微积分习题答案
习题答案
习题1-1
1. (1) [-3,3];(2) (-
∞,0)∪(2,
+∞);
(3) (-2,1);(4) (-1.01,-1)∪(-1,0.99)
2. (1) [-1,0)∪(0,1);(2) (1,2];
(3) [-6,1).
3. (1) (-∞,1)∪(1,2],f(0)=0,f(2)=1.当a<0时,f(a)=1a,当0≤a≤1时,f(a)=2a,当1<a≤2
时,f(a)=1.
(2) (-2,2),f(0)=1,f((-a)2,当1<a<2时,f(a)=a2-1.
4. 1.
5. (1) 偶函数;(2) 非奇非偶函数;(3) 奇函数
.
8. (1) y=13arcsinx2;(2) y=lo
g2x1-x
(3) f-1(x)=12(x+1), -1
≤x≤1,
2-2-x, 1<x≤2.
9. (1) y=101+x2(-∞,+∞);(2) y=
sinxln2,(-∞,+∞);
(3) y=arctana2+x2(-∞,+∞).
习题1-2
1. (1) y=3u,u=arcsinv,v=ax;(2) y=u
3,u=sinv,v=lnx;
(3) y=au,u=tanv,v=x2;(4) y=lnu,u=v2,v=lnw,w=t3
,t=lnx.
2. (1) [-1,1],(2) [2kπ,(2k+1)π
],k∈Z;
(3) [-a,1-a];(4) (-∞,-1].
3. (1) φ(x)=6+x-x2;(2) g(x)=(1+x)2+(1+x)+1;
(3) f(x)=x2-2.
习题1-3
1. R(x)=4x-12x2.
2. R(x)≈130x,
117x+9100, 0≤x≤700,
700<x≤1000.
3. L=L(Q)=-15Q2+8Q-50,
=-Q5+8-50Q
.
习题2-1
略.
习题2-2
2. f(x)=-1,
1, x≤0
x>0,则limx→0f(x)
=1,但limx→0-f(x)=-1,limx
→0+f(x)=1,故limx→0f(x)不存在.
3. limx→0(x2+a)=a,limx
→0-e1x
=0,a=0.
习题2-3
2. ,,,,,,,.
3. (1)无穷大量.
(2) x→0+时为无穷大量,x→1时为无穷小量.x→+∞时为无穷大量.
(3) x→0+时为无穷大量,x→0-时为无穷小量.
(4) 无穷小量.
(5) 无穷小量.
(6) 无穷小量.
习题2-4
5. (1)3/5;(2) 0;(3) ∞; (4) 1/3;
(5) 4/3
6. (1) 16;(2) ∞;(3) 3;
(4) -22;
(5) 3x2.(6) 43;(7) n(n+1)2;(8)
(9) 1;(10) -1;(11) 0.
习题2-5
1.53;2. 25;
3. 1;4. 22;
5. 212;6. e-1;
7. e3;8. lna;
9. 2lna; 10. 0;11. e-12;12. 1;
13. 1;14. 1;15. e1
e;16. e-1.
习题2-6
1;
3. tanx-sinx=O(x3)
4. (1) ab;(2) k22;(3) 2;(4) 24;
(5) 1;(6) 1;(7) 49;(8) 3.
习题2-7
4. (1) x=1(可去),定义f(1)=2;x=2(第二类);
(2) x=0(可去),定义f(0)=1;x=kπ,k≠0,为整数(第二类);
(3) x=0(第一类;
(4) x=2(第二类);x=-2(可去),定义f(-2)=0;
(5) x=0(可去),定义f(0)=0.
6. f(x)=sgnx,x=0(第一类),f(x)∈C[(-∞,0)∪(0,+∞)]
7. (1) 12;(2) 3;(3) 0;(4) π3;
(5) 1.
习题3-1
1. 29.
2. -1x20.
3. 4x-y-4=0,8x-y-16=0
4. (1) -f′(x0);(2) -f′(x0);(3) 2f′(x0)
5. (1) 12x;(2)
-23x-53;
(3) 16x-56.
6. 连续但不可导.
8. (1) f′
(2) f′12,f′
9. f′(x)=cosx,
1, x<0,
x≥0.
10. a=2,b=-1.
11. (1) 在x=0处连续,不可导;(2) 在x=0处连续且可导;
(3) 在x=1必连续,不可导.
13. (1) -0.78m/s;(2) 10-gt;(3) 10g(s).
14. dQdtt=t0.
15. (1) limΔT→0Q(T+ΔT)
-Q(T)ΔT;(2) a+2bT.
习题3-2
1. (1) 3t;(2) xx+12xlnx;
(3) 2xsin2x-2xsinx+cosx-x2cosx-sin2x+x2sin
2x.
(4) 1-sinx-cosx(1-cosx)2;(5) sec
2x;
(6) xsecxtanx-secxx2-3secx²tanx
;(7) 1x1-2ln
10+3ln2;
(8) -1+2x(1+x+x2)2.
2. (1) 241+π2;(2) f′(0)=
325,f′(2)=1715;
(3) f′(1)=5.
3. 略.
4. (1) 3e3x;(2) 2x1+x4;
(3) 12x+1e2x+1;
(4) 2xln(x+1+x2)+1+x2;
(5) 2x²sin1x2-2x
cos1x2;(6) -3ax2sin2ax3;
(7) xx2²x2-1;(8) 2arcsinx24-x2;
(9) lnxx²1+ln2x;(10) nsinn-1x²cos(n+1)x;
(11) 11-x2+1-x2;(12
) -1(1+x)2x(1-x);
(13) -thx;(14) a2-x2.
5. 13.
6. 2x+3y-3=0; 3x-2y+2=0; x=-1; y=0.
7. (1) 2xf′(x2);(2) sin2x[f′(sin2x)-f′(cos
2x)].
8. (1) -x2-ayy2-ax;(2) 1-yx(lnx+lny+1);
(3) -ey+yexxey+ex;(4)
x+yx-y;
(5) ex+y-yx-ex+y.
9. (1) x+2(3-x)4(x+1)512(x+2)-43-x-5x+1;
(2) sinxcosxcos2xsinx-sinxln sinx;
(3) e2x(x+3)(x+5)(x-4)2+1x+1-12(x+5)-12(x-4).
10. (1) sinat+cosbtcosat-sinbt;(2) cosθ-θsinθ1-sinθ-θcosθ.
11. 3-2.
习题3-3
1. f(n)(x)=(-1)n-1(n-1)!(1+x)n.
2. y(n)=(-1)n²an²n!²(ax+b)-(n+1).
f(n)(x)=(-1)n2·n!·1(x
-1)n+1-1(x+1)n+1
3. (1) 0;(2) 4e,8e;(3) 7200,720.
4. (1) -b4a2y3;(2) e
2y(3-y)(2-y)3;
(3) -2csc2(x+y)cot3(x+y);(4) 2x2y[3(y2+1)
2+2x4(1-y2)](y2+1)3.
5. (1) -1a(1-cost)2;(2) 1f″(t).
6. (1) 4x2f″(x2)+2f′(x2);(2) f″(x
)f(x)-[f′(x)]2f.
习题3-4
1. (1) sint;(2) -1ωcosωt;
(3) ln(1+x);(4) -12e-2x;
(5) 2x;(6) 13tanx;(7) ln2x2;(8) -1-x2.
2. (1) 0.21,0.2,0.01;(2)
.02,0.0001.
3. (1) (x+1)exdx;(2) 1-lnx〖
〗x2dx;
(3) -12xsinxdx;(4) 2ln5²5ln tanx²1sin2xdx;
(5) -12cscx2dx;(6) 8[xx(1+lnx)-12e2x]dx;
(7) 121-x2arcsinx
+2arctanx1+x2d
x.
4. (1) ey1-xeydx;(2)
-b2xa2ydx;
(3) 22-cosyds;(4)
1-y21+2y²1-y2dx. 0.0201,0
5. (1) 2.0083;(2) -0.01;(3) 0.7954.
习题3-5
1. (1) 1.1;(2) 650;(3) 650-50
129.
2. (1) 96.56;(2) 是,提高2.
3. (1) a,axax+b,aax+b;(2) abebx,bx,b;
(3) axa-1,a,ax.
4. 提高8%;提高16%.
5. 5.9.
习题4-1
1. ξ=π2.
2. (1) 满足,有ξ=0;(2) 不满足第二个条件,没有;
(3) 不满足第一和第三个条件,有ξ=π2.
3. 有分别位于区间(1,2),(2,3),(3,4)内的三个根.
4. ξ=33.
习题4-2
1. (1) -35;(2)
12;(3) mnam-n;(
4) 1a
(5) 0;(6) 0;(7) 1;(8) 32;
(9) e;(10) e-2π
;(11) 1e;(12) ∞
(13) 13;(14) e-12.
2. m=-4,n=3
4. f″(x);
习题4-3
1. xex=x+x2+x32!+…+xn(n-1)!
+1(n+1)!(n+1+θx)eθxxn+1(
0<θ<1).
2. 1x=-1-(x+1)-(x+1)2-…-(x+1)n+(-1)n+1(x+1)n+1[-1+θ(x+1)]n+2(0<θ<1).
3. f(x)=-56+21(x-4)+37(x-4)2+11(x-4)3+(x-4)4.
4. (1) 16(提示:只要将sinx展开成三次多
项式即可).
(2) 12(提示:令u=1x,再将ln(1+u
)展开成二次多项式).
习题4-4
1. (1) (-∞,-1)和(3,+∞)为增区间,(-1,3)为减区间,f(-1)=
3为极大值,f(3)=-61为极小值.
(2) (1,+∞)为增区间,(0,1)为减区间,f(1)=1为极小值.
(3) (-∞,2)为增区间,(2,+∞)为减区间,f(2)=1为极大值.
(4) (-∞,0)和(0,2)为增区间,(2,+∞)为减区间,f(2)=-4为极大值.
5. 当a=2时,f(x)在x=π3取极大值3.
习题4-5
1. 15元
2. x=αcPQ11-α
3. (1) Q=3;(2) MC==6
4. (1) 1000件;(2) 6000件
5. (1) 431.325吨(2) 12次(3) 30.452天(4) 1366
43.9元
6. α=23(3-6)π.
7. t=14r2.
8. v=320000≈27.14(km/h)
习题4-6
1. (1) 在-∞,13下凸,13,+∞上凸,拐点1
3,227;
(2) 在(-∞,-1)上凸,(-1,1)下凸,(1,+∞)上凸,拐点(-1,ln2)及(1,ln 2);
(3) 在(-∞,-2)上凸,(-2,+∞)下凸,拐点(-2,-2e-2);
(4) 在(-∞,+∞)下凸,无拐点;
(5) 在(-∞,-3)上凸,(-3,6)上凸,(6,+∞)下凸,拐点6,227;
(6) 在-∞,12上凸,12,+∞下凸,拐点12,earct
an12.
3. a=-32,b=92.
4. (1) 垂直渐近线x=0;(2) 水平渐近线y=0;
(3) 水平渐近线y=0,垂直渐近线x=3;
(4) 垂直渐近线x=12,斜渐近线y=12x+1〖
〗4.
5. (1) 定义域(-∞,+∞),极大值f(1)=12
,极小值f(-1)=-12,拐点3,34,-3,-34,渐近线y=0;
(2) 定义域(-∞,+∞),极大值f(-1)=π2-1,极小值f(1
)=1-π2,拐点(0,0),渐近线y=x+π,y=x-π;
(3) 定义域(0,+∞),极大值f(1)=2e,拐点,2,4e2,渐近线y=0.
习题5-1
1. (1) 27x7〖
〗2-103x32
+C;(2) 2x-43x
32+25x52+C;
(3) 3xex1+ln3+C;(4) x+sinx2+C;
(5) 2x-523
xln2-ln3+C;(6) -(cotx+tanx
)+C.
2. (1) y=x2-2x+1;(2) cosx+C;
(3) x-sinx;(4) Q=100013P
习题5-2
1. (1) 1a;(2) 17;(3)
110;(4) -12;
(5) 112;(6) 12;(7) -2;(
8) 15;
(9) -1;(10) -1;(11) 13;(12) 1
2;
(13) -1;(14) 32.
2. (1) 15e5t+C;(2) -18(3-2x)4+C;
(3) -12ln1-2
x+C;(4)
-12(2-3x)23+C;
(5) -2cost+C;(6) ln
lnlnx+C;
(7) 111tan11x+C;(8)
-12e-x2+C;
(9) lntanx+C;(10)
-lncos1+x2+C;
(11) arctanex+C;(12) -13
(2-3x2)12+C;
(13) -34ln1-
x4+C;(1
4) 12cos2x+C;
(15) 12arcsin2x3+
149-4x2+C;(1
6) x22-92ln(x2
+9)+C;
(17) 122ln2x-12x+1+C;(18) 13lnx-2x+1+C;
(19) t2+14ωsin2(ωt+φ)+C;(20) -13ωcos3(ωt+φ)+C;
(21) 12cosx-110
cos5x+C;(22) 13sin
3x2+sinx2+C;
(23) 14sin2x-124
sin12x+C;(24) 13sec3x-
secx+C;
(25) (arctanx)2+C;(26) -1arcsinx+C;
(27) 12(lntanx)2+C;(28)
-1xlnx+C;
(29) a22(arcsinxa
-xa2a2-x2)+C;(30) x1+x2+C;
(31) x9-9-3arccos3
x+C;(32) 12(arcsinx+ln
x+1-x2)+C;
(33) arcsinx-x1+1-x2+C;(34) arcsinxa-a
2-x2+C;
(35) -4-x2x-arcsinx2+C;
(36) ln1+x+x2+2x-2xx
2+2x+C;
(37) -11+tanx+C;(38) x+
lnx1+xex+C.
习题5-3
1. (1) -xcosx+sinx+C;(2) -(x+1)e-x
+C;
(3) xarcsinx+1-x2+C;(4) sin
x-cosx2e-x+C;
(5) -217e-2xx
2+4sinx2+C;(6) -12x2+xtanx+lncos
x+C;
(7) -t2+14
e-2t+C;
(8) x(arcsinx)2+21-x2
arcsinx-2x+C;
(9) 12-15sin2x-110cos2x)ex+C;
(10) 3e3x(3x2-23x+2+C;
(11) x2(coslnx+sinlnx)+C;
(12) -12x2-32cos2x+x2sin2x+C;
(13) 12(x2-1)ln(x-1)-14x2-12x+C;
(14) x36+12x2sinx+xcosx-sinx+C;
(15) -1x(ln3x+3ln2x+6lnx+6)+C;
(16) -14xcos2x+18sin2x
+C;
(17) -12xcot2x-12x-12cotx+C;(18) 12x2e
x2+C;
(19) xlnlnx+C;(20) (1+ex)ln(1+ex)-ex+C;
(21) 12tanxsecx-12ln
secx+tanx+C;
(22) -ln(x+1+x22(1+x2)+x22+x2
+C;
(23) ex1+x+C;(24) x-121+x2earctanx+C.
习题5-4
(1) lnx+1x2-x+1+3arctan2x-13+C;
(2) x33+x22+x+8lnx-3lnx-1-4lnx+1+C;
(3) x-tanx+secx+C;
(4) 14lntanx2-18tan2x2+C.
习题6-1
1. 13(b3-a3)+b-a.
2. (1) 1;(2) 14πa2.
3. (1) ∫10x2dx较大;(2) ∫10exdx较大.
4. (1) 6≤∫41(x2+1)dx≤51;(2)
π9≤∫31
3xarctanxdx≤23π;
(3) 2ae-a2<∫a-ae-x2dx<2a;(4)
-2e2≤∫02ex2-xdx≤-2e-1〖
〗4.
习题6-2
1. (1) 2x1+x4;(2) x5e
-3x;
(3) (sinx-cosx)cos(πsin2x);(4) sinx-
xcosxx2.
2. (1) -12;(2) 6;(3) 2.
3. cosxsinx-1.
4. 当x=0时.
5. (1) 23(8-33);(2) 16;(3) 1+π8;(4) 203.
6. -32.
习题6-3
1. (1) 0;(2) 51512;(
3) 16;(4) 14
;
(5) π6-38
;(6) 2(3-1);(7) 2-233;
(8) π2;
(9) 12ln32;
(10) ln2-13ln5;(11) 7
ln2-6ln(62+1);
(12) 43.
2. (1) 0;(2) 0;(3) 32π.
习题6-4
2. (1) 1-2e;(2) 14(e2+1);
(3) 4(2ln2-1);(4) 14-
133π+12ln32;
(5) 15(eπ-2);(6) 2-34
ln2;
(7) π36-π4;(8) 12(esin1-ecos1+1);
(9) ln2-12;(10) 12-38ln3.
3. 0.
习题6-5
1. (1) 1;(2) 2;(3) 4
3;(4) 76;
(5) 12+ln2;(6) 1
6;(7) e+1e-2;(
8) b-a.
2. (1) Vy=2π;(2) Vx=1287
π,Vy=12.8π;
(3) Vy=310π;(4) Vx=pa2π;(5)
Vy=4π2.
3. (1) a=1e,(x0,y0)=(e2,1);(2) S=16e2-12.
4. 12ln2提示:f(x)=0,
x1+x2, x≥0
x<0.
5. a=-4,b=6,c=0.
6. 50;100.
7. (1) Q=2.5,L=6.25;(2) 0.25.
8. 96.73
习题6-6
1. (1) 13;(2) 发散;(3) 1a;(4)
(5) 发散;(6) π;(7) 83;(8) 1;
(9) π2;(10) -1;(11
) 发散;(12) 1.
2. 当k>1时收敛于1(k-1)(ln2)12-1;
当k≤1时发散;当k=1-1lnln2时取得最小值.
3. n!.
发散;
4. (1) π4;(2) π2
5. In=-(2n)!!(2n+1)!!=22n(n!)2〖
〗(2n+1)!(n=0,1,2,…).
6. (1) 1nΓ1n;(2) Γ(α+1);
(3) 1nΓm+1n;(4) 12Γn+1
2.
习题7-1
1. 略.
2. (1) (a,b,-c),(-a,b,c),(a,-b,c);
(2) (a,-b,-c),(-a,b,-c),(-a,-b,c);
(3) (-a,-b,-c).
3. 坐标面: (x0,y0,0),(0,y0,z0),(x0,0,z0);
坐标轴: (x0,0,0),(0,y0,0),(0,0,z0).
4.x轴: 34, y轴: 41, z轴: 5.
5. (0,1,-2).
6. 略.
习题7-2
1. MA→=-12
(a+b);MB→=12(a-b);MC→=12(a+b);MD
→=12(b-a).
2. 略.
3. (2,1,1).
4. (16,0,-20).
5. M1M2→=(
1,-2,-2),M1M2→=3.
13,-23,-23或-13
,23,23.
习题7-3
1. (1) 1;(2) 4;(3) 28.
2. (1) 3,5i+j+7k;(2) -18,10i+2j+14k;
(3) -10i-2j-14k.
3. -32.
4. ±(62,82,0).
5. 14.
6. 略.
7. 45j-35
k或-45j+
35k.
8. ∠A=76°22′,∠B=79°2′,∠C=24°36′.
习题7-4
1. 3x-2y+5z-22=0.
2. 2x+9y-6z=121.
3. 略.
4. x+z-1=0.
5. x+y+z-2=0.
6. 2x+3y+z-6=0.
7. (1) x=2;(2) x+3y=0;(3) x-y=0.
8. 13,23,-2
3.
9. (1) 互相垂直;(2) 互相平行;
(3) 斜交(相交但不垂直).
习题7-5
1. (1) x-23=y-31
=z-11;(2) x-31=y-42=z+4-1;
(3) x-21=y-20=z+1〖
〗0;(4) x2=y-31=z+23.
2. x+3-5=y=z-25, [JB({〗x=-3-5t,
y=t,
z=2+5t.
3. x-2=y-23=z-4〖
〗1.
4. x-21=y+22=z3
.
5. x-10=y+37=z+2〖
〗16.
6. 461,661,-361.
7. B=1,D=-9.
8. x-3-1=y-31=z
1.
9. φ=arcsin1310.
10. 4x-y-2z-1=0.
11. y-z+3=0,
x-y-z+1=0.
12. 5.
13. (1)垂直,(2) 平行,(3) 重合.
习题7-6
1. (x+1)2+(y+3)2+(z-2)2=32.
2. 以点(1,-2,-1)为球心,半径等于6的球面.
3. (1) x23+y24+z24=1; x23+y2
4+z23=1;
(2) x2-y2-z2=1; x2+y2-z2=1.
4. (1) 母线平行于z轴的椭圆柱面;(2) 母线平行于x轴的抛物 柱面;
(3) 椭圆锥面;(4) 旋转椭球面;
(5) 双叶双曲面;(6) 圆锥面.
5. 3y2-z2=16, 3x2+2z2=16
6. x2+y2+(1-x)2=9,
z=0; (1-z)2+y2+z2=9,
x=0; x+z=1,
y=0.
7. (1) 椭圆;(2) 双曲线;(3) 抛物线.
8. 略.
习题8-1
1. (1) (x,y)x2a2+y2b2≤1;
(2) {(x,y)x>y,且x-y≠1};
(3) (x,y)-1≤yx≤1,且x≠0={x>0,-x≤y≤x;x<0,x≤y≤-x};
(4) {(x,y)x≥y,x2+y2
≤1,y≥0}.
2. (1) 31;(2) 1x3
-4xy+12y2;
(3) (x+y)3-2(x2-y2)+3(x-y)2.
3. f(x)=(x+2)x,
F(x,y)=y+x-1.
4. 略
习题8-2
1. (1) 不存在,(2) 存在.
2. (1) 0,(2) 1,(3) 2,(4)
3. {(x,y)y2=2x,x∈R}
.
习题8-3
1. (1) z′x=y(1+x)y-1,z′y=(1+x)yln(1+x);
(2) z′x=-yx2coty
x²sec2yx,z′
y=1xcotyx²sec2yx;
(3) z′x=-yx2+y2,z′
y=xx2+y2;
(4) u′x=-zlnyx2²y
zx,u′y=zx
²yzx-1,u′z=1xyzx²
lny.
2. -1,2.
3. 1,1+π6.
4. 略. 0.
5. 偏导数存在.
6. α=π4.
7. Δz=-0.12,dz=-0.1.
8. (1) du=dx-dy;
(2) dz=-xy(x2+y2)3/2dx+xy(x2+y2)3/2dy.
习题8-4
1. (1) 2e2cost+3t2[3t-sint];
(2) 3-4t-3+32t
12sec23t+2
t2+t32.
2. (1) z′u=(2xy-y2)cosv+(x2-2xy)
sinv;
(2) z′v=-(2xy-y2)usinv,z′y=
euvx2+y2(ux+vy).
3. (1) ux=1y
f′
1,uy=-x
y2f′1+1zf′2,uz=-yz2f′2;
(2) zx=2xf′,z
y=2yf′;
(3) ux=f′1+yf
′2+yzf′3,uy
=xf′2+xzf′3,uz=xyf′2.
4. 略.
5. (1) dz=(x2+y2)sin(2x+y)2sin(2x+y)x2+y2(xdx+yd y)
+cos(2x+y)ln(x2+y2)(2dx+dy);
(2) du=1f(x2+y2-z2)dy-yf′(x2+y2-z2)f(x2+y2-z2)(2xd x+2ydy-2zdz).
6. (1) z′x=ex+y+y
zez-xy,z′y=ex+y
+xzez-xy;
(2) zx=zx+z,
zy=z2y(x+z).
7. 略.
8. zx=(vcosv-usinv)e
-u,zy=(ucosv+vsinv)e-u
.
9. dudx=f′x+y2f′y1-xy+zf′zxz-x.
习题8-5
1. (1) 2zx2=12x
2-8y2,2zy2=12y2-8x2
,2zxy=-16xy;
(2) 2zx2=2xy(x
2+y2)2,2zy2=-xy(x2+y2)2,2zx
y=y2-x2(x2+y2)2;
(3) 2zx2=yxln2y,
2z
y2=x(x-1)yx-2;2zx
y=yx-1(1+xlny);
(4) 2zx=1x,2zy2=-xy2,2zxy=1y.
2. (1) 2zx2=4x
f″(x2+y2)+2f′(x2+y2),
2zy2=4yf″(x2+y2)+2f′(
x2+y2);
(2) 2zx2=y2f″11+2yf″12+f″22,2zy2=x2f″ 11+4xf″12+4f″22,
2zxy=xyf″
11+2yf′12+f′1+xf″
21+2f″22.
3. 2zx2=z(2x-2-z
2)x2(z-1)3,2zy2=z(2z-2-z2)y2(z-1)3,
2zxy=-zxy(z-1)3.
习题8-6
1. 1+23.
2. 23.
3. α=π4时取得最大值2
;α=5π4时取得最小值-2;
α=7π4时,方向导数为零.
习题8-7
1. (1) 极大值f(0,0)=3;(2) 极小值f1
2,-1=-e2;
(3) 极大值fa3,a3
=a327 (a>0),
极小值fa3,a3=a327 (a<0).
2. 极大值z(4,1)=7,最小值z43+223,-1≈-11.67
.
3. 极小值z(2,2)=4.
4. a≥12,最小距离为a-14;
a≤12,最小距离为a.
5. a的分法是三等分时,乘积最大为a327.
6. x=100,y=25,f(100,25)=1250.
7. x=70,y=30,λ=-72,L=145(万元).
习题8-8
1. (1) ∫1-1dx∫3-3f(x,y)dy, ∫
3-3dy∫1-1f(x,y)dx;
(2) ∫40dx∫2xxf(x,y)dy, ∫4
0dy∫y14y2f(x,y)dx;
(3) ∫r-rdx∫r2-x20f(x,y)
dy, ∫r0dy∫r2-y2-r
2-y2f(x,y)dx.
2. (1) ∫10dx∫xx2f(x,y)dy;(2) ∫a0dy∫a+a2-y2a-a2-y2f(x,y)dx;
(3) ∫10dy∫2-yyf(x,y)dx.
3. (1) e-1e
2;(2) 2915;(3) -12;(4) 23;
(5) 1-2π;(6) 2πR22+R3;(7) 3
64π2;(8) 2-π2.
4. 5144.
5. π.
6. 8π.
7. SD=12e-1,VD=12
e2-e-12.
习题9-1
1. (1) a>1收敛;0<a≤1发散;(2) 发散;
(3) 发散;(4) 收敛;(5) 发散;(6) 发散;
(7) 发散;(8) 发散.
2. (1) 收敛,s=32;(2) 收
敛,s=14;(3) 发散;(4) 发散.
习题9-2
1. (1) 收敛;(2) 发散;(3) 发散;(4) 收敛;
(5) a>1,收敛;0<a≤1发散;(6) 发散;
(7) 发散;(8) 收敛;(9) 发散;(10) 发散;
(11) 收敛;(12) 收敛;(13) 收敛;(14) 收敛;
(15) 收敛;(16) 收敛.
习题9-3
1. (1) 条件收敛;(2) 绝对收敛;(3) 绝对收敛;(4)
绝对收敛;
(5) 绝对收敛;(6) 条件收敛;(7) 绝对收敛;(8) 条件收敛.
习题9-4
1. (1) (-∞,+∞);(2) (-e,e);(3) (-2,2);
(4) (-1,1);
(5) (-4,0);(6) 12,3〖
〗2.
2. (1) -ln(1+x);x<1
;(2) 2x(1-x2)2,x<1;
(3) 当x≠0且x<
1时,s(x)=1+1x
-1ln(1-x);当x=0时,s(x)=0;
(4) 1+x(1-x)2,x<1.
3. (1) 1532;(2) 1
2ln(1+2);(
3) 109;(4) 4.
习题9-5
1. (1) 1-x22²2!+x
42²4!-…+(-1)nx2n2²(2n)!+…(-∞<x<+∞);
(2) ∑∞n=1(-1)n-1(2n-
1)!x22n-1
(-∞<x<+∞);
(3) ∑∞n=1(-1)n-1x2n-1〖
〗(n-1)!(-∞<x<+∞);
(4) ∑∞n=0x2n, x<1;
(5) 22∑∞n=0
(-1)nx2n(2n)!+x2n+1(2n+1)!(-∞<x<+∞).
2. (1) ∑∞n=012n+1
(x-1)n(-1<x<3);
(2) ∑∞n=0[JB((〗(-1)n2²x-π3
2n(2n)!+(-1)n+132 x-π22n+1(2n+1)!(-∞< x<+∞);
(3) ∑∞n=0(-1)n12
n+2-122n+3(x
-1)n(-1<x<3);
(4) ∑∞n=0(-1)n3n+1(x-3)n(0<x<6).
3. (1) 2.71828;(2) 0.25049.
习题10-1
1. (1) 一阶,(2) 二阶,(3) 三阶,(4) 一阶.
2. 略.
3. y′=y-xx.
4. y′=y-x+1.
习题10-2
1. (1) (1-x)(1+y)=C(C为任意常数,以下C,C1,C2 …均为任意常数);
(2) 1-x2=lny+C;(3) y2=C(1-x2)-1;
(4) secx+tany=C;(5) 2y3+3y2-2x3-3x
2=5;
(6) (y+1)e-y=12(1+x2);(7) ey=12(e2x+1).
2. T=T0e-kt+α(1-e-kt), k为比例系数.
3. (1) y+x2+y2=Cx2;(2) y=2xa
rctan(Cx);
(3) x3+y3=Cx2;(4) y=2x1+x2;
(5) y=xe1-x;(6) (x+3)2+(y+1)2=Ce
-arctanyx;
(7) x+3y+2ln2-x-y=C.
4. (1) y=Cex-12(sinx+co
sx);(2) y=xn(C+ex);
(3) x=2(y-1)+Ce-y;(4) x=y+Ccosy;
(5) y=(x+1)ex;(6) y=2(1+x3)3(1+x2);
(7) y=2lnx-x+2;(8) y=(1+sinx-xcosx)²e-x2;
(9) y3=Cx3+3x4;(10) 1x2=1-y2+C
e-y2.
5. y′=3yx2-2²yx,y-x=-x3y.
6. x=ab+x0-abe-bt.
7. f(x)=-2e-3x-1.
8. C(x)=(x+1)[C0+ln(x+1)].
9. x=ab(C0x0-a)
1b+1²x0.
习题10-3
1. (1) y=(x-3)ex+12C1x2+C
2x+C3;
(2) y=xarctanx-12ln(1+x2)+C1x+C2;
(3) y=C1arctanx+C2;
(4) y=-lnx+c1
+c2;
(5) 1+C1x2=(C2t+C2)2;
(6) lny=C1(y-x)+C2.
2. (1) y=16x3lnx-11
36(x3-1);(2) y=lnx+12ln2x;
(3) y=x.
3. C1+C2ex+x.
4. (1) y=(C1+C2x)e2x;(2) y=
C1e-x+C2e2x;
(3) y=9e-2x-8e-3x;(4) y=-13exxcos3x.
5. (1) y=(1-12x)e-2x+C1e-5x +C2e2x;(2) y=(x+1)2+C1e2x
+C2e4x;
(3) y=118cosx+4sinx-1
8cos3x;(4) y=x+12x2e4x.
6. f(x)=2(ex-x).
7. a=-3,b=2,α=-1;y=C1ex+C2e2x+xex.
8. φ(x)=12(sinx+cosx+ex).
9. y=23e2x-23e-x-xe-x.
10. y=-7e-2x+8e-x+(3x2-6x)e-x
.
11. s=mgkt-m2gk2(1
-e-kmt).
习题10-4
1. C(x)=3ex(1+2e3x)-1.
2. R=abs0(ebt-1),S(t)=s0
e-bt.
3. Y(t)=Y0eγt,D(t)=αY0γ
eγt+βt+D0-αY0γ,limt→+∞D(t)Y(t)=α〖 〗γ.
4. (1) Y(t)=(Y0-Ye)eμt
+Ye,Ye=b1-a,μ=1
-aka,
C(t)=a(Y0-Ye)eμt+Ye,
I(t)=(1-a)(Y0-Ye)eμt;
(2) limt→+∞Y(t)I(t)
=11-a.
5. y(6)=50001+11.5e-3(ln11.5- ln8).
习题11-1
1. (3),(4).
2. (1) 一阶;(2) 五阶;(3) 三阶;(4) 六阶;
(5) 二阶.
3. (1) Δ2yt=2;(2) Δyt=(e-1) 2et;
(3) Δ2yt=6(t+1),Δ3yt=6;(4) Δ 2yt=lnt2+4t+3t2+4t+4.
4. 略.
习题11-2
1. yA(t)=A1+A2t+1(A1,A2为
任意常数.以下A,A1,A2…均为任意常数).
2. a(t)=-1+15,f(t)=1-1t²2t.
3. 略.
4. (1) yt=A-13t+1;(2) yt=A-12t+79+13t ;
(3) yt=A(-1)t+13²2t;(4
) yt=A-13²2tcosπt.
5. (1) yt=0.1³38t+0.1;(2) yt=12t-2+t;
(3) yt=2t-t+4;(4) yt=(-4)t+sinπt.
6. yt=A(-a)t+b1+a.
7. (1) 略;
(2) yt=1y0-
bC-aaCt+bC-a)-1,
1y0+bat-1,当C≠a时,
当C=a时.
(3) yt=12
t+1+32-1.
习题11-3
1. (1) yA(t)=A1(-1)t+A212t;
(2) yA(t)=(3)t(A1cosωt+A2sin
ωt),tanω=-2;
(3) yA(t)=(A1+A2t)²4t;
(4) yA(t)=A1cosπ3t+A2sin
π3t;
(5) yA(t)=A1(1.8)t+A2(2.1)t;
(6) yA(t)=A1[2(a+1)+2a+1)]t+A 2[2(a+1)-2a+1]t.
2. (1) yt=A15+172t+A25-172t-1;
(2) yt=2tA1cosπ3t+
A2
sinπ3t+13(a+bt);
(3) yt=A1+A2²2t+14³5t;
(4) yt=A1+A2-13t+cosπ2t-2sinπ2 t;
(5) yt=A1(-1)t+A2(-2)t+t2-t+3;
(6) yt=A1(-2)tt+A2²3t²t11
5t-225.
3. (1) t=25t2+125t+64125+1 86125(-4)t;
(2) t=4t+43(-2)t-43;
(3) t=195130-20〖
〗130(-4)t-92613t;
(4) t=4+32
12t+12-72t.
习题11-4
1. Yt=(Y0-Ye)αt+Ye,Ye
=1+β1-α;
Ct=(Y0-Ye)αt+αI+β1-α.
2. Yt=(Y0-Ye)²λt+Ye,其中λ=
1+r(1-α),Ye=β1-α;
Ct=α(Y0-Ye)λt+Ye;It=(1-α)(Y0 -Ye)λt.
3. Yt=Y0+βα²λt-βα,其中λ=δrδr-α;
St=(αY0+β)²λt;It=1δ(αY0+
β)²λt.
4. Dn(t)=A1λt1+A2λt2,其中λ1,2=2[( ab+1)±1+2ab].
5. Yt=(β)t(A1cosωt+A2s
i
nωt)+α1-β,其中ω=arctan1β-1,0<ω<π
Uπ=(β)t+1[A1cosω(t-1)
+A2sinω(t-1)]+αβ1-β;
St=(β)t{A1[βcosω(t
-1)-cosω(t-2)]+A2[βsinω(t-1)- sinω(t-2)]}.