高数同济6版课后答案

高数同济6版课后答案（通用2篇）

篇1：高数同济6版课后答案

同济六版上册高数总结

微分公式与积分公式

(tgx)secx

(ctgx)csc2x

(secx)secxtgx(cscx)cscxctgx(ax)axlna

1(logax)xlna2(arcsinx)1x21(arccosx)x21(arctgx)1x21(arcctgx)1x2tgxdxlncosxCctgxdxlnsinxCsecxdxlnsecxtgxCcscxdxlncscxctgxCdx1xarctgCa2x2aa

dx1xalnx2a22axaCdx1axa2x22alnaxCdxxarcsinCa2x2a



2ndx2cos2xsecxdxtgxCdx2csc2sinxxdxctgxCsecxtgxdxsecxCcscxctgxdxcscxCaxadxlnaCxshxdxchxCchxdxshxCdxx2a2ln(xx2a2)C

2Insinxdxcosnxdx00n1In2n







x2a22xadxxaln(xx2a2)C22x2a2222xadxxalnxx2a2C22x2a2x222axdxaxarcsinC22a22

三角函数的有理式积分：

2u1u2x2du

sinx，cosx，utg，dx

21u21u21u2

两个重要极限：

公式1lim

sinx

1公式2lim(1x)1/xe

x0x0x

有关三角函数的常用公式

和差角公式：

和差化积公式：

sinsin2sin

sin()sincoscossincos()coscossinsintg()

tgtg1tgtgctgctg

1ctg()

ctgctg



22

sinsin2cossin

22

coscos2coscos

22

coscos2sinsin

cos



三倍角公式:半角公式：

sin(3α)=3sinα-4sin^3(α)sin(α/2)=±√(1-cosα)/2cos(3α)=4cos^3(α)-3cosαCos(α/2)=±√(1+cosα)/2

降幂公式:万能公式：

sin^2(α)=(1-cos(2α))/2=versin(2α)/2sinα=2tan(α/2)/[1+tan^2(α/2)]cos^2(α)=(1+cos(2α))/2=covers(2α)/2cosα=[1-tan^2(α/2)]/[1+tan^2(α/2)]tan^2(α)=(1-cos(2α))/(1+cos(2α))tanα=2tan(α/2)/[1-tan^2(α/2)]

推导公式

tanα+cotα=2/sin2αtanα-cotα=-2cot2α1+cos2α=2cos^2α1-cos2α=2sin^2α1+sinα=(sinα/2+cosα/2)^2

abc

2R正弦定理：

sinAsinBsinC

余弦定理： c2a2b22abcosC反三角函数性质：arcsinxarccosx



arctgxarcctgx



(特别要注意这两个恒等式，证明的话，只需做出左边的函数的导数为0即可）

高阶导数公式——莱布尼兹（Leibniz）公式：

(uv)

(n)

k(nk)(k)

Cnuvk0n

u(n)vnu(n1)v

n(n1)(n2)n(n1)(nk1)(nk)(k)

uvuvuv(n)

2!k!

中值定理与导数应用：

拉格朗日中值定理：f(b)f(a)f()(ba)f(b)f(a)f()



F(b)F(a)F()

当F(x)x时，柯西中值定理就是拉格朗日中值定理。

曲率：

弧微分公式：dsy2dx,其中ytg平均曲率：K



:从M点到M点，切线斜率的倾角变化量；s：MM弧长。s

yd

M点的曲率：Klim.23s0sds(1y)

直线：K0;1

半径为a的圆：K.a

定积分的近似计算：

b

f(x)

ab

ba

(y0y1Lyn1)n

ba1

[(y0yn)y1Lyn1] n2

f(x)

a

定积分应用相关公式：

功：WFs

水压力：FpA

mm

引力：Fk122,k为引力系数

r

b1

函数的平均值：yf(x)dxbaa12f(t)dtbaa

b

微分方程的相关概念：

一阶微分方程：yf(x,y)或　P(x,y)dxQ(x,y)dy0

可分离变量的微分方程：一阶微分方程可以化为g(y)dyf(x)dx的形式，解法：

g(y)dyf(x)dx得：G(y)F(x)C称为隐式通解。

dyy

f(x,y)(x,y)，即写成的函数，解法：dxx

ydydududxduy设u，则ux，u(u)，代替u，xdxdxdxx(u)ux齐次方程：一阶微分方即得齐次方程通解。

一阶线性微分方程：

dy

1P(x)yQ(x)

dx

P(x)dx

当Q(x)0时,为齐次方程，yCe

当Q(x)0时，为非齐次方程，y(Q(x)edy

2P(x)yQ(x)yn，(n0,1)

dx

P(x)dx

dxC)e

P(x)dx

全微分方程：

如果P(x,y)dxQ(x,y)dy0中左端是某函数的全微分方程，即：

uu

du(x,y)P(x,y)dxQ(x,y)dy0P(x,y)Q(x,y)

xyu(x,y)C应该是该全微分方程的通解。

二阶微分方程：

f(x)0时为齐次d2ydy

P(x)Q(x)yf(x)2

dxdxf(x)0时为非齐次

二阶常系数齐次线性微分方程及其解法：

(*)ypyqy0，其中p,q为常数；求解步骤：

1、写出特征方程：()r2prq0，其中r2，r的系数及常数项恰好是(*)式中y,y,y的系数；

2、求出()式的两个根r1,r23、根据r1,r2的不同情况，按下表写出(*)式的通解：

二阶常系数非齐次线性微分方程

ypyqyf(x)，p,q为常数f(x)exPm(x)型，为常数；f(x)ex[Pl(x)cosxPn(x)sinx]型

篇2：高数同济6版课后答案

Step 1 (for section A)

Ⅰ. 1. beef2. medium3. kind4. small

5. help6. juice7. bowl8. would; tea

9. reason 10. Friday

Ⅱ. 1~5 DABDA 6~10 BBCDA

Ⅲ. 1~5 BDBAA 6~8 BDB

Step 2 (for section B)

Ⅰ. 1. a large bowl of noodles2. telephone number 3. a bowl of tomato soup 4. a cup of green tea

5. a piece of beef 6. 一些特色菜

7. 多大碗的8. 想做某亊 9. 什么种类的

10. ……的理由

Ⅱ. 1~5 DCBCC 6~10 CBBCD

Ⅲ. 1~5 DBCDD 6~10 BCCBA

Ⅳ. 1. different2. countries3. twelve

4. each5. dumplings6. with

7. Everyone8. new 9. dont10. life

Step 3 (for Self Check)

Ⅰ. 1. large 2. juice3. would4. soup

5. dumplings6. medium 7. beef

8. potatoes 9. chickens; chicken

10. ordered

Ⅱ. 1~5ADABA 6~10BDCCB

Ⅲ. 1~5CABCC

Step 4 能力拓展专版

Ⅰ. 1~5 BBCAB

Ⅱ. 1~5 BACAA6~10 BBACB

11~15 DCBBC

Ⅲ. 1. drinking 2. family 3. even 4. make

5. kinds6. Sometimes7. money

8. favorite 9. quite 10. with

【新目标英语七年级(下)Unit 9 Step by Step随堂通】

Step 1（for Section A）

Ⅰ. 1. What did; did her homework

2. Did; clean; did; When did; clean

3. How was; played computer games

Ⅱ. 1. speak2. playing3. reading4. swimming

5. cleaned6. watched7. is8. Dont turn

Ⅲ. 1~5 DBCAA6~10 CCDBB

Step 2（for Section B）

Ⅰ. 1. watched TV2. played tennis

3. went to the movies4. went to the mountains

5. played chess6. had a party

Ⅱ. 1~5 CCCBD6~10 BDABD

Ⅲ. 1~5 CBADA6~10 CACCD

Step 3 (for Self-check)

Ⅰ. 1. homework 2. clean3. enjoy4. movies

5. played6. geography7. busy

8. went 9. reading 10. watched

Ⅱ. 1. think; likes; stay; home2. sat; for

3. Last; went; with4. How was; It

5. does; night6. is; kind of7. wants; an

Ⅲ. 1. had2. was3. had4. got5. went

Ⅳ. 1. for2. left 3. afraid4. took

5. minutes6. paper7. there8. another

9. place 10. said

Step 4 能力拓展专版

Ⅰ. 1~5 CBCCB

Ⅱ. 1. 各自付账。

2. Playing a joke on me. / 戏弄我。

3. 24. In the first circle, 56+79 divided by 5=27.

The same formula applies to circles two and three.

4. F. Because they represent the places First,

Second, Third and Fourth.

5. CZ.

【新目标英语七年级（下）Unit 8单元要点检测题参考答案及录音稿】

Ⅰ. 听录音，选出你所听到的单词或词组，读一遍。 （每小题1分，共5分）

Text 1 Id like some cabbages, mutton and dumplings， please.

Text 2 I dont like ice cream， cabbages or oranges.

Text 3 What size and what kind of drink would you

like?

Text 4 I want to order some food, please.

Text 5 We have two great new specials.

Ⅱ. 听五段小对话，找出问题的正确答语。每段对话

读一遍。（每小题1分，共5分）

Text 6Can I help you?

Text 7What kind of noodles would you like?

Text 8What size bowl of dumplings would you like?

Text 9Whats your address?

Text 10Whats your phone number?

Ⅲ. 听下面两段大对话，每段对话后有几个小题，从

题中所给的三个选项中选出最佳选项。每段对

话读两遍。（每小题1分，共5分）

听第11段对话, 回答第11至12小题。

Text 11

M: Can I help you?

W: Yes. Id like a large bowl of noodles.

M: And what kind of noodles would you like?

W: Id like beef and broccoli noodles.

听第12段对话, 回答第13至15小题。

Text 12

M: Hello, House of Dumplings!

W: Hello! I want to order some food, please.

M: Sure.

W: Id like beef and onions, please.

M: Uh-huh.

W: And fourteen dumplings.

M: What kind of dumplings would you like?

W: Mutton and cabbage dumplings, please.

M: OK. Anything else?

W: Oh, yes. Id like some soup.

M: OK. What kind of soup would you like?

W: Tomato soup.

M: Is that all?

W: Yes.

M: Thatll be RMB 56.

Ⅳ. 听短文，选出最佳答案。读两遍。（每小题1分，共5分）

Text 13

At the House of Noodles we have some great specials! Special 1 has beef and tomatoes, and is just RMB 12 for 16. Special 2 is only RMB 8, and has onions and chicken. Coca-cola is only RMB 2 for each glass. The noodle and hotdog lunch special is RMB 7. Come and get your great noodles today! Our number is 86457939!

1~5 ABABC6~10 AABBB

11~15 ACBCA16~20 BCACB

21. order22. kinds23. specials24. juice25. green

26. porridge27. number28. address29. medium30. bowl

31~35 DDBCB36~40 BADCB41~45 DACEB

46. is47. to eat48. has49. drinks50. potato

51~55 BAAAA56~60 CCCAC

61~65 BDBCD66~70 BADCC

71. dinner72. nine73. bread74. milk75. tea

76. First77. vegetables78. fruit79. like80. middle

One possible version:

What kind of dessert would you like? At our Desert House we have some great new specials! Strawberry and banana ice cream special is just RMB 10 for 15. Special 2 is only RMB 8 for 15, and has apple and orange ice cream. Orange juice drinks are only RMB 2. The cake and ice cream special is RMB 11. Come and get your desserts today!

【新目标英语七年级（下）Unit 9单元要点检测题参考答案及录音稿】

Ⅰ. 听录音，选出你所听到的图片，读一遍。（每小题

1分，共5分）

Text 1 Lucy went swimming last weekend.

Text 2 My sister played computer games last Saturday.

Text 3 My good friend Ann is reading an English book.

Text 4 I played soccer with my friends yesterday

afternoon.

Text 5 Tony was ill yesterday. He went to see a doctor.

Ⅱ. 听五段小对话，找出问题的正确答语。每段对话

读一遍。（每小题1分，共5分）

Text 6

M: What did you do last weekend, Emma?

W: I went to the mountains.

Q: Where did Emma go last weekend?

Text 7

M:How was your weekend, Sarah?

W: It was pretty good.

Q: How was Sarahs weekend?

Text 8

M: What was the weather like yesterday?

W: It was rainy. But its sunny and hot today.

Q: What was the weather like yesterday?

Text 9

M: How was your weekend, Carol?

W: It wasnt very good. I stayed at home all day.

Q: How was the girlsweekend?

Text 10

W: What did your friends do, Jim?

M: Sorry, I dont know.

Q: What did Jimsfriends do?

Ⅲ. 听下面两段大对话，每段对话后有几个小题，从

题中所给的三个选项中选出最佳选项。每段对

话读两遍。（每小题1分，共5分）

听第11段对话, 回答第11至13小题。

Text 11

M: What did you do last Sunday, Sally?

W: I watched TV and did my homework. What about

you, Jim?

M: I visited my grandfather in the morning. In the

afternoon I went to the library to read some books.

W: What book did you read?

M: Harry Potter.

W: Oh, thats my favorite.

听第12段对话, 回答第14至15小题。

Text 12

W: How was your weekend, Tony?

M: It was great.

W: What did you do last weekend?

M: On Saturday morning, I did my homework. On

Saturday afternoon, I studied for the math test. On

Sunday afternoon, I went to the movies with my

friends.

Ⅳ. 听短文，选出最佳答案。读两遍。（每小题1分，

共5分）

Text 13

It was a fine day yesterday. Jim and his friends went to the park. There were a lot of people in the park. They played games in the park. After that, they went shopping. Jim got a new pencil case. Tom didnt get anything. Bob got a lot of food. Li Ping got some fruit. Emma got a beautiful hat. And Lucy got some picture books. They had a happy Sunday.

1～5 CBEAD6～10 ABCCA

11～15 ACABC16～20 BCCBA

21. because22. sat23. mountains24. spend

25. middle 26. talk27. had28. wrote29. songs

30. visit

31～35 ABBAC36～40 BDCAD

41. What did42. How was43. did; go

44. didnt clean 45. to have

46～50 BCADB51～55 AABBB

56～60 BDACA61～65 DDBAB

66. sunny67. playing68. went 69. enjoy70. next

71. boring 72. helped73. had 74. walk 75. him

One possible version:

Last Saturday, Sally had a busy day. But it was not very bad. On Saturday morning, she got up early and did some morning exercises. Then she did her homework. On Saturday afternoon, she helped her parents to do some housework. After that, she watched TV at home. On Saturday evening, she went out to have pizza with her friends. She had a good time.

【新目标英语七年级（下）Units 8~9阶段测试题参考答案】

Ⅰ. A）1. shopping2. height3. potatoes4. specials

5. stayed

B）6. geography 7. wrote8. test9. spend

10. practices

Ⅱ. 11~15 DCDAD 16~20 ABBAD

21~25 BBCCD26~30 BCAAD

Ⅲ. 31~35 BCDCA 36~40 BCDBC

Ⅳ. 41~45 BDBCC 46~50 BCADC

Ⅴ. 51. restaurants52. ask53. wait 54. themselves

55. out56. window57. through58. cars

59. popular60. busy

Ⅵ. One possible version:

George had a very busy weekend last week.

On Saturday morning, he cleaned his room and washed his clothes. He did his homework on Saturday afternoon. He visited his aunt on Saturday evening.