\u03b5\u03bd\u03b7 \u03c4\u03b7\u03c2 Cf \u03c3\u03c4\u03bf \u039f(0,0).ii) \u0007\u03a4\u03b1 \u03ba\u03bf\u03b9\u03bd\u03ac \u03c3\u03b7\u03bc\u03b5\u03af\u03b1 \u03c4\u03b7\u03c2 Cf \u03ba\u03b1\u03b9 \u03c4\u03b7\u03c2 \u03b5\u03c5\u03b8\u03b5\u03af\u03b1\u03c2 y = 0 \u03c0\u03c1\u03bf\u03ba\u03cd\u03c0\u03c4\u03bf\u03c5\u03bd \u03b1\u03c0\u03cc \u03c4\u03b7 \u03bb\u03cd\u03c3\u03b7 \u03c4\u03b7\u03c2 \u03b5\u03be\u03af\u03c3\u03c9\u03c3\u03b7\u03c2 f(x) = 0.\u25cf \u0393\u03b9\u03b1 x \u2260 0 \u03b5\u03af\u03bd\u03b1\u03b9f (x) = 0 \u21d4 x2\u03b7\u00b5 1 = 0 \u21d4 \u03b7\u00b5 1 = 0 xx \u21d4 1 = \u03ba\u03c0 , \u03ba \u2208 \uf05a * \u21d4 x = 1 , \u03ba \u2208 \uf05a * . (1) x \u03ba\u03c0\u25cf \u0393\u03b9\u03b1 x = 0 \u03b5\u03af\u03bd\u03b1\u03b9 f(0) = 0.\u0386\u03c1\u03b1, \u03c4\u03b1 \u03ba\u03bf\u03b9\u03bd\u03ac \u03c3\u03b7\u03bc\u03b5\u03af\u03b1 \u03b5\u03af\u03bd\u03b1\u03b9 \u03ac\u03c0\u03b5\u03b9\u03c1\u03b1 \u03c4\u03bf \u039f(0,0) \u03ba\u03b1\u03b9 \u03c4\u03b1 \u03c5\u03c0\u03cc\u03bb\u03bf\u03b9\u03c0\u03b1 \u03ad\u03c7\u03bf\u03c5\u03bd \u03c4\u03b5\u03c4\u03bc\u03b7\u03bc\u03ad\u03bd\u03b5\u03c2\u03c0\u03bf\u03c5 \u03b4\u03af\u03bd\u03bf\u03bd\u03c4\u03b1\u03b9, \u03b3\u03b9\u03b1 \u03c4\u03b9\u03c2 \u03b4\u03b9\u03ac\u03c6\u03bf\u03c1\u03b5\u03c2 \u03c4\u03b9\u03bc\u03ad\u03c2 \u03c4\u03bf\u03c5 \u03ba \u2208 \uf05a * \u03b1\u03c0\u03cc \u03c4\u03b7 \u03c3\u03c7\u03ad\u03c3\u03b7 (1). (\u0395\u03af\u03bd\u03b1\u03b9\u03c0\u03c1\u03bf\u03c6\u03b1\u03bd\u03ad\u03c2 \u03cc\u03c4\u03b9 \u03b3\u03b9\u03b1 \u03bc\u03b5\u03b3\u03ac\u03bb\u03b5\u03c2 \u03ba\u03b1\u03c4\u2019 \u03b1\u03c0\u03cc\u03bb\u03c5\u03c4\u03b7 \u03c4\u03b9\u03bc\u03ae \u03c4\u03bf\u03c5 \u03ba, \u03bf\u03b9 \u03c4\u03b9\u03bc\u03ad\u03c2 \u03c4\u03bf\u03c5 x \u03b5\u03af\u03bd\u03b1\u03b9 \u03c0\u03bf\u03bb\u03cd\u03bc\u03b9\u03ba\u03c1\u03ad\u03c2 \u03ba\u03b1\u03b9 \u03c0\u03bb\u03b7\u03c3\u03b9\u03ac\u03b6\u03bf\u03c5\u03bd \u03c4\u03bf 0).iii) \u0391\u03c1\u03ba\u03b5\u03af \u03bd\u03b1 \u03b4\u03b5\u03af\u03be\u03bf\u03c5\u03bc\u03b5 \u03cc\u03c4\u03b9 lim ( f (x) \u2212 x) = 0 \u03ba\u03b1\u03b9 lim ( f (x) \u2212 x) = 0 x \u2192 +\u221e x \u2192 \u2212\u221e\u25cf \u0388\u03c7\u03bf\u03c5\u03bc\u03b5: lim ( f (x) \u2212 x) = lim \uf8eb x2\u03b7\u00b5 1 \u2212 x \uf8f6 \uf8ec\uf8ed x \uf8f8\uf8f7 x \u2192 +\u221e x \u2192 +\u221e = lim \uf8eb 1 \u03b7\u00b5t \u2212 1\uf8f6 \uf8ed\uf8ec t2 t \uf8f7\uf8f8 t\u21920 = lim \u03b7\u00b5t \u2212 t \uf8eb \u00b5\u03bf\u03c1\u03d5\u03ae 0 \uf8f6 t2 \uf8ed 0\uf8f8 t\u21920 = lim \u03c3\u03c5\u03bdt \u22121 \uf8eb \u00b5\u03bf\u03c1\u03d5\u03ae 0 \uf8f6 t\u21920 2t \uf8ed 0\uf8f8 = lim \u2212\u03b7\u00b5t = 0. t\u21920 2 167","\u0393\u0395\u039d\u0399\u039a\u0395\u03a3 \u0391\u03a3\u039a\u0397\u03a3\u0395\u0399\u03a3 2\u03bf\u03c5 \u039a\u0395\u03a6\u0391\u039b\u0391\u0399\u039f\u03a5 \u0393\u2019 \u039f\u039c\u0391\u0394\u0391\u03a3\u25cf \u039f\u03bc\u03bf\u03af\u03c9\u03c2, \u03ad\u03c7\u03bf\u03c5\u03bc\u03b5 lim ( f (x) \u2212 x) = 0. x \u2192 \u2212\u221e11. \u0391. i) H \u03c3\u03c5\u03bd\u03ac\u03c1\u03c4\u03b7\u03c3\u03b7 \u03c8 \u03b5\u03af\u03bd\u03b1\u03b9 \u03c0\u03b1\u03c1\u03b1\u03b3\u03c9\u03b3\u03af\u03c3\u03b9\u03bc\u03b7 \u03c3\u03c4\u03bf R \u03bc\u03b5 \u03c8\u2032(x) = 2\u03c6\u2032(x)\u03c6\u2033(x) + 2\u03c6(x)\u2219\u03c6\u2032(x) = 2\u03c6\u2032(x)( \u03c6\u2033(x) + \u03c6(x)) = 2\u03c6\u2032(x)\u22190 (\u03b1\u03c0\u03cc \u03c5\u03c0\u03cc\u03b8\u03b5\u03c3\u03b7) = 0.\u0395\u03c0\u03bf\u03bc\u03ad\u03bd\u03c9\u03c2, \u03b7 \u03c8 \u03b5\u03af\u03bd\u03b1\u03b9 \u03c3\u03c4\u03b1\u03b8\u03b5\u03c1\u03ae \u03c3\u03c4\u03bf R. \u0395\u03c0\u03b5\u03b9\u03b4\u03ae \u03c8 (0) = (\u03d5\u2032(0))2 + (\u03d5(0))2 = 0 + 0 = 0,\u03b5\u03af\u03bd\u03b1\u03b9 \u03c8 (x) = 0, x \u2208 Rii) \u0395\u0007 \u03c0\u03b5\u03b9\u03b4\u03ae \u03c8(x) = 0, \u03b5\u03af\u03bd\u03b1\u03b9 (\u03d5\u2032(x))2 + (\u03d5(x))2 = 0 \u03b3\u03b9\u03b1 \u03ba\u03ac\u03b8\u03b5 x \u2208 R. \u0386\u03c1\u03b1 \u03c6\u2032(x) = 0 \u03ba\u03b1\u03b9 \u03c6(x) = 0 \u03b3\u03b9\u03b1 \u03ba\u03ac\u03b8\u03b5 x \u2208 R, \u03bf\u03c0\u03cc\u03c4\u03b5 \u03c6(x) = 0, \u03b3\u03b9\u03b1 \u03ba\u03ac\u03b8\u03b5 x \u2208 R.\u0392. \u0395\u03af\u03bd\u03b1\u03b9 \u03d5\u2032(x) = f \u2032(x) \u2212 \u03c3\u03c5\u03bdx \u03ba\u03b1\u03b9 \u03d5\u2032\u2032(x) = f \u2032\u2032(x) + \u03b7\u00b5x.\u0386\u03c1\u03b1 \u03d5\u2032\u2032(x) + \u03d5(x) = f \u2032\u2032(x) + \u03b7\u00b5x + f (x) \u2212 \u03b7\u00b5x = f \u2032\u2032(x) + f (x) = 0 (\u03b1\u03c0\u03cc \u03c5\u03c0\u03cc\u03b8\u03b5\u03c3\u03b7)\u0395\u03c0\u03af\u03c3\u03b7\u03c2 \u03d5(0) = f (0) \u2212 \u03b7\u00b50 = 0\u03ba\u03b1\u03b9 \u03d5\u2032(0) = f \u2032(0) \u2212 \u03c3\u03c5\u03bd0 = 1\u22121 = 0.\u0395\u03c0\u03bf\u03bc\u03ad\u03bd\u03c9\u03c2 \u03b7 \u03c6 \u03b9\u03ba\u03b1\u03bd\u03bf\u03c0\u03bf\u03b9\u03b5\u03af \u03c4\u03b9\u03c2 \u03c5\u03c0\u03bf\u03b8\u03ad\u03c3\u03b5\u03b9\u03c2 (1) \u03c4\u03bf\u03c5 \u03b5\u03c1\u03c9\u03c4\u03ae\u03bc\u03b1\u03c4\u03bf\u03c2 (\u0391).\u039f\u03bc\u03bf\u03af\u03c9\u03c2, \u03ad\u03c7\u03bf\u03c5\u03bc\u03b5: \u03c8 \u2032(x) = g\u2032(x) + \u03b7\u00b5x 168","\u0393\u0395\u039d\u0399\u039a\u0395\u03a3 \u0391\u03a3\u039a\u0397\u03a3\u0395\u0399\u03a3 2\u03bf\u03c5 \u039a\u0395\u03a6\u0391\u039b\u0391\u0399\u039f\u03a5 \u0393\u2019 \u039f\u039c\u0391\u0394\u0391\u03a3\u03ba\u03b1\u03b9 \u03c8 \u2032\u2032(x) = g\u2032\u2032(x) + \u03c3\u03c5\u03bdx.\u0386\u03c1\u03b1, \u03c8 \u2032\u2032(x) +\u03c8 (x) = g\u2032\u2032(x) + \u03c3\u03c5\u03bdx + g(x) \u2212 \u03c3\u03c5\u03bdx = g\u2032\u2032(x) + g(x) = 0 (\u03b1\u03c0\u03cc \u03c5\u03c0\u03cc\u03b8\u03b5\u03c3\u03b7).\u0395\u03c0\u03af\u03c3\u03b7\u03c2 \u03c8 (0) = g(0) \u2212 \u03c3\u03c5\u03bd0 = 1\u22121 = 0 \u03c8 \u2032(0) = g\u2032(0) + \u03b7\u00b50 = 0.\u0395\u03c0\u03bf\u03bc\u03ad\u03bd\u03c9\u03c2, \u03b7 \u03c3\u03c5\u03bd\u03ac\u03c1\u03c4\u03b7\u03c3\u03b7 y \u03b9\u03ba\u03b1\u03bd\u03bf\u03c0\u03bf\u03b9\u03b5\u03af \u03c4\u03b9\u03c2 \u03c5\u03c0\u03bf\u03b8\u03ad\u03c3\u03b5\u03b9\u03c2 (1) \u03c4\u03bf\u03c5 \u03b5\u03c1\u03c9\u03c4\u03ae\u03bc\u03b1\u03c4\u03bf\u03c2 \u0391.ii) \u0007\u0391\u03c6\u03bf\u03cd \u03bf\u03b9 \u03c3\u03c5\u03bd\u03b1\u03c1\u03c4\u03ae\u03c3\u03b5\u03b9\u03c2 \u03c6, \u03c8 \u03b9\u03ba\u03b1\u03bd\u03bf\u03c0\u03bf\u03b9\u03bf\u03cd\u03bd \u03c4\u03b9\u03c2 \u03c5\u03c0\u03bf\u03b8\u03ad\u03c3\u03b5\u03b9\u03c2 (1) \u03c4\u03bf\u03c5 \u03b5\u03c1\u03c9\u03c4\u03ae\u03bc\u03b1\u03c4\u03bf\u03c2 \u0391, \u03c3\u03cd\u03bc\u03c6\u03c9\u03bd\u03b1 \u03bc\u03b5 \u03c4\u03bf \u03b5\u03c1\u03ce\u03c4\u03b7\u03bc\u03b1 (\u0391), \u03b9\u03c3\u03c7\u03cd\u03b5\u03b9 \u03c6(x) = 0 \u03ba\u03b1\u03b9 \u03c8(x) = 0 \u03b3\u03b9\u03b1 \u03ba\u03ac\u03b8\u03b5 x \u2208 R, \u03bf\u03c0\u03cc\u03c4\u03b5 f(x) = \u03b7\u03bcx \u03ba\u03b1\u03b9 g(x) = \u03c3\u03c5\u03bdx \u03b3\u03b9\u03b1 \u03ba\u03ac\u03b8\u03b5 x \u2208 R.12. i) \u039f\u0007 \u03b9 \u03c3\u03c5\u03bd\u03c4\u03b5\u03c4\u03b1\u03b3\u03bc\u03ad\u03bd\u03b5\u03c2 \u03c4\u03bf\u03c5 \u03c3\u03b7\u03bc\u03b5\u03af\u03bf\u03c5 \u039c \u03b5\u03af\u03bd\u03b1\u03b9 (\u03c3\u03c5\u03bd\u03b8,\u03b7\u03bc\u03b8). \u03a4\u03b1 \u03b4\u03b9\u03b1\u03bd\u03cd\u03c3\u03bc\u03b1\u03c4\u03b1\u03a1\u039c = (\u03c3\u03c5\u03bd\u03b8 \u2212 x, \u03b7\u00b5\u03b8 ) \u03ba\u03b1\u03b9 \u03a1\u039d = (1\u2212 x,\u03b8 ) \u03b5\u03af\u03bd\u03b1\u03b9 \u03c3\u03c5\u03b3\u03b3\u03c1\u03b1\u03bc\u03bc\u03b9\u03ba\u03ac.\u0395\u03c0\u03bf\u03bc\u03ad\u03bd\u03c9\u03c2, det(\u03a1\u039c , \u03a1\u039d ) = 0\u21d4 \u03c3\u03c5\u03bd\u03b8 \u2212 x \u03b7\u00b5\u03b8 1\u2212 x \u03b8 =0 \u21d4 \u03b8 (\u03c3\u03c5\u03bd\u03b8 \u2212 x) \u2212 \u03b7\u00b5\u03b8 (1\u2212 x) = 0 \u21d4 \u03b8 \u03c3\u03c5\u03bd\u03b8 \u2212 \u03b7\u00b5\u03b8 = x\u03b8 \u2212 x\u03b7\u00b5\u03b8 \u21d4 x = \u03b8 \u03c3\u03c5\u03bd\u03b8 \u2212 \u03b7\u00b5\u03b8 = x(\u03b8 ). \u03b8 \u2212 \u03b7\u00b5\u03b8ii) \u0388\u03c7\u03bf\u03c5\u03bc\u03b5 lim x(\u03b8 ) = lim \u03b8 \u03c3\u03c5\u03bd\u03b8 \u2212 \u03b7\u00b5\u03b8 \u03b8 \u21920 \u03b8 \u21920 \u03b8 \u2212 \u03b7\u00b5\u03b8 = lim \u03c3\u03c5\u03bd\u03b8 \u2212\u03b8 \u03b7\u00b5\u03b8 \u2212 \u03c3\u03c5\u03bd\u03b8 \u03b8 \u21920 1\u2212 \u03c3\u03c5\u03bd\u03b8 = lim \u2212\u03b8 \u03b7\u00b5\u03b8 \u03b8 \u21920 1\u2212 \u03c3\u03c5\u03bd\u03b8 = lim \u2212\u03b7\u00b5\u03b8 \u2212\u03b8 \u03c3\u03c5\u03bd\u03b8 \u03b8 \u21920 \u03b7\u00b5\u03b8 169","\u0393\u0395\u039d\u0399\u039a\u0395\u03a3 \u0391\u03a3\u039a\u0397\u03a3\u0395\u0399\u03a3 2\u03bf\u03c5 \u039a\u0395\u03a6\u0391\u039b\u0391\u0399\u039f\u03a5 \u0393\u2019 \u039f\u039c\u0391\u0394\u0391\u03a3 = lim \u2212\u03c3\u03c5\u03bd\u03b8 \u2212 \u03c3\u03c5\u03bd\u03b8 +\u03b8 \u03b7\u00b5\u03b8 \u03b8 \u21920 \u03c3\u03c5\u03bd\u03b8 = \u22122 = \u22122. 113. A. i) \u2014\u03a4\u03bf \u03bc\u03ae\u03ba\u03bf\u03c2 s \u03c4\u03bf\u03c5 \u03c4\u03cc\u03be\u03bf\u03c5 \u0391\u03a0 \u03b5\u03af\u03bd\u03b1\u03b9 s = 2\u03c0\u03c1 \u03b8 = \u03b8\u03c1 = 2\u03b8 , \u03bf\u03c0\u03cc\u03c4\u03b5 2\u03c0 \u03b8 = s. 2\u2014\u0391\u03bd \u039f\u2206 \u22a5 \u0391\u03a0 , \u03b1\u03c0\u03cc \u03c4\u03bf \u03c4\u03c1\u03af\u03b3\u03c9\u03bd\u03bf \u039f\u0391\u0394 \u03ad\u03c7\u03bf\u03c5\u03bc\u03b5 \u03b7\u00b5 \u03b8 = \u0391\u2206 = l \/ 2 = l 22 24\u03bf\u03c0\u03cc\u03c4\u03b5 l = 4\u03b7\u00b5 \u03b8 . 2ii) \u0395\u0007 \u03c0\u03b5\u03b9\u03b4\u03ae \u03bf \u03c0\u03b5\u03b6\u03bf\u03c0\u03cc\u03c1\u03bf\u03c2 \u03b2\u03b1\u03b4\u03af\u03b6\u03b5\u03b9 \u03bc\u03b5 \u03c4\u03b1\u03c7\u03cd\u03c4\u03b7\u03c4\u03b1 \u03c5 = 4 km\/h, \u03c4\u03b7 \u03c7\u03c1\u03bf\u03bd\u03b9\u03ba\u03ae \u03c3\u03c4\u03b9\u03b3\u03bc\u03ae t\u03b8\u03b1 \u03ad\u03c7\u03b5\u03b9 \u03b4\u03b9\u03b1\u03bd\u03cd\u03c3\u03b5\u03b9 \u03b4\u03b9\u03ac\u03c3\u03c4\u03b7\u03bc\u03b1 s = 4t. \u0391\u03c6\u03bf\u03cd \u03b8 = l, \u03b5\u03af\u03bd\u03b1\u03b9 2 \u03b8 = 4t = 2t \u03ba\u03b1\u03b9 l = 4\u03b7\u00b5 2t = 4\u03b7\u00b5t. 22\u0392. \u0395\u03af\u03bd\u03b1\u03b9 l\u2032(t) = 4\u03c3\u03c5\u03bdt, \u03bf\u03c0\u03cc\u03c4\u03b5:\u03b1) \u038c\u03c4\u03b1\u03bd \u03b8 = 2\u03c0 , \u03b5\u03af\u03bd\u03b1\u03b9 3 2\u03c0 t= 3 =\u03c0 \u03ba\u03b1\u03b9 l\u2032 \uf8eb \u03c0 \uf8f6 = 4\u03c3\u03c5\u03bd \u03c0 = 4\u22c5 1 = 2 km\/h. 2 3 \uf8ec\uf8ed 3 \uf8f8\uf8f7 3 2\u03b2) \u038c\u03c4\u03b1\u03bd \u03b8 = \u03c0, \u03b5\u03af\u03bd\u03b1\u03b9 t=\u03c0 \u03ba\u03b1\u03b9 l \u2032 \uf8ec\uf8ed\uf8eb \u03c0 \uf8f6 = 4\u03c3\u03c5\u03bd \u03c0 =0 km\/h 2 2 \uf8f8\uf8f7 2\u03b3) \u038c\u03c4\u03b1\u03bd \u03b8 = 4\u03c0 , \u03b5\u03af\u03bd\u03b1\u03b9 3 t = 2\u03c0 \u03ba\u03b1\u03b9 l\u2032 \uf8eb 2\u03c0 \uf8f6 = 4\u03c3\u03c5\u03bd 2\u03c0 = \u22122 km\/h. 3 \uf8ed\uf8ec 3 \uf8f8\uf8f7 3 170","\u0393\u0395\u039d\u0399\u039a\u0395\u03a3 \u0391\u03a3\u039a\u0397\u03a3\u0395\u0399\u03a3 2\u03bf\u03c5 \u039a\u0395\u03a6\u0391\u039b\u0391\u0399\u039f\u03a5 \u0393\u2019 \u039f\u039c\u0391\u0394\u0391\u03a314. \u0388\u0007 \u03c3\u03c4\u03c9 \u03cc\u03c4\u03b9 \u03bf \u03b1\u03b3\u03c1\u03cc\u03c4\u03b7\u03c2 \u03b8\u03b1 \u03c0\u03c1\u03bf\u03c3\u03bb\u03ac\u03b2\u03b5\u03b9 x \u03b5\u03c1\u03b3\u03ac\u03c4\u03b5\u03c2, \u03ba\u03b1\u03b9 \u03c4\u03bf\u03bd \u03b5\u03c0\u03b9\u03c3\u03c4\u03ac\u03c4\u03b7 \u03ba\u03b1\u03b9 \u03ad\u03c3\u03c4\u03c9 \u03cc\u03c4\u03b9 \u03c7\u03c1\u03b5\u03b9\u03ac\u03b6\u03bf\u03bd\u03c4\u03b1\u03b9 t \u03ce\u03c1\u03b5\u03c2 \u03b3\u03b9\u03b1 \u03bd\u03b1 \u03bc\u03b1\u03b6\u03b5\u03c5\u03c4\u03bf\u03cd\u03bd \u03bf\u03b9 \u03bd\u03c4\u03bf\u03bc\u03ac\u03c4\u03b5\u03c2. \u0391\u03c6\u03bf\u03cd \u03ba\u03ac\u03b8\u03b5 \u03b5\u03c1\u03b3\u03ac\u03c4\u03b7\u03c2 \u03bc\u03b1\u03b6\u03b5\u03cd\u03b5\u03b9 125 \u03ba\u03b9\u03bb\u03ac \u03bd\u03c4\u03bf\u03bc\u03ac\u03c4\u03b5\u03c2 \u03c4\u03b7\u03bd \u03ce\u03c1\u03b1, \u03c3\u03b5 t \u03ce\u03c1\u03b5\u03c2 \u03bf\u03b9 x \u03b5\u03c1\u03b3\u03ac\u03c4\u03b5\u03c2 \u03b8\u03b1 \u03bc\u03b1\u03b6\u03ad\u03c8\u03bf\u03c5\u03bd \u03cc\u03bb\u03b5\u03c2 \u03c4\u03b9\u03c2 \u03bd\u03c4\u03bf\u03bc\u03ac\u03c4\u03b5\u03c2.\u0386\u03c1\u03b1 125xt = 12500 \u21d4 xt = 100 \u21d4 t = 100 . (1) x\u0391\u03bd \u039a \u03b5\u03af\u03bd\u03b1\u03b9 \u03c3\u03c5\u03bd\u03bf\u03bb\u03b9\u03ba\u03cc \u03ba\u03cc\u03c3\u03c4\u03bf\u03c2 \u03b3\u03b9\u03b1 \u03c4\u03b7\u03bd \u03b5\u03c1\u03b3\u03b1\u03c3\u03af\u03b1, \u03c4\u03cc\u03c4\u03b5 \u03ad\u03c7\u03bf\u03c5\u03bc\u03b5 \u039a = 6t \u22c5 x +10t +10(x +1).\u0388\u03c4\u03c3\u03b9, \u03bb\u03cc\u03b3\u03c9 \u03c4\u03b7\u03c2 (1), \u03b5\u03af\u03bd\u03b1\u03b9 \u039a (x) = 6 \u22c5100 \u22c5 x +10 \u22c5100 +10(x +1) xx\u03b4\u03b7\u03bb\u03b1\u03b4\u03ae \u039a (x) = 600 + 1000 +10x +10. x\u0397 \u03c3\u03c5\u03bd\u03ac\u03c1\u03c4\u03b7\u03c3\u03b7 \u039a \u03b5\u03af\u03bd\u03b1\u03b9 \u03c0\u03b1\u03c1\u03b1\u03b3\u03c9\u03b3\u03af\u03c3\u03b9\u03bc\u03b7 \u03b3\u03b9\u03b1 x > 0 \u03bc\u03b5 \u039a \u2032( x) = \u2212 1000 + 10 = 10 x 2 \u22121000 = 10(x2 \u2212100) . x2 x2 x2\u0395\u03af\u03bd\u03b1\u03b9 \u039a \u2032(x) = 0 \u21d4 x2 \u2212100 = 0 \u21d4 x = 10, \u03b1\u03c6\u03bf\u03cd x > 0.\u03a4\u03bf \u03c0\u03c1\u03cc\u03c3\u03b7\u03bc\u03bf \u03c4\u03b7\u03c2 \u039a\u2032, \u03ba\u03b1\u03b8\u03ce\u03c2 \u03b7 \u03bc\u03bf\u03bd\u03bf\u03c4\u03bf\u03bd\u03af\u03b1 \u03ba\u03b1\u03b9 \u03c4\u03b1 \u03b1\u03ba\u03c1\u03cc\u03c4\u03b1\u03c4\u03b1 \u03c4\u03b7\u03c2 \u039a \u03c6\u03b1\u03af\u03bd\u03bf\u03bd\u03c4\u03b1\u03b9 \u03c3\u03c4\u03bf\u03bd\u03c0\u03af\u03bd\u03b1\u03ba\u03b1. x0 10 +\u221e \u039a\u2032(x) \u2013 0+ \u039a(x) K(10) min\u0386\u03c1\u03b1, \u03b3\u03b9\u03b1 x = 10 \u03b7 \u03c3\u03c5\u03bd\u03ac\u03c1\u03c4\u03b7\u03c3\u03b7 \u03ad\u03c7\u03b5\u03b9 \u03b5\u03bb\u03ac\u03c7\u03b9\u03c3\u03c4\u03bf, \u03c4\u03bf \u039a (10) = 600 + 1000 +10 \u22c511 = 810. 10\u0395\u03c0\u03bf\u03bc\u03ad\u03bd\u03c9\u03c2, \u03bf \u03b1\u03b3\u03c1\u03cc\u03c4\u03b7\u03c2 \u03c0\u03c1\u03ad\u03c0\u03b5\u03b9 \u03bd\u03b1 \u03c0\u03c1\u03bf\u03c3\u03bb\u03ac\u03b2\u03b5\u03b9 10 \u03b5\u03c1\u03b3\u03ac\u03c4\u03b5\u03c2. \u03a4\u03bf \u03bc\u03b9\u03ba\u03c1\u03cc\u03c4\u03b5\u03c1\u03bf \u03b4\u03c5\u03bd\u03b1\u03c4\u03cc\u03ba\u03cc\u03c3\u03c4\u03bf\u03c2 \u03b5\u03af\u03bd\u03b1\u03b9 810 \u03b5\u03c5\u03c1\u03ce, \u03b5\u03bd\u03ce \u03c7\u03c1\u03b5\u03b9\u03ac\u03b6\u03bf\u03bd\u03c4\u03b1=\u03b9 t 1=00 10 \u03ce\u03c1\u03b5\u03c2 \u03b3\u03b9\u03b1 \u03bd\u03b1 \u03bc\u03b1\u03b6\u03b5\u03c5\u03c4\u03bf\u03cd\u03bd x\u03bf\u03b9 \u03bd\u03c4\u03bf\u03bc\u03ac\u03c4\u03b5\u03c2. 171","","\u039a\u0395\u03a6\u0391\u039b\u0391\u0399\u039f 3 \u039f\u039b\u039f\u039a\u039b\u0397\u03a1\u03a9\u03a4\u0399\u039a\u039f\u03a3 \u039b\u039f\u0393\u0399\u03a3\u039c\u039f\u03a33.1 \u0391\u0384 \u039f\u039c\u0391\u0394\u0391\u03a31. \u0388\u03c7\u03bf\u03c5\u03bc\u03b5:i) \u222b (x3 + \u03b7\u00b5x + \u03c3\u03c5\u03bdx)dx = \u222b x3dx + \u222b \u03b7\u00b5xdx + \u222b \u03c3\u03c5\u03bdxdx = x4 \u2212 \u03c3\u03c5\u03bdx + \u03b7\u00b5x + c 4ii) \u222b x2 + x +1 dx = \u222b xdx + \u222b1dx + \u222b 1 dx x x = x2 + x + ln x + c 2\u222b \u222biii) 3x 3 = 3 x 3 +1 + c = 6 5 +c xdx = 3 3 2 5 x 2 dx x2 +1 2iv) \u222b x3 + 8 dx = \u222b x2dx + 2\u222b xdx + \u222b 4dx x+2 = x3 + x2 + 4x + c 3v) \u222b \uf8eb e x \u2212 3 + \u03c3\u03c5\u03bd2x \uf8f6 dx = \u222b exdx \u2212 3\u222b dx + \u222b \u03c3\u03c5\u03bd2xdx \uf8ed\uf8ec x \uf8f7\uf8f8 x = \u222b ex dx \u2212 3\u222b dx + 1 \u222b (\u03b7\u00b52 x)\u2032dx x 2 = ex \u2212 3ln x + 1 \u03b7\u00b52x + c. 2 \uf8eb 1 1 \uf8f6 1 1 \uf8ec \u03c3\u03c5\u03bd2 \u03b7\u00b52 \uf8f7 \u03c3\u03c5\u03bd \u03b7\u00b52\u222b \u222b \u222bvi)\uf8ed x \u2212 x \uf8f8 dx = 2 x dx \u2212 x dx = \u03b5\u03d5x + \u03c3\u03d5x + c 173","3.1vii) \u222b x+ 3 dx = \u222b x + 2 +1 dx = \u222b1dx + \u222b dx x+ 2 x+2 x+2 = x + ln x + 2 + c.2. \u0395\u03c0\u03b5\u03b9\u03b4\u03ae \u222b f \u2032(x)dx = f (x) + c, \u03ad\u03c7\u03bf\u03c5\u03bc\u03b5 \u03b4\u03b9\u03b1\u03b4\u03bf\u03c7\u03b9\u03ba\u03ac \u222b 1 dx = f (x) + c x \u22121 \u222b x 2 dx = f (x) + c 1 x2 = f (x) + c, 1 2 f (x) = 2 x \u2212 c.\u0395\u03c0\u03b5\u03b9\u03b4\u03ae f(9) = 1, \u03ad\u03c7\u03bf\u03c5\u03bc\u03b5 2 9 \u2212 c = 1, \u03bf\u03c0\u03cc\u03c4\u03b5 c = 5. \u0395\u03c0\u03bf\u03bc\u03ad\u03bd\u03c9\u03c2 f (x) = 2 x \u2212 5.3. \u0395\u03c0\u03b5\u03b9\u03b4\u03ae \u222b f \u2032\u2032(x)dx = f \u2032(x) + c, \u03ad\u03c7\u03bf\u03c5\u03bc\u03b5 \u03b4\u03b9\u03b1\u03b4\u03bf\u03c7\u03b9\u03ba\u03ac: \u222b 3dx = f \u2032(x) + c, f \u2032(x) = 3x \u2212 c.\u0395\u03c0\u03b5\u03b9\u03b4\u03ae f \u2032(1) = 6 \u03ad\u03c7\u03bf\u03c5\u03bc\u03b5 3 \u2013 c = 6, \u03bf\u03c0\u03cc\u03c4\u03b5 c = \u2013 3. \u0395\u03c0\u03bf\u03bc\u03ad\u03bd\u03c9\u03c2 f \u2032(x) = 3x + 3.\u0395\u03c0\u03b5\u03b9\u03b4\u03ae \u222b f \u2032(x)dx = f (x) + c, \u03ad\u03c7\u03bf\u03c5\u03bc\u03b5 \u03b4\u03b9\u03b1\u03b4\u03bf\u03c7\u03b9\u03ba\u03ac: \u222b (3x + 3)dx = f (x) + c f (x) = 3 x2 + 3x \u2212 c. 2\u0395\u03c0\u03b5\u03b9\u03b4\u03ae f(0) = 4 \u03ad\u03c7\u03bf\u03c5\u03bc\u03b5 3 \u22c5 0 + 3 \u22c5 0 \u2212 c1 = 4, \u03bf\u03c0\u03cc\u03c4\u03b5 c1 = \u22124. \u0395\u03c0\u03bf\u03bc\u03ad\u03bd\u03c9\u03c2 2 f \u2032(x) = 3 x2 + 3x + 4. 24. \u0388\u03c7\u03bf\u03c5\u03bc\u03b5 \u03b4\u03b9\u03b1\u03b4\u03bf\u03c7\u03b9\u03ba\u03ac: \u222b f \u2032\u2032(x)dx = f \u2032(x) + c 174","3.1 \u222b (12x2 + 2)dx = f \u2032(x) + c 4x3 + 2x = f \u2032(x) + c, f \u2032(x) = 4x3 + 2x \u2212 c.\u0395\u03c0\u03b5\u03b9\u03b4\u03ae f \u2032(1) = 3, \u03ad\u03c7\u03bf\u03c5\u03bc\u03b5 4 + 2 \u2013 c = 3, \u03bf\u03c0\u03cc\u03c4\u03b5 c = 3. \u0395\u03c0\u03bf\u03bc\u03ad\u03bd\u03c9\u03c2 f \u2032(x) = 4x3 + 2x \u2212 3.\u0395\u03c0\u03af\u03c3\u03b7\u03c2 \u03ad\u03c7\u03bf\u03c5\u03bc\u03b5 \u03b4\u03b9\u03b1\u03b4\u03bf\u03c7\u03b9\u03ba\u03ac \u222b f \u2032(x)dx = f (x) + c1 \u222b (4x3 + 2x \u2212 3)dx = f (x) + c1 x4 + x2 \u2212 3x = f (x) + c1, f (x) = x4 + x2 \u2212 3x \u2212 c1.\u0395\u03c0\u03b5\u03b9\u03b4\u03ae \u03c4\u03bf \u0391(1,1) \u03b5\u03af\u03bd\u03b1\u03b9 \u03c3\u03b7\u03bc\u03b5\u03af\u03bf \u03c4\u03b7\u03c2 \u03b3\u03c1\u03b1\u03c6\u03b9\u03ba\u03ae\u03c2 \u03c0\u03b1\u03c1\u03ac\u03c3\u03c4\u03b1\u03c3\u03b7\u03c2 \u03c4\u03b7\u03c2 f, \u03ad\u03c7\u03bf\u03c5\u03bc\u03b5: f (1) = 1 \u21d4 1+1\u2212 3 \u2212 c1 = 1 \u21d4 c1 = \u22122.\u0395\u03c0\u03bf\u03bc\u03ad\u03bd\u03c9\u03c2 f (x) = x4 + x2 \u2212 3x + 2.5. \u0395\u03c0\u03b5\u03b9\u03b4\u03ae \u039d \u2032(t) = 1 t \u03ad\u03c7\u03bf\u03c5\u03bc\u03b5 \u03b4\u03b9\u03b1\u03b4\u03bf\u03c7\u03b9\u03ba\u03ac 20 e20 , \u222b \u039d \u2032(t)dt = \u039d (t) + c \u222b 1 et\/20dt = \u039d (t) + c 20 et\/20 = \u039d (t) + c \u039d (t) = et\/20 \u2212 c\u0395\u03c0\u03bf\u03bc\u03ad\u03bd\u03c9\u03c2, \u03b7 \u03b1\u03cd\u03be\u03b7\u03c3\u03b7 \u03c4\u03bf\u03c5 \u03c0\u03bb\u03b7\u03b8\u03c5\u03c3\u03bc\u03bf\u03cd \u03c3\u03c4\u03b1 \u03c0\u03c1\u03ce\u03c4\u03b1 60 \u03bb\u03b5\u03c0\u03c4\u03ac, \u03b5\u03af\u03bd\u03b1\u03b9 \u03af\u03c3\u03b7 \u03bc\u03b5: \u039d (60) \u2212 \u039d (0) = (e60\/20 \u2212 c) \u2212 (e0 \u2212 c) = e3 \u22121 \u2245 19 \u03b5\u03ba\u03b1\u03c4\u03bf\u03bc.6. \u0391\u0007 \u03bd \u039a(x) \u03c4\u03bf \u03ba\u03cc\u03c3\u03c4\u03bf\u03c2, \u03c3\u03b5 \u03b5\u03c5\u03c1\u03ce, \u03c4\u03b7\u03c2 \u03b5\u03b2\u03b4\u03bf\u03bc\u03b1\u03b4\u03b9\u03b1\u03af\u03b1\u03c2 \u03c0\u03b1\u03c1\u03b1\u03b3\u03c9\u03b3\u03ae\u03c2 x, \u03c4\u03cc\u03c4\u03b5 \u039a \u2032(x) = x2 + 5x, \u03bf\u03c0\u03cc\u03c4\u03b5 \u03ad\u03c7\u03bf\u03c5\u03bc\u03b5 \u222b \u039a \u2032(x)dx = \u039a (x) + c \u03ae \u222b (x2 + 5x)dx = \u039a (x) + c, 175","3.1\u03bf\u03c0\u03cc\u03c4\u03b5 \u039a (x) = x3 + 5 x2 \u2212 c. 32\u0391\u03c0\u03cc \u03c4\u03b1 \u03b4\u03b5\u03b4\u03bf\u03bc\u03ad\u03bd\u03b1 \u03c4\u03bf\u03c5 \u03c0\u03c1\u03bf\u03b2\u03bb\u03ae\u03bc\u03b1\u03c4\u03bf\u03c2 \u03ad\u03c7\u03bf\u03c5\u03bc\u03b5 \u039a(0) = 100, \u03bf\u03c0\u03cc\u03c4\u03b5 \u2013 c = 100 \u03ba\u03b1\u03b9 \u03ac\u03c1\u03b1c = \u2013 100. \u0395\u03c0\u03bf\u03bc\u03ad\u03bd\u03c9\u03c2, \u03b7 \u03c3\u03c5\u03bd\u03ac\u03c1\u03c4\u03b7\u03c3\u03b7 \u03ba\u03cc\u03c3\u03c4\u03bf\u03c5\u03c2 \u03c4\u03b7\u03c2 \u03b5\u03b2\u03b4\u03bf\u03bc\u03b1\u03b4\u03b9\u03b1\u03af\u03b1\u03c2 \u03c0\u03b1\u03c1\u03b1\u03b3\u03c9\u03b3\u03ae\u03c2 \u03b5\u03af\u03bd\u03b1\u03b9:7. \u0388\u03c7\u03bf\u03c5\u03bc\u03b5 \u03b4\u03b9\u03b1\u03b4\u03bf\u03c7\u03b9\u03ba\u03ac: \u039a (x) = x3 + 5x2 +100. 32 \u222b R\u2032(t)dt = R(t) + c \u222b \uf8eb 20 + 10t \u2212 3 t2 \uf8f6 dt = R(t) + c, \uf8ec\uf8ed 4 \uf8f8\uf8f7 R(t) = 20t + 5t2 \u2212 1 t3 \u2212 c. 4\u03a0\u03c1\u03bf\u03c6\u03b1\u03bd\u03ce\u03c2 R(0) = 0, \u03bf\u03c0\u03cc\u03c4\u03b5 c = 0 \u03ba\u03b1\u03b9 \u03ac\u03c1\u03b1 R(t) = 20t + 5t2 \u2212 1 t3. 4\u0395\u03c0\u03bf\u03bc\u03ad\u03bd\u03c9\u03c2 \u03c4\u03b1 \u03b2\u03b1\u03c1\u03ad\u03bb\u03b9\u03b1 \u03c0\u03bf\u03c5 \u03b8\u03b1 \u03b1\u03bd\u03c4\u03bb\u03b7\u03b8\u03bf\u03cd\u03bd \u03c3\u03c4\u03bf\u03c5\u03c2 \u03c0\u03c1\u03ce\u03c4\u03bf\u03c5\u03c2 8 \u03bc\u03ae\u03bd\u03b5\u03c2 \u03b5\u03af\u03bd\u03b1\u03b9: R(8) = 20 \u22c58 + 5 \u22c582 \u2212 1 83 = 160 + 5 \u22c582 \u2212 2 \u22c582 = 352 \u03c7\u03b9\u03bb\u03b9\u03ac\u03b4\u03b5\u03c2. 43.1 \u0392\u0384 \u039f\u039c\u0391\u0394\u0391\u03a31. \u0395\u03c0\u03b5\u03b9\u03b4\u03ae T \u2032(t) = \u2212k\u03b1 e\u2212kt ,\u03ad\u03c7\u03bf\u03c5\u03bc\u03b5 \u03b4\u03b9\u03b1\u03b4\u03bf\u03c7\u03b9\u03ba\u03ac: \u222bT \u2032(t)dt = T (t) + c \u222b \u2212k\u03b1 e\u2212kt dt = T (t) + c \u222b\u03b1 (e\u2212kt )\u2032dt = T (t) + c, T (t) = \u03b1 e\u2212kt \u2212 c.\u0395\u03c0\u03b5\u03b9\u03b4\u03ae \u03a4(0) + \u03b1 \u03ba\u03b1\u03b9 T (0) = \u03b1 e\u2212k\u22c50 \u2212 c = \u03b1 \u2212 c, \u03ad\u03c7\u03bf\u03c5\u03bc\u03b5\u0395\u03c0\u03bf\u03bc\u03ad\u03bd\u03c9\u03c2 T0 + \u03b1 = \u03b1 \u2212 c \u21d4 c = \u2212T0. T (t) = \u03b1 e\u2212kt + T0. 176","3.12. \u0388\u03c7\u03bf\u03c5\u03bc\u03b5 \u03b4\u03b9\u03b1\u03b4\u03bf\u03c7\u03b9\u03ba\u03ac \u222b P\u2032(x)dx = P(x) + c \u2212x \u222b 5,8e 2000 dx = P(x) + c \u222b5,8\u22c5 (\u22122000) \u2212x )\u2032dx = P(x) + c (e 2000 \u2212x P(x) = \u221211.600e 2000 \u2212 c.\u03a4\u03bf \u03c3\u03c5\u03bd\u03bf\u03bb\u03b9\u03ba\u03cc \u03ba\u03ad\u03c1\u03b4\u03bf\u03c2 \u03c0\u03bf\u03c5 \u03bf\u03c6\u03b5\u03af\u03bb\u03b5\u03c4\u03b1\u03b9 \u03c3\u03c4\u03b7\u03bd \u03b1\u03cd\u03be\u03b7\u03c3\u03b7 \u03c4\u03b7\u03c2 \u03b5\u03c0\u03ad\u03bd\u03b4\u03c5\u03c3\u03b7\u03c2 \u03b1\u03c0\u03cc 4.000.000\u03c3\u03b5 6.000.000 \u03b5\u03af\u03bd\u03b1\u03b9: = 11600(e\u22122 \u2212 e\u22123 ) = 11600 \uf8eb e \u22121 \uf8f6 \uf8ed\uf8ec e3 \uf8f8\uf8f7 \u2245 11600 \u22c5 0, 086 = 997, 6 \u03c7\u03b9\u03bb\u03b9\u03ac\u03b4\u03b5\u03c2 \u03b5\u03c5\u03c1\u03ce = 997.600 \u03b5\u03c5\u03c1\u03ce3. \u0388\u03c3\u03c4\u03c9 P(t) \u03c4\u03bf \u03ba\u03ad\u03c1\u03b4\u03bf\u03c2 \u03c4\u03b7\u03c2 \u03b5\u03c4\u03b1\u03b9\u03c1\u03b5\u03af\u03b1\u03c2 \u03c3\u03c4\u03b9\u03c2 \u03c0\u03c1\u03ce\u03c4\u03b5\u03c2 t \u03b7\u03bc\u03ad\u03c1\u03b5\u03c2. \u03a4\u03cc\u03c4\u03b5 \u03a1(t) = \u0395(t) \u2013 \u039a(t),\u03bf\u03c0\u03cc\u03c4\u03b5 P\u2032(t) = \u0395 \u2032(t) \u2212 \u039a \u2032(t) = 1000 + 0,3t \u2212 800 + 0, 6t = 200 + 0,9t.\u0388\u03c4\u03c3\u03b9 \u03ad\u03c7\u03bf\u03c5\u03bc\u03b5 \u03b4\u03b9\u03b1\u03b4\u03bf\u03c7\u03b9\u03ba\u03ac: \u222b P\u2032(t)dt = P(t) + c \u222b (200 + 0,9t)dt = P(t) + c P(t) = 200t + 0, 9 t2 + c1. 2\u03a4\u03bf \u03c3\u03c5\u03bd\u03bf\u03bb\u03b9\u03ba\u03cc \u03ba\u03ad\u03c1\u03b4\u03bf\u03c2 \u03c4\u03b7\u03c2 \u03b5\u03c4\u03b1\u03b9\u03c1\u03b5\u03af\u03b1\u03c2 \u03b1\u03c0\u03cc \u03c4\u03b7\u03bd 3\u03b7 \u03ad\u03c9\u03c2 \u03c4\u03b7\u03bd 6\u03b7 \u03b7\u03bc\u03ad\u03c1\u03b1 \u03b5\u03af\u03bd\u03b1\u03b9: P(6) \u2212 P(2) = 200 \u22c5 6 + 0, 9 62 + c1 \u2212 200 \u22c5 2 \u2212 0, 9 22 \u2212 c1 2 2 = 1216, 2 \u2212 401,8 = 814, 4 \u03b5\u03c5\u03c1\u03ce.4. i) \u0391\u03c0\u03cc \u03c4\u03b7\u03bd \u03b9\u03c3\u03cc\u03c4\u03b7\u03c4\u03b1 f \u2033(x) = g\u2033(x) \u03ad\u03c7\u03bf\u03c5\u03bc\u03b5 \u03b4\u03b9\u03b1\u03b4\u03bf\u03c7\u03b9\u03ba\u03ac (1) f \u2032(x) = g\u2032(x) + c1 f (x) = g(x) + c1x + c2. 177","3.2\u0393\u03b9\u03b1 x = 0 \u03b5\u03af\u03bd\u03b1\u03b9 f(0) = g(0) + 0 + c2, \u03bf\u03c0\u03cc\u03c4\u03b5 c2 = 0, \u03b1\u03c6\u03bf\u03cd f(0) = g(0). (2)\u0395\u03c0\u03bf\u03bc\u03ad\u03bd\u03c9\u03c2 f (x) = g(x) + c1x.\u0393\u03b9\u03b1 x = l, \u03b1\u03c0\u03cc \u03c4\u03b7\u03bd (2), \u03ad\u03c7\u03bf\u03c5\u03bc\u03b5 f(1) = g(1) + c1, \u03bf\u03c0\u03cc\u03c4\u03b5 c1 = 1, \u03b1\u03c6\u03bf\u03cd f(1) = g(1)+ 1. \u0388\u03c4\u03c3\u03b9 \u03b1\u03c0\u03cc \u03c4\u03b7 (2) \u03c0\u03c1\u03bf\u03ba\u03cd\u03c0\u03c4\u03b5\u03b9 f (x) = g(x) + x.ii) H f(x) \u03b5\u03af\u03bd\u03b1\u03b9 \u03c3\u03c5\u03bd\u03b5\u03c7\u03ae\u03c2 \u03c3\u03c4\u03bf [\u03b1,\u03b2] \u03ba\u03b1\u03b9 \u03b9\u03c3\u03c7\u03cd\u03b5\u03b9 f (\u03b1 ) = g(\u03b1 ) + \u03b1 = 0 + \u03b1 = \u03b1 < 0 f (\u03b2 ) = g(\u03b2 ) + \u03b2 = 0 + \u03b2 = \u03b2 > 0.\u0386\u03c1\u03b1, f(\u03b1) f(\u03b2) < 0, \u03bf\u03c0\u03cc\u03c4\u03b5, \u03c3\u03cd\u03bc\u03c6\u03c9\u03bd\u03b1 \u03bc\u03b5 \u03c4\u03bf \u03b8\u03b5\u03ce\u03c1\u03b7\u03bc\u03b1 Bolzano, \u03c5\u03c0\u03ac\u03c1\u03c7\u03b5\u03b9\u03c4\u03bf\u03c5\u03bb\u03ac\u03c7\u03b9\u03c3\u03c4\u03bf\u03bd \u03bc\u03af\u03b1 \u03c1\u03af\u03b6\u03b1 \u03c3\u03c4\u03bf (\u03b1,\u03b2).3.2 \u0391\u0384 \u039f\u039c\u0391\u0394\u0391\u03a31. \u0388\u03c7\u03bf\u03c5\u03bc\u03b5 \u222b \u222bi) x2e\u2212xdx = \u2212 x2 (e\u2212x )\u2032dx \u222b \u222b= \u2212x2e\u2212x + 2xe\u2212xdx = \u2212x2e\u2212x \u2212 2 x(e\u2212x )\u2032dx \u222b= \u2212x2e\u2212x \u2212 2xe\u2212x + 2 e\u2212xdx = \u2212x2e\u2212x \u2212 2xe\u2212x \u2212 2e\u2212x + c = \u2212e\u2212x (x2 + 2x + 2) + c.\u222b \u222bii) (3x2 \u2212 2x +1)e2xdx = 1 (3x2 \u2212 2x +1)(e2x )\u2032dx 2 \u222b= 1 (3x2 \u2212 2x +1)e2x \u2212 1 (6x \u2212 2)e2xdx 22 \u222b= 1 (3x2 \u2212 2x +1)e2x \u2212 1 (6x \u2212 2)(e2x )\u2032dx 24 \u222b= 1 (3x2 \u2212 2x +1)e2x \u2212 1 (6x \u2212 2)e2x + 1 6e2xdx 2 44 = 1 e2x (6x2 \u2212 4x + 2 \u2212 6x + 2) + 3 e2x + c 44 = 1 e2x (6x2 \u221210x + 7) + c 4 178","3.2iii) \u222b x3 ln xdx = \u222b \uf8eb x4 \uf8f6\u2032 ln xdx \uf8ec 4 \uf8f7 \uf8ed \uf8f8 \u222b= x4 ln x \u2212 x4 (ln x)\u2032dx 44 \u222b= 1 x4 ln x \u2212 1 x3dx = 1 x4 ln x \u2212 x4 + c 4 4 4 16iv) \u222b 2x2\u03b7\u00b52xdx = \u222b x2 (\u03c3\u03c5\u03bd2x)\u2032dx = \u2212x2\u03c3\u03c5\u03bd2x + \u222b 2x\u03c3\u03c5\u03bd2xdx = \u2212x2\u03c3\u03c5\u03bd2x + \u222b x(\u03b7\u00b52x)\u2032dx = \u2212x2\u03c3\u03c5\u03bd2x + x\u03b7\u00b52x \u2212 \u222b \u03b7\u00b52xdx = \u2212x2\u03c3\u03c5\u03bd2x + x\u03b7\u00b52x + 1 \u03c3\u03c5\u03bd2x + c 2 = \uf8eb \u2212x2 + 1 \uf8f6 \u03c3\u03c5\u03bd2x + x\u03b7\u00b52x + c \uf8ec\uf8ed 2 \uf8f7\uf8f8v) \u222b 4x\u03c3\u03c5\u03bd2xdx = \u222b 2x \u22c5 (\u03b7\u00b52x)\u2032dx = 2x\u03b7\u00b52x \u2212 \u222b 2\u03b7\u00b52xdx = 2x\u03b7\u00b52x + \u03c3\u03c5\u03bd2x + cvi) \u222b ln xdx = \u222b (x)\u2032ln xdx = x ln x \u2212 \u222b1dx = x ln x \u2212 x + cvii) \u222b ln x dx = \u2212\u222b \uf8eb 1 \uf8f6\u2032 ln xdx = \u2212 1 ln x+ \u222b 1 dx = \u2212 ln x \u2212 1 +c x2 \uf8ed\uf8ec x \uf8f7\uf8f8 x x2 x x\u222b \u222bviii) I = ex\u03c3\u03c5\u03bd2xdx = ex\u03c3\u03c5\u03bd2x + 2 ex\u03b7\u00b52xdx \u222b= ex\u03c3\u03c5\u03bd2x + 2ex\u03b7\u00b52x \u2212 4 ex\u03c3\u03c5\u03bd2xdx. 179","3.2 \u0386\u03c1\u03b1 I = ex (\u03c3\u03c5\u03bd2x + 2\u03b7\u00b52x) \u2212 4I 5I = ex (\u03c3\u03c5\u03bd2x + 2\u03b7\u00b52x) I = 1 ex (\u03c3\u03c5\u03bd2x + 2\u03b7\u00b52x) + c. 52. i) \u0398\u03ad\u03c4\u03bf\u03c5\u03bc\u03b5 u = 3x, \u03bf\u03c0\u03cc\u03c4\u03b5 du = 3dx \u03ba\u03b1\u03b9 \u03ac\u03c1\u03b1 dx = 1 du. \u0395\u03c0\u03bf\u03bc\u03ad\u03bd\u03c9\u03c2, 3 \u222b \u03b7\u00b53xdx = 1 \u222b \u03b7\u00b5udu = \u2212 1 \u03c3\u03c5\u03bdu + c = \u2212 1 \u03c3\u03c5\u03bd3x + c 3 3 3 ii) \u0398\u03ad\u03c4\u03bf\u03c5\u03bc\u03b5 u = 4x2 \u221216x + 7, \u03bf\u03c0\u03cc\u03c4\u03b5 du = (8x \u221216)dx = 8(x \u2212 2)dx. \u0395\u03c0\u03bf\u03bc\u03ad\u03bd\u03c9\u03c2 \u222b \u222b(4x2 \u221216x + 7)3 (x \u2212 2)dx = 1 u3du = 1 u4 + c 8 84 = 1 (4x2 \u221216x + 7)4 + c. 32 iii) \u0398\u03ad\u03c4\u03bf\u03c5\u03bc\u03b5 u = x2 + 6x, \u03bf\u03c0\u03cc\u03c4\u03b5 du = (2x + 6)dx = 2(x + 3)dx. \u0395\u03c0\u03bf\u03bc\u03ad\u03bd\u03c9\u03c2, \u222b \u222b \u222bx + 3 dx = 1 du = 1 u\u22124du = 1 u\u22123 + c 2 u4 2 2 \u22123 (x2 + 6x)4 = \u2212 1 1 + c = \u22121 + c. 6 u3 6(x2 + 6x)3 iv) \u0398\u03ad\u03c4\u03bf\u03c5\u03bc\u03b5 u = 2 + x3, \u03bf\u03c0\u03cc\u03c4\u03b5 du = 3x2dx. \u0395\u03c0\u03bf\u03bc\u03ad\u03bd\u03c9\u03c2, \u222b \u222b \u222bx2 \u22121 1 2 1 dx = 1 du = 1 = 2 + c = (2 + + c. u 2 du u2 x3 )2 2 + x3 3 u3 33 v) \u0398\u03ad\u03c4\u03bf\u03c5\u03bc\u03b5 u = x + l, \u03bf\u03c0\u03cc\u03c4\u03b5 du = dx \u03ba\u03b1\u03b9 x = u \u2013 1. \u0395\u03c0\u03bf\u03bc\u03ad\u03bd\u03c9\u03c2, 31 \u222b x x +1dx = \u222b (u \u22121) udu = \u222b u 2 du \u2212 \u222b u 2 du = 2 u 5 \u2212 2 u 3 +c 2 2 53 = 2u 3 \uf8eb 1 u \u2212 1 \uf8f6 + c 2 \uf8ec\uf8ed 5 3 \uf8f7\uf8f8 180","3.2 = 2 3 15 (x +1)2 (3x \u2212 2) + c.3. i) \u0398\u03ad\u03c4\u03bf\u03c5\u03bc\u03b5 u = ex, \u03bf\u03c0\u03cc\u03c4\u03b5 du = exdx. \u0395\u03c0\u03bf\u03bc\u03ad\u03bd\u03c9\u03c2, \u222b \u222bex\u03b7\u00b5exdx = \u03b7\u00b5udu = \u2212\u03c3\u03c5\u03bdu + c = \u2212\u03c3\u03c5\u03bdex + cii) \u0398\u03ad\u03c4\u03bf\u03c5\u03bc\u03b5 u = ex + 1, \u03bf\u03c0\u03cc\u03c4\u03b5 du = exdx. \u0395\u03c0\u03bf\u03bc\u03ad\u03bd\u03c9\u03c2, \u222b \u222bex 1 dx = du = ln u + c = ln(ex +1) + c u ex +iii) \u0398\u03ad\u03c4\u03bf\u03c5\u03bc\u03b5 u = ln x, \u03bf\u03c0\u03cc\u03c4\u03b5 du = 1 dx. \u0395\u03c0\u03bf\u03bc\u03ad\u03bd\u03c9\u03c2, x 1 \u222b \u222b \u222bdx = du = \u22121 = u2 +c u 1 x ln x u 2 du 2 = 2 u + c = 2 ln x + civ) \u0398\u03ad\u03c4\u03bf\u03c5\u03bc\u03b5 u = ln(ex + 1), \u03bf\u03c0\u03cc\u03c4\u03b5 du = e ex 1 dx. \u0395\u03c0\u03bf\u03bc\u03ad\u03bd\u03c9\u03c2, x+ ex du = ln u + c +1) ln(ex u \u222b \u222b(ex dx = + 1) = ln ln(ex +1) + c = ln(ln(ex +1)) + c \u03b1\u03c6\u03bf\u03cd ln(ex + 1) > ln1 = 0.v) \u0398\u03ad\u03c4\u03bf\u03c5\u03bc\u03b5 u = 1 , \u03bf\u03c0\u03cc\u03c4\u03b5 du = \u2212 1 dx. \u0395\u03c0\u03bf\u03bc\u03ad\u03bd\u03c9\u03c2, x x2 \u03b7\u00b5 \uf8eb 1 \uf8f6 1 \uf8ed\uf8ec x \uf8f8\uf8f7 x \u222b dx = \u2212\u222b \u03b7\u00b5udu = \u03c3\u03c5\u03bdu + c = \u03c3\u03c5\u03bd + c. x2 181","3.23.2 \u0392\u0384 \u039f\u039c\u0391\u0394\u0391\u03a31. i) \u0398\u03ad\u03c4\u03bf\u03c5\u03bc\u03b5 u = 1+ \u03c3\u03c5\u03bd2 x, \u03bf\u03c0\u03cc\u03c4\u03b5 du = \u2013 2\u03b7\u03bcx\u03c3\u03c5\u03bdxdx \u03ae du = \u2013 \u03b7\u03bc2xdx. \u0395\u03c0\u03bf\u03bc\u03ad\u03bd\u03c9\u03c2, \u222b 1 \u03b7\u00b52x x dx = \u2212\u222b du = \u2212 ln u +c = \u2212 ln(1+ \u03c3\u03c5\u03bd2 x) + c + \u03c3\u03c5\u03bd2 uii) \u0398\u03ad\u03c4\u03bf\u03c5\u03bc\u03b5 u = ln(\u03c3\u03c5\u03bdx), \u03bf\u03c0\u03cc\u03c4\u03b5 du = \u2212 \u03b7\u00b5x dx = \u2212\u03b5\u03d5xdx. \u0395\u03c0\u03bf\u03bc\u03ad\u03bd\u03c9\u03c2, \u03c3\u03c5\u03bdx \u222b \u03b5\u03d5x ln(\u03c3\u03c5\u03bdx)dx = \u2212\u222b udu = \u2212 u2 +c = \u2212 1 [ln(\u03c3\u03c5\u03bdx)]2 +c 2 2iii) \u0398\u03ad\u03c4\u03bf\u03c5\u03bc\u03b5 u = \u03b7\u03bcx, \u03bf\u03c0\u03cc\u03c4\u03b5 du = \u03c3\u03c5\u03bdxdx. \u0395\u03c0\u03bf\u03bc\u03ad\u03bd\u03c9\u03c2, \u222b \u222b\u03c3\u03c5\u03bdxe\u03b7\u00b5xdx = eu du = eu + c = e\u03b7\u00b5x + c.2. i) \u0398\u03ad\u03c4\u03bf\u03c5\u03bc\u03b5 u = x3 + 1 , \u03bf\u03c0\u03cc\u03c4\u03b5 du = 3x2 \u22c5 x3 \u2212 3x2 (x3 +1) dx = \u22123 dx. \u0395\u03c0\u03bf\u03bc\u03ad\u03bd\u03c9\u03c2, x3 x6 x4 \u222b \u222b \u222bx3 +1 1 x3 u 2 du 1 dx = \u2212 1 udu = \u2212 1 x4 3 3 1 +1 3 3 = \u22121 u 2 + c = \u2212 1 u2 +c = \u2212 2 \uf8eb x3 +1\uf8f62 + c. 3 1 3 3 9 \uf8ec \uf8f7 + 1 \uf8ed x3 \uf8f8 22ii) \u0398\u03ad\u03c4\u03bf\u03c5\u03bc\u03b5 u = x2 +1, \u03bf\u03c0\u03cc\u03c4\u03b5 du = 2x dx = x dx. \u0395\u03c0\u03bf\u03bc\u03ad\u03bd\u03c9\u03c2, 2 x2 +1 x2 +1 \u222b x dx = \u222b du = u + c = x2 +1 + c. x2 +1iii) \u0398\u03ad\u03c4\u03bf\u03c5\u03bc\u03b5 u = x2 +1, \u03bf\u03c0\u03cc\u03c4\u03b5 du = 2xdx, \u03bf\u03c0\u03cc\u03c4\u03b5 \u03ad\u03c7\u03bf\u03c5\u03bc\u03b5 \u222b x ln( x 2 + 1)dx = 1 \u222b ln udu = 1 \u222b (u)\u2032 ln udu 2 2 = 1 u ln u \u2212 1 \u222b du 2 2 = 1 u ln u \u2212 1 u + c = 1 (x2 +1) ln(x2 +1) \u2212 1 (x2 +1) + c. 222 2 182","3.2 \uf8eb x3 \uf8f6\u20323. i) \u0388\u03c7\u03bf\u03c5\u03bc\u03b5: \uf8ec 3 \uf8f7 \uf8ed \uf8f8 \u222b \u222bx2 ln x2dx = ln x 2 dx \u222b= x3 ln x2 \u2212 x3 1 (x2 )\u2032dx 3 x2 3 \u222b= 1 x3 ln x2 \u2212 2 x2dx = x3 ln x2 \u2212 2 x3 + c 33 39 x3ii) \u0388\u03c7\u03bf\u03c5\u03bc\u03b5 \u222b (ln t)2 dt = \u222b (t)\u2032(ln t)2 dt = t(ln t)2 \u2212 \u222b t2 ln t(ln t)\u2032dt = t(ln t)2 \u2212 2\u222b ln tdt = t(ln t)2 \u2212 2\u222b (t)\u2032ln tdt = t(ln t)2 \u2212 2t ln t + 2\u222b t 1 dt t = t(ln t)2 \u2212 2t ln t + 2t + ciii) \u0398\u03ad\u03c4\u03bf\u03c5\u03bc\u03b5 u = ex, \u03bf\u03c0\u03cc\u03c4\u03b5 du = exdx. \u0395\u03c0\u03bf\u03bc\u03ad\u03bd\u03c9\u03c2 \u222b \u222b \u222be2x\u03c3\u03c5\u03bdexdx = ex\u03c3\u03c5\u03bdex exdx = u\u03c3\u03c5\u03bdudu = \u222b u(\u03b7\u00b5u)\u2032du = u\u03b7\u00b5u \u2212 \u222b \u03b7\u00b5udu = u\u03b7\u00b5u + \u03c3\u03c5\u03bdu + c = ex\u03b7\u00b5ex + \u03c3\u03c5\u03bdex + c.4. i) \u0388\u03c7\u03bf\u03c5\u03bc\u03b5 \u03b7\u00b5x (\u03c3\u03c5\u03bdx)\u2032 \u03c3\u03c5\u03bdx \u03c3\u03c5\u03bdx \u222b \u03b5\u03d5xdx = \u222b dx = \u2212\u222b dx = \u2212 ln \u03c3\u03c5\u03bdx + c\u03ba\u03b1\u03b9 x \u03c3\u03c5\u03bd2 \u222b x dx = \u222b x(\u03b5\u03d5x)\u2032dx = x\u03b5\u03d5x \u2212 \u222b \u03b5\u03d5xdx = x\u03b5\u03d5x + ln \u03c3\u03c5\u03bdx + c1.ii) \u0398\u03ad\u03c4\u03bf\u03c5\u03bc\u03b5 u = \u03b7\u03bcx, \u03bf\u03c0\u03cc\u03c4\u03b5 du = \u03c3\u03c5\u03bdxdx. \u0395\u03c0\u03bf\u03bc\u03ad\u03bd\u03c9\u03c2, \u222b \u03c3\u03c5\u03bdx dx = \u222b du = \u2212 1 +c = \u2212 1 + c. \u03b7\u00b52 x u2 u \u03b7\u00b5x 183","3.2 \u0395\u03c0\u03af\u03c3\u03b7\u03c2 \u03ad\u03c7\u03bf\u03c5\u03bc\u03b5 \u222b 1+ \u03c3\u03c5\u03bdx dx = \u222b 1 x dx + \u222b \u03c3\u03c5\u03bdx dx \u03b7\u00b52 x \u03b7\u00b52 \u03b7\u00b52 x = \u2212\u03c3\u03d5x \u2212 1 + c. \u03b7\u00b5xiii) \u0388\u03c7\u03bf\u03c5\u03bc\u03b5 \u222b \u03b7\u00b53xdx = \u222b \u03b7\u00b52 x\u03b7\u00b5xdx = \u222b (1\u2212 \u03c3\u03c5\u03bd2 x)\u03b7\u00b5xdx. \u0398\u03ad\u03c4\u03bf\u03c5\u03bc\u03b5 u = \u03c3\u03c5\u03bdx, \u03bf\u03c0\u03cc\u03c4\u03b5 du = \u2013 \u03b7\u03bcxdx. \u0395\u03c0\u03bf\u03bc\u03ad\u03bd\u03c9\u03c2, \u222b \u03b7\u00b53xdx = \u2212\u222b (1\u2212 u2 )du = \u222b u2du \u2212 \u222b1du = u3 \u2212 u + c = \u03c3\u03c5\u03bd2 x \u2212 \u03c3\u03c5\u03bdx + c. 33 \u0395\u03c0\u03af\u03c3\u03b7\u03c2 \u03ad\u03c7\u03bf\u03c5\u03bc\u03b5 \u222b \u03c3\u03c5\u03bd3xdx = \u222b \u03c3\u03c5\u03bd2 x\u03c3\u03c5\u03bdxdx = \u222b (1\u2212 \u03b7\u00b52 x)\u03c3\u03c5\u03bdxdx. \u0398\u03ad\u03c4\u03bf\u03c5\u03bc\u03b5 u = \u03b7\u03bcx, \u03bf\u03c0\u03cc\u03c4\u03b5 du = \u03c3\u03c5\u03bdxdx. \u0395\u03c0\u03bf\u03bc\u03ad\u03bd\u03c9\u03c2, \u222b \u222b\u03c3\u03c5\u03bd3 xdx = (1\u2212 u2 )du = u \u2212 u3 + c = \u03b7\u00b5x \u2212 \u03b7\u00b53x + c. 335. \u0388\u03c7\u03bf\u03c5\u03bc\u03b5i) \u222b \u03b7\u00b52 xdx = \u222b 1 \u2212 \u03c3\u03c5\u03bd2x dx = 1 x\u2212 1 \u222b \u03c3\u03c5\u03bd2xdx 2 2 2 = 1 x \u2212 1 \u03b7\u00b52x + c 24ii) \u222b \u03c3\u03c5\u03bd2 xdx = \u222b 1 + \u03c3\u03c5\u03bd2 x dx = 1 x+ 1 \u03b7\u00b52x + c 2 2 4\u222b \u222biii) \u03b7\u00b52 x\u03c3\u03c5\u03bd2 xdx = 1 \u03b7\u00b52 2xdx 4 = 1 \u222b 1\u2212 \u03c3\u03c5\u03bd4x dx 4 2 = 1 x \u2212 1 \u222b \u03c3\u03c5\u03bd4xdx 8 8 = 1 x \u2212 1 \u03b7\u00b54x + c. 8 32 184","3.26. \u0388\u03c7\u03bf\u03c5\u03bc\u03b5i) \u222b \u03b7\u00b5x\u03c3\u03c5\u03bd2xdx = 1 \u222b[\u03b7\u00b5(\u2212x) + \u03b7\u00b53x]dx 2 = \u2212 1 \u222b \u03b7\u00b5xdx + 1 \u222b \u03b7\u00b53xdx 2 2 = 1 \u03c3\u03c5\u03bdx \u2212 1 \u03c3\u03c5\u03bd3x + c 26ii) \u222b \u03c3\u03c5\u03bd3x \u03c3\u03c5\u03bd5xdx = 1 \u222b (\u03c3\u03c5\u03bd2x + \u03c3\u03c5\u03bd8x)dx 2 = 1 \u222b \u03c3\u03c5\u03bd2xdx + 1 \u222b \u03c3\u03c5\u03bd8xdx 2 2 = 1 \u03b7\u00b52x + 1 \u03b7\u00b58x + c 4 16iii) \u222b \u03b7\u00b52x\u03b7\u00b54xdx = 1 \u222b (\u03c3\u03c5\u03bd2x \u2212 \u03c3\u03c5\u03bd6x)dx 2 = 1 \u03b7\u00b52x \u2212 1 \u03b7\u00b56x + c. 4 127. i) \u0388\u03c7\u03bf\u03c5\u03bc\u03b5: 2x \u22123 (x2 \u2212 3x + 2)\u2032 x2 \u2212 3x + x2 \u2212 3x + 2 \u222b \u222bx2 2 dx = dx = ln \u2212 3x +2 + c.ii) \u0388\u03c7\u03bf\u03c5\u03bc\u03b5: 3x + 2 2 = (x 3x + 2 2) , x \u2208R \u2212{1, 2}. x2 \u2212 3x + \u22121)(x \u2212 \u0391\u03bd\u03b1\u03b6\u03b7\u03c4\u03bf\u03cd\u03bc\u03b5 \u03c0\u03c1\u03b1\u03b3\u03bc\u03b1\u03c4\u03b9\u03ba\u03bf\u03cd\u03c2 \u03b1\u03c1\u03b9\u03b8\u03bc\u03bf\u03cd\u03c2 \u0391, \u0392 \u03ad\u03c4\u03c3\u03b9 \u03ce\u03c3\u03c4\u03b5: 3x + 2 = \u0391 + \u0392 , \u03b3\u03b9\u03b1 \u03ba\u03ac\u03b8\u03b5 x \u2208 R \u2212{1, 2}. (1) (x \u22121)(x \u2212 2) x \u22121 x \u2212 2 \u039c\u03b5 \u03b1\u03c0\u03b1\u03bb\u03bf\u03b9\u03c6\u03ae \u03c0\u03b1\u03c1\u03bf\u03bd\u03bf\u03bc\u03b1\u03c3\u03c4\u03ce\u03bd \u03ad\u03c7\u03bf\u03c5\u03bc\u03b5 \u03c4\u03b5\u03bb\u03b9\u03ba\u03ac: (\u0391 + \u0392)x \u2013 (2\u0391 + \u0392) = 3x + 2, \u03b3\u03b9\u03b1 \u03ba\u03ac\u03b8\u03b5 x \u2208R \u2212{1, 2} \u0397 (1) \u03b9\u03c3\u03c7\u03cd\u03b5\u03b9 \u03b3\u03b9\u03b1 \u03ba\u03ac\u03b8\u03b5 x \u2208 R \u2212{1, 2}, \u03b1\u03bd \u03ba\u03b1\u03b9 \u03bc\u03cc\u03bd\u03bf \u03b1\u03bd \uf8f1\u0391 + \u0392 = 3 2 \u21d4 \u0391 = \u22125 \u03ba\u03b1\u03b9 \u0392 = 8. \uf8f3\uf8f2\u22122\u0391 \u2212 \u0392 = 185","3.2\u0395\u03c0\u03bf\u03bc\u03ad\u03bd\u03c9\u03c2 \u222b x2 3x + 2 2 dx = \u222b \u22125 dx + \u222b x 8 2 dx \u2212 3x + x \u22121 \u2212 = \u22125ln x \u22121 + 8ln x \u2212 2 + c.iii) \u0391\u03c0\u03cc \u03c4\u03b7 \u03b4\u03b9\u03b1\u03af\u03c1\u03b5\u03c3\u03b7 (x3 \u2013 2x):(x2 + 3x + 2) \u03b2\u03c1\u03af\u03c3\u03ba\u03bf\u03c5\u03bc\u03b5: x3 \u2212 2x = (x2 + 3x + 2)(x \u2212 3) + 5x + 6\u03bf\u03c0\u03cc\u03c4\u03b5 x3 \u2212 2x 2 = x \u2212 3+ 5x + 6 2 (1) x2 + 3x + x2 + 3x + (2)\u0395\u03be\u03ac\u03bb\u03bb\u03bf\u03c5 \u03ad\u03c7\u03bf\u03c5\u03bc\u03b5: (1) 5x + 6 2 = (x 5x + 6 2) . x2 \u2212 3x + +1)(x +\u0391\u03bd\u03b1\u03b6\u03b7\u03c4\u03bf\u03cd\u03bc\u03b5 \u03c0\u03c1\u03b1\u03b3\u03bc\u03b1\u03c4\u03b9\u03ba\u03bf\u03cd\u03c2 \u03b1\u03c1\u03b9\u03b8\u03bc\u03bf\u03cd\u03c2 \u0391, \u0392 \u03ad\u03c4\u03c3\u03b9, \u03ce\u03c3\u03c4\u03b5 5x + 6 = \u0391 + \u0392 \u03b3\u03b9\u03b1 \u03ba\u03ac\u03b8\u03b5 x \u2208 R \u2212{\u22121, \u22122}. (x +1)(x + 2) x +1 x + 2\u039c\u03b5 \u03b1\u03c0\u03b1\u03bb\u03bf\u03b9\u03c6\u03ae \u03c0\u03b1\u03c1\u03bf\u03bd\u03bf\u03bc\u03b1\u03c3\u03c4\u03ce\u03bd, \u03ad\u03c7\u03bf\u03c5\u03bc\u03b5 \u03c4\u03b5\u03bb\u03b9\u03ba\u03ac (\u0391 + \u0392 )x + 2\u0391 + \u0392 = 5x + 6.\u0397 (2) \u03b9\u03c3\u03c7\u03cd\u03b5\u03b9 \u03b3\u03b9\u03b1 \u03ba\u03ac\u03b8\u03b5 x \u2208R \u2212{\u22121, \u22122}, \u03b1\u03bd \u03ba\u03b1\u03b9 \u03bc\u03cc\u03bd\u03bf \u03b1\u03bd \uf8f1\u0391 + \u0392 = 5 \u21d4 \u0391 =1 \u03ba\u03b1\u03b9 \u0392 = 4. \uf8f2\uf8f32\u0391 + \u0392 = 6\u0395\u03c0\u03bf\u03bc\u03ad\u03bd\u03c9\u03c2 \u03bb\u03cc\u03b3\u03c9 \u03ba\u03b1\u03b9 \u03c4\u03b7\u03c2 (1) \u03ad\u03c7\u03bf\u03c5\u03bc\u03b5: \u222b x3 \u2212 2x 2 dx = \u222b ( x \u2212 3)dx + \u222b dx + \u222b x 4 2 dx x2 + 3x + x +1 + = x2 \u2212 3x + ln x +1 + 4 ln x + 2 + c. 2iv) \u0388\u03c7\u03bf\u03c5\u03bc\u03b5 2 = \u0391+ \u0392, \u03b3\u03b9\u03b1 \u03ba\u03ac\u03b8\u03b5 x\u2208R \u2212{1, \u22121}. x2 \u22121 x \u22121 x +1\u039c\u03b5 \u03b1\u03c0\u03b1\u03bb\u03bf\u03b9\u03c6\u03ae \u03c0\u03b1\u03c1\u03bf\u03bd\u03bf\u03bc\u03b1\u03c3\u03c4\u03ce\u03bd \u03ad\u03c7\u03bf\u03c5\u03bc\u03b5 (\u0391 + \u0392)x + \u0391 \u2013 \u0392 = 2. 186","3.3\u0397 (1) \u03b9\u03c3\u03c7\u03cd\u03b5\u03b9 \u03b3\u03b9\u03b1 \u03ba\u03ac\u03b8\u03b5 x \u2208R \u2212{1, \u22121}, \u03b1\u03bd \u03ba\u03b1\u03b9 \u03bc\u03cc\u03bd\u03bf \u03b1\u03bd \uf8f1\u0391 + \u0392 = 0 \u21d4 \u0391 = 1 \u03ba\u03b1\u03b9 \u0392 = \u2013 1. \uf8f3\uf8f2\u0391 \u2212 \u0392 = 2\u0395\u03c0\u03bf\u03bc\u03ad\u03bd\u03c9\u03c2, \u03ad\u03c7\u03bf\u03c5\u03bc\u03b5 \u222b x 2 1 dx = \u222b 1 dx \u2212 \u222b 1 dx = ln x \u22121 \u2212 ln x +1 + c. 2\u2212 x \u22121 x +13.3 \u0391\u0384 \u039f\u039c\u0391\u0394\u0391\u03a31. i) \u0397 \u03b5\u03be\u03af\u03c3\u03c9\u03c3\u03b7 \u03b3\u03c1\u03ac\u03c6\u03b5\u03c4\u03b1\u03b9 \u03b4\u03b9\u03b1\u03b4\u03bf\u03c7\u03b9\u03ba\u03ac: dy = \u22124xy2 dx dy = \u22124xdx y2 \u222b dy = \u22124\u222b xdx y2 \u2212 1 = \u22122x2 + c1 y y = 2 1 c , c \u2208R. x2 +ii) \u0397 \u03b5\u03be\u03af\u03c3\u03c9\u03c3\u03b7 \u03b3\u03c1\u03ac\u03c6\u03b5\u03c4\u03b1\u03b9 y dy = x dx ydy = xdx \u222b ydy = \u222b xdx y2 = x2 + c1 2 2 y2 = x2 + 2c1 y2 \u2212 x2 = c, y = c + x2 , \u03b1\u03c6\u03bf\u03cd y > 0 (c \u2208R). 187","3.3iii) \u0397 \u03b5\u03be\u03af\u03c3\u03c9\u03c3\u03b7 \u03b3\u03c1\u03ac\u03c6\u03b5\u03c4\u03b1\u03b9 \u03b4\u03b9\u03b1\u03b4\u03bf\u03c7\u03b9\u03ba\u03ac: 1 dy = 2 xy dx dy = 2xdx y \u222b 1 dy = \u222b 2xdx y ln y = x2 + c1 y = ex2 +c1 y = ec1 ex2 y = \u00b1ec1 ex2 y = cex2 , \u03cc\u03c0\u03bf\u03c5 c = \u00b1ec1 . iv) \u0397 \u03b5\u03be\u03af\u03c3\u03c9\u03c3\u03b7 \u03b3\u03c1\u03ac\u03c6\u03b5\u03c4\u03b1\u03b9 \u03b4\u03b9\u03b1\u03b4\u03bf\u03c7\u03b9\u03ba\u03ac: dy = e\u2212 y\u03c3\u03c5\u03bdx dx eydy = \u03c3\u03c5\u03bdxdx \u222b eydy = \u222b \u03c3\u03c5\u03bdxdx ey = \u03b7\u00b5x + c y = ln(\u03b7\u00b5x + c), c \u2208R .2. i) \u0007\u039c\u03af\u03b1 \u03c0\u03b1\u03c1\u03ac\u03b3\u03bf\u03c5\u03c3\u03b1 \u03c4\u03b7\u03c2 \u03b1(x) = 2 \u03b5\u03af\u03bd\u03b1\u03b9 \u03b7 \u0391(x) = 2x. \u03a0\u03bf\u03bb\u03bb\u03b1\u03c0\u03bb\u03b1\u03c3\u03b9\u03ac\u03b6\u03bf\u03c5\u03bc\u03b5 \u03ba\u03b1\u03b9 \u03c4\u03b1 \u03b4\u03cd\u03bf \u03bc\u03ad\u03bb\u03b7 \u03c4\u03b7\u03c2 \u03b5\u03be\u03af\u03c3\u03c9\u03c3\u03b7\u03c2 \u03bc\u03b5 e2x \u03ba\u03b1\u03b9 \u03ad\u03c7\u03bf\u03c5\u03bc\u03b5 \u03b4\u03b9\u03b1\u03b4\u03bf\u03c7\u03b9\u03ba\u03ac: y\u2032e2x + 2e2x y = 3e2x ( ye2x )\u2032 = 3e2x \u222b \u222b( ye2x )\u2032dx = 3e2xdx ye2x = 3 e2x + c 2 188","3.3 y = 3 + ce\u22122x , c\u2208 R. 2ii) \u0007\u039c\u03af\u03b1 \u03c0\u03b1\u03c1\u03ac\u03b3\u03bf\u03c5\u03c3\u03b1 \u03c4\u03b7\u03c2 \u03b1(x) = 2 \u03b5\u03af\u03bd\u03b1\u03b9 \u03b7 A(x) = 2x. \u03a0\u03bf\u03bb\u03bb\u03b1\u03c0\u03bb\u03b1\u03c3\u03b9\u03ac\u03b6\u03bf\u03c5\u03bc\u03b5 \u03bc\u03b5 e2x \u03bf\u03c0\u03cc\u03c4\u03b5 \u03ad\u03c7\u03bf\u03c5\u03bc\u03b5 \u03b4\u03b9\u03b1\u03b4\u03bf\u03c7\u03b9\u03ba\u03ac y\u2032e2x + 2 ye2x = ex ( ye2x )\u2032 = ex \u222b \u222b( ye2x )\u2032dx = exdx ye2x = ex + c y = e\u2212x + ce\u22122x , c\u2208 R.iii) \u039c\u0007 \u03af\u03b1 \u03c0\u03b1\u03c1\u03ac\u03b3\u03bf\u03c5\u03c3\u03b1 \u03c4\u03b7\u03c2 \u03b1(x) = l \u03b5\u03af\u03bd\u03b1\u03b9 \u03b7 A(x) = x. \u03a0\u03bf\u03bb\u03bb\u03b1\u03c0\u03bb\u03b1\u03c3\u03b9\u03ac\u03b6\u03bf\u03c5\u03bc\u03b5 \u03bc\u03b5 ex, \u03bf\u03c0\u03cc\u03c4\u03b5 \u03ad\u03c7\u03bf\u03c5\u03bc\u03b5 \u03b4\u03b9\u03b1\u03b4\u03bf\u03c7\u03b9\u03ba\u03ac y\u2032ex + yex = ex \u22c5 2x ( yex )\u2032 = 2ex x \u222b ( yex )\u2032dx = 2\u222b ex \u22c5 xdx \u222byex + c1 = 2xex \u2212 2 exdx yex = 2xex \u2212 2ex + c y = 2x \u2212 2 + ce\u2212x , c\u2208 R.iv) \u039c\u0007 \u03af\u03b1 \u03c0\u03b1\u03c1\u03ac\u03b3\u03bf\u03c5\u03c3\u03b1 \u03c4\u03b7\u03c2 \u03b1(x) = 2x \u03b5\u03af\u03bd\u03b1\u03b9 \u03b7 A(x) = x2. \u03a0\u03bf\u03bb\u03bb\u03b1\u03c0\u03bb\u03b1\u03c3\u03b9\u03ac\u03b6\u03bf\u03c5\u03bc\u03b5 ex2 , \u03bf\u03c0\u03cc\u03c4\u03b5 \u03ad\u03c7\u03bf\u03c5\u03bc\u03b5 \u03b4\u03b9\u03b1\u03b4\u03bf\u03c7\u03b9\u03ba\u03ac y\u2032ex2 + 2xex2 y = xex2 ( yex2 )\u2032 = xex2 \u222byex2 + c1 = xex2 dx yex2 = 1 ex2 + c 2 y = 1 + ce\u2212x2 , c\u2208 R. 2 189","3.33. \u0397 \u03b5\u03be\u03af\u03c3\u03c9\u03c3\u03b7 \u03b3\u03c1\u03ac\u03c6\u03b5\u03c4\u03b1\u03b9 \u03b4\u03b9\u03b1\u03b4\u03bf\u03c7\u03b9\u03ba\u03ac: dy = 2x2 y2 dx dy = 2 x 2 dx y2 \u222b \u222by\u22122dy = 2 x2dx \u2212 1 = 2 x3 + c1 y 3 \u2212 1 = 2x3 + 3c1 y3 y = \u22123 c . 2x3 +\u0395\u03c0\u03b5\u03b9\u03b4\u03ae y(0) = \u2013 3, \u03ad\u03c7\u03bf\u03c5\u03bc\u03b5 \u22123 = \u22123, \u03bf\u03c0\u03cc\u03c4\u03b5 c = 1. \u0386\u03c1\u03b1 c y = 2 \u22123 1 . x3 +4. \u0007\u0397 \u03b5\u03be\u03af\u03c3\u03c9\u03c3\u03b7 \u03b3\u03c1\u03ac\u03c6\u03b5\u03c4\u03b1\u03b9 y\u2032 + 3y = 2. \u039c\u03b9\u03b1 \u03c0\u03b1\u03c1\u03ac\u03b3\u03bf\u03c5\u03c3\u03b1 \u03c4\u03b7\u03c2 \u03b1(x) = 3 \u03b5\u03af\u03bd\u03b1\u03b9 \u03b7 \u0391(x) = 3x, \u03bf\u03c0\u03cc\u03c4\u03b5 \u03ad\u03c7\u03bf\u03c5\u03bc\u03b5 \u03b4\u03b9\u03b1\u03b4\u03bf\u03c7\u03b9\u03ba\u03ac y\u2032e3x + 3ye3x = 2e3x y\u2032e3x + y(e3x )\u2032 = 2e3x ( ye3x )\u2032 = 2e3x \u222b \u222b( ye3x )\u2032dx = 2 e3xdx ye3x = 2 e3x + c 3 y = 2 + c . 3 e3x\u0395\u03c0\u03b5\u03b9\u03b4\u03ae y(0) = 2 \u03ad\u03c7\u03bf\u03c5\u03bc\u03b5 2 = 2 + c , \u03bf\u03c0\u03cc\u03c4\u03b5 c = 0. \u0386\u03c1\u03b1 y = 2. 3 3 3 e0 3 190","3.35. i) \u0007\u039c\u03b9\u03b1 \u03c0\u03b1\u03c1\u03ac\u03b3\u03bf\u03c5\u03c3\u03b1 \u03c4\u03b7\u03c2 \u03b1 ( x) = 1 \u03b5\u03af\u03bd\u03b1\u03b9 \u03b7 \u0391(x) = \u03b5\u03c6x. \u03a0\u03bf\u03bb\u03bb\u03b1\u03c0\u03bb\u03b1\u03c3\u03b9\u03ac\u03b6\u03bf\u03c5\u03bc\u03b5 \u03c3\u03c5\u03bd2 x \u03bc\u03b5 e\u03b5\u03d5x , \u03bf\u03c0\u03cc\u03c4\u03b5 \u03ad\u03c7\u03bf\u03c5\u03bc\u03b5 \u03b4\u03b9\u03b1\u03b4\u03bf\u03c7\u03b9\u03ba\u03ac: y\u2032e\u03b5\u03d5x + e\u03b5\u03d5x 1 y = 1 e\u03b5\u03d5x \u03c3\u03c5\u03bd2 x \u03c3\u03c5\u03bd2 x ( ye\u03b5\u03d5x )\u2032 = e\u03b5\u03d5x 1 \u03c3\u03c5\u03bd2 x \u222bye\u03b5\u03d5x + c1 = e\u03b5\u03d5x 1 dx \u03c3\u03c5\u03bd2 x \u222bye\u03b5\u03d5x + c1 = e\u03b5\u03d5x (\u03b5\u03d5x)\u2032dx ye\u03b5\u03d5x = e\u03b5\u03d5x + c y = 1+ ce\u2212\u03b5\u03d5x . \u0395\u03c0\u03b5\u03b9\u03b4\u03ae y(0) = \u2013 3, \u03ad\u03c7\u03bf\u03c5\u03bc\u03b5 \u2013 3 = 1 + c, \u03bf\u03c0\u03cc\u03c4\u03b5 c = \u2013 4. \u0386\u03c1\u03b1 y = 1\u2212 4 . e\u03b5\u03d5x ii) \u0395\u03c0\u03b5\u03b9\u03b4\u03ae x > 0, \u03b5\u03af\u03bd\u03b1\u03b9 x + l > 0, \u03bf\u03c0\u03cc\u03c4\u03b5 \u03b7 \u03b5\u03be\u03af\u03c3\u03c9\u03c3\u03b7 \u03b3\u03c1\u03ac\u03c6\u03b5\u03c4\u03b1\u03b9 y\u2032 + 1 y = 1 ln x. x +1 x +1 \u039c\u03af\u03b1 \u03c0\u03b1\u03c1\u03ac\u03b3\u03bf\u03c5\u03c3\u03b1 \u03c4\u03b7\u03c2 \u03b1 ( x) = 1 \u03b5\u03af\u03bd\u03b1\u03b9 \u03b7 \u0391(x) = ln(x + 1). \u03a0\u03bf\u03bb\u03bb\u03b1\u03c0\u03bb\u03b1\u03c3\u03b9\u03ac\u03b6\u03bf\u03c5\u03bc\u03b5 x +1 \u03bc\u03b5 eln(x+1) = x +1, \u03bf\u03c0\u03cc\u03c4\u03b5 \u03ad\u03c7\u03bf\u03c5\u03bc\u03b5 \u03b4\u03b9\u03b1\u03b4\u03bf\u03c7\u03b9\u03ba\u03ac y\u2032 \u22c5 (x +1) + y = ln x ( y(x +1))\u2032 = ln x y(x +1) + c1 = \u222b ln xdx y(x +1) = x ln x \u2212 x + c y = x ln x \u2212 x + c . x +1 \u0395\u03c0\u03b5\u03b9\u03b4\u03ae y(1) = 10, \u03ad\u03c7\u03bf\u03c5\u03bc\u03b5 \u22121+ c = 10, \u03bf\u03c0\u03cc\u03c4\u03b5 c = 21. \u0395\u03c0\u03bf\u03bc\u03ad\u03bd\u03c9\u03c2 2 y = x ln x \u2212 x + 21. x +1 191","3.33.3 \u0392\u0384 \u039f\u039c\u0391\u0394\u0391\u03a31. \u0007\u039c\u03af\u03b1 \u03c0\u03b1\u03c1\u03ac\u03b3\u03bf\u03c5\u03c3\u03b1 \u03c4\u03b7\u03c2 \u03b1(t) = 1 \u03b5\u03af\u03bd\u03b1\u03b9 \u03b7 A(t) = t. \u03a0\u03bf\u03bb\u03bb\u03b1\u03c0\u03bb\u03b1\u03c3\u03b9\u03ac\u03b6\u03bf\u03c5\u03bc\u03b5 \u03c4\u03b1 \u03bc\u03ad\u03bb\u03b7 \u03c4\u03b7\u03c2 \u03b5\u03be\u03af\u03c3\u03c9\u03c3\u03b7\u03c2 \u03bc\u03b5 et \u03ba\u03b1\u03b9 \u03ad\u03c7\u03bf\u03c5\u03bc\u03b5 \u03b4\u03b9\u03b1\u03b4\u03bf\u03c7\u03b9\u03ba\u03ac: I \u2032(t)et + I (t)et = et\u03b7\u00b5t (I (t)et )\u2032 = et\u03b7\u00b5t \u222bI (t)et + c1 = et\u03b7\u00b5tdt (1) \u222b \u222bet\u03b7\u00b5tdt = et\u03b7\u00b5t \u2212 et\u03c3\u03c5\u03bdtdt\u0395\u03be\u03ac\u03bb\u03bb\u03bf\u03c5 \u03ad\u03c7\u03bf\u03c5\u03bc\u03b5 \u222b= et\u03b7\u00b5t \u2212 \uf8ee\uf8f0et\u03c3\u03c5\u03bdt + et\u03b7\u00b5tdt \uf8f9\uf8fb ,\u03bf\u03c0\u03cc\u03c4\u03b5 \u222b2 et\u03b7\u00b5tdt = et (\u03b7\u00b5t \u2212 \u03c3\u03c5\u03bdt) + c1.\u0386\u03c1\u03b1 \u222b et\u03b7\u00b5tdt = 1 et (\u03b7\u00b5t \u2212 \u03c3\u03c5\u03bdt) + c, 2\u03bf\u03c0\u03cc\u03c4\u03b5 \u03b1\u03c0\u03cc \u03c4\u03b7\u03bd (1) \u03c0\u03c1\u03bf\u03ba\u03cd\u03c0\u03c4\u03b5\u03b9 \u03cc\u03c4\u03b9\u0393\u03b9\u03b1 t = 0 \u03ad\u03c7\u03bf\u03c5\u03bc\u03b5 I (t)et = 1 et (\u03b7\u00b5t \u2212 \u03c3\u03c5\u03bdt) + c. 2 I (0)e0 = 1 e0 (\u03b7\u00b50 \u2212 \u03c3\u03c5\u03bd0) + c 2 0 = \u22121 +c 2\u0388\u03c4\u03c3\u03b9, \u03c4\u03b5\u03bb\u03b9\u03ba\u03ac \u03b5\u03af\u03bd\u03b1\u03b9 c = 1. 2 I (t)et = 1 et (\u03b7\u00b5t \u2212 \u03c3\u03c5\u03bdt) + 1 22 I (t) = 1 (\u03b7\u00b5t \u2212 \u03c3\u03c5\u03bdt) + 1 e\u2212t . 22 192","3.32. \u0397 \u03b5\u03be\u03af\u03c3\u03c9\u03c3\u03b7 \u03b3\u03c1\u03ac\u03c6\u03b5\u03c4\u03b1\u03b9 \u03b4\u03b9\u03b1\u03b4\u03bf\u03c7\u03b9\u03ba\u03ac: ye y2 dy = e2x dx ye y2 dy = e2xdx\u0386\u03c1\u03b1 \u222b \u222byey2 dy = e2xdx 1 ey2 = 1 e2x + c1 2 2 e y2 = e2x + 2c1 ey2 = e2x + c, c\u2208 R.\u0395\u03c0\u03b5\u03b9\u03b4\u03ae y(2) = 2, \u03ad\u03c7\u03bf\u03c5\u03bc\u03b5 e4 = e4 + c, \u03bf\u03c0\u03cc\u03c4\u03b5 c = 0. \u0395\u03c0\u03bf\u03bc\u03ad\u03bd\u03c9\u03c2 ey2 = e2x , \u03bf\u03c0\u03cc\u03c4\u03b5 y2 =2x \u03ba\u03b1\u03b9 \u03ac\u03c1\u03b1 y = 2x, \u03b1\u03c6\u03bf\u03cd \u03c0\u03b5\u03c1\u03bd\u03ac\u03b5\u03b9 \u03b1\u03c0\u03cc \u03c4\u03bf \u03c3\u03b7\u03bc\u03b5\u03af\u03bf \u0391(2,2).3. \u0007\u039c\u03af\u03b1 \u03c0\u03b1\u03c1\u03ac\u03b3\u03bf\u03c5\u03c3\u03b1 \u03b1 (x) = \u2212 1 \u03b5\u03af\u03bd\u03b1\u03b9 \u03b7 \u0391(x) = \u2212 ln x. \u03a0\u03bf\u03bb\u03bb\u03b1\u03c0\u03bb\u03b1\u03c3\u03b9\u03ac\u03b6\u03bf\u03c5\u03bc\u03b5 \u03bc\u03b5 xe\u2212 ln x ln 1 = 1, \u03bf\u03c0\u03cc\u03c4\u03b5 \u03ad\u03c7\u03bf\u03c5\u03bc\u03b5 \u03b4\u03b9\u03b1\u03b4\u03bf\u03c7\u03b9\u03ba\u03ac x =e x y\u2032 1 \u2212 1 y = x1 x x2 x \uf8eb y \u22c5 1 \uf8f6\u2032 = 1 \uf8ec\uf8ed x \uf8f7\uf8f8 y\u22c51 = x+c x y = x2 + cx, c\u2208 R.4. \u0399\u03c3\u03c7\u03cd\u03b5\u03b9 y\u2032 = xy, y > 0, \u03bf\u03c0\u03cc\u03c4\u03b5 \u03ad\u03c7\u03bf\u03c5\u03bc\u03b5 \u03b4\u03b9\u03b1\u03b4\u03bf\u03c7\u03b9\u03ba\u03ac: dy = xdx y \u222b dy = \u222b xdx y ln y = x2 + c1, y>0 2 193","3.3 y = x2 + c1 e2 x2 y = c \u22c5 e 2 , c = ec1 > 0.\u0395\u03be\u03ac\u03bb\u03bb\u03bf\u03c5 \u03b9\u03c3\u03c7\u03cd\u03b5\u03b9 y(0) = 1, \u03bf\u03c0\u03cc\u03c4\u03b5 c = 1. \u0386\u03c1\u03b1 x2 y=e2.5. i) \u039c\u03af\u03b1 \u03c0\u03b1\u03c1\u03ac\u03b3\u03bf\u03c5\u03c3\u03b1 \u03c4\u03b7\u03c2 \u03b1(t) = \u03b1 \u03b5\u03af\u03bd\u03b1\u03b9 \u03b7 A(t) = \u03b1t, \u03bf\u03c0\u03cc\u03c4\u03b5 \u03ad\u03c7\u03bf\u03c5\u03bc\u03b5 \u03b4\u03b9\u03b1\u03b4\u03bf\u03c7\u03b9\u03ba\u03ac y\u2032e\u03b1t + \u03b1 e\u03b1t y = \u03b2 e\u2212\u03bbt \u22c5 e\u03b1t ( ye\u03b1t )\u2032 = \u03b2 e(\u03b1 \u2212\u03bb)t \u222bye\u03b1t + c1 = \u03b2 e(\u03b1 \u2212\u03bb)t dt ye\u03b1t = \u03b2 e(\u03b1 \u2212\u03bb)t + c \u03b1 \u2212\u03bb y = \u03b1 \u03b2 \u03bb e\u2212\u03bbt + c . \u2212 e\u03b1 t\u0386\u03c1\u03b1 y(t) = \u03b2 1 + c , c\u2208 R. \u03b1 \u2212\u03bb e\u03bbt e\u03b1 tii) \u0395\u03c0\u03b5\u03b9\u03b4\u03ae \u03b1 > 0, \u03bb > 0 \u03b9\u03c3\u03c7\u03cd\u03b5\u03b9 lim 1 =0 \u03ba\u03b1\u03b9 lim c = 0, \u03bf\u03c0\u03cc\u03c4\u03b5 e\u03bbt e\u03b1 t t \u2192+\u221e t \u2192+\u221e lim y(t) = 0. t \u2192+\u221e6. \u0395\u03c0\u03b5\u03b9\u03b4\u03ae \u03b8 \u2013 \u03a4 > 0 \u03b7 \u03b5\u03be\u03af\u03c3\u03c9\u03c3\u03b7 \u03b3\u03c1\u03ac\u03c6\u03b5\u03c4\u03b1\u03b9 \u03b4\u03b9\u03b1\u03b4\u03bf\u03c7\u03b9\u03ba\u03ac: d\u03b8 = \u2212\u03ba dt \u03b8 \u2212\u03a4 \u222b \u03b8 d\u03b8 = \u2212\u03bat + c1, \u2212\u03a4 ln(\u03b8 \u2212\u03a4 ) = \u2212\u03bat + c1 \u03b8 \u2212\u03a4 = e\u2212\u03bat +c1 \u03b8 (t) = T + ce\u2212\u03bat , c = ec1 . 194","3.3\u0395\u03be\u03ac\u03bb\u03bb\u03bf\u03c5 \u03b8 (0) = \u03b80 \u21d4 \u03b80 = T + c \u22c5 e0 \u21d4 c = \u03b80 \u2212 T.\u0386\u03c1\u03b1 \u03b8 (t) = T + (\u03b80 \u2212 T )e\u2212\u03bat .7. i) \u0388\u0007 \u03c3\u03c4\u03c9 \u03a11(t) \u03bf \u03c0\u03bb\u03b7\u03b8\u03c5\u03c3\u03bc\u03cc\u03c2 \u03c4\u03b7\u03c2 \u03c7\u03ce\u03c1\u03b1\u03c2, \u03b1\u03bd \u03b4\u03b5\u03bd \u03c5\u03c0\u03ae\u03c1\u03c7\u03b5 \u03b7 \u03bc\u03b5\u03c4\u03b1\u03bd\u03ac\u03c3\u03c4\u03b5\u03c5\u03c3\u03b7 \u03ba\u03b1\u03b9 P2(t) \u03bf \u03c0\u03bb\u03b7\u03b8\u03c5\u03c3\u03bc\u03cc\u03c2 \u03c0\u03bf\u03c5 \u03ad\u03c7\u03b5\u03b9 \u03bc\u03b5\u03c4\u03b1\u03bd\u03b1\u03c3\u03c4\u03b5\u03cd\u03c3\u03b5\u03b9 \u03bc\u03ad\u03c7\u03c1\u03b9 \u03c4\u03b7 \u03c7\u03c1\u03bf\u03bd\u03b9\u03ba\u03ae \u03c3\u03c4\u03b9\u03b3\u03bc\u03ae t. \u03a4\u03cc\u03c4\u03b5 \u03bf \u03c0\u03bb\u03b7\u03b8\u03c5\u03c3\u03bc\u03cc\u03c2 \u03c4\u03b7\u03c2 \u03c7\u03ce\u03c1\u03b1\u03c2 \u03b5\u03af\u03bd\u03b1\u03b9 P(t) = P1(t) \u2212 P2 (t)\u03bf\u03c0\u03cc\u03c4\u03b5 P\u2032(t) = P1\u2032 (t) \u2212 P2\u2032 (t). (1)\u0395\u03af\u03bd\u03b1\u03b9 \u03cc\u03bc\u03c9\u03c2 P1\u2032 (t) = k \u22c5 P(t), k > 0,\u03b1\u03c6\u03bf\u03cd \u03ad\u03c7\u03bf\u03c5\u03bc\u03b5 \u03c1\u03c5\u03b8\u03bc\u03cc \u03b1\u03cd\u03be\u03b7\u03c3\u03b7\u03c2 \u03c4\u03bf\u03c5 \u03a11(t) \u03b1\u03bd\u03ac\u03bb\u03bf\u03b3\u03bf \u03c4\u03bf\u03c5 P(t).\u0395\u03c0\u03af\u03c3\u03b7\u03c2 \u03b5\u03af\u03bd\u03b1\u03b9 P2\u2032 (t) = m, \u03bf\u03c0\u03cc\u03c4\u03b5 \u03b7 (1) \u03b3\u03c1\u03ac\u03c6\u03b5\u03c4\u03b1\u03b9 P\u2032(t) = kP(t) \u2212 m,\u03ae \u03b9\u03c3\u03bf\u03b4\u03cd\u03bd\u03b1\u03bc\u03b1 P\u2032 \u2212 kP = \u2212m.\u039c\u03af\u03b1 \u03c0\u03b1\u03c1\u03ac\u03b3\u03bf\u03c5\u03c3\u03b1 \u03c4\u03b7\u03c2 \u03b1(t) = \u2013 k \u03b5\u03af\u03bd\u03b1\u03b9 \u03b7 A(t) = \u2013 kt. \u03a0\u03bf\u03bb\u03bb\u03b1\u03c0\u03bb\u03b1\u03c3\u03b9\u03ac\u03b6\u03bf\u03c5\u03bc\u03b5 \u03bc\u03b5 e\u2013kt\u03c4\u03b1 \u03bc\u03ad\u03bb\u03b7 \u03c4\u03b7\u03c2 \u03b5\u03be\u03af\u03c3\u03c9\u03c3\u03b7\u03c2, \u03bf\u03c0\u03cc\u03c4\u03b5 \u03ad\u03c7\u03bf\u03c5\u03bc\u03b5 \u03b4\u03b9\u03b1\u03b4\u03bf\u03c7\u03b9\u03ba\u03ac: P\u2032e\u2212kt \u2212 ke\u2212kt P = \u2212me\u2212kt (Pe\u2212kt )\u2032 = \u2212me\u2212kt \u222bPe\u2212kt + c1 = \u2212m e\u2212kt dt Pe\u2212kt = m e\u2212kt + c k P(t) = m + cekt . k\u0395\u03c0\u03b5\u03b9\u03b4\u03ae P(0) = P0, \u03ad\u03c7\u03bf\u03c5\u03bc\u03b5 P0 = m + c, \u03bf\u03c0\u03cc\u03c4\u03b5 c = P0 \u2212 m. \u0386\u03c1\u03b1 k k 195","3.3 P(t) = m + \uf8eb P0 \u2212 m \uf8f6 ekt , k>0 k \uf8ec\uf8ed k \uf8f8\uf8f7iii) \u0395\u03af\u03bd\u03b1\u03b9 P\u2032(t) = (kP0 \u2212 m)ekt \u2014 \u03b1\u03bd m < kP0 \u03c4\u03cc\u03c4\u03b5 P\u2032(t) > 0, \u03bf\u03c0\u03cc\u03c4\u03b5 \u03bf \u03c0\u03bb\u03b7\u03b8\u03c5\u03c3\u03bc\u03cc\u03c2 \u03b1\u03c5\u03be\u03ac\u03bd\u03b5\u03c4\u03b1\u03b9. \u2014 \u03b1\u03bd m > kP0 \u03c4\u03cc\u03c4\u03b5 P\u2032(t) < 0, \u03bf\u03c0\u03cc\u03c4\u03b5 \u03bf \u03c0\u03bb\u03b7\u03b8\u03c5\u03c3\u03bc\u03cc\u03c2 \u03bc\u03b5\u03b9\u03ce\u03bd\u03b5\u03c4\u03b1\u03b9. \u2014 \u03b1\u03bd m = kP0 \u03c4\u03cc\u03c4\u03b5 P\u2032(t) > 0, \u03bf\u03c0\u03cc\u03c4\u03b5 \u03bf \u03c0\u03bb\u03b7\u03b8\u03c5\u03c3\u03bc\u03cc\u03c2 \u03b5\u03af\u03bd\u03b1\u03b9 \u03c3\u03c4\u03b1\u03b8\u03b5\u03c1\u03cc\u03c2.8. i) \u039f \u03cc\u03b3\u03ba\u03bf\u03c2 \u03c4\u03bf\u03c5 \u03bd\u03b5\u03c1\u03bf\u03cd \u03c4\u03b7\u03c2 \u03b4\u03b5\u03be\u03b1\u03bc\u03b5\u03bd\u03ae\u03c2 \u03c4\u03b7 \u03c7\u03c1\u03bf\u03bd\u03b9\u03ba\u03ae \u03c3\u03c4\u03b9\u03b3\u03bc\u03ae t \u03b5\u03af\u03bd\u03b1\u03b9 V (t) = \u03c0 r2 y(t) = \u03c0 y(t),\u03cc\u03c0\u03bf\u03c5 r = 1m \u03b7 \u03b1\u03ba\u03c4\u03af\u03bd\u03b1 \u03c4\u03bf\u03c5 \u03ba\u03c5\u03bb\u03af\u03bd\u03b4\u03c1\u03bf\u03c5, \u03bf\u03c0\u03cc\u03c4\u03b5 V \u2032(t) = \u03c0 y\u2032(t).\u0395\u03be\u03ac\u03bb\u03bb\u03bf\u03c5 \u03ad\u03c7\u03bf\u03c5\u03bc\u03b5 \u2212\u03b1 2gy = \u2212\u03c0 \u22c5 0,12 20 y = \u22120, 02\u03c0 5y.\u0388\u03c4\u03c3\u03b9 \u03bf \u03bd\u03cc\u03bc\u03bf\u03c2 \u03c4\u03bf\u03c5 Torricelli \u03b3\u03c1\u03ac\u03c6\u03b5\u03c4\u03b1\u03b9 \u03c0 y\u2032 = \u22120, 02\u03c0 5y ,\u03ae \u03b9\u03c3\u03bf\u03b4\u03cd\u03bd\u03b1\u03bc\u03b1 y\u2032 = \u2212 5 y (1) 50ii) \u03a0\u03c1\u03bf\u03c6\u03b1\u03bd\u03ce\u03c2 \u03c4\u03bf y = 0 \u03b1\u03c0\u03bf\u03c4\u03b5\u03bb\u03b5\u03af \u03bb\u03cd\u03c3\u03b7 \u03c4\u03b7\u03c2 (1). \u0393\u03b9\u03b1 y > 0 \u03b7 \u03b5\u03be\u03af\u03c3\u03c9\u03c3\u03b7 \u03b3\u03c1\u03ac\u03c6\u03b5\u03c4\u03b1\u03b9 dy = \u2212 5 dt, y 50\u03bf\u03c0\u03cc\u03c4\u03b5 \u03ad\u03c7\u03bf\u03c5\u03bc\u03b5 \u03b4\u03b9\u03b1\u03b4\u03bf\u03c7\u03b9\u03ba\u03ac: \u222b y\u22121\/2dy = \u2212 5 t + c 50 2 y1\/2 = \u2212 5 t + c 50 y1\/2 = \u2212 5 t + c 100 2 y = \uf8eb \u22125 t + c \uf8f62 . \uf8ec\uf8ec\uf8ed 100 2 \uf8f7\uf8f8\uf8f7 196","3.3\u038c\u03bc\u03c9\u03c2 \u03b9\u03c3\u03c7\u03cd\u03b5\u03b9 y(0) = 36 dm, \u03bf\u03c0\u03cc\u03c4\u03b5 36 = \uf8eb c \uf8f62 , \u03c3\u03c5\u03bd\u03b5\u03c0\u03ce\u03c2 c = 12. \u0386\u03c1\u03b1 \uf8ed\uf8ec 2 \uf8f7\uf8f8 y(t ) = \uf8eb \u2212 5 t + 6 \uf8f62 \uf8ec\uf8ec\uf8ed 100 \uf8f7\uf8f7\uf8f8iii) H \u03b4\u03b5\u03be\u03b1\u03bc\u03b5\u03bd\u03ae \u03b1\u03b4\u03b5\u03b9\u03ac\u03b6\u03b5\u03b9 \u03c4\u03b5\u03bb\u03b5\u03af\u03c9\u03c2, \u03cc\u03c4\u03b1\u03bd y(t) = 0. \u0388\u03c4\u03c3\u03b9 \u03ad\u03c7\u03bf\u03c5\u03bc\u03b5: y(t) = 0 \u21d4 \u2212 5 t + 6 = 0 \u21d4 t = 600 = 600 5 = 120 5 sec. 100 5 59. \u0397 \u0395 = 0 \u03b1\u03c0\u03bf\u03c4\u03b5\u03bb\u03b5\u03af \u03bc\u03af\u03b1 \u03c0\u03c1\u03bf\u03c6\u03b1\u03bd\u03ae \u03bb\u03cd\u03c3\u03b7 \u03c4\u03b7\u03c2 \u03b4\u03b9\u03b1\u03c6\u03bf\u03c1\u03b9\u03ba\u03ae\u03c2 \u03b5\u03be\u03af\u03c3\u03c9\u03c3\u03b7\u03c2. \u0393\u03b9\u03b1 \u0395 > 0 \u03b7 \u03b5\u03be\u03af\u03c3\u03c9\u03c3\u03b7 \u03b3\u03c1\u03ac\u03c6\u03b5\u03c4\u03b1\u03b9 \u03b4\u03b9\u03b1\u03b4\u03bf\u03c7\u03b9\u03ba\u03ac: d\u0395 = \u2212 1 dt \u0395 RC ln \u0395 =\u2212 1 t + c1 RC \u0395 (t) = \u2212 t + c1 RC e \u2212t k = ec1 . \u0395 (t) = k \u22c5 e RC ,\u0395\u03be\u03ac\u03bb\u03bb\u03bf\u03c5 \u2212 t1 t1 \u0395 (t1) = \u0395 0 \u21d4 \u0395 0 = ke RC \u21d4 k = \u0395 0e RC .\u0386\u03c1\u03b1 t1 \u2212t E(t) = E0e RC .10. i) \u03b1) \u0007\u0391\u03bd \u03b1\u03bd\u03c4\u03b9\u03ba\u03b1\u03c4\u03b1\u03c3\u03c4\u03ae\u03c3\u03bf\u03c5\u03bc\u03b5 \u03c4\u03b9\u03c2 \u03c4\u03b9\u03bc\u03ad\u03c2 \u03c4\u03c9\u03bd R, L \u03ba\u03b1\u03b9 \u0395, \u03ba\u03b1\u03bd\u03cc\u03bd\u03b1\u03c2 \u03c4\u03bf\u03c5 Kirchhoff \u03b3\u03c1\u03ac\u03c6\u03b5\u03c4\u03b1\u03b9 4I \u2032 +12I = 60, \u03ae \u03b9\u03c3\u03bf\u03b4\u03cd\u03bd\u03b1\u03bc\u03b1 I \u2032 = 3I +15. (1) \u039c\u03af\u03b1 \u03c0\u03b1\u03c1\u03ac\u03b3\u03bf\u03c5\u03c3\u03b1 \u03c4\u03b7\u03c2 \u03b1(t) = 3 \u03b5\u03af\u03bd\u03b1\u03b9 \u03b7 A(t) = 3t. \u03a0\u03bf\u03bb\u03bb\u03b1\u03c0\u03bb\u03b1\u03c3\u03b9\u03ac\u03b6\u03bf\u03c5\u03bc\u03b5 \u03c4\u03b1 \u03bc\u03ad\u03bb\u03b7 (1) \u03bc\u03b5 e3t, \u03bf\u03c0\u03cc\u03c4\u03b5 \u03ad\u03c7\u03bf\u03c5\u03bc\u03b5 \u03b4\u03b9\u03b1\u03b4\u03bf\u03c7\u03b9\u03ba\u03ac: I \u2032e3t + 3e3t I = 15e3t (Ie3t )\u2032 = 15e3t 197","3.3 \u222bIe3t + c1 = 5 3e3t dt Ie3t = 5e3t + c. I (t) = 5 + c . e3t\u03b2) \u0395\u03af\u03bd\u03b1\u03b9 lim I (t) = lim \uf8eb 5 + c \uf8f6 = 5. \uf8ed\uf8ec e3t \uf8f8\uf8f7 t \u2192+\u221e t \u2192+\u221e\u0391\u03c0\u03cc \u03c4\u03b7\u03bd \u03b9\u03c3\u03cc\u03c4\u03b7\u03c4\u03b1 lim I (t) = 5 \u03c3\u03c5\u03bc\u03c0\u03b5\u03c1\u03b1\u03af\u03bd\u03bf\u03c5\u03bc\u03b5 \u03cc\u03c4\u03b9 \u03b3\u03b9\u03b1 \u00ab\u03bc\u03b5\u03b3\u03ac\u03bb\u03b5\u03c2\u00bb \u03c4\u03b9\u03bc\u03ad\u03c2 \u03c4\u03bf\u03c5 t \u03b7 t \u2192+\u221e\u03ad\u03bd\u03c4\u03b1\u03c3\u03b7 \u03b3\u03af\u03bd\u03b5\u03c4\u03b1\u03b9 \u03c3\u03c4\u03b1\u03b8\u03b5\u03c1\u03ae \u03ba\u03b1\u03b9 \u03b7 \u03b3\u03c1\u03b1\u03c6\u03b9\u03ba\u03ae \u03c0\u03b1\u03c1\u03ac\u03c3\u03c4\u03b1\u03c3\u03b7 \u03c4\u03b7\u03c2 y = I(t) \u03ad\u03c7\u03b5\u03b9 \u03b1\u03c3\u03cd\u03bc\u03c0\u03c4\u03c9\u03c4\u03b7\u03c4\u03b7\u03bd \u03b5\u03c5\u03b8\u03b5\u03af\u03b1 y = 5.ii) \u0391\u03bd \u0395 = 60\u03b7\u03bc3t \u03bf \u03ba\u03b1\u03bd\u03cc\u03bd\u03b1\u03c2 \u03c4\u03bf\u03c5 Kirchhoff \u03b3\u03c1\u03ac\u03c6\u03b5\u03c4\u03b1\u03b9 \u03b4\u03b9\u03b1\u03b4\u03bf\u03c7\u03b9\u03ba\u03ac: I \u2032 + 3I = 15\u03b7\u00b53t I \u2032e3t + 3e3t I = 15e3t \u03b7\u00b53t (Ie3t )\u2032 = 15e3t\u03b7\u00b53t \u222bIe3t + c1 = 5 3e3t\u03b7\u00b53tdt. (2)\u222b\u0398\u03ad\u03c4\u03bf\u03c5\u03bc\u03b5 J = 3e3t\u03b7\u00b53tdt, \u03bf\u03c0\u03cc\u03c4\u03b5 \u222b \u222bJ = (e3t )\u2032\u03b7\u00b53tdt = e3t\u03b7\u00b53t \u2212 3 e3t\u03c3\u03c5\u03bd3tdt \u222b= e3t\u03b7\u00b53t \u2212 (e3t )\u2032\u03c3\u03c5\u03bd3tdt \u222b= e3t\u03b7\u00b53t \u2212 \uf8f0\uf8eee3t\u03c3\u03c5\u03bd3t + 3 e3t\u03b7\u00b53tdt \uf8f9\uf8fb = e3t (\u03b7\u00b53t \u2212 \u03c3\u03c5\u03bd3t) \u2212 3J.\u0386\u03c1\u03b1 1 4 J = e3t (\u03b7\u00b53t \u2212 \u03c3\u03c5\u03bd3t ) + c1 , c1 \u2208 R .\u039b\u03cc\u03b3\u03c9 \u03c4\u03b7\u03c2 (2) \u03ad\u03c7\u03bf\u03c5\u03bc\u03b5 Ie3t = 5 e3t (\u03b7\u00b53t \u2212 \u03c3\u03c5\u03bd3t) + c 4\u0386\u03c1\u03b1 5 c 4 e3t I (t) = (\u03b7\u00b53t \u2212 \u03c3\u03c5\u03bd3t) + . 198","3.43.4 \u0391\u0384 \u039f\u039c\u0391\u0394\u0391\u03a31. \u0388\u03c7\u03bf\u03c5\u03bc\u03b5\u222b \u222bi) 3 f (x)dx = \u2212 4 f (x)dx = \u221211 43ii) 8 f (x)dx = 1 f (x)dx + 8 f (x)dx = \u2212 4 f (x)dx + 8 f (x)dx = \u22129 +13 = 4\u222b \u222b \u222b \u222b \u222b4 4 1 11iii) 3 f (x)dx = 4 f (x)dx + 3 f (x)dx = 9 \u2212 4 f (x)dx = 9 \u221211 = \u22122\u222b \u222b \u222b \u222b1 1 4 3iv) 8 f (x)dx = 4 f (x)dx + 1 f (x)dx + 8 f (x)dx \u222b \u222b \u222b \u222b3 3 4 1 \u222b= 11\u2212 4 f (x)dx +13 = 24 \u2212 9 = 15. 12. \u0388\u03c7\u03bf\u03c5\u03bc\u03b5 \u222b \u222b \u222b \u222b1ln 1 dt = \u03b5 \u03b5t 1(ln1\u2212 ln t)dt = \u2212 1ln tdt = \u03b5 \u03b51 ln tdt.\u222b \u222b3. \u0397 \u03b9\u03c3\u03cc\u03c4\u03b7\u03c4\u03b1kx2 \u22124 dx \u2212 1 5 1 dx =3 \u03b3\u03c1\u03ac\u03c6\u03b5\u03c4\u03b1\u03b9 \u03b4\u03b9\u03b1\u03b4\u03bf\u03c7\u03b9\u03ba\u03ac: 1 x2 +1 k x2 + \u222b \u222bkx2 \u22124 dx + k x 5 1 dx = 3 1 x2 +1 1 2+ \u222bk x2 \u2212 4+ 5 dx = 3 1 x2 +1 \u222b k1dx = 3 1 (k \u22121) = 3 k = 4.4. \u0388\u03c7\u03bf\u03c5\u03bc\u03b5\u222b \u222b \u222bi) 3[2 f (x) \u2212 6g(x)]dx = 2 3 f (x)dx \u2212 6 3 g(x)dx = 2 \u22c5 5 \u2212 6 \u22c5 (\u22122) = 22 1 11ii) 1[2 f (x) \u2212 g(x)]dx = 2 1 f (x)dx \u2212 1 g(x)dx \u222b \u222b \u222b3 3 3 \u222b \u222b3 3 = \u22122 f (x)dx + g(x)dx 11 = \u22122 \u22c55 \u2212 2 = \u221212. 199"]