เอกสารประกอบการสอน คณิตศาสตร์...

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Mr.Jaranawit Kongkaew : Mathermatics II

เอกสารประกอบการสอน คณตศาสตร 2

โดย

นายจรณวชณ กองแกว

แผนกคณตศาสตร – วทยาศาสตร คณะชางอตสาหกรรม

โรงเรยนพายพเทคโนโลยและบรหารธรกจ ปการศกษา 2554

@ลขสทธโรงเรยนพายพเทคโนโลยและบรหารธรกจ

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ค าน า

การศกษาระดบประกาศนยบตรวชาชพชนสง (ปวส.) มหลากหลายสาขาวชา ใหนกศกษาไดเลอกเรยนตามความถนดของตนเอง ซงจะมวชาสามญหลายวชามาเกยวของ อาท คณตศาสตร ภาษาไทย ภาษาองกฤษ เปนตน โดยจะพบกบปญหาเกยวกบความรและกระบวนการคด การฝกท าแบบทดสอบ ใบงาน รวมถงการทบทวนเนอหา

เนอหาในหนงสอเลมน ผจดท าไดเรยบเรยงบทสรปเนอหา สตรเกยวกบการค านวณ ฟงกชนเอกซโปแนนเซยล ฟงกชนลอการทม ทฤษฏบทวนาม เมตรกซ ดเทอรมแนนต สมการเชงเสน ตรโกณม ต และภาคตดกรวย ระดบ ปวส. ตามหลกสตรของกระทรวงศกษาธการ

จงหวงวาหนงสอเลมนจะเปนประโยชนแกนกศกษา ครผสอน และอกหลาย ๆ ทานทสนใจ หากมขอบกพรองและค าแนะน าประการใด ผเรยบเรยงขอนอมรบไวเพอปรบปรงใหสมบรณยงขนตอไป สวนคณงามความดของหนงสอเลมนขอมอบใหนกศกษาทกทาน

นายจรณวชณ กองแกว

โรงเรยนพายพเทคโนโลยและบรหารธรกจ 20 กนยายน 2554

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สารบญ

ค าน า ใบงานท 1 เลขยกก าลง ใบงานท 2 รากหรอกรณฑ ใบงานท 3 การเขยนจ านวนในรปสญกรณ ใบงานท 4 การแกสมการเลขยกก าลง ใบงานท 5 ฟงกชนลอการทม ใบงานท 6 การแกสมการฟงกชนลอการทม ใบงานท 7 แฟกทอเรยล และสมประสทธทวนาม ใบงานท 8 ทฤษฏบททวนามสามเหลยมปาสคาล ใบงานท 9 การบวก ลบ เมทรกซ ใบงานท 10 การคณเมทรกซ ใบงานท 11 ไมเนอรและโคแฟกเตอร ใบงานท 12 ดเทอรมแนนต ใบงานท 13 อนเวอรสของเมทรกซ ใบงานท 14 การแกสมการเชงเสนโดยวธของคราเมอร ใบงานท 15 การแกสมการเชงเสนโดยวธของเกาส ใบงานท 16 องศากบเรเดยน ใบงานท 17 ฟงกชนตรโกณมต ใบงานท 18 ฟงกชนตรโกณมตของมมรอบจดศนยกลาง ใบงานท 19 เอกลกษณของฟงกชนตรโกณมต ใบงานท 20 ภาคตดกรวย บรรณานกรม

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1. จดประสงค 1.1 บอกความหมายของเลขยกก าลงทมเลขชก าลงเปนจ านวนเตมได 1.2 อธบายความหมายของเลขฐาน และเลขชก าลงได 1.3 หาผลบวก ลบของเลขยกก าลงทมฐานเทากน และมเลขชก าลงเปนจ านวนเตมได 1.4 หาผลคณ ผลหารของเลขยกก าลงทมฐานเทากน และมเลขชก าลงเปนจ านวนเตมได

2. เนอหาโดยสงเขป ถา a เปนจ านวนใด ๆ และ a เปนจ านวนเตมบวก “a ยกก าลง n “ หรอ “

a ก าลง n “ เขยนแทนดวย na มความหมายดงน

aaaaa n ...

n ตว

เรยก na วา เลขยกก าลงทม a เปนฐาน และ n เปนเลขชก าลง

คณสมบตของเลขก าลง

ให เปนจ านวนจรงใด ๆ และ

1) =

2) =

3) ( ) =

4) ( ) =

5) ( )

=

6) =

7) =

8) 1a = a

9) n ma =

เรอง เลขยกก าลง

ใบงาน 1 รายวชา คณตศาสตร 2

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Ex.1 จงเขยนจ านวนตอไปนในรปเลขยกก าลง

ขอ จ านวนทก าหนด

การกระจาย เลขยกก าลง

ฐาน เลขชก าลง

1. 2.

2,187 -128

3x3x3x3x3x3x3 (-2)x(-2)x(-2)x(-2)x(-2)x(-2)x(-2)

37 (-2)7

3 -2

7 7

การบวก ลบ เลขยกก าลงทมฐานเทากน และเลขยกก าลงเทากน ท าไดโดยน าสมประสทธของ

เลขยกก าลงเหลานนมาบวก ลบกน เชน

222 759 xxx = 2)759( x = 27x

การคณ หาร เลขยกก าลงทมฐานเทากน และเลขยกก าลงเทากน ท าไดโดยน าเลขชก าลงมาบวก ลบกน เชน

3528 ××× aaaa = 3528 a

= 8a

14

124

2

22 = 141242

= 22

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3. แบบฝกหด

I. จงเตมค าตอบลงในชองวาง

ขอ จ านวน การกระจาย เลขยกก าลง ฐาน เลขชก าลง 1. 2. 3. 4. 5. 6. 7. 8. 9. 10.

12527

…………… …………… …………… ……………

-2.197 47.61

1331

64

……………

7,776

……………………… 0.4 x 0.4 x 0.4 x 0.4

……………………… ……………………… ……………………… ……………………… ……………………… ……………………… ……………………… ………………………

…………….. ……………..

105

…………….. …………….. …………….. ……….…….……….…….……….…….………..……

……… ……… ………

-0.2 3

-1.3 -6.9

……… 0.01

6

…………… …………… ……………

8 6

…………… ……………

3 2

……………

II. จงกาเครองหมาย หนาขอทถกและกาเครองหมาย หนาขอทผด

………….. 1. {5 – 3 + 2 – 4}0 = 1 ………….. 2. 2 x 2m = 4m ………….. 3. 53m = 5m x 5m x 5m

………….. 4. 104n = 104 x 10m

………….. 5. a x ak = ak+1

………….. 6. 1523

723

k

k = 238

………….. 7. aa

77

1 = 70

………….. 8. 286 = 4 x 76 ………….. 9. 295 294 x 290 = 29

………….. 10. 32

32

yx

yx = 4

6

x

y

………….. 11. (2m4n2)3 = 8m12n6

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………….. 12. (0.5x-1y)2 = 2

2

4 x

y

………….. 13. 2

5342

428

ba

ba = 2

2

2b

a

………….. 14. (25)5 = 552

………….. 15. (p + q)9 = p9 + q9

………….. 16. 8

44

m = 182

8)4( m

………….. 17. 1

122

3

)(3

dc = 2

2

c

d

………….. 18. 3

3

323

17

51

yx = 3

2yx

………….. 19. 510714

25132735

= 2

………….. 20. 262

4312

)2()52(

)2()5(

= 2

III. จงท าใหอยในรปอยางงาย

1) 810

410 = ……………………………………………………………………

2) 74

31624 = ……………………………………………………………………

3) 52

12832 = ……………………………………………………………………

4) 10533-- zyx = ……………………………………………………………………

5) 2

ab

a = ……………………………………………………………………

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6) 23

77

a

ba = ……………………………………………………………………

7) baba 352 = ……………………………………………………………………

8)

2

225

3

ba

ab = ……………………………………………………………………

9) 55

101010

25

842

bb

bbb

= ……………………………………………………………………

10) 5

44

4

26

xy

xyxy = ……………………………………………………………………

11) xxxx 634 353 = ……………………………………………………………………

12) 1

23

a

aa = ……………………………………………………………………

13) 32

23

yxy

yxx

= ……………………………………………………………………

14) 2

233

ab

baba = ……………………………………………………………………

15) 32

2543

mn

nmnm = ……………………………………………………………………

16) 1

4

ma

ma = ……………………………………………………………………

17) 21

44 )2()3( y = ……………………………………………………………………

18) 2

5

13

32

n

n

n

n

x

x

x

x = ……………………………………………………………………

19)

41

22

3

12

16

8

yx

xy = ……………………………………………………………………

20) n

nn

nn

1

112

1213

33

33

= ……………………………………………………………………

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1. จดประสงค 2.1 มความเขาใจเกยวกบรากหรอกรณฑ 2.2 สามารถบวก ลบ คณ หารรากหรอกรณฑได

2. เนอหาโดยสงเขป

นยาม : ให เปนจ านวนจรง และ เปนจ านวนเตมบวกทมากกวา 1 เปนรากท ของ กตอเมอ

ให เปนจ านวนจรง และ เปนจ านวนเตมบวกทมากกวา 1 แลว

1) n ma = n

m

a

2) n a = na

1

3) n

n a

= a

4) n bn a = n ab

5) n b

n a = nb

a

6) m n a = nm a เมอ 2m

Ex. จงเขยน 6x ใหอยในรปเลขยกก าลง

6x = 21

6x = 3x

Ex. จงหาคาของ 32 32 = 32 = 6

เรอง รากหรอกรณฑ


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การคณ หาร บวก ลบกรณฑ รากหรอกรณฑ ทจะน ามา บวก ลบกนได ตองเปนรากหรอกรณฑทมอนดบเทากน และม

จ านวนทอยภายในกรณฑเทากน เชน 2425 =

29 สวน รากหรอกรณฑ ทจะน ามา คณ หารกนได ตองเปนรากหรอกรณฑทมอนดบเทากน เชน 6868 = 66686888

= 8 - 6 = 2

3. แบบฝกหด I. จงเขยนจ านวนของรปตอไปนใหอยในรปเลขยกก าลง

1) 6x = ……………………………………………………………………

2) 8y = ……………………………………………………………………

3) 410 yx = ……………………………………………………………………

= ……………………………………………………………………

4) 3 6x = ……………………………………………………………………

= ……………………………………………………………………

5) 3 93yx = ……………………………………………………………………

= ……………………………………………………………………

6) 4 8)( yx = ……………………………………………………………………

= ……………………………………………………………………

7) 3

2

6

327

b

a = ……………………………………………………………………

= ……………………………………………………………………

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II. จงหาคาตอไปน 1) 32 = ……………………………………………………………………

= ……………………………………………………………………

= ……………………………………………………………………

2) 3 23 4 = ……………………………………………………………………

= ……………………………………………………………………

= ……………………………………………………………………

3) 3 9

3 27

= ……………………………………………………………………

= ……………………………………………………………………

= ……………………………………………………………………

4) 4 4

4 324 = ……………………………………………………………………

= ……………………………………………………………………

= ……………………………………………………………………

5) 3 6a = ……………………………………………………………………

= ……………………………………………………………………

= ……………………………………………………………………

6) 5 3 30a = ……………………………………………………………………

= ……………………………………………………………………

= ……………………………………………………………………

= ……………………………………………………………………

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III. จงหาคาของ 1) 3212372 = ………………………………………………………………

= ………………………………………………………………

2) 333 5425016 = ………………………………………………………………

= ………………………………………………………………

3) 54335332 = ………………………………………………………………

= ………………………………………………………………

= ………………………………………………………………

4) 3263

23

= ………………………………………………………………

= ………………………………………………………………

= ………………………………………………………………

5) xy

yx28 = ………………………………………………………………

= ………………………………………………………………

6) 13

2

= ………………………………………………………………

= ………………………………………………………………

= ………………………………………………………………

7) 243 812 baba = ………………………………………………………………

= ………………………………………………………………

8) 8

5 = ………………………………………………………………

= ………………………………………………………………

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1. จดประสงค 1.1 มความเขาใจเกยวกบการเขยนจ านวนในรปสญกรณทางวทยาศาสตรได 1.2 สามารถบวก ลบ คณ หารจ านวนทอยในรปสญกรณทางวทยาศาสตร

2 เนอหาโดยสงเขป

การเขยนจ านวนใหอยในรปสญกรณทางวทยาศาสตร (A x10n) เมอ 1 A < 10 และ n เปนจ านวนเตม เมอโจทยก าหนดให 1 A < 10 แสดงวา A มคาไดตงแต 1.0 ถง 9.999… นนคอ จ านวนเตมทอยใน A ตองเปนเลขหลกหนวยเทานน

Ex. จงเขยนจ านวนทก าหนดทก าหนดใหอยในร (A x10n) เมอ 1 A < 10 และ n เปน

จ านวนเตม 1. 3,490,000 = 3.49 x 106 2. 0.00078 = 7.8 x 10- 4 3. 42 x 1011 = 4.2 x 1012

4. 12

9

107

1035.0

= 3107

35.0

= 0.05 x 10- 3 = 5.0 x 10- 2 x 10- 3 = 5.0 x 10- 5

6. 002.063

1031042 68

= 3

2

10263

10342

= 1.0 x 105 (0.002 = 2 x 10- 3 ,

263

342

= 1.0)

7. 7.4 x 105 + 1.6 x 105 = (7.4 + 1.6) x 105 = 9.0 x 105

8. 7 x 107 – 4.2 x 106 = 70 x 106 – 4.2 x 106 = (70- 4.2) x 106 = 65.8 x 106

เรอง การเขยนจ านวนในรปสญกรณ


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3. แบบฝกหด I. จงเขยนจ านวนตอไปนในรปสญกรณวทยาศาสตร

1.) 32,000,000 = …………………………………….

2.) 138,830 = …………………………………….

3.) 711,000,000 = …………………………………….

4.) 4,040,000 = …………………………………….

5.) 99,990,000 = …………………………………….

6.) 123,000 = …………………………………….

7.) 1,010,000 = …………………………………….

8.) 543,210,000 = …………………………………….

9.) 22,222,000 = …………………………………….

10.) 789,000 = …………………………………….

11.) 0.000202 = …………………………………….

12.) 0.00123 = …………………………………….

13.) 0.7890 = …………………………………….

14.) 0.0123 = …………………………………….

15.) 0.9876 = …………………………………….

16.) 0.000011 = …………………………………….

17.) 0.0009 = …………………………………….

18.) 0.00000099 = …………………………………….

19.) 0.000501 = …………………………………….

20.) 0.0707 = …………………………………….

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II. จงเขยนตวเลขในแตละขอตอไปนโดยไมใชสญกรณวทยาศาสตร

1.) 3.0 x 108 = …………………………………….

2.) 1 x 108 = …………………………………….

3.) 9.99 x 109 = …………………………………….

4.) 3.45 x 106 = …………………………………….

5.) 4.44 x 104 = …………………………………….

6.) 110 x 1010 = …………………………………….

7.) 501 x 105 = …………………………………….

8.) 7.65 x 104 = …………………………………….

9.) 2 x 103 = …………………………………….

10.) 2.0 x 105 = …………………………………….

11.) 3.0 x 10-8 = …………………………………….

12.) 7.05 x 10-4 = …………………………………….

13.) 9.99 x 10-3 = …………………………………….

14.) 3.45 x 10-6 = …………………………………….

15.) 4.44 x 10-4 = …………………………………….

16.) 11.0 x 10-1 = …………………………………….

17.) 5.01 x 10-2 = …………………………………….

18.) 7.65 x 10-3 = …………………………………….

19.) 2 x 10-3 = …………………………………….

20.) 2.0 x 10-5 = …………………………………….

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III. จงตอบค าถามตอไปนและเขยนค าตอบในรปสญกรณวทยาศาสตร 1. จากจ านวน 7.54 x 10-6 ถาสลบเลขโดด 5 และ 4 จะไดจ านวนใหมทมคามาก

หรอนอยกวาจ านวนเดมเทาไร ................................................................................................................................. ................................................................................................................................. ................................................................................................................................. ................................................................................................................................. .................................................................................................................................

2. แสงมความเรว 3.0 x 108 เมตรตอวนาท ผเสอบนดวยความเรว 0.5 x 10-2 เมตรตอวนาท ถามวาแสงมความเรวมากกวาผเสอกเทา ................................................................................................................................. ................................................................................................................................. ................................................................................................................................. ................................................................................................................................. .................................................................................................................................

3. สารชนดทหนงมความหนาแนน 0.928 กโลกรมตอลกบาศกเมตร สารชนดทสองหนาแนนเปน 0.6 เทาของสารชนดทหนง ดงนนสารชนดทสองมความหนาแนน กกโลกรมตอลกบาศกเมตร ................................................................................................................................. ................................................................................................................................. ................................................................................................................................. ................................................................................................................................. .................................................................................................................................

4. บรษทแหงหนงมเงนทนส ารองอยในธนาคาร 25 x 1010 บาท ถาตองน าเงนสวนนไปใชในการขยายกจการ 25 % จะยงคงเหลอเงนทนส ารองในธนาคารจ านวนกบาท ................................................................................................................................. ................................................................................................................................. ................................................................................................................................. ................................................................................................................................. .................................................................................................................................

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1. จดประสงค 1.1 มความเขาใจเกยวกบการแกสมการเลขยกก าลง 1.2 สามารถค านวณการแกสมการเลขยกก าลงได 1.3 สามารถค านวณหาคาตวแปรในระบบสมการเลขยกก าลงได


นยาม : สมการเลขยกก าลง หมายถง สมการทมตวแปรเปนเลขยกก าลง

การแกสมการเลขยกก าลง อาจท าได 2 วธ คอ วธท 1 โดยการเทยบเลขชก าลง มกลงการดงน

1) เขยนเลขชก าลงใหมฐานเทากน 2) น าเลขชก าลงมาเทากนแลวแกสมการหาคาของตวแปร

วธท 2 โดยการใชลอการทม

Ex. จงแกสมการตอไปน 1) 1

1

21

xx

น า 1x คณตลอดสมการ 2)1( x = 1x

1x = 1x 2)1( x = 1x

122 xx = 1x xx 32 = 0 )3( xx = 0

x = 0 , 3 แทนคาจะไดค าตอบของสมการคอ x = 3

เรอง การแกสมการเลขยกก าลง


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2) 0644 2 x x24 = 64 x24 = 34

x2 = 3 x =

2

3

3) 11 68 xx ฐานไมเทากน แตเลขยกก าลงเทากน ใหเลขยกก าลง เทากบ 0

1x = 0 x = 1

3. แบบฝกหด I. จงเขยนแกสมการเลขยกก าลงตอไปน

1.) 3x + 8 = 2x – 10 ................................................................................................................................. ................................................................................................................................. ................................................................................................................................. .................................................................................................................................

2.) 4x2 + 7x - 2 = 0 ................................................................................................................................. ................................................................................................................................. ................................................................................................................................. .................................................................................................................................

3.) 6x2 + 13x - 5 = 0 ................................................................................................................................. ................................................................................................................................. ................................................................................................................................. .................................................................................................................................

4.) 8x + 2 = 4x

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5.) 43x - 1 = x16

1

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6.) 22x – 2x + 1+ 1 = 0 ................................................................................................................................. ................................................................................................................................. ................................................................................................................................. .................................................................................................................................

7.) 32x – (4)3x + 1+ 27 = 0 ................................................................................................................................. ................................................................................................................................. ................................................................................................................................. .................................................................................................................................

8.) 4x = 161

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9.) 3x + 2 = 81 ................................................................................................................................. ................................................................................................................................. ................................................................................................................................. .................................................................................................................................

10.) 25x + 2 = x

1251

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11.) 23 - x = 8x + 2 ................................................................................................................................. ................................................................................................................................. ................................................................................................................................. ................................................................................................................................. .................................................................................................................................

12.) 32x - 3x + 2 = 0 ................................................................................................................................. ................................................................................................................................. ................................................................................................................................. ................................................................................................................................. .................................................................................................................................

13.) 32x - 3x + 2 = 0 ................................................................................................................................. ................................................................................................................................. ................................................................................................................................. ................................................................................................................................. .................................................................................................................................

14.) 33

33

xx

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15.) 23

14

x

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1. จดประสงค 1.1 มความเขาใจเกยวกบฟงกชนลอการทม 1.2 สามารถบอกคณสมบตของฟงกชนลอการทมได 1.3 สามารถค านวณคาฟงกชนลอการทมได

3. เนอหาโดยสงเขป ฟงกชนลอการทม คอ อนเวอรสของฟงกชนเอกซโปเนนเซยลอยในรป

Exponential :

1,0,),( aaxayRRyxf

Log :

1,0,),(1 aayaxRRyxf

นยาม : ฟงกชนลอการทม หมายถง ฟงกชนทเขยนในรป

1,0,log),( aaxayRRyxf ฟงกชนลอการทมเปนอนเวอรสของฟงกชนเอกซโปเนนเซยล

1,0,),( aaxayRRyxf

ดงนน ความสมพนธ xay เขยนแทนดวย yax " xalog " อานวา “ ลอการทมเอกซฐานเอ ” หรอ “ ลอกเอกซฐานเอ ” เนองจาก f (ฟงกชนเอกซโปเนนเซยล) เปนฟงกชน 1 – 1 ดงนน จงเปน

ฟงกชนและเปนฟงกชน 1 – 1 ดวย

กราฟแสดงฟงกชนลอการทม

เรอง ฟงกชนลอการทม


Log1.7 x

Loge x

Log10 x

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คณสมบตของฟงกชนลอการทม

1. MNalog =

NM aa loglog

2.

N

Malog =

NM aa loglog

3. aalog =

1 4. n

aMlog =

Man log 5. 1loga =

0

6. balog = a

b

log

log

7. xaa log =

x

8.

N

1loga =

Nalog

Ex. จงรวมพจนลอการทม ตอไปน

1) log 2 3 + log 2 4 + log 2 6 = log 2 ( 3 x 4 x 6 ) = log 2 72

2) log 2 5 - log 2 10

= log 2

10

5

= log 2

2

1

Ex. จงหาคาของ 12log39

=

12log2 33

=

212log33

=

212log33 = 122 = 144

Ex. ก าหนดให log 1.358 = 0.1329 จงหาคาของ log 1358 log 1358 = log (1.358 )

= log 1.358 + log = 0.1329 + 3 = 3.1329

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3. แบบฝกหด I. จงหาคาฟงกชนลอการทม ตอไปน

1) log 100 + log 10 + log 1

.................................................................................................................................

.................................................................................................................................

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2) ( log 1000) ( log 105 )

.................................................................................................................................

.................................................................................................................................

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3) log 2 32 + log 5 25 + log 3 81

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4) log 3 log 2 log 3 log 2 512

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5) log3 3 33

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6) log2 16

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7) 81log

3

1

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8)

81

1log

9

1

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9) log7 343

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10) log16 2

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11) log8 32

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12) log12 4 + log12 3

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13) 3log2 33

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14) )251

log(01.0

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15) ก าหนดให 3010.02log , 4771.03log , 6990.05log และ 8451.07log จงหาคาของ 125.0log420log

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1. จดประสงค 1.1 มความเขาใจเกยวกบการแกสมการฟงกชนลอการทม 1.2 สามารถอธบายการแกสมการฟงกชนลอการทมได 1.3 สามารถค านวณหาคาสมการฟงกชนลอการทมได


นยาม : ฟงกชนลอการทม หมายถง ฟงกชนทเขยนในรป 1,0,log),( aaxayRRyxf

ฟงกชนลอการทมเปนอนเวอรสของฟงกชนเอกซโปเนนเซยล

1,0,),( aaxayRRyxf

คณสมบตของฟงกชนลอการทม

3. MNalog =

NM aa loglog

4.

N

Malog =

NM aa loglog

5. aalog =

1

6. naMlog =

Man log

7. 1loga =

0

8. balog = a

b

log

log

9. xaa log =

x

10.

N

1loga =

Nalog

เรอง การแกสมการฟงกชนลอการทม


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Ex. จงหาคาสมการตอไปน

1) log2 )3( x - log2 )2( x = 3

log2

2

3

x

x = 3

2

3

x

x = 32

2

3

x

x = 8

3x = 8 )2( x

3x = 168 x

x7 = -19

x =

7

19

2) log3 )4( x + log3 )4( x = 2

log3 )4()4( xx = 2

)4()4( xx = 23

162 x = 9

2x = 25

x = 5

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3. แบบฝกหด I. จงแกสมการลอการทมตอไปน

1) log 3 )8(x log 3 x = 2

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2) log 5 x log 5 )4( x = 1

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3) log 49 16

1x

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4) log 2 log 2 log 2 x = 0

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5) ln 2ln xe = xe 4lnln

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6) log x log )8( x = log )3(x log )4( x

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7) ก าหนดให log 2 = 0.3010 จงแกสมการ 02.0)2.0( x

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8) log 8 x + log 8 )2( x = log 8 )32( x

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9) ln (2 ln x + 3 ) = ln 3

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10) ln 102xe = )22ln( xxe

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11) x2log3 = 5log1 3

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12) )13(log6 x = 2log10log 66

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13) 5loglog 2x = x2log7log

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14) zyx zy 27logloglog = 5log3

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1. จดประสงค

1.1 มความเขาใจเกยวกบแฟกทอเรยล และสมประสทธทวนาม

1.2 สามารถความหมายแฟกทอเรยล และสมประสทธทวนามได

1.3 สามารถค านวณหาคาแฟกทอเรยล และสมประสทธทวนามได

2. เนอหาโดยสงเขป แฟกทอเรยลของ n เขยนแทนดวย !n อานวา เอนแฟกทอเรยล

นยามท 1 แฟกทอเรยล n เมอ n เปนจ านวนเตมบวก คอ !n = n )1n( )2n( )3n( . . . . 123

ถา n = 0 จะก าหนดให 0! = 1 ซงแสดงใหเหนไดดงน จาก !n = n )!1n(

)!1n( = n

n!

แทน n = 1

)!11( = 1

!1

0! = 1

Ex. จงหาคาของ !3

!6

!3!6 =

123123456

= 456

Ex. จงเขยน 2526272829 ใหอยในรปของแฟกทอเรยล

2526272829 = !24

!242526272829

= !24!29

เรอง แฟกทอเรยลและสมประสทธทวนาม


of 93


สมประสทธทวนาม

สมประสทธทวนาม เปนจ านวนทคณกบพจนของทวนามทกระจายออกเปน

พจนยอย ๆ ซงเขยนเปนสญลกษณ

rn โดย

rn หมายถง ทวนามยกก าลง n และ

พจนทสมประสทธก ากบอยคอ พจนท r + 1 ซงสมประสทธดงกลาว ตามนยามท 2 คอ

นยามท 2 เมอ n , r เปนจ านวนเตม และ 0 r n แลว

rn =

)!rn(!r!n

Ex. จงหาคาสมประสทธทวนาม ตอไปน

1)

59

2)

49

วธท า

1)

59

= )!59(!5

!9

= !4!5

!56789

= 126

2)

49

= )!49(!4

!9

= !5!4

!56789

= 126

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3. แบบฝกหด I. จงเขยนใหอยในรปแฟกทรอเรยล

1) 54321 = ……………………………………………………………………

2) 321 = ……………………………………………………………………

3) 7654 = ……………………………………………………………………

4) 3231302928 = ……………………………………………………………………

5) ( 4321 ) + ( 654 ) = ……………………………………………………………………

II. จงหาคาของแฟกทรอเรยล

1) 4! = ……………………………………………………………………

2) 7! = ……………………………………………………………………

3) !3!6

= ……………………………………………………………………

4) !3!8!11

= ……………………………………………………………………

5) 5! + (2! + 3!) = ……………………………………………………………………

6) !3!6

= ……………………………………………………………………

7) !3!8!11

= ……………………………………………………………………

8) 5! + (2! + 3!) = ……………………………………………………………………

9) !4!9

+ 3! = ……………………………………………………………………

10) 3! +

!7!3!5

= ……………………………………………………………………

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III. จงหาสมประสทธทวนาม

1)

36 = ……………………………………………………………………

= ……………………………………………………………………

= ……………………………………………………………………

2)

59 = ……………………………………………………………………

= ……………………………………………………………………

= ……………………………………………………………………

3)

05 = ……………………………………………………………………

= ……………………………………………………………………

= ……………………………………………………………………

4)

58 = ……………………………………………………………………

= ……………………………………………………………………

= ……………………………………………………………………

5)

47 = ……………………………………………………………………

= ……………………………………………………………………

= ……………………………………………………………………

6)

312 = ……………………………………………………………………

= ……………………………………………………………………

= ……………………………………………………………………

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IV. จงหาคาของ

1)

24

26 = ……………………………………………………………………

= ……………………………………………………………………

= ……………………………………………………………………

2)

45

47 = ……………………………………………………………………

= ……………………………………………………………………

= ……………………………………………………………………

3) !4!9 +

25 = ……………………………………………………………………

= ……………………………………………………………………

= ……………………………………………………………………

4) )!1n()!3n(

เมอ n = 4 = ……………………………………………………………………

= ……………………………………………………………………

= ……………………………………………………………………

5) 563n

จงหาคา n = ……………………………………………………………………

= ……………………………………………………………………

= ……………………………………………………………………

6)

610 +

!2!3!5

= ……………………………………………………………………

= ……………………………………………………………………

= ……………………………………………………………………

of 93



1.1 มความเขาใจเกยวกบทฤษฏบททวนามสามเหลยมปาสคาล

1.2 สามารถความหมายทฤษฏบททวนามและสามเหลยมปาสคาลได

1.3 สามารถค านวณโดยใชทฤษฏบททวนามและสามเหลยมปาสคาลได

2. เนอหาโดยสงเขป สามเหลยมปาสคาล

การกระจาย n)ba( เมอ a , b เปนจ านวนจรงใด ๆ และ n เปนจ านวนเตมบวก เมอกระจายดวยวธการคณแลวจะได

0)ba( = 1 1)ba( = a + b 2)ba( = 22 bab2a 3)ba( = 3223 bab3ba3a 4)ba( = 432234 bab4ba6ba4a 5)ba( = 54322345 bab5ba10ba10ba5a

.

.

.

= .

.

.

จากการกระจาย nba )( ถาเราน าเฉพาะสมประสทธมาเขยน จะมลกษณะเปนรปสามเหลยม ดงน

แถวท 1 1 แถวท 2 1 1 แถวท 3 1 2 1 แถวท 4 1 3 3 1 แถวท 5 1 4 6 4 1 แถวท 6 1 5 10 10 5 1 แถวท 7 1 6 15 20 15 6 1

เรอง ทฤษฏบททวนามและสามเหลยมปาสคาล


of 93


Ex. ของสามเหลยมปาสคาล และการกระจาย 6)ba( วธท า แถวท 6 1 5 10 10 5 1 แถวท 7 1 6 15 20 15 6 1

6)ba( มสมประสทธ คอ แถวท 7 ของสามเหลยมปาสคาล จะได

6)ba( =

60514233241506 ba)1(ba)6(ba)15(ba)20(ba)15(ba)6(ba)1(

= 6542332456 bab6ba15ba20ba15ba6a ทฤษฏบททวนาม

ถา n และ r เปนจ านวนเตม โดยท และ 0 r n แลว

n)ba( =

n

0r

rrn barn

= n1nrrn1nn bnn

ab1n

n...ba

rn

...ba1n

a0n

ขอสงเกต 1. พจนท r + 1 กระจายไดเปน rrn barn

2. สมประสทธของพจนท r + 1 คอ

rn =

)!rn(!r!n

Ex. จงกระจาย 5)ba( โดยใชทฤษฏบททวนาม วธท า

5)( ba =

54322345 b55

ab45

ba35

ba25

ba15

a05

หาสมประสทธทวนาม

05 =

55 =

!5!0!5 = 1

15 =

45 =

!4!1!5 = 5

25 =

35 =

!3!2!5 = 10

จะได 5)ba( = 54322345 510105 babbababaa

of 93


3. แบบฝกหด I. จงกระจายทวนามตอไปน โดยใชสามเหลยมปาสกาล

1) 2ba = …………………………………………………………………… = …………………………………………………………………… = ……………………………………………………………………

2) 3ba = …………………………………………………………………… = …………………………………………………………………… = ……………………………………………………………………

3) 4ba = …………………………………………………………………… = …………………………………………………………………… = ……………………………………………………………………

4) 6ba = …………………………………………………………………… = …………………………………………………………………… = ……………………………………………………………………

5) 52 y3x = …………………………………………………………………… = …………………………………………………………………… = …………………………………………………………………… = ……………………………………………………………………

6) 52 yx = …………………………………………………………………… = …………………………………………………………………… = …………………………………………………………………… = ……………………………………………………………………

7) 52x = …………………………………………………………………… = …………………………………………………………………… = …………………………………………………………………… = ……………………………………………………………………

8) 432 yx = …………………………………………………………………… = …………………………………………………………………… = …………………………………………………………………… = ……………………………………………………………………

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= …………………………………………………………………… 9) 3x21 = ……………………………………………………………………

= …………………………………………………………………… = …………………………………………………………………… = ……………………………………………………………………

10) 24x = …………………………………………………………………… = …………………………………………………………………… = …………………………………………………………………… = ……………………………………………………………………

II. จงกระจายคาตอไปน โดยใชทฤษฏทวนาม

1) 5ba = …………………………………………………………………… = …………………………………………………………………… = …………………………………………………………………… = ……………………………………………………………………

2) 52 yx = …………………………………………………………………… = …………………………………………………………………… = …………………………………………………………………… = ……………………………………………………………………

3) 52 3yx = …………………………………………………………………… = …………………………………………………………………… = …………………………………………………………………… = ……………………………………………………………………

4) 52x = …………………………………………………………………… = …………………………………………………………………… = …………………………………………………………………… = ……………………………………………………………………

5) 331 x = …………………………………………………………………… = …………………………………………………………………… = …………………………………………………………………… = ……………………………………………………………………

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III. จงหาหาคาตอไปน โดยใชทฤษฏทวนาม

1) จงหาพจนท 5 ของ 103y2x ................................................................................................................................. ................................................................................................................................. ................................................................................................................................. ................................................................................................................................. ................................................................................................................................. ................................................................................................................................. .................................................................................................................................

2) จงหาพจนท 6 ของ 102y3x ................................................................................................................................. ................................................................................................................................. ................................................................................................................................. ................................................................................................................................. ................................................................................................................................. ................................................................................................................................. .................................................................................................................................

3) จงหาพจนท 8 ของ 153x ................................................................................................................................. ................................................................................................................................. ................................................................................................................................. ................................................................................................................................. ................................................................................................................................. ................................................................................................................................. .................................................................................................................................

3) จงหาพจนท 5 ของ 532 x ................................................................................................................................. ................................................................................................................................. ................................................................................................................................. ................................................................................................................................. ................................................................................................................................. ................................................................................................................................ .................................................................................................................................

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1. จดประสงค 1.1 มความเขาใจเกยวกบการบวก ลบ เมทรกซ 1.2 สามารถอธบายการบวก ลบ เมทรกซได 1.3 สามารถค านวณการบวก ลบ เมทรกซได


เชน

A =

342

301 มมต 2 × 3 หรอเปน 2 × 3 เมตรกซ

ในกรณทว ๆ ไป ถา A มมต M X N สญลกษณทวไปของ A เปนดงน

A =

mnmm

n

aaa

naaa

aaa

...

.

...

2...

...

21

2221

11211

อาจเขยนอยางยอวา A = ija m x n

การบวก ลบ เมตรกซ สามารถกระท าไดภายใตเงอนไข 1. เมตรกซ ทงสองตองมมตเทากน 2. น าสมาชกทอยต าแหนงเดยวกนบวกหรอลบกน

เรอง การบวก ลบ เมทรกซ


บทนยาม A = [a1j]mn B = [bij]mn A + B = [a1j + bij]mn A - B = [a1j - bij]mn

ถา A เปนเมตรกซทม M แถว และม n หลก จะเรยก A วามมต M × N (อานวา เอมคณเอน)

of 93


Ex. A =

35

12 B =

62

43

A+B =

6325

4132

=

97

35

คณสมบตการบวกของเมตรกซ

S เปนเซตของเมตรกซ M x N A,B,C อยใน S 1. ปดการบวก A + B = S 2. สลบทการบวก A + B = B + A 3. เปลยนกลม ( A + B ) + C = A + ( B + C ) 4. เอกลกษณการบวก A + 0 = A 0 เปนเอกลกษณการบวก 5. อนเวอรสการบวก A+ (-A ) = 0 -A เปนอนเวอรสการบวกของ A

ทรานสโพส (Transpose)

Ex. ก าหนดให A =

540

221 จงหา At

จะได At =

53

42

01

คณสมบตของทรานสโพส 1. ถา A เปนเมตรกซทมมต m x n แลว ( At ) t = A 2. ถา A เปนเมตรกซทมมต m x n แลว k เปนจ านวนจรงแลว ( kA) t = kA t 3. ถา A และ Bเปนเมตรกซทมมต m x n แลว (A + B) t = At + Bt 4. ถา A เปนเมตรกซทมมต m x n และ B เปนเมตรกซทมมต n x k แลว (AB)t = Bt At

นยาม A = ia m x n ทรานสโพสของ A แทนดวย “A1” คอ เมตรกซ ซงมมตเปน m x n โดยทมหลกท 1 เทากบแถวท 1 ของเมตรกซ A

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3. แบบฝกหด I. จงหาคาของเมทรกซ ตอไปน

1. ก าหนดให A =

021

411 และ B =

426

301

จงหาคาของ 1) A+B 2) 2A-B 3) 2B+A 4) 2B-2A

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02

30

11

, B =

30

12

21

และ C =

42

10

35

จงหาคาของ 1) 2(A+B)-C 2) 3A-B 3) 2A+(2B-3C) ................................................................................................................................. ................................................................................................................................. ................................................................................................................................. ................................................................................................................................. ................................................................................................................................. ................................................................................................................................. ................................................................................................................................. ................................................................................................................................. ................................................................................................................................. ................................................................................................................................. ................................................................................................................................. .................................................................................................................................

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323

421

245

, B =

214

503

410

จงหาคาของ 21

A - 4B

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24

32

03

, B =

12

30

12

จงหาคาของ -3A + 21

B

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5. ให

wzyx

wzyx 2 =

41

53 จงหาคาของ x + y + z + w

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1. จดประสงค 1.1 มความเขาใจเกยวกบหลกการคณเมทรกซ 1.2 สามารถอธบายวธการคณเมทรกซได 1.3 สามารถค านวณหาคาเมทรกซโดยวธการคณเมทรกซได


การคณเมทรกซ ดวย สเกลาร ก าหนด k เปนสเกลาร ใด ๆ แลว

kA =

gc

hb

da

k =

kgkc

khkb

kdka


24

32

03

จงหา 2A

2A = 2

24

32

03

=

2242

)3222

0232

(

=

48

64

06

เรอง การคณเมทรกซ


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การคณเมทรกซ ดวยเมทรกซ

เมทรกซ จะคณกนไดกตอเมอ จ านวนหลกของเมทรกซตวตงเทากบจ านวนแถวของเมตรกซตวคณ

ถา A , B ,C เปนเมตรกซ A มมต m n B มมต n p และ AB = C แลว C มมต m p การคณตามผงทแสดงกลาวคอ แถวของตวตงไปคณกบหลกของตวคณ โดยคณ

สมาชกทสมนยกนเปนค ท าเชนนเรอย ๆ จนครบทกหลกและเรมทแถวทสองตอไป

Ex. ก าหนด A =

43

21 B =

34

12

วธท า AB =

43

21

34

12

=

)3)(4()1)(3()4)(4()2)(3(

)3)(2()1)(1()4)(2()2)(1(

=

)12()3()16()6(

)6()1()8()2(

=

922

610

Am x n Bn x p = Cm x p

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3. แบบฝกหด I. จงหาคาของเมทรกซ ตอไปน


3

2

1

, B = 321 และ C =

42

31

จงหาคาของ 1) AB 2) 2AB 3) 2A 2C ................................................................................................................................. ................................................................................................................................. ................................................................................................................................. ................................................................................................................................. ................................................................................................................................. ................................................................................................................................. ................................................................................................................................. ................................................................................................................................. ................................................................................................................................. ................................................................................................................................. ................................................................................................................................. ................................................................................................................................. ................................................................................................................................. ................................................................................................................................. .................................................................................................................................


75

14 , B =

574

242 จงหาคาของ AB และ BA

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32

10

12

,B =

05

43

12

,C =

352

701 และD =

1

0

2

จงแสดงวาเมตรกซตอไปนเทากนหรอไม 1) (AC) 2D กบ A 3(CD) 2) 3C 2(A+B) กบ 2(CA) + (CB) 3) 3(A+B) 2C กบ (AC) + 2(BC) ................................................................................................................................. ................................................................................................................................. ................................................................................................................................. ................................................................................................................................. ................................................................................................................................. ................................................................................................................................. ................................................................................................................................. ................................................................................................................................. ................................................................................................................................. ................................................................................................................................. ................................................................................................................................. ................................................................................................................................. ................................................................................................................................. ................................................................................................................................. ................................................................................................................................. ................................................................................................................................. ................................................................................................................................. ................................................................................................................................. ................................................................................................................................. ................................................................................................................................. ................................................................................................................................. ................................................................................................................................. ................................................................................................................................. ................................................................................................................................. ................................................................................................................................. ................................................................................................................................. .................................................................................................................................

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1.1 มความเขาใจเกยวกบไมเนอรและโคแฟกเตอร

1.2 สามารถอธบายวธการหลกการของไมเนอรและโคแฟกเตอรได

1.3 สามารถค านวณหาคาไมเนอรและโคแฟกเตอรได


ไมเนอร (Minor) A = ija

, n > 2 Minor ของ แทนดวย “ Mij ”

คอ Determinant ของ Matrix ซงเกดจากการตวแถวท 1 และหลกท j ของ Matrix ออก

Ex. A =

333231

232221

131211

aaa

aaa

aaa

จงหา M11

M11 =

333231

232221

131211

aaa

aaa

aaa

; ตดแถวท 1 และหลกท 1 ออก

=

3332

2322

aa

aa

โคแฟกเตอร (Cofactor) A = ija

n x n , n > 2 Cofactor ของ aij แทนดวย “ Cij ” คอ (-1) i+ j Mij

เชน C11 = (-1)2 M11 = M11

C12 = (-1)3 M12 = -M12

เรอง ไมเนอรและโคแฟกเตอร


of 93



323

421

245

จงหา 22

M22 =

33

25 = (5×3) – (3×2) = 15 – 6 = 9


I. จงหาคาไมเนอร ของเมทรกซ ตอไปน

1) ก าหนด A =

25

31จงหาคาของ 22

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.................................................................................................................................


1210

2152

3423

3211

จงหาคาของ 12 , 13 , 22 , 31

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22

12จงหาคาของ 12 , 21

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II. จงหาคาโคแฟกเตอร ของเมทรกซ ตอไปน


734

225

123

จงหาคาของ C12 , C 21

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570

684

312

จงหาคาของ C12 , C 23 , C 32

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4) ให A =

1210

2152

3423

3211

จงหาคาของ C12 , C13 , C 22 , C 32 ,

(C33+C11) ................................................................................................................................. ................................................................................................................................. ................................................................................................................................. ................................................................................................................................. ................................................................................................................................. ................................................................................................................................. ................................................................................................................................. ................................................................................................................................. ................................................................................................................................. ................................................................................................................................. ................................................................................................................................. ................................................................................................................................. ................................................................................................................................. ................................................................................................................................. ................................................................................................................................. ................................................................................................................................. ................................................................................................................................. ................................................................................................................................. ................................................................................................................................. ................................................................................................................................. ................................................................................................................................. ................................................................................................................................. ................................................................................................................................. ................................................................................................................................. ................................................................................................................................. ................................................................................................................................. ................................................................................................................................. ................................................................................................................................. .................................................................................................................................

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1. จดประสงค 1.1 มความเขาใจเกยวกบดเทอรมแนนต 1.2 สามารถอธบายวธการหลกการของดเทอรมแนนตได 1.3 สามารถค านวณหาคาดเทอรมแนนตได


ดเทอรมแนนท (Determinant) เปนคาทไดจากการค านวณจากเมตรกซทก าหนดให A เปน nn เมตรกซ ดเทอร

มแนนทของเมตรกซ A เขยนแทนดวย det(A) หรอ A

การหา det กรณท 1 โดยวธเพมหลก กรณท 2 โดยวธ โคแฟกเตอร

*** 1. det(A) ทมมต 33 เมตรกซ จะเพม 2 หลกแรก และหาคาโดยวธใชลกศร 2 det(At) =det(A)

3. det(An) = (det(A))n 4. det(AB) = det(A)det(B)

ก าหนดให A =

333231

232221

131211

aaa

aaa

aaa

จะได det A =

3231

2221

1211

333231

232221

131211

aa

aa

aa

aaa

aaa

aaa

det A =

)( 322113312312332211 aaaaaaaaa - )( 122133112332132231 aaaaaaaaa

เรอง ดเทอรมแนนต


of 93



734

225

123

จงหา det A

วธท า กรณท 1 โดยวธเพมหลก

det A =

34

25

23

734

225

123

det A =

)3)(5)(1()4)(2)(2()7)(2)(3( - )2)(5)(7()3)(2)(3()1)(2)(4(

=

151642 – 70188

=

11 – 80

=

– 91

กรณท 2 โดยวธ โคแฟกเตอร

det A = 131312121111 CaCaCa

=

34

25)1)(1(

74

25)1)(2(

73

22)1)(3( 312111

=

)2)(4()3)(5()1)(1()2)(4()7)(5()1)(2()2)(3()7)(2()1)(3(

=

)23)(1()27)(2()20)(3(

=

– 91

of 93


3. แบบฝกหด I. จงหาคาดเทอรมแนนต ตอไปน

1) A =

323

421

245

2) B =

570

684

312

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3) K =

1210

2152

3423

3211

4) G =

12121

2152

344321

211

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5) ให H =

3211

115321

48321

2121

และ P =

62

43 จงหา det H – det P

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5) ก าหนดให H =

574

043

0020

และ D =

920

710

5120

ถา HD = B จงหา det B

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1. จดประสงค 1.1 มความเขาใจเกยวกบอนเวอรสของเมทรกซ 1.2 สามารถอธบายการหาอนเวอรสของเมทรกซได 1.3 สามารถค านวณหาคาอนเวอรสของเมทรกซได


นยาม : เมทรกซ B ซงมมต nn เปนอนเวอรสการคณของเมทรกซ A ซงมมต nn ก ตอเมอ AB = BA = I เมอ I คอ เมทรกซเอกลกษณมต nn เขยนแทน B ซงเปนอนเวอรสของ A ดวย A-1

ถาเมทรกซ A =

dc

ba และ 0 bcad

จะได B หรอ A-1 =

ac

bd

bcad1

แตถาเมทรกซ A มมต nn เมอ n ≥ 2

จะได A-1 = A adj Adet

1

โดย adj A =

t

mnmm

n

n

CCC

CCC

CCC

21

22221

11211

เรอง อนเวอรสของเมทรกซ


of 93



53

24 จงหา A-1

วธท า เนองจาก (4)(5) – (-2)(3) ≠ 0

ดงนน A-1 =

43

25

)3)(2()5)(4(1

=

132

263

262

265


011

421

062

จงหา A-1

วธท า เนองจาก det A = (0 + 24 + 0) - (0 + 8 + 0)

= 16 ( มคา ≠ 0 ดงนนหา A-1 ได )

ดงนน A-1 = A adj Adet

1

=

t

21

62

41

02

42

0611

62

01

02

01

0611

21

01

41

01

42

161

=

t

10824

400

344

161

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=

1043

804

2404

161

=

85

41

163

21

041

23

041

3. แบบฝกหด I. จงหาคาของ A-1 ตอไปน

1) A =

12

36 2) A =

34

12

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3) A =

23

36 4) A =

38

12

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5) A =

151

743

412

6) A =

131

543

012

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7) A =

1210

2152

3423

3211

8) A =

3 8210

2142

3723

4201

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of 93


1. จดประสงค 1.1 มความเขาใจเกยวกบกฎของคราเมอร 1.2 สามารถอธบายหลกการแกระบบสมการโดยใชกฎของคราเมอรได 1.3 สามารถค านวณแกระบบสมการโดยใชกฎของคราเมอรได

2. เนอหาโดยสงเขป เปนการแกระบบสมการเชงเสนโดยใชดเทอรมเนนตหาค าตอบของระบบสมการ

เชงเสน โดยทมจ านวนสมการเทากบจ านวนตวแปร เรยกอกนยหนงวา กฎของคราเมอร (Cramer’s Rules) ซงมวธการหาดงน

ก าหนดระบบสมการเชงเสนมจ านวน m สมการ และ n ตวแปร ซงเขยนอยในรปสมการเมทรกซได AX = B

ให A เปนเมทรกซสมประสทธ B เปนเมทรกซตวแปร C เปนเมทรกซคงท 2.1 ระบบสมการเชงเสน 2 ตวแปร

รปทวไปของสมการ 2 ตวแปร คอ a1x + b1y = c1 a2x + b2y = c2

น ามาเขยนในรปเมทรกซ AX = B ไดดงน

y

x

ba

ba

22

11 =

2

1c

c

ถาให D เปนดเทอรมแนนตของเมทรกซมประสทธของ x และ y ทงหมด และถา D ≠ 0 แลว ระบบสมการนจะมรากเพยงรากเดยว

D1 เปนดเทอรมแนนตทเกดจากการตดสมประสทธของ x ออกแลวน า c1 , c2 มาแทนท

เรอง การแกสมการเชงเสนโดยวธของคราเมอร


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D2 เปนดเทอรมแนนตทเกดจากการตดสมประสทธของ y ออกแลวน า c1 , c2 มาแทนท

ดงนนจะไดดงน

D = 22

11ba

ba

D1 = 22

11bc

bc

D2 = 22

11ca

ca

แลวค าตอบของสมการ คอ x = D

D1 , y = D

D2

2.2 ระบบสมการเชงเสน 3 ตวแปร

รปทวไปของสมการ 3 ตวแปร คอ a1x + b1y + c1y = d1 a2x + b2y + c2y = d2

น ามาเขยนในรปเมทรกซ AX = B ไดดงน

z

y

x

cba

cba

cba

333

222

111 =

3

2

1

d

d

d

ดงนนจะไดดงน

D =

333

222

111

cba

cba

cba

D1 =

333

222

111

cbd

cbd

cbd

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D2 =

333

222

111

cda

cda

cda

D3 =

333

222

111

dba

dba

dba


D1 , y = D

D2 , z = D

D3

Ex. จงแกระบบสมการโดยใชวธของคราเมอร

3x + 2y = -3

4x - 3y = 13

วธท า เขยนในรปเมทรกซ AX = B ไดดงน

y

x

34

23 =

13

3

จะได D =

34

23 = (3)(-3) – (4)(2) = -17

D1 =

313

23 = (-3)(-3) – (13)(2) = -17

D2 =

134

33 = (3)(13) – (4)(-3) = 51


D1 = 17-17-

= 1

y = D

D2 = 17-51

= -3

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I. จงแกระบบสมการตอไปน โดยใชวธของคราเมอร 1) 2x + y = 4 2) 5x + 2y = 0

3x - 2y = 12 4x - 3y = 23 ................................................................................................................................. ................................................................................................................................. ................................................................................................................................. ................................................................................................................................. ................................................................................................................................. ................................................................................................................................. .................................................................................................................................

3) 3x - 4y = -23 4) – y + z = 10 2x + 3y = -4 -2y + z = 9

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5) 2x + y – z = 5 6) -x - y + 2z = 1 3x - 2y + 2z = -3 2x + y - 2z = -3 x - 3y - 3z = -2 x + y - z = 0

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7) 3x + 4y - 2z = 4 8) x - 5y + 7z = 8 -3x + 5y - 2z = -10 4x - y + 9z = 13 2x - y - 3z = -3 5x + y + z = -2

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9) x + 2y + z = 0 10) x + y - z = 6 3x + y = -11 x - y + z = -4 2x + z = -7 x + y + z = 12

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11) x + 2y + z = 9 12) 2x + y - z = 8 x - y = -3 x - 2y + z = -5 x + 2z = 11 x + y + 2z = 10 จงหา x + y + z จงหา x - y + z

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1. จดประสงค 1.1 มความเขาใจเกยวกบการแกระบบสมการโดยวธของเกาส 1.2 สามารถอธบายหลกการแกระบบสมการโดยวธของเกาสได 1.3 สามารถค านวณแกระบบสมการโดยวธของเกาสได

2. เนอหาโดยสงเขป a11x1 + a12x2 + a13x3 + . . . + a1nxn = b1

a21x1 + a22x2 + a23x3 + . . . + a2nxn = b2

an1x1 + an2x2 + an3x3 + . . . + annxn = bn แทนระบบสมการนดวยเมทรกซสมประสทธของตวแปรและคาคงตวของ

สมการ ดงน

BA =

nnnnn

n

n

b

b

b

aaa

aaa

aaa

.

...

...

....

............

...

...

2

1

21

22221

11211

วธการแกระบบสมการใหน า BA มาลดรป ใหเปลยนเปนเมทรกซสามเหลยมบนหรอสามเหลยมลาง โดยใชการด าเนนการตามแถวขนมลฐาน (Elementary Row Operation : E.R.O.) ซงมวธการอย 3 ขอ คอ

1) สลบ 2 แถวใด ๆ ของเมทรกซได 2) น าจ านวนใด ๆ ทไมเทากบ 0 คณแถวใดแถวหนงของเมทรกซได 3) คณแถวใดแถวหนงดวยจ านวนคงท และน าผลลพธไปบวกกบอกแถวหนง

ได

** เมอท าการลดรปแลวกแทนทยอนกลบ ค านวณหาค าตอบของระบบสมการ

เรอง การแกสมการเชงเสนโดยวธของเกาส


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Ex. จงหาค าตอบของระบบสมการ 2x + 3y + z = 11

2x + 2y + 3z = 16

4x – y + 3z = 11

วธท า BA =

11

15

11

314

322

132

=

313

212R2)R(R ;

R1)R(R ;

11

4

11

170

210

132

=

323 R7)R(R ;

39

4

11

1300

210

132

=

33

22

)R131

(R ;

1)R(R ;

3

4

11

100

210

132

จากเมทรกซลดรปสดทาย แทนคากลบเปนสมการได ดงน z = 3

y - 2z = -4 y – 2(3) = -4

y = 2 2x + 3y + z = 11

2x + 3(2) + 3 = 11 x = 1

ดงนน ค าตอบของระบบสมการ คอ x = 1, y = 2 และ z = 3

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I. จงแกระบบสมการตอไปน โดยใชวธของเกาส 1) 2x + y = 4 2) 5x + 2y = 0

3x - 2y = 12 4x - 3y = 23 ................................................................................................................................. ................................................................................................................................. ................................................................................................................................. ................................................................................................................................. ................................................................................................................................. ................................................................................................................................. .................................................................................................................................

3) 3x - 4y = -23 4) – y + z = 10 2x + 3y = -4 -2y + z = 9

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.................................................................................................................................

5) 2x + y – z = 5 6) -x - y + 2z = 1 3x - 2y + 2z = -3 2x + y - 2z = -3 x - 3y - 3z = -2 x + y - z = 0

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7) 3x + 4y - 2z = 4 8) x - 5y + 7z = 8 -3x + 5y - 2z = -10 4x - y + 9z = 13 2x - y - 3z = -3 5x + y + z = -2

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9) x + 2y + z = 0 10) x + y - z = 6 3x + y = -11 x - y + z = -4 2x + z = -7 x + y + z = 12

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11) x + 2y + z = 9 12) 2x + y - z = 8 x - y = -3 x - 2y + z = -5 x + 2z = 11 x + y + 2z = 10 จงหา x + y + z จงหา x - y + z

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1. จดประสงค 1.1 มความเขาใจเกยวกบการแปลงคาองศากบเรเดยน 1.2 สามารถอธบายความหมาย ความแตกตางระหวางองศากบเรเดยนได 1.3 สามารถค านวณหาคาองศา และเรเดยนได

2. เนอหาโดยสงเขป 2.1 การวดมมเปนองศา

มมทเกดจากการหมนสวนของเสนตรงไปครบหนงรอบมขนาด 360 องศา (360o) ซงในแตละ 1 องศาจะแบงเปนหนวยยอย 60 ลปดา (60') ในแตละ 1 ลปดา จะแบงหนวยยอยเปน 60 ฟลปดา (60'')

นนคอ 1 องศา = 60 ลปดา 1 ลปดา = 60 ฟลปดา

หรอ 1 องศา = 3,600 ฟลปดา

2.2 การวดมมเปนเรเดยน ขนาดของมมเรเดยนมคาเทากบอตราสวนของความยาวของสวนโคงทรองรบมม

นน กบความยาวของรศมวงกลม เมอหมนรศมของวงกลมไปครบ 1 รอบ มมรอบจดศนยกลางในหนวยเรดยน

จะเทากบ ππ

2r

r2

เรเดยน

นนคอ 360 องศา = π2 เรเดยน 180 องศา = π เรเดยน

1 องศา = 180π

เรเดยน

เรอง องศากบเรเดยน


y

x

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Ex. จงเปลยนมม 30 องศา ใหอยในรปของหนวยเรเดยน

วธท า เนองจาก มม 1 องศา = 180π

เรเดยน

มม 30 องศา = 180π

30 เรเดยน

= 60π

เรเดยน

= 601

722

เรเดยน (π =7

22)

ดงนน มม 30 องศา

0.5238 เรเดยน

Ex. จงเปลยนมม 5π เรเดยน ใหอยในรปของหนวยองศา

วธท า เนองจาก มม 1 เรเดยน = π

180 องศา

มม 5π เรเดยน =

π

1805π

องศา

= 36 องศา


I. จงเปลยนมมองศา ใหอยในรปของหนวยเรเดยน 1) 160o 2) 40o

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3) 60o 4) 206o ................................................................................................................................. ................................................................................................... .............................. ................................................................................................................................. ................................................................................................................................. .................................................................................................................................

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5) 144o 6) 336o ................................................................................................................................. ................................................................................................................................. ................................................................................................................................. ................................................................................................................................. .................................................................................................................................

II. จงเปลยนมมเรเดยน ใหอยในรปของหนวยองศา

1) 8π

2) 3

5π

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3) 4π

4) 5

2π

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5) 115π

6) 6

5π

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III. จงเปลยนมมตอไปน

1) จงเปลยนมม 48ππ

เรเดยน ใหอยในรปของหนวยองศา

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2) จงเปลยนมม

43

25

ππ เรเดยน ใหอยในรปของหนวยองศา

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3) จงเปลยนมม 9,680 ฟลปดา ใหอยในรปของหนวยเรเดยน ................................................................................................................................. ................................................................................................................................. ................................................................................................................................. ................................................................................................................................. ................................................................................................................................. ................................................................................................................................. .................................................................................................................................

4) จงเปลยนมม 690,350 ฟลปดา ใหอยในรปของหนวยองศา ................................................................................................................................. ................................................................................................................................. ................................................................................................................................. ................................................................................................................................. ................................................................................................................................. ................................................................................................................................. .................................................................................................................................

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1. จดประสงค 1.1 มความเขาใจเกยวกบฟงกชนตรโกณมต 1.2 สามารถหาคาฟงกชนตรโกณมตของมม 30o 45o 60o ได 1.3 สามารถค านวณหาคา sin cos tan cot และฟงกชนตรโกณมตอน ๆ ได

2. เนอหาโดยสงเขป พจารณารปสามเหลยมมมฉาก ABC โดยมมม C เปนมมฉาก

เมอพจารณามม A มอตราสวนของดาน ซงเปนอตราสวนตรโกณมต คอ

sin A = มมมฉาก ดานตรงขา

A มมม ดานตรงขา =

ba

cos A = มมมฉาก ดานตรงขา

A มม ดานประชด =

bc

tan A = A มม ดานประชดA มมม ดานตรงขา

= ca

cosec A = A มมม ดานตรงขา

มมมฉาก ดานตรงขา =

ab

sec A = A มม ดานประชด

มมมฉาก ดานตรงขา =

cb

เรอง ฟงกชนตรโกณมต


c a

b

B

C A

c2 = a2 + b2

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cot A = A มมม ดานตรงขาA มม ดานประชด

= ac

อตราสวนตรโกณมตทง 6 อตราสวนมความสมพนธทเปนสวนกลบกบ ดงน

tan A = A cosAsin

; cot A = Asin A cos

cosec A = Asin

1 ; sec A =

A cos1

มม 30o 45o 60o

sin 21

23

2

1

cos 23

21

2

1

tan 3

1 3 1

cot 3 31

1

sec 3

2 2 2

cosec 2 3

2 2

Ex. จงหาคาของ sin 45 o sec 45 o + tan 60 o cos 30 o

วธท า sin 45 o sec 45 o + tan 60 o cos 30 o =

23

322

1

= 23

1

= 25

1

2 2 3 3

1

2 1

1

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I. จงหาคาตอไปน 1) sin 45 o sec 60 o + cot 60 o cos 30 o

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2) cos 45 o cosec 60 o - cot 60 o cos 45 o ................................................................................................................................. ................................................................................................................................. .................................................................................................................................

3) 2cos 45 o + cosec 45 o + cot 60 o cot 30 o – 2sin 60 o ................................................................................................................................. ................................................................................................................................. .................................................................................................................................

4) oo

oo

60 cosec45 cot

60sin 45 cos

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5) oo

oo

30 cosec30 cot

30sin 45 cot+ cot 45 o – 2sin 30 o

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6)

oo

2oo

30 sec60 cot

60sin 45tan - cot 60 o – 2cosec 30 o

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II. จงหาคาตอไปน 1)

2)

3)

4)

5)

sin A =.........................................................................

cos A = .........................................................................

tan A = .........................................................................

9

8 6

B A

C

sin A =.........................................................................

cos A = .........................................................................

tan A = .........................................................................

7

4

B A

C

sin A =.........................................................................

cos A = .........................................................................

tan A = .........................................................................

2

5 1

B A

C

Sec A =.........................................................................

cosec A = .........................................................................

tan A = .........................................................................

sin A =.........................................................................

cos A = .........................................................................

cot A = .........................................................................

3

3

B A

C

22

5

B A

C

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III. ก าหนดใหรปสามเหลยม ABC มมม B เปนมมฉาก และ a , b , c เปนความยาวดาน ตรงขามมม A มม B มม C ตามล าดบ

1) sin A = 65

, a = 15 หนวย จงหา b

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2) tan A = 125

, c = 24 หนวย จงหา a

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3) a = 12 หนวย , c = 5 หนวย จงหา sin C ................................................................................................................................. ................................................................................................................................. ................................................................................................................................. ................................................................................................................................. .................................................................................................................................

4) cos C = 32

, a = 10 หนวย จงหา b

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5) a = 4 หนวย , c = 6 หนวย จงหา tan A ................................................................................................................................. ................................................................................................................................. ................................................................................................................................. ................................................................................................................................ .................................................................................................................................

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1. จดประสงค 1.1 มความเขาใจเกยวกบฟงกชนตรโกณมตของมมรรอบจดศนยกลาง 1.2 สามารถหาคาฟงกชนตรโกณมตของมมรรอบจดศนยกลางได 1.3 สามารถค านวณหาคา sin cos tan cot และฟงกชนตรโกณมตอน ๆ ได


หมายเหต ; 360 องศา = π2 เรเดยน 180 องศา = π เรเดยน

1 องศา = 180π

เรเดยน

1 องศา = 60 ลปดา 1 ลปดา = 60 ฟลปดา

1 องศา = 3,600 ฟลปดา

เรอง ฟงกชนตรโกณมตของมมรอบจดศนยกลาง


y

x

45o 30o

60o

√

√

√

√

(0,0)

(0,-1)

(1,0) (-1,0)

(0,1)

sin มคาเปน (+) sin มคาเปน (+) cos มคาเปน (-) cos มคาเปน (+)

sin มคาเปน (-) sin มคาเปน (-) cos มคาเปน (-) cos มคาเปน (+)

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Ex. จงหาคาของ sin 405 o วธท า sin 405 o = sin ( 360o + 45o )

= sin 45o

= 2

1

Ex. จงหาคาของ tan 3

13

วธท า tan 3

13 = tan

34

= tan 3

= 3


I. จงหาคาตอไปน 1) 2sin2 30 o – 4cos2 60 o + 3tan2 45o

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2) o30 coseco60 sec3o45 4cot

o602tano30 23cos -o45 2sin

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3) tan2 3

+ 2tan2 4

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4) 2cosec2 4

- 3sec2 6

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5) oo

oo

30 cosec30 cot

30sin 45 cot+ cot

6 – 2sin 30 o

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6)

oo

2oo

30 sec330 cot

150sin 135tan - cot 60 o – 2cosec 270 o

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7) cot2 4

+ cos2 3

- sin2 3

- 43

cot2 3

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8) cos (-300 o ) – cot (-690 o ) ................................................................................................................................. ................................................................................................................................. ................................................................................................................................. .................................................................................................................................

9) sin 150 o + cot (-600 o ) ................................................................................................................................. ................................................................................................................................. ................................................................................................................................. .................................................................................................................................

10) cos 585 o + tan

32

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11) ถา A cos60 sec45 cot

30 cosec60 cos60tan oo

ooo จงหาคา 6 tan2 A + 8 sin 2A cos A

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12) ถา cos A = 0.8 แลวคาของ tan A มคาเทาใด ................................................................................................................................. ................................................................................................................................. ................................................................................................................................. ................................................................................................................................. .................................................................................................................................

13) ก าหนดให 13 cos A = 12 คาของ Ctan

1A cos

1 มคาเทาใด

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14) 2 sin 30 o – 6 cot + 3 tan 45o ................................................................................................................................. ................................................................................................................................. .................................................................................................................................

15) o30 coseco60 sec32 cot4

o60tan o30 cos 3 -sin

co

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16) cot2 3

+ 2cosec 4

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1.1 มความเขาใจเกยวกบเอกลกษณของฟงกชนตรโกณมต

1.2 สามารถอธบายเกยวกบเอกลกษณของฟงกชนตรโกณมตได

1.3 สามารถพสจนหาเอกลกษณของฟงกชนตรโกณมตได

2. เนอหาโดยสงเขป เอกลกษณของฟงกชนตรโกณมต คอ ความสมพนธของฟงกชนตรโกณมตท

สามารถเขยนแสดงใหอยในรปของสตรได และสามารถน าความสมพนธนไปหาความสมพนธอน ๆ ไดอก ซงความสมพนธสามารถสรปเปนสตรได ดงน

1. sin A cosec A = 1

2. cos A sec A = 1

3. tan A cot A = 1

4. tan A = AcosAsin

5. cot A = AsinAcos

6. sin2 A + cos2 A = 1

7. sec2 A - tan2 A = 1

8. cosec2 A - cot2 A = 1

9. sin (A + B) = sin A cos B + cos A sin B

sin (A - B) = sin A cos B - cos A sin B

10. cos (A + B) = cos A cos B - sin A sin B

cos (A - B) = cos A cos B + sin A sin B

เรอง เอกลกษณของฟงกชนตรโกณมต


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11. tan (A + B) = BtanAtan1BtanAtan

tan (A - B) = BtanAtan1

BtanAtan

12. sin 2A = 2 sin A cos A

13. cos 2A = cos2 A – sin2 A

cos 2A = 1 – 2 sin2 A

cos 2A = 2 cos2 A – 1

14. tan 2A = Atan1

Atan22

15. sin2 A = 21

– 21

cos 2A

16. cos2 A = 21

+ 21

cos 2A

17. sin A cos B = B)sin(AB)sin(A21

18. sin A sin B = B)cos(AB)cos(A21

19. cos A cos B = B)cos(AB)cos(A21

Ex. จงพสจนเอกลกษณ (1 – cos2 A) cosec2 A = 1

วธท า (1 – cos2 A) cosec2 A = (1 – cos2 A) Asin

12

= Asin

Acos

Asin

12

2

2

= cosec2 A - cot2 A

= 1

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I. จงพสจนเอกลกษณ ตอไปน 1) sec4 A – 1 = 2 tan2 A + tan4 A

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2) sin (A + B) sin (A – B) = cos2 B – sin2 A ................................................................................................................................. ................................................................................................................................. ................................................................................................................................. ................................................................................................................................. ................................................................................................................................. ................................................................................................................................. .................................................................................................................................

3) cosec A cos A tan A = 1 ................................................................................................................................. ................................................................................................................................. ................................................................................................................................. ................................................................................................................................. ................................................................................................................................. ................................................................................................................................. .................................................................................................................................

4) (1 – cos2 A)(1 + tan2 A) = tan2 A ................................................................................................................................. ................................................................................................................................. ................................................................................................................................. ................................................................................................................................. ................................................................................................................................. ................................................................................................................................. .................................................................................................................................

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2) (tan A + cot A)2 = sec2 A + cosec2 A ................................................................................................................................. ................................................................................................................................. ................................................................................................................................. ................................................................................................................................. ................................................................................................................................. .......................................................................................... ....................................... .................................................................................................................................

6) (1 – sin2 A) sec2 A = 1 ................................................................................................................................. ................................................................................................................................. ................................................................................................................................. ................................................................................................................................. ................................................................................................................................. ................................................................................................................................. .................................................................................................................................

7) A cos1Asin

Asin A cos1

= 2scosec A

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8) A secA cos

A cosecAsin

= 1

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9) A cosAsin AcosAsin

2Asin 21

133

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10) (sec A – tan A)(sec A + tan A)2 = A cos

Asin 1

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11) cos4 A – sin4 A = cos 2A ................................................................................................................................. ................................................................................................................................. ................................................................................................................................. ................................................................................................................................. ................................................................................................................................. ................................................................................................................................. .................................................................................................................................

12) (1 – cos2 A)(1 + cot2 A) = 1 ................................................................................................................................. ................................................................................................................................. ................................................................................................................................. ................................................................................................................................. ................................................................................................................................. ................................................................................................................................. .................................................................................................................................

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1. จดประสงค 1.1 มความเขาใจเกยวกบวงกลม พาราโบลา วงร 1.2 สามารถอธบายเกยวกบนยาม ของวงกลม พาราโบลา วงรได 1.3 สามารถค านวณหาคาตาง ๆ ของวงกลม พาราโบลา วงรได

2. เนอหาโดยสงเขป 2.1 วงกลม

วงกลมรปมาตรฐาน วงกลมรปทวไป

สมการ (x – h)2 + (y – k)2 = r2 ; r 0 x2 + y2 + Dx + Ey + F = 0

ถา D2+E2-4F 0

จดศนยกลาง (h, k)

2E

,2D

รศม r 4F-E+D21 22

2.2 วงร

จากรปคอ แกนเอกขนานกบแกน x แตถาขนานกบแกน y รปจะเปนแนวตง สมการทวไปของวงร คอ Ax2 + By2 + Cx + Dy + E = 0 โดยท A ≠ B ≠ 0

Y

V' F' C F V

X

(-a,0) (-c,0) (h, k) (c,0) (a,0)

เรอง ภาคตดกรวย


โดย C(h, k) เปนจดศนยกลาง r เปนรศม P(x, y) เปนจดใด ๆ บนเสนรอบ

วงกลม

r

C(h, k)

P(x, y)

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สมการ 1b

y

a

x2

2

2

2 ; a b สมการ 1

a

y

b

x2

2

2

2 ; a b

1. จดศนยกลาง C(0, 0) 2. จดยอด V(a, 0) , V' (-a, 0) 3. จดโฟกส F(c, 0) , F' (-c, 0) 4. แกนเอกยาว 2a และทบแกน x 5. แกนโทยาว 2b และทบแกน y

6. เลตสเรกตม (L.R) ยาว a

2b 2

1. จดศนยกลาง C(0, 0) 2. จดยอด V(0, a) , V' (0, -a) 3. จดโฟกส F(0, c) , F' (0, -c) 4. แกนเอกยาว 2a และทบแกน y 5. แกนโทยาว 2b และทบแกน x

เลตสเรกตม (L.R) ยาว a

2b 2

สมการ 1b

k)(y

a

h)(x2

2

2

2

สมการ 1

a

k)(y

b

h)(x2

2

2

2

1. จดศนยกลาง C(h, k) 2. จดยอด V(h + a, k) , V' (h - a, k) 3. จดโฟกส F(h + c, k) , F' (h - c, k) 4. แกนเอกยาว 2a และขนานแกน x 5. แกนโทยาว 2b และขนานแกน y


2b 2

1. จดศนยกลาง C(h, k) 2. จดยอด V(h, k + a) , V' (0, k - a) 3. จดโฟกส F(h, k + c) , F' (h, k - c) 4. แกนเอกยาว 2a และขนานแกน y 5. แกนโทยาว 2b และขนานแกน x


2b 2

2.3 พาราโบลา

สมการทวไปของพาราโบลา คอ รปเปดดานบน หรอดานลาง x2 + Dx + Ey + F = 0 โดยท E ≠ 0 รปเปดดานขวา หรอดานซาย y2 + Dy + Ex + F = 0 โดยท E ≠ 0

สวนสมการแบบมาตรฐาน เปนดงน สมการ (y – k)2 = 4c(x – h) สมการ (y – k)2 = –4c(x – h)

ไดเรกทรกซ ไดเรกทรกซ

C F(h + c,0) F(h - c,k) C(h, k)

x = h - c x = h + c

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สมการ (x - h)2 = 4c(y – k) สมการ (x - h)2 = -4c(y – k) ไดเรกทรกซ y = k + c

F(h, k+c)

F(h, k-c)

ไดเรกทรกซ y = k - c

สมการ รปแบบ จดโฟกส ไดเรกทรกซ แกนพาราโบลา (x - h)2 = 4c(y – k) เปดดานบน F(h, k + c) y = k - c x = h (x - h)2 = -4c(y – k) เปดดานลาง F(h, k - c) y = k + c x = h (y – k)2 = 4c(x – h) เปดดานขวา F(h + c, k) y = h - c y = k (y – k)2 = –4c(x – h) เปดดานซาย F(h - c, k) y = h + c y = k

Ex. ก าหนดใหสมการวงกลมมจดศนยกลางอยทจดก าเนด และมรศมยาว 2 หนวย จงหาสมการวงกลม

วธท า จากสมการ (x – h)2 + (y – k)2 = r2 แทนรศม r = 2 จะได

(x – h)2 + (y – k)2 = 22

(x – h)2 + (y – k)2 = 4

Ex. จงหาสมการพาราโบลาทมจดยอดอยท (2, -3) และจดโฟกสอยท (5, -3) วธท า พาราโบลารปเปดขวา มสมการรปมาตรฐาน คอ

(y – k)2 = 4c(x – h) (y – (-3))2 = 4(3)(x – 2) โดย 5 – 2 = 3

(มาจากการลบคา x ของจด C กบ F) (y + 3)2 = 12(x – 2) y2 + 6y + 9 = 12x – 24

y2 + 6y – 12x + 9 + 24 = 0 y2 + 6y – 12x + 33 = 0

C(h, k) C(h, k)

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I. จงตอบค าถามตอไปน 1) ก าหนดใหสมการวงกลมมจดศนยกลางอยท (1, 2) และมรศมยาว √ หนวย จงหาสมการวงกลม

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2) จงหาจดศนยกลางและรศมของวงกลม x2 + y2 – 4x + 6y + 4 = 0 ................................................................................................................................... ................................................................................................................................... ................................................................................................................................... ................................................................................................................................... ................................................................................................................................... ...................................................................................................................................

3) จงหาจดยอด จดโฟกส สมการไดเรกทรกซ ความยาวของเสนเลตสเรกตมของสมการพาราโบลา y2 = 16x ................................................................................................................................... ................................................................................................................................... ................................................................................................................................... ................................................................................................................................... ................................................................................................................................... ................................................................................................................................... ...................................................................................................................................

4) จงหาสมการพาราโบลาทมจดยอดอยท (3, 3) และมสมการไดเรกทรกซคอ y = 1 ................................................................................................................................... ................................................................................................................................... ................................................................................................................................... ................................................................................................................................... ................................................................................................................................... ................................................................................................................................... ...................................................................................................................................

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5) จงหาสมการวงรทมจดศนยกลางอยทจดก าเนดมจดยอดท (4,0)และจดโฟกส (3,0) ................................................................................................................................... ................................................................................................................................... ................................................................................................................................... ................................................................................................................................... ................................................................................................................................... ...................................................................................................................................

6) จงหาจดศนยกลาง จดโฟกส จดยอด ความยาวของเลตสเรกตมของสมการวงร

19

y4

x 22

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7) จงหาสมการวงรทมจดศนยกลางอยท (-1, 2) แกนโทยาวเทากบ 6 หนวย แกนเอกยาว 10 หนวย และขนานกบแกน y ................................................................................................................................... ................................................................................................................................... ................................................................................................................................... ................................................................................................................................... ................................................................................................................................... ................................................................................................................................... ...................................................................................................................................

8) จงหาจดศนยกลาง จดโฟกส จดยอด ความยาวแกนโท ความยาวแกนเอก และความยาวของเสนเลตสเรกตมของวงร 16x2 + 25y2 – 128x + 250y + 481 = 0 ................................................................................................................................... ................................................................................................................................... ................................................................................................................................... ................................................................................................................................... ................................................................................................................................... ................................................................................................................................... ................................................................................................................................... ...................................................................................................................................

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บรรณานกรม

บรบรณ ศรมาชย. คณตศาสตร 2.กรงเทพฯ: ส านกพมพเอมพนธ จ ากด, 2546 มนส ประสงค.คณตศาสตร 2. .กรงเทพฯ: ส านกพมพสงเสรมวชาการ, 2537

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