复旦大学第三版数学分析答案---副本.pdf

1 14 14 1C 1.?Vg 1. )e?xx? (1) 2 1 x + 2 (2) (x 1)(x + 2)(x 3) |x| (|x1| + + |xn|) y (1) |x|y| xyK(x y)2 (|x| |y|)2u|x y| |x| |y| (2) 8By. (i) ?n = 2d|x1+ x2| 6 |x1| + |x2|?(. (ii) b?n = k(=k|x1+ x2+ x3+ + xk| 6 |x1| + |x2| + + |xk|. K?n = k + 1|x1+ x2+ x3+ + xk+1| 6 |x1+ x2+ x3+ + xk| + |xk+1| 6 |x1| + |x2| + + |xk| + |xk+1| ng,n|x1+ x2+ x3+ + xn| 6 |x1| + |x2| + + |xn|. (3) |x + x1+ + xn| |x| |x1+ x2+ x3+ + xn| |x| (|x1| + + |xn|) 3. )e?xx? (1) |x| |x + 1| (2) 2 1 |x| A (4) |x a| 0 (5) ? ? ? ? x 2 x + 1 ? ? ? ? x 2 x + 1 (6) 2 1 |x + 2| 0Kx 2x 6 1. 7. ?f(x) = (|x| + x)(1 x)ve?x (1) f(0) = 0 (2) f(x) 0Kf(x) |V !?R0CR|?.3?mSA,B:m?V w .6ICR?. )d9n?V = I(R0+ R). 9. 3?/NS?,MT?/N?.aph?M?px1-6. T M?NV xm?XV = V (x)?. )d?V = a2x?0,h1,a2h 10. ,/Y?F/X1-7.2?45oCDLY?ABCD? SY?h?X. )d9?S = h(h + 2). 11. k?H?XR?zl?lSL-./? lsmt?X1-8. )d9?s = H Rt ? t ? 0, H t ? 12. ?y = f(x) = ? 1 + x2,x 0 f(2),f(1),f(0),f(1)f ? 1 2 ? . )d?f(2) = 5,f(1) = 2,f(0) = 1,f(1) = 0,f ? 1 2 ? = 1 2. 5 13. ?x(t) = 0,0 6 t 1minx2k = 1?infxn = 1. 6. yN?ke.?7k4. yduynke.?yn7ke( de(.?k(i)yn (n = 1,2,3,)(ii) 0?kyN ynyN Nkyn Nk0 6 yn 0?kn0 Z+?|xn0| M. ?M = 1K73n1? ? ? ?x (1) n1 ? ? ? 1M = 2K73n2? ? ? ?x (1) n2 ? ? ? 2M = KK7 44 3nK nK1? ? ? ?x (1) nK ? ? ? K. K?f? n x(1) nk o M Z+?K = MK?k Kk ? ? ?x (1) nk ? ? ? M?k lim k x(1) nk= . dxnKd?M0 0N Z+?km Z+?m N k|xm| m0?|xm1| 6 M0 2?N = m1K?km2 m1?|xm2| 6 M0 Xd?1e?K?mtm1 m NkM|q|m+1 1 1 |q| 0”? 1 2du|xm xn| = ? ? ? ? sin(n + 1) 2n+1 + sin(n + 2) 2n+2 + + sinm 2m ? ? ? ? 6 1 2n+1 + 1 2n+2 + + 1 2m = 1 2n+1 ? 1 + 1 2 + + 1 2mn1 ? = 1 2n+1 1 ? 1 2 ?mn 1 1 2 1 2n e|xm xn| Nk|xm xn| 0k Z+du|xn+kxn| = ? ? ? ? (1)n+2 n + 1 + (1)n+3 n + 2 + + (1)n+k1 n + k ? ? ? ?= ? ? ? ? 1 n + 1 1 n + 2 + + (1)k1 n + k ? ? ? ?= 1 n + 1 ? 1 n + 2 1 n + 3 + + (1)k n + k ? 1 n + 1 1 ne|xn+k xn| Nk|xn+k xn| 0?3O(x0,0) kxn?k. K 03O(x0,)kxn?. ?n= 1 nw,3O(x0,n)kxn?K3xn?xn1 O(x0,1)q? xn2 O ? x0, 1 2 ? (n2 n1)Xd?1e?xn?f?xnk,|xnk x0| Kk|xnk x0| 1 k 1 K a. dunk (k )?k7knk Nl?xnk xN aua = lim k xnk xN ag. 2y lim n xn= a. 0,k0 ? ? ?x nk0 a ? ? ? = a xnk0 Nkxn xnk0= xNl?k|a xn| = a xn6 a xnk0 (x0) =f(x0) + (x0) ? = x0 x0 0f(x)3(a,b)NO? x0 x 0=x0 0?x0,x00 Xk|f(x0) f(x00)| 0,X 0?x Xk|f(x) A| Xk|f(x0) A| 2,|f(x 00) A| 0?x0,x00 Xk|f(x0) f(x00)| 0,X 0?x0,x00 Xk|f(x0) f(x00)| 0N Z+?n Nkxn Xl?n,m N kxn X,xm X?k|f(xn) f(xm)| 03 0?0 0?0 0?0 |x0 x0| 0?x0,x00 D(f)?0 Nk0 0?|x0 x0| 0?|x x0| 0?|x0 x0| 0?|x x0| 0?x1,x2 (a,b)|x1 x2| x2 0x1,x2 (,+),|f(x1)f(x2)| = |x2 1x 2 2| = |x1+x2|x1x2| = (x1+x2)(x1 x2) 2x2(x1 x2)30 0 0?x2= 20 ,x1= x2+ 2 w,kx1 x2 0|x1 x2| = 2 2x2(x1 x2) = 2 20 2 = 20 0 l?f(x) = x23(,+)Y. (2) f(x) = x23(l,l)(l 0)Y. f(x)3l,l(l 0)Y?Kdxn?f(x)3l,lYl?f(x) = x23(l,l) Y. 6. ef(x)3(a,b)Sk(a,b)S?x3x?,?Ox?f(x)3OxSkf(x)3(a,b)S k.qe(a,b)Ua,bX y 48 (1) f(x)3(a,b)k .f(x) = 1 x3(0,1)Skx (a,b)Y?7k.=3x?Ox(O(x,x) ?3Ox(O(x,x)Sk.?3(0,1)S k.f(x) = sinx3 ? 0, 2 ? k ? 0, 2 ? S?x3x?,?Ox?f(x)3OxSk .f(x)3 ? 0, 2 ? k.0 0,1 0mXS?:x0,x00|x0 x00| 1k|f(x0) f(x00)| 0,2 0mXS?:x0,x00|x0 x00| 0|f(x)| 0mXS?:x0,x00|x0 x00| 0mXS?:x0,x00|x0 x00| x?1 y y0= 1 x + 1 2(1 x) 1 2(1 + x) Ky 0 = 1 x x2 (1 + x)1 x2 (0 0K lny = n P i=1 iln|x i|x?1 y y0= n X i=1 i x i Ky 0 = n P i=1 i x i n Y i=1 (x i)i(x D)D = ? n Q i=1(x i)i 0 ? (4) y = (x + 1 + x2)nK lny = nln(x +1 + x2)x?1 y y0= n 1 + x 1 + x2 x + 1 + x2= n 1 + x2Ky 0 = n 1 + x2(x + p 1 + x2)n (5) y = xmmxKlny = mln|x| + xlnmx?1 y y0= m x + lnmKy 0 = xm1mx+1+ xmmxlnm 6. ?f(x)x?dy dx. (1) y = f(x2) (2) y = f(ex) ef(x) (3) y = f(f(f(x) ) (1) dy dx = 2xf0(x2) (2) dy dx = exf0(ex) ef(x)+ f0(x)f(ex)ef(x)= ef(x)(exf0(ex) + f(ex)f0(x) (3) dy dx = f0(f(f(x)f0(f(x)f0(x) 7. ?(x),(x)x?dy dx. (1) y = p2(x) + 2(x) (2) y = arctan (x) (x)(x) 6= 0) (3) y = (x) p(x)(x) 6= 0,(x) 0) (4) y = log(x)(x)(x) 0,(x) 6= 0) ) (1) dy dx = (x)0(x) + (x)0(x) p2(x) + 2(x) (2) dy dx = 0(x)(x) 0(x)(x) 2(x) + 2(x) (3) dy dx = (x) p (x) ? 0(x) (x)(x) 0(x)ln(x) 2(x) ? 60 (4) dy dx = 0(x) (x) ln(x) 0(x) (x) ln(x) (ln(x)2 = 0(x) (x)ln(x) 0(x)ln(x) (x)(ln(x)2 = log(x)(x) ? 0(x) (x)ln(x) 0(x) (x)ln(x) ? 8. 4-7-Y?w$?. )s = p l2 r2sin2t rcost?v = s 0 = r sint r2 sin2t 2 p l2 r2sin2t . 9. -y = 1 x23x =1 2?. )y 0 = x 1 x2K3x = 1 2?k = 3 3 uy 3 2 = 3 3 ? x 1 2 ? =x + 3y 2 = 0 y 3 2 = 3?x 1 2 ? =3x y = 0. 10. -y = ex?:LT:?y = ex1?T:?. )k = y 0 = ex= eKx = 1KL(1,e):?y = ex1LT:? y e = 1 e (x + 1)=x ey + e2+ 1 = 0. 11. -y = 1 x2?Y. )k = y 0 = x 1 x2= 0Kx = 0ud-3(0,1)?Yy = 1. 12. -y = 1 2(1 + 2x 2 p 1 + 4x2)Ix = U?:?.-?u? )y 0 = 2x 2x 1 + 4x2K-3x = U?k = 2U 2U 1 + 4U2ud-3 :(U, 1 2(1 + 2U 2 p 1 + 4U2)?y 1 2(1 + 2U 2 p 1 + 4U2) = (2U 2U 1 + 4U2)(x U) =2U(1 + 4U2 1)x 1 + 4U2y 1 2 + 1 2(1 2U 2)p1 + 4U2 = 0d-?u U( 1 + 4U2 1) 1 + 4U2, 1 2 1 + 2U2(1 + 4U2 1)2 1 + 4U2 s 1 + 4U2(1 + 4U2 1)2 1 + 4U2 . 13. ?y = f(x)3x0?P(t) = f(x0+ at)a0(0). )ea = 0K(t) = f(x0)K0(0) = 0 ea 6= 0K0(x) = lim t0 (x) (0) t = lim t0 f(x0+ at) f(x0) t = a lim t0 f(x0+ at) f(x0) at = af0(x0). 61 5.9$ 1. e?3:? (1) y = anxn+ an1xn1+ + a0dy(0),dy(1) (2) y = secx + tanxdy(0),dy ? 4 ? ,dy() (3) y = 1 a arctan x a dy(0),dy(a) (4) y = 1 x + 1 x2 dy(0.1),dy(0.01) ) (1) dy = nanxn1+ (n 1)an1xn2+ + a1dxKdy(0) = a1dx,dy(1) = n P i=1 iaidx (2) dy = (tanxsecx + sec2x)dxKdy(0) = dx,dy ? 4 ? = (2 + 2)dx,dy() = dx (3) dy = dx a2+ x2 dxKdy(0) = dx a2 dx,dy(a) = dx 2a2 dx (4) y = x + 2 x3 dxKdy(0.1) = 2100dx,dy(0.01) = 2010000dx 2. e?y = y(x)? (1) y = x 1 2x 2 + 1 3x 3 1 4x 4 (2) y = x2sinx (3) y = x 1 x2 (4) y = xlnx x (5) y = (1 x2)n (6) y = x + lnx 1 x (7) y = lntanx (8) y = sinaxcosbx (9) y = eaxcosbx (10) y = arcsin 1 x2 ) (1) dy = (1 x + x2 x3)dx (2) dy = (2xsinx + x2cosx)dx (3) dy = 1 + x2 (1 x2)2 dx (4) dy = lnxdx (5) dy = 2nx(1 x2)n1dx (6) dy = x + 2x + 1 x 3 2 dx (7) dy = 2 sin2xdx (8) dy = (acosaxcosbx bsinaxsinbx)dx (9) dy = eax(acosbx bsinbx)dx (10) dy = x |x|1 x2 dx 3. e?y? (1) y = sin2t,t = ln(3x + 1) 62 (2) y = ln(3t + 1),t = sin2x (3) y = e3u,u = 1 2 lnt,t = x2 2x + 5 (4) y = arctanu,u = (lnt)2,t = 1 + x2 cotx ) (1) dy = 3sin(2ln(3x + 1) 3x + 1 dx (2) y = 3sin2x 3sin2x + 1dx (3) y = 3(3x2 2) 2(x3 2x + 5)e 3 2 ln(x22x+5)dx (4) y = 2ln(1 + x2 cotx)(2x + csc2x) 1 + (ln(1 + x2 cotx)4(1 + x2 cotx)dx 4. eu,v,wx?y?dy (1) y = u v w (2) y = u w v2 (3) y = 1 u2 + v2 (4) y = ln u2 + v2 (5) y = arctan u v ) (1) dy = (u0 v w + u v0 w + u v w0)dx (2) dy = v2(u0w + uw0) 2uvv0w v4 dx (3) dy = uu0+ vv0 (u2+ v2) 3 2 dx(u2+ v2 0) (4) dy = uu0+ vv0 u2+ v2 dx (5) dy = u0v uv0 u2+ v2 dx(v 6= 0) 63 6.9L? 1. e?dy dx (1) x2 a2 + y2 b2 = 1a,b (2) y2= 2pxp (3) x2+ xy + y2= a2a (4) x3+ y3 xy = 0 (5) y = x + 1 2 siny (6) x 2 3+ y 2 3= a 2 3a (7) y cos(x + y) = 0 (8) y = x + arctany (9) y = 1 ln(x + y) + ey (10) arctan y x = ln p x2+ y2 ) (1) 3x?5?yx?k2x a2 + 2yy 0 b2 = 0Ky 0 = b2x a2y (y 6= 0). (2) 3x?5?yx?k2yy 0 = 2pKy 0 = p y (y 6= 0). (3) 3x?5?yx?k2x + xy 0 + y + 2yy 0 = 0Ky 0 = 2x + y x + 2y . (4) 3x?5?yx?k3x2+ 3y2y 0 xy 0 y = 0Ky 0 = 3x2 y x 3y2 . (5) 3x?5?yx?ky 0 = 1 + y 0 2 cosyKy 0 = 2 2 cosy . (6) 3x?5?yx?k2 3x 1 3+ 2 3y 1 3y0= 0Ky0= 3 rx y . (7) 3x?5?yx?ky 0+(1+y0)sin(x+y) = 0Ky0 = sin(x + y) 1 + sin(x + y). (8) 3x?5?yx?ky 0 = 1 + y 0 1 + y2 Ky 0 = 1 + y2 y2 . (9) 3x?5?yx?ky 0 = 1 + y 0 x + y +y 0eyKy0 = 1 (x + y)ey x y 1. (10) 3x?5?yx?kxy 0 y x2+ y2 = x + yy 0 x2+ y2 Ky 0 = x + y x y . 2. e?3:?dy dx (1) y = cosx + 1 2 siny: ? 2 ,0 ? (2) yex+ lny = 1:(0,1) ) (1) 3x?5?yx?ky 0 = sinx + y 0 2 cosyKy 0 = 2sinx cosy 2u3 : ? 2 ,0 ? ?y 0 = 2. (2) 3x?5?yx?kex(y + y 0) +y 0 y = 0Ky 0 = y2ex yex+ 1u3 :(0,1)?y 0 = 1 2. 64 3. -x 3 2+ y 3 2= 163:(4,4)?. )3x?5?yx?k3 2x 1 2+ 3 2y 1 2y0= 0Ky0= rx y uy 0| x=4 y=4 = 1l?y 4 = (x 4)=x + y 8 = 0 y 4 = x 4=x = y. 4. e?3:? (1) ? x =acost y =bsint 3t = 3 4 ? (2) ? x =t sint y =1 cost 3t = 2 ,? (3) ? x =1 t2 y =t t3 3t = 2 2 , 3 3 ? (4) ? x =a(t sint) y =a(1 cost) (a)3t = 0, 2 ? ) (1) x 0(t) = asint,y0(t) = bcostKdy dx = y 0(t) x0(t) = b a cottu?t = 3 y 0 = 3b 3a ?t = 4 y 0 = b a (2) x 0(t) = 1 cost,y0(t) = sintKdy dx = y 0(t) x0(t) = sint 1 costu?t = 2 y 0 = 1?t = y 0 = 0 (3) x 0(t) = 2t,y0(t) = 1 3t2Kdy dx = y 0(t) x0(t) = 3t2 1 2t u?t = 2 2 y 0 = 2 4 ?t = 3 3 y 0 = 0 (4) x 0(t) = a(1 cost),y0(t) = asintKdy dx = y 0(t) x0(t) = cot t 2u?t = 0y 0?t = 2 y = 1 5. e? (1) ? x =acosht y =bsinht (2) ? x =sin2t y =cos2t (3) ? x =acos3t y =asin3t (4) ? x =e2tcos2t y =e2tsin2t ) (1) dy dx = y 0(t) x0(t) = asinht bcosht = a b cotht (2) dy dx = y 0(t) x0(t) = 2costsint 2sintcost = 1 (3) dy dx = y 0(t) x0(t) = 3sin2tcost 3cos2tsint = !tant (4) dy dx = y 0(t) x0(t) = e2t(2sin2t + 2sintcost) e2t(2cos2t 2costsint) = tant sint + cost cost sint 6. ?I/N?10?44-11 (1) /YY?NV Yph?Cz (2) NV N?R?Cz. 65 )NV N?RYph?XV = 1 3R 2hd?R 4 = h 10=h = 5 2Ru (1) V = 1 3 ? 2 5h ?2 h = 4 75h 3l?dV dh = 4 25h 2 (2) V = 1 3R 2 5 2R = 5 6R 3l?dV dR = 5 2R 2. 7. ?I/N.X?2arctan 3 48p?,N (1) ?Nr3O?dr dt 1 4NO? dV dt ? (2) ?N6NO?24O? )NV Nr?XV = 4 9r 3V,rmt?t?dV dt = 4 9(3r 2)dr dt =dV dt = 4 3r 2dr dt K (1) ?r = 3, dr dt = 1 4 dV dt = 3 (2) ddr dt = 3 4r2 dV dt ?r = 6, dV dt = 24dr dt = 1 2 . 8. Ylp18f!.6f?I/65f?/S.Y?12f Y?e1f/d?Y?,. )?lmYt?I/M?xf?/?Y,p?yf“d ?M?N1 3 6 2 18 1 3 ? x 18 6 ?2 x = 216 27x 3f?/5? M?N 52 y = 25yf“K25y = 216 27x 3?y = 1 25 ? 216 x3 27 ? u dy dt = dy dx dx dt = 1 675 3x2 dx dt = 1 225x 2 dx dt “?x = 12(f)dx dt = 1(f/)ud? Y?,dy dx = 1 225 122(1) = 16 25 = 0.64(f/). 9. 4-12P = i2R6i = U r + R.?N?CRP?Cz dP dR. )P = i2R,i = U r + RK dP dR = 2iR di dR + i2= 2U2R (r + R)3 + U2 (r + R)2 = U2(r R) (r + R)3 66 7.? 1. e?3:x0?f0(x0)m?f0+(x0) (1) y = ? x2,x 6 0, xex,x 0, x0= 0 (2) y = ( x 1 + e 1 x ,x 6= 0, 0,x = 0, x0= 0 (3) y = ( x2sin 1 x, x 6= 0, 0,x = 0, x0= 0 ) (1) f0+(x0) = lim x+0 xex 0 x = 1f0(x0) = lim x0 x2 0 x = 0. (2) f0+(x0) = lim x+0 x 1 + e 1 x 0 x = lim x+0 1 1 + e 1 x = 0 f0(x0) = lim x0 1 1 + e 1 x = 1. (3) f0+(x0) = lim x+0 x2sin 1 x 0 x = 0f0(x0) = lim x0 x2sin 1 x 0 x = 0. 2. e?3?3?:?!m? (1) y = |ln|x| (2) y = |tanx| (3) y = 1 cosx ) (1) y = |ln|x| = ln(x),x 6 1 ln(x),1 0)k?f0(x) =1 3 3 x2lim x0 f0(x) = ?lim x0 f(x) = 0. 68 6. e (1) f(x)3x = g(x0)k?g(x)3x0:vk? (2) f(x)3x = g(x0)vk?g(x)3x0:k? (3) f(x)3x = g(x0)vk?g(x)3x0:vk? KEF(x) = f(g(x)3x0:? ) (1) EF(x) = f(g(x)3x0:U?. (i) ?f(u) = u2,g(x) = |x|,x0= 0f(u) = u23u0= 0 = g(x0)?f0(0) = 0g(x) = |x|3x0= 0?F(x) = f(g(x) = |x|2= x23x0= 0?F0(0) = 0 (ii) ?f(u) = u,g(x) = |x|,x0= 0f(u) = u3u0= 0 = g(x0)?f0(0) = 1g(x) = |x|3x0= 0?F(x) = f(g(x) = |x|3x0= 0?. (2) EF(x) = f(g(x)3x0:U?. (i) ?f(u) = |u|,g(x) = x2,x0= 0f(u) = |u|3u0= 0 = g(x0)?g(x) = x23x0= 0? g0(0) = 0F(x) = f(g(x) = |x2| = x23x0= 0?F0(0) = 0 (ii) ?f(u) = |u|,g(x) = x,x0= 0f(u) = |u|3u0= 0 = g(x0)?g(x) = x3x0= 0? g0(0) = 1F(x) = f(g(x) = |x|3x0= 0?. (3) EF(x) = f(g(x)3x0:U?. (i) ?f(u) = 2u+|u|,g(x) = 2 3x |x| 3 ,x0= 0f(u) = 2u+|u|3u0= 0 = g(x0)?g(x) = 2 3x |x| 3 3x0= 0?F(x) = f(g(x) = 2 ? 2 3x |x| 3 ? + ? ? ? ? 2 3x |x| 3 ? ? ? ? = ? x,x 0 x,x 0) (2) (cosx)(n)= cos ? x + n 2 ? (3) (lnx)(n)= (1)n1 (n 1)! xn y (1)(i) ?n = 1(ax)0= axlna = ax(lna)1Kn = 1. (ii) b?n = k=(ax)(k)= ax(lna)k K?n = k + 1(ax)(k+1)= h (ax)(lna)(k) i0 = (lna)k(ax)0= (lna)k axlna = ax(lna)k+1u ?n = k + 1. n?n?g,(ax)(n)= ax (lna)n(a 0). 70 (2)(i) ?n = 1(cosx)0= sinx = cos ? x + 2 ? Kn = 1. (ii) b?n = k=(cosx)(k)= cos ? x + k 2 ? K?n = k + 1(cosx)(k+1)= h (cosx)(k) i0 = h cos ? x + k 2 ?i0 = sin ? x + k 2 ? = cos ? x + (k + 1) 2 ? u?n = k + 1. n?n?g,(cosx)(n)= cos ? x + n 2 ? . (3)(i) ?n = 1(lnx)0= 1 x = (1)11(1 1)! x1 Kn = 1. (ii) b?n = k=(lnx)(k)= (1)k1 (k 1)! xn K?n = k + 1(lnx)(k+1)= h (lnx)(k) i0 = ? (1)k1 (k 1)! xk ?0 = k (1)k1 (k 1)! xk+1 = (1)k k! xk+1 = (1)k+11 (k + 1 1)! xk+1 u?n = k + 1. n?n?g,(lnx)(n)= (1)n1 (n 1)! xn . 5. n? (1) y = 1 x(1 x) (2) y = 1 x2 2x 8 (3) y = amxm+ am1xm1+ + a0 (4) y = cos2x (5) y = ex x (6) y = 2x lnx (7) y = eaxpn(x)pn(x)ng. ) (1) y = 1 x(1 x) = 1 x + 1 1 xKy (n) = ? 1 x 1 1 x ?(n) = ? 1 x ?(n) + ? 1 1 x ?(n) = (x1)(n)+(1 x)1(n)= (1)n n! xn1+ (1)2n n! (1 x)n1= n!(1)nxn1+ (1 x)n1 (2) y = 1 x2 2x 8 = 1 (x + 2)(x 4) = 1 6 ? 1 x 4 1 x + 2 ? Ky(n)= 1 6 ? 1 x 4 1 x + 2 ?(n) = 1 6(x4) 1)(n)(x+2)1)(n) =1 6(1) nn!(x4)n1(1)nn!(n+2)n1 =(1)n 6 n! ? 1 (x 4)n+1 1 (x + 2)n+1 ? (3) ?n mx(n)= (x2)(n)= = (xm)(n)= 0Ky(n)= 0 ?n = mx(n)= (x2)(n)= = (xm1)(n)= 0,(xm)(n)= (xm)(m)= m!Ky(n)= am m! ?n 0)d2y (5) y = lnu(x)d3y (6) y = sin(u(x)d3y ) (1) dy = (u0(x)v(x) + u(x)v0(x)dx, d2y = u00(x)v(x) + 2u0(x)v0(x) + u(x)v00(x)dx2 (2) dy = u0(x)v(x) u(x)v0(x) v2(x) dx, d2y = ? u00(x) v(x) u(x)v00(x) + 2u0(x)v0(x) v2(x) + 2u(x)(v0(x)2 v3(x) ? dx2 75 (3) dy = mum1(x)vn(x)u0(x) + num(x)vn1(x)v0(x)dx, d2y = m(m 1)um2(x)vn(x)(u0(x)2+ 2mnum1(x)vn1(x)u0(x)v0(x) + mum1(x)vn(x)u00(x) + n(n 1)um(x)vn2(x)(v0(x)2+ num(x)vn1(x)v00(x)dx2 (4) dy = au(x)lna u0(x)dx, d2y = au(x)lnalna(u0(x)2+ u00(x)dx2 (5) dy = u0(x) u(x) dx,d2y = ? u00(x) u(x) (u0(x)2 u2(x) ? dx2, d3y = ? u000(x) u(x) 3u0(x)u00(x) u2(x) + 2(u0(x)3 u3(x) ? dx3 (6) dy = cos(u(x)u0(x)dx,d2y = cos(u(x)u00(x) sin(u(x)(u0(x)2dx2, d3y = cos(u(x)u000(x) 3sin(u(x)u0(x)u00(x) cos(u(x)(u0(x)3dx3. 76 1?n9A 1.n 1. 3?nex0m?:(. )y = x3m1,1k?3:x0= 1?=x 1,1kf(x) 6 f(x0) = 1,?y0|x=1= 1 6= 0. 2. ux0 (a,b)ef0(x0) 0K3?!m?O(x0,),O+(x0,)?x O(x0,)?f(x0) f(x)?x O+(x0,)?f(x0) 0?45?3x0?( 0)?O(x0,) (a,b) ?x O(x0,)kf(x) f(x0) x x0 0l?x O(x0,)=x x0 f(x) ?x O+(x0,)=x x0 0kf(x0) 0,f0(x0) f(x0). yf0+(x0) =lim xx0+0 f(x) f(x0) x x0 0Kdm45?73x0?1(1 0)m?O+(x0,1) ?x O+(x0,1)=0 0Kf(x)3(a,b)S?k“:. yf0(a) f0(b) 0”?f0(a) 0,f0(b) 0(f0(a) 0)m?O+(a,1) ?x O+(a,1)=0 0 qf0(b) = f0(b) =lim xb0 f(x) f(b) x b 0Kd45?73b?2(2 0)?O(b,2) ?x O(b,2)=0 f(x) ?x2 O(b,1)Kkf(x2) 0Kx x 1) (3) f0(x) = ex,f0(x + x) = ex+x Kex+xx = f(x + x) f(x) = ex+x ex= ex(ex 1)l?exx = ex 1u = 1 x ln ex 1 x ?y (0,1) 6. ?f(x)3ma,bSY3(a,b)?| (x) = ? ? ? ? ? ? xf(x)1 bf(b)1 af(a)1 ? ? ? ? ? ? y.KFQ(x)?A. y(x) = ? ? ? ? ? ? xf(x)1 bf(b)1 af(a)1 ? ? ? ? ? ? = (a b)f(x) + (f(b) f(a)x + bf(a) af(b) qf(x)3ma,bSYKdY?oK$K(x)3a,bY qf(x)3(a,b)?Kd?oK$K(x)3(a,b)?. q(a) = ? ? ? ? ? ? af(a)1 bf(b)1 af(a)1 ? ? ? ? ? ? = 0,(b) = ? ? ? ? ? ? bf(b)1 bf(b)1 af(a)1 ? ? ? ? ? ? = 0Kd?n?3(a,b)S?k: 0() = 0. ?0(x) = (a b)f0(x) + f(b) f(a)K0 = 0() = (a b)f0() + f(b) f(a)=f0() = f(b) f(a) b a . (x)?An?/S= 1 2 ? ? ? ? ? ? x1y11 x2y21 x3y31 ? ? ? ? ? ? (x1,y1),(x2,y2),(x3,y3)L:I K(x)LA(x,f(x),B(a,f(a),C(b,f(b):?n?/?. 7. e?.KFf(b) f(a) = f0(c)(b a)c. (1) f(x) = x3,x 0,1 (2) f(x) = arctanx,x 0,1 ) (1) f0(x) = 3x2K3c2(1 0) = 13 03=3c2= 1qc (0,1)?c = 3 3 . (2) f0(x) = 1 1 + x2 K 1 1 + c2 (1 0) = arctan1 arctan0= 1 1 + c2 = 4 qc (0,1)?c = r 4 1. 8. e?f(b) f(a) g(b) g(a) = f0(c) g0(c) c. (1) f(x) = sinx,g(x) = cosx,x h 0, 2 i (2) f(x) = x2,g(x) = x,x 1,4