数学分析讲义第四版刘玉琏傅沛仁著高等教育出版社课后答案第一单元.pdf

8 111?1 ?SSS1.1(1)?SSS1.2(5) ?SSS1.3(9) 111?444? 13 ?SSS2.1(13)?SSS2.2(21) ?SSS2.3(36)?SSS2.4(40) ?SSS2.5(51) 111nnn?YYY 57 ?SSS3.1(57)?SSS3.2(64) 111ooo?YYY555 75 ?SSS4.1(75)?SSS4.2(83) 111? 95 ?SSS5.1(95)?SSS5.2(100) ?SSS5.3(109)?SSS5.4(113) ?SSS5.5(114) 111888?nnn444?AAA.125 ?SSS6.1(125)?SSS6.2(136) ?SSS6.3(142)?SSS6.4(148) 111? . 170 ?SSS7.1(170)?SSS7.2(172) ?SSS7.3(179)?SSS7.4(187) 111lll? . 194 ?SSS8.2(194)?SSS8.3(204) ?SSS8.4(210)?SSS8.5(232) 111? . 244 ?SSS9.1(?)(244)?SSS9.1(?)(252) ?SSS9.1(nnn)(266)?SSS9.2(?)(273) ?SSS9.2(?)(290)?SSS9.3(298) ?SSS9.4(309) 1 111?.323 ?SSS10.1(323)?SSS10.2(132) ?SSS10.3(334)?SSS10.4(352) 111? . 366 ?SSS11.1(366)?SSS11.2(375) ?SSS11.3(378)?SSS11.4(389) 111?222?CCC? 393 ?SSS12.1(393)?SSS12.2(400) ?SSS12.3(406) 111?nnn? . 425 ?SSS13.1(?)(425)?SSS13.1(?)(430) ?SSS13.2(493) 111?ooo?.465 2 1? ? S K1.1 (56 1 ? ?1 10 ?) 5.(e? (5) y = 1 |x| x ) ?|x| x 6= 0|x| 6= 0,=x 0,?4x 0, 1 2 0k Z,2k ? a 4 ? ?A?P = ?a 4 ?2 A ? ?“?AX. ?n?/?r r?n?/?n?O?a,b,c,=a + b + c = r)?A? Q = s r 2 ?r 2 a ?r 2 b ?r 2 c ? S?(?a,b,ck),AX?. 10y:e(x) = ln 1 x 1 + x,K(a) + (b) = ? a + b 1 + ab ? . y d(x) = ln 1 x 1 + x?,(x)?|x| n?/?1,?n?/?:?n?/,q?n ?/?:?n?/, Xd?UYe?,?n?/?. ) ?n?/?1= 1,X1.a.(?:?n?/?2 ?n?/1= 1?1 4.?n2(?n?/?:?n?/ 3= 1 42 Xd?UYe?,u?n?/? n 1 4n1 o : 1, 1 4, 1 42 , 1 43 , , 1 4n1 , , 14. y?e?an?,?an 0,n = 1,2, ,K?lnan?. y fi ?an?,?n N, an+1 an = a(a?,?),k lnan+1 lnan= ln an+1 an = lna,() =lnan?lnan?. 15. fi ?y = f(x)3ma,b?(3ma,b?x?,k?:? ?,k?:?K?), e?: (1) y1= |f(x)|. ) y1= |f(x)| = ? f(x),?f(x) 0, f(x),?f(x) 0,x A,k |f(x)| M?|(x)| M. u|f(x) + (x)| |f(x)| + |(x)| M + M = 2M,=f(x) + (x)3Ak ?y,f(x) (x)?f(x)(x)?3Ak 4. ?e?”?” (6) ln(x + 1 + x2) ). ?.fl ?, ln(x + p 1 + (x)2) = ln (1 + x2 x)(1 + x2+ x) x + 1 + x2 = ln 1 x + 1 + x2= ln(x + p 1 + x2). (7) ln 1 x 1 + x. ). ?.fl ?, ln 1 (x) 1 + (x) = ln 1 + x 1 x = ln 1 x 1 + x. 5. y?f(x) = 1 x3m(0,1) y. b 1,?xb (0,1),?“ 1 xb b.l?“ 1 xb b)? 1 xb 1,xb (0,1) ? xb b, 3 =f(x) = 1 x3(0,1).,l?f(x) = 1 x3(0,1). 6. ?e?=?,ek?,?: (1) y = sin2x. ). y = sin2x = 1 2 (1 cos2x).fi ?cos2x?,x R,k y = sin2(x ) = 1 21 cos2(x ) = 1 2(1 cos2x) = sin 2 x. u,sin2?. (2) y = sinx2 ). ?. ?y.b?y = sinx2,?T 0,Kx R,k sin(x + T)2= sinx2. -x = 0,ksinT2= sin0 = 0,Kn0 N,T2= n0. 2-2T(?o?-?e?y),k sin(2 + 1)T2= sin2T2= sin2n0 = 0, Km0 N,(2 + 1)T2= m0(2 + 1)2= m0 T2 = m0 n0 = m0 n0 . ?“?k?(2 + 1)2n,?k? m0 n0 ,g.u, y = sinx2?. (6) y = D(x)()|?5). fl ?,?knr,x R,k D(x r) = ? 1?xkn 0?xn ? = D(x). u,?knr?D(x)?,?kn8vk?,D(x)vk?. (8) y = sin x 2 + sin x 5. ). y = sin x 2+sin x 5 .fi ?sin x 2?sin x 5?O?4?10,?4?10 ?20.x R ,k sin 1 2(x 20) + sin 1 5(x 20) = sin x 2 + sin x 5. u,?20. 8. y:f(x)3mI?N x1,x2,x3 I,?x1 f(x2)g.3R?3?O?. 11. ?Le?9?: (1) f(x)3R?. ? ?x R,kf(x) = f(x). ?x0 R,x06= 0,kf(x0) 6= f(x0). (2) f(x)3R?. ? l 0,x R,kf(x l) = f(x). ?l,x0 R,kf(x0+ l) 6= f(x0)f(x0 l) 6= f(x0). (3) f(x),(a,b)?O?O. 5 ? ?Ox1,x2 (a,b),?x1 0,x A,k f(x l) + g(x l) = f(x nT1) + g(x mT2) = f(x) + g(x) ?f(x l)g(x l) = f(x nT1)g(x nT2) = f(x)g(x), =f(x) + g(x)?f(x)g(x)?3A? ? S K1.3 (56?1 31 ?) 1. ?e?3?m?9? (4) ax + b cx + d,a,b,c,d?ad bc 6= 0. ) fi ?ad bc 6= 0,Ka?c?“. ec = 0,a 6= 0d?d 6= 0,l?y = a d x + b d )?x = d a y b a.u,?y = d a x b a,?R; ec 6= 0,l?y = a d x + b d )?x = b dy cy a. u,?y = b dx cy a.?R na c o . (5) y = shx = 1 2(e x ex),x R. ) l?y = shx = 1 2(e x ex)e2x 2yex 1)?ex= y py2 + 1(?U? ?)x = ln(y + py2 + 1).u, ?y = ln(x + py2 + 1).?R. (6) y = xx (,1), x2x 1,4, 2xx (4,+). ) ?x (,1)?,y = x?x = y,y (,1) ?x 1,4?y = x?x = y,y 1,16. 6 ?x 4,?y = 2x?x = log2y,y (16,+).u,?: y = xx (,1), x x 1,16, log2xx (16,+). 2.y :ey = (x)38A?,Ky = (x)?3?x = 1(y) ,? x = 1(y)3(A)?. y ky(x)?3?,=y (A),U?X1A?x A.?y,b ?y0 (A),U?X1?A?x1,x2 A,?x16= x2,y0= (x1) = (x2),? y = (x)?g.u,y = (x)?3?x = 1(y) y (A). ?gy:?x = 1(y)3(A)?. y1,y2 (A),?y1 x2 1(y1) 1(y2), =?x = 1(y)3A?. 5.y :ef(x),g(x),h(x)?NO?,? f(x) g(x) h(x),Kff(x) gg(x) hh(x). y ?fi ?“f(x) g(x)?g(x) h(x)?xOO?f(x)?h(x),k fg(x) gg(x)?gh(x) hh(x) qfi ?f(x)?g(x)?NO?,k ff(x) fg(x)?gg(x) gh(x). u,ff(x) fg(x) gg(x) gh(x) hh(x), =ff(x) gg(x) hh(x). 7. ?f(x)x?g,?f(0) = 1,f(x + 1) f(x) = 2x,f(x). ) ?f(x) = ax2+ bx + c(f(x)?Xa,b,c). fi ?f(0) = c = 1,l?f(x) = ax2 + bx + 1.qfi ? f(x + 1) f(x) =a(x + 1)2+ b(x + 1) + 1 (ax2+ bx + 1) =2ax + a + b = 2x. ?“?“,?g?X7?,k 2a = 2?a + b = 0,)?a = 1,b = 1. u,?gf(x) = x2 x + 1 8. y:f (g h) = (f g) h. y ?AX()?,?x,k 7 f (g h)(x) = f(g h)(x) = fgh(x). ? (f g) h(x) = (f g)h(x) = fgh(x). ?“?“?m?,l?3?,=f (g h) = (f g) h. 9. ?f(x + 1 x) = x 2 + 1 x2,f(x). ) ?y = x + 1 x,kx 2 + 1 x2 = ? x + 1 x ?2 2 = y2 2,=f(y) = y2 2.?gC?yL ?x,kf(x) = x2 2. 10. ?f(x) = x 1