New ËŒÍ§ Á.4/ àÅ¢·Õè Math For Physics 1. ¡c¬¶ ¬q · 2016. 5. 11. ·...

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Math For Physics

1. ¾×é¹°Ò¹·ÑèÇä»·Ò§¤³ÔµÈÒÊµÃ

1.1 àÈÉÊ‹Ç¹ ·È¹ÔÂÁ ÃŒÍÂÅÐËÃ×Íà»ÍÃà«ç¹µ àÈÉÊ‹Ç¹ (fraction) ã¹·Ò§àÅ¢¤³Ôµ ËÁÒÂ¶Ö§ ¨íÒ¹Ç¹·ÕèÍÂÙ‹ã¹ÃÙ»¢Í§

𝐴𝐴𝐵𝐵

â´Â A àÃÕÂ¡Ç‹ÒàÈÉ áÅÐ B àÃÕÂ¡Ç‹ÒÊ‹Ç¹

1.2 ÍÑµÃÒÊ‹Ç¹ ÍÑµÃÒ ÊÑ́ Ê‹Ç¹ ÍÑµÃÒÊ‹Ç¹ (Ratio) à»š¹¡ÒÃà»ÃÕÂºà·ÕÂºÊÍ§»ÃÔÁÒ³â´Â¡ÒÃËÒÃ«Öè§¨ÐàËÁ×Í¹àÈÉÊ‹Ç¹

ÍÑµÃÒ (Rate) à»š¹ÍÑµÃÒÊ‹Ç¹ÃÐËÇ‹Ò§ÊÍ§»ÃÔÁÒ³·ÕèÁÕË¹‹ÇÂµ‹Ò§¡Ñ¹

ÊÑ́ Ê‹Ç¹ (Proportion) à»š¹ÊÁ¡ÒÃ/¢ŒÍ¤ÇÒÁ·ÕèáÊ´§¡ÒÃà·‹Ò¡Ñ¹¢Í§ÍÑµÃÒÊ‹Ç¹¢Í§ÊÍ§ÍÑµÃÒÊ‹Ç¹

2. ¾Õª¤³Ôµ

2.1 àÅ¢ªÕé¡íÒÅÑ§ àÅ¢ªÕé¡íÒÅÑ§ (Exponent) ËÁÒÂ¶Ö§ µÑÇàÅ¢ËÃ×ÍÊÑÞÅÑ¡É³·Õèà¢ÕÂ¹äÇŒ´ŒÒ¹º¹¢ÇÒ¢Í§¨íÒ¹Ç¹ËÃ×Í¹Ô¾¨¹ã´æ

ÊÁºÑµÔ¢Í§àÅ¢ªÕé¡íÒÅÑ§

1. 𝒂𝒂𝟎𝟎 = 𝟏𝟏2. 𝒂𝒂𝟏𝟏 = 𝒂𝒂3. 𝟏𝟏

𝒂𝒂𝒑𝒑= 𝒂𝒂−𝒑𝒑

4. 𝟏𝟏𝒂𝒂−𝒑𝒑

= 𝒂𝒂𝒑𝒑

5. 𝒂𝒂𝒓𝒓.𝒂𝒂𝒓𝒓 = 𝒂𝒂𝒓𝒓+𝒔𝒔

6. 𝒂𝒂𝒓𝒓

𝒂𝒂𝒔𝒔= 𝒂𝒂𝒓𝒓−𝒔𝒔

7. (𝒂𝒂𝒓𝒓)𝒔𝒔 = 𝒂𝒂𝒓𝒓.𝒔𝒔

8. (𝒂𝒂𝒓𝒓𝒃𝒃𝒓𝒓)𝒔𝒔 = 𝒂𝒂𝒓𝒓𝒔𝒔𝒃𝒃𝒓𝒓𝒔𝒔

ÊÑÞ¡Ã³ÇÔ·ÂÒÈÒÊµÃ

¡ÒÃà¢ÕÂ¹àÅ¢Â¡¡íÒÅÑ§áººÁÒµÃ°Ò¹ (standard form)

àÁ×èÍ

𝑨𝑨 × 𝟏𝟏𝟎𝟎𝒏𝒏 หลักการเปล่ียนตวัเลขเปนเลขยกกําลัง 1. เลื่อนจุดทศนิยมไป ทางขวา เลขช้ีกําลังเปน ลบ

2. เลื่อนจุดทศนิยมไป ทางซาย เลขช้ีกําลังเปน บวก𝟏𝟏 ≤ 𝑨𝑨 < 𝟏𝟏𝟎𝟎

หลักการเปล่ียนเลขยกกาํลังเปนตวัเลข

1. เลขช้ีกําลังเปน ลบ เลื่อนจุดทศนิยมไป ทางซาย

2. เลขช้ีกําลังเปน บวก เลื่อนจุดทศนิยมไป ทางขวา

BASIC MATH FOR PHYSICS ฟิสิกส์ ครูโก๊ด PAGE 1

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ฟิสิกส์ ครูโก๊ด

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áºº½ƒ¡·Õè 1 ¨§¤íÒ¹Ç³áÅÐá»Å§µÑÇàÅ¢ãËŒÍÂÙ‹ã¹ÃÙ»ÊÔºÂ¡¡íÒÅÑ§

1. 2345600

2. 4321000000

3. 123456789

4. 0.00000432

5. 0.0030003

6. 0.00000456

7. 1023 × 1032

8. 10−21 × 10−2

9. 10−5 × 107

10. 1012

105

11. 1015

10−9

12. 10−4

104

13. 103×104

105×106




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14. 102×10−3×103

102

15. 1012×10−7

108× 10−5

16. 1.4412

17. 6600 ÷ 0.00011

18. 24

8000

19. 200000 × 0.0005

2.2 ¡ÒÃá¡ŒÊÁ¡ÒÃ 2.2.1 ¡ÒÃ¨Ñ́ ÃÙ»ÊÁ¡ÒÃ ãËŒ¨Ñ´¡ÃÐ·íÒ¡ÑºÊÁ¡ÒÃâ´ÂÍÒÈÑÂËÅÑ¡¡ÒÃä´Œá¡‹

- ¡ÒÃºÇ¡ËÃ×ÍÅº´ŒÇÂàÅ¢ËÃ×ÍÊÑÞÅÑ¡É³ - ¡ÒÃ¤Ù³ËÃ×ÍËÒÃ´ŒÇÂµÑÇàÅ¢ËÃ×ÍÊÑÞÅÑ¡É³ - ¡ÒÃÂ¡¡íÒÅÑ§ËÃ×ÍãÊ‹ÃÒ¡·ÕèÊÍ§´ŒÇÂµÑÇàÅ¢ËÃ×ÍÊÑÞÅÑ¡É³ - Ç§àÅçº

⟨[ºÇ¡, Åº]|[¤Ù³, ËÒÃ ]|[Â¡¡íÒÅÑ§, ¶Í´ÃÙ·] |[Ç§àÅçº]⟩ ¡ÒÃ¨Ñ´¡ÃÐ·íÒ´Ñ§¡Å‹ÒÇµŒÍ§·íÒ·Ñé§ÊÍ§¢ŒÒ§¢Í§ÊÁ¡ÒÃà¾×èÍãËŒ·Ñé§ÊÍ§¢ŒÒ§¢Í§ÊÁ¡ÒÃ ÂÑ§¤§à·‹Ò¡Ñ¹

¨¹¡ÃÐ·Ñè§ä´ŒµÑÇäÁ‹ÃÙŒ¤‹ÒÍÂÙ‹¢ŒÒ§«ŒÒÂ¢Í§à¤Ã×èÍ§ËÁÒÂà·‹Ò¡Ñº




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áºº½ƒ¡·Õè 2 ¨§ËÒ¤‹Ò X ¨Ò¡ÊÁ¡ÒÃ

¡. 𝑥𝑥 + 6 = 2 ¢. 4𝑥𝑥 = 12 ¤. 3𝑥𝑥 + 6 = 21

2.2.2 ¡ÒÃ¡ÃÐ¨ÒÂáÅÐ¡ÒÃ´Ö§µÑÇÃ‹ÇÁ à¹×èÍ§¨Ò¡à¤Ã×èÍ§ËÁÒÂÇ§àÅçº à»š¹µÑÇºÑ§¤ÑºãËŒ¤Ô´àÅ¢ã¹Ç§àÅçº¡‹Í¹´Ñ§¹Ñé¹à¾×èÍ¤ÇÒÁÊÐ´Ç¡ã¹¡ÒÃ¤íÒ¹Ç³

àÃÒÊÒÁÒÃ¶á»Å§ÃÙ»¨Ò¡Ç§àÅçºà»š¹äÁ‹ÁÕÇ§àÅçºËÃ×Í¨Ò¡äÁ‹ÁÕÇ§àÅçºà»š¹ÁÕÇ§àÅçºä´Œ

àª‹¹ 𝑎𝑎(𝑏𝑏 + 𝑐𝑐) = … … … … … … . .. ËÃ×Í (𝑎𝑎𝑏𝑏)𝑐𝑐 = … … … … … … . .. ¢ŒÍÊÑ§à¡µ

ÁÕá¤‹¡ÒÃ¡ÃÐ¨ÒÂ¨Ò¡¤Ù³,ËÒÃà¢ŒÒä»ã¹ºÇ¡Åº áÅÐ ¡ÃÐ¨ÒÂ¨Ò¡Â¡¡íÒÅÑ§,¶Í´ÃÙ·à¢ŒÒä» ã¹¤Ù³ËÒÃà·‹Ò¹Ñé¹

2.2.3 ÊÙµÃá»Å§ÃÙ»ÊÁ¡ÒÃ

¼Åµ‹Ò§¡íÒÅÑ§ÊÍ§

¼Åµ‹Ò§áÅÐ¼ÅºÇ¡¡íÒÅÑ§ÊÒÁ

¡íÒÅÑ§ÊÍ§ÊÁºÙÃ³

áºº½ƒ¡·Õè 3 ¨§¨Ñ́ ÃÙ»ÊÁ¡ÒÃã¹áµ‹ÅÐ¢ŒÍãËŒÍÂÙ‹ã¹ÃÙ» x ã¹à·ÍÁ¢Í§ y

1. 𝑥𝑥𝑥𝑥 = 2

2. (𝑥𝑥 − 2)(𝑥𝑥 − 1) = 1




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3. 𝑥𝑥 = 1𝑥𝑥−1

4. 𝑥𝑥 = 2𝑥𝑥−3𝑥𝑥+1

5. 𝑥𝑥 = 𝑥𝑥+3𝑥𝑥−1

6. 𝑥𝑥 = 𝑥𝑥−3𝑥𝑥+1

2.2.4 ¡ÒÃá¡ŒÊÁ¡ÒÃ·ÕèÁÕµÑÇäÁ‹ÃÙŒ¤‹Ò 1 µÑÇ ã¹¡ÒÃá¡ŒÊÁ¡ÒÃ·ÕèÁÕµÑÇäÁ‹ÃÙŒ¤‹Ò 1 µÑÇ àª‹¹ 2𝑥𝑥 + 5 = 8 ãËŒá¡ŒÊÁ¡ÒÃâ´Â¡ÒÃ¨Ñ´ÃÙ»ÊÁ¡ÒÃãËŒàËÅ×ÍµÑÇá»Ã·ÕèäÁ‹ÃÙŒ¤‹Òà¾ÕÂ§µÑÇá»Ãà´ÕÂÇ

áºº½ƒ¡·Õè 4 ¨§ËÒ¤‹Ò X ¨Ò¡ÊÁ¡ÒÃµ‹Íä»¹Õé

1. 𝑥𝑥 + 2 = 1

2. 𝑥𝑥2

= 5




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3. 2𝑥𝑥 + 1 = 3

4. 4𝑥𝑥 + 2 − 𝑥𝑥 = 5𝑥𝑥 − 1

5. 2𝑥𝑥2 = 32

6. √𝑥𝑥 − 1 + 6 = 8

2.2.5 ¡ÒÃá¡ŒÊÁ¡ÒÃ¡íÒÅÑ§ÊÍ§ à»š¹¡ÒÃ¤Ô´ÂŒÍ¹¡ÅÑº¡ÒÃ¤Ù³¡ÒÃ¡ÃÐ¨ÒÂ ÁÕ 3 áºº ¤×Í

1. 𝒙𝒙𝟐𝟐 + (𝒂𝒂 + 𝒃𝒃)𝒙𝒙+ 𝒂𝒂𝒃𝒃 = (𝒙𝒙+ 𝒂𝒂)(𝒙𝒙+ 𝒃𝒃) 2. 𝒙𝒙𝟐𝟐 − (𝒂𝒂 + 𝒃𝒃)𝒙𝒙+ 𝒂𝒂𝒃𝒃 = (𝒙𝒙 − 𝒂𝒂)(𝒙𝒙 − 𝒃𝒃) 3. 𝒙𝒙𝟐𝟐 + (𝒂𝒂 − 𝒃𝒃)𝒙𝒙 − 𝒂𝒂𝒃𝒃 = (𝒙𝒙+ 𝒂𝒂)(𝒙𝒙 − 𝒃𝒃)

µÑÇÍÂ‹Ò§·Õè 1 ¨§ËÒ¤‹Ò X ¨Ò¡ÊÁ¡ÒÃ

1. 𝒙𝒙𝟐𝟐 + 𝟓𝟓𝒙𝒙 + 𝟔𝟔 = (𝒙𝒙+. … . . )(𝒙𝒙+. … . . )

2. 𝒙𝒙𝟐𝟐 − 𝟐𝟐𝒙𝒙 − 𝟑𝟑𝟓𝟓 = (𝒙𝒙−. … . . )(𝒙𝒙+. … . . )

¶ŒÒäÁ‹ÊÒÁÒÃ¶áÂ¡µÑÇ»ÃÐ¡Íºä´Œ ¨Ñ́ ÃÙ»ãËŒà»š¹ 𝑎𝑎𝒙𝒙𝟐𝟐 + 𝒃𝒃𝒙𝒙 + 𝒄𝒄 áÅŒÇãËŒãªŒÊÁ¡ÒÃ

𝒙𝒙 =−𝒃𝒃 ± √𝒃𝒃𝟐𝟐 − 𝟒𝟒𝒂𝒂𝒄𝒄

𝟐𝟐𝒂𝒂




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áºº½ƒ¡·Õè 5 ¨§ËÒ¤‹Ò X ¨Ò¡ÊÁ¡ÒÃµ‹Íä»¹Õé

1. 𝒙𝒙𝟐𝟐 + 𝟐𝟐𝒙𝒙 + 𝟏𝟏 = 𝟎𝟎

2. 𝒙𝒙𝟐𝟐 + 𝟒𝟒𝒙𝒙 = −𝟑𝟑

3. −𝟓𝟓𝒙𝒙 = 𝒙𝒙𝟐𝟐 + 𝟔𝟔

4. 𝒙𝒙𝟐𝟐 − 𝟏𝟏𝟐𝟐𝒙𝒙 + 𝟐𝟐𝟎𝟎 = 𝟎𝟎

5. 𝒙𝒙𝟐𝟐 − 𝟕𝟕𝒙𝒙 + 𝟏𝟏𝟒𝟒 = 𝟐𝟐

6. 𝟓𝟓𝒙𝒙 = 𝒙𝒙𝟐𝟐 + 𝟔𝟔

7. 𝒙𝒙𝟐𝟐 + 𝒙𝒙 − 𝟓𝟓𝟔𝟔 = 𝟎𝟎

8. 𝒙𝒙𝟐𝟐 − 𝒙𝒙 − 𝟓𝟓𝟔𝟔 = 𝟎𝟎




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9. 𝒙𝒙𝟐𝟐 = 𝟐𝟐𝟒𝟒 + 𝟓𝟓𝒙𝒙

10. 𝟑𝟑𝟓𝟓𝒙𝒙 + 𝟐𝟐𝒙𝒙𝟐𝟐 = 𝒙𝒙𝟑𝟑

11. 𝒙𝒙𝟐𝟐 = 𝟑𝟑𝒙𝒙

12. 𝒙𝒙𝟐𝟐 + 𝒙𝒙 − 𝟏𝟏 = 𝟎𝟎

2.2.6 ¡ÒÃá¡ŒÊÁ¡ÒÃËÅÒÂµÑÇá»Ã µŒÍ§á»Å§ãËŒà»š¹ÊÁ¡ÒÃµÑÇá»Ãà´ÕÂÇ¡‹Í¹¨Ö§¨ÐÊÒÁÒÃ¶ËÒ¤íÒµÍºä´Œâ´ÂÁÕ 2 ÇÔ Õ̧¤×Í

1. ¡ÃÐ·íÒÊÍ§¢ŒÒ§àËÁ×Í¹æ ¡Ñ¹ 2. ¨Ñ´ÃÙ»µÑÇá»Ãã´ ãËŒÍÂÙ‹ã¹à·ÍÁ¢Í§ÍÕ¡µÑÇá»Ã




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áºº½ƒ¡·Õè 6 ¨§ËÒ¤‹Ò¢Í§µÑÇá»Ã x áÅÐµÑÇá»Ã y

1. 𝑥𝑥 = 2 áÅÐ 2𝑥𝑥 + 𝑥𝑥 = 1

2. 3𝑥𝑥 − 2𝑥𝑥 = 0 áÅÐ 3𝑥𝑥 + 𝑥𝑥 = 4

3. 𝑥𝑥 + 𝑥𝑥 − 1 = 0 áÅÐ 2𝑥𝑥 + 𝑥𝑥 − 3 = 0

4. 𝑥𝑥𝑥𝑥 = 3 áÅÐ 𝑥𝑥 + 2𝑥𝑥 = 5




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2.2.7 ¡ÒÃá»Ã¼Ñ¹áÅÐÊÁ¡ÒÃ (Proportionality and equations) ã¹¤Œ¹ËÒ¤ÇÒÁÃÙŒ·Ò§ÇÔ·ÂÒÈÒÊµÃ ¨Ð¾ºÇ‹Ò àÁ×èÍ»ÃÔÁÒ³·Ò§¡ÒÂÀÒ¾Ë¹Öè§ÁÕ¡ÒÃà»ÅÕèÂ¹á»Å§¨ÐÊ‹§¼Åµ‹ÍÍÕ¡

»ÃÔÁÒ³¡ÒÂÀÒ¾Í×è¹ »̃ÞËÒÊíÒ¤ÑÞ»ÃÐ¡ÒÃË¹Öè§¡ç¤×Í ¡ÒÃËÒÇ‹Ò»ÃÔÁÒ³µ‹Ò§æ àËÅ‹Ò¹Ñé¹ÁÕ¤ÇÒÁÊÑÁ¾Ñ¹ ̧¡Ñ¹ÍÂ‹Ò§äÃ

- á»Ã¼Ñ¹µÃ§ (Proportional)

- á»Ã¼Ñ¹¡ÅÑº ËÃ×Í á»Ã¼¡¼Ñ¹ (Inverse Proportional)

- ¤ÇÒÁªÑ¹¢Í§¡ÃÒ¿

- ¾×é¹·ÕèãµŒ¡ÃÒ¿

áºº½ƒ¡·Õè 7 ¨§ËÒ¤‹ÒµÑÇá»Ã·Õèâ¨·Â¶ÒÁ

1. 𝑥𝑥 𝛼𝛼 𝑥𝑥 , 𝑥𝑥1 = 2, 𝑥𝑥2 = 5, 𝑥𝑥1 = 3, 𝑥𝑥2 =?

2. 𝑥𝑥 𝛼𝛼 𝑥𝑥 , 𝑥𝑥2 = 6, 𝑥𝑥1 = 7, 𝑥𝑥2 = 8, 𝑥𝑥1 =?




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3. 𝑥𝑥 𝛼𝛼 1𝑥𝑥 , 𝑥𝑥1 = 1, 𝑥𝑥2 = 2, 𝑥𝑥2 = 4, 𝑥𝑥1 =?

4. 𝑥𝑥 𝛼𝛼 1𝑥𝑥 , 𝑥𝑥1 = 12, 𝑥𝑥1 = 13, 𝑥𝑥2 = 14, 𝑥𝑥2 =?

5. 𝑃𝑃𝑃𝑃 = 𝑛𝑛𝑛𝑛𝑛𝑛 ,ãËŒ P, n, R à»š¹¤‹Ò¤§·Õè 𝑃𝑃1 = 3, 𝑛𝑛1 = 4, 𝑃𝑃2 = 5, 𝑛𝑛2 =?

6. 𝑃𝑃𝑃𝑃 = 𝑛𝑛𝑛𝑛𝑛𝑛 ,ãËŒ n, R,𝑛𝑛 à»š¹¤‹Ò¤§·Õè 𝑃𝑃1 = 6, 𝑃𝑃1 = 5, 𝑃𝑃2 = 4, 𝑃𝑃2 =?




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àÃÕÂ¡Ç‹Ò Co - ordinate ¨Ø´µ‹Ò§ æ ·Õè¾ÅÍµÅ§ã¹¡ÃÒ¿ ¨Ö§àÃÕÂ¡Ç‹Ò CoOrdinate

¢ŒÍ¤ÇÃÃÙŒà¡ÕèÂÇ¡Ñº¡ÃÒ¿

1. ¡ÒÃËÒ¤íÒµÍº¢Í§ÊÙµÃ¨Ò¡¾×é¹·ÕèãµŒ¡ÃÒ¿ ¾×é¹·ÕèãµŒ¡ÃÒ¿ = 𝑥𝑥.𝑥𝑥 “àÁ×èÍà¨ÍÊÙµÃ·Ò§¿ÊÔ¡Ê¤Ù³¡Ñ¹¤íÒµÍº·Õèä´Œ¤×Í¨Ò¡ÊÙµÃ¹Ñé¹ËÒä´Œ¨Ò¡¾×é¹·ÕèãµŒ¡ÃÒ¿”

µÑÇÍÂ‹Ò§·Õè 1 ¨Ò¡¡ÃÒ¿¨§ËÒÇ‹Ò ÃÐÂÐ·Ò§(𝑠𝑠)ÁÕ¤‹Ò¡ÕèàÁµÃ (ËÒ¨Ò¡¾×é¹·Õè)

2. ¡ÒÃËÒ¤íÒµÍº¢Í§ÊÙµÃ¨Ò¡¤ÇÒÁªÑ¹¢Í§¡ÃÒ¿

¤ÇÒÁªÑ¹¢Í§¡ÃÒ¿(Slope) = ∆𝑦𝑦∆𝑥𝑥

= 𝑦𝑦2−𝑦𝑦1𝑥𝑥2−𝑥𝑥1

= ¼Åµ‹Ò§á¡¹ 𝑦𝑦¼Åµ‹Ò§á¡¹ 𝑥𝑥

“àÁ×èÍà¨ÍÊÙµÃ·Ò§¿ÊÔ¡ÊËÒÃ¡Ñ¹ ¤íÒµÍº·Õèä´Œ¤×Í¨Ò¡ÊÙµÃ¹Ñé¹ËÒä´Œ̈ Ò¡¤ÇÒÁªÑ¹¢Í§¡ÃÒ¿”

ËÃ×Í¨Ñ́ ãËŒÍÂÙ‹ã¹ÃÙ»ÊÁ¡ÒÃàÊŒ¹µÃ§ 𝑥𝑥 = 𝑚𝑚𝑥𝑥 + 𝑐𝑐

¡Ã³Õ¡ÃÒ¿à»š¹àÊŒ¹â¤Œ§ ¤ÇÒÁªÑ¹¨ÐÁÕËÅÒÂ¤‹Òà¾ÃÒÐ¤ÇÒÁªÑ¹ËÒä´Œ¨Ò¡àÊŒ¹ÊÑÁ¼ÑÊàÊŒ¹â¤Œ§ áÅÐµŒÍ§´Ù´ŒÇÂÇ‹Ò

¤ÇÒÁªÑ¹¢Í§â¤Œ§¹Ñé¹¡íÒÅÑ§à¾ÔèÁ¢Öé¹ËÃ×Í¡íÒÅÑ§Å´Å§




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µÑÇÍÂ‹Ò§·Õè 2 ¨Ò¡¡ÃÒ¿¨§ËÒÇ‹Ò ¤ÇÒÁàÃ‹§ (𝑎𝑎)ÁÕ¤‹Ò¡ÕèàÁµÃµ‹ÍÇÔ¹Ò·Õ (ËÒ¨Ò¡¤ÇÒÁªÑ¹)

¡ÃÒ¿·ÕèÁÑ¡¾ºã¹ÇÔªÒ¿ÊÔ¡ÊÊ‹Ç¹ãËÞ‹ä´Œá¡‹




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ËÅÑ¡ã¹¡ÒÃà¢ÕÂ¹ÊÁ¡ÒÃàÊŒ¹µÃ§

áºº½ƒ¡·Õè 8 ¨§à¢ÕÂ¹ÊÁ¡ÒÃàÊŒ¹µÃ§¨Ò¡¢ŒÍÁÙÅ·ÕèãËŒ

1. ÁÕ¤ÇÒÁªÑ¹ −53 áÅÐ¼‹Ò¹¨Ø´ (0,0)

2. ÁÕ¤ÇÒÁªÑ¹ 12 áÅÐ¼‹Ò¹¨Ø´ (-3,7)

3. ¼‹Ò¹¨Ø´ (1,2) áÅÐ (4,5)

4. ¼‹Ò¹¨Ø´ (-2,3) áÅÐ (-3,-3)




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5.

6.

áºº½ƒ¡·Õè 9 ¨§ÇÒ´¡ÃÒ¿µÒÁ¢ŒÍÁÙÅ·ÕèãËŒ áÅÐËÒ¤‹Òµ‹Ò§æ ¨Ò¡¡ÃÒ¿ ·Õèâ¨·Â¶ÒÁ

1. ¡ÃÒ¿àÊŒ¹µÃ§àÊŒ¹à´ÕÂÇ àÃÔèÁ¨Ò¡¨Ǿ (0,0) áÅÐ¼‹Ò¹¨Ø´ (6,8) ãËŒËÒ¤ÇÒÁªÑ¹áÅÐ¾×é¹·ÕèãµŒ¡ÃÒ¿¨Ò¡ x=0 ¶Ö§ x=6

2. ¡ÃÒ¿àÊŒ¹µÃ§àÊŒ¹à´ÕÂÇ àÃÔèÁ¨Ò¡¨Ǿ (0,7) áÅÐ¼‹Ò¹¨Ø´ (10,0) ãËŒËÒ¤ÇÒÁªÑ¹áÅÐ¾×é¹·ÕèãµŒ¡ÃÒ¿¨Ò¡ x=0 ¶Ö§ x=10




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3. ¨Ò¡ÊÁ¡ÒÃ y=2x-3 ãËŒËÒ¤ÇÒÁªÑ¹áÅÐ¨Ø´µÑ́ á¡¹ y

4. ¨Ò¡ÊÁ¡ÒÃ 4x+3y=1 ãËŒËÒ¤ÇÒÁªÑ¹áÅÐ¨Ø´µÑ´á¡¹ y

5. ¡ÃÒ¿àÊŒ¹µÃ§àÃÔèÁ¨Ò¡¨Ø´ (0,0) ,ÁÕ¤ÇÒÁªÑ¹à·‹Ò¡Ñº 3 àÁ×èÍ¶Ö§ x=3 ¤ÇÒÁªÑ¹à»ÅÕèÂ¹à»š¹ 0 ä»µÅÍ´ãËŒËÒ¾×é¹·ÕèãµŒ¡ÃÒ¿¨Ò¡ x=0 ¶Ö§ x=10

ÅÑ¡É³ÐÊÁ¡ÒÃ¢Í§

¡ÃÒ¿¾ÒÃÒâºÅÒ

“µÑÇá»Ã x áÅÐ y ¨ÐÍÂÙ‹¤¹ÅÐ

½̃›§«Öè§µÑÇá»ÃË¹Öè§¨ÐÁÕ¡íÒÅÑ§

ÁÒ¡¡Ç‹ÒÍÕ¡ µÑÇá»ÃË¹Öè§ÍÂÙ‹

2 à·‹Ò áÅÐÁÑ¹¨ÐÊÁÁÒµÃá¡¹

·ÕèÁÕ¡íÒÅÑ§¹ŒÍÂ¡Ç‹Ò”




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µÑÇÍÂ‹Ò§·Õè 3

ÊÁ¡ÒÃ 𝐸𝐸𝑘𝑘 =12𝑚𝑚𝑣𝑣2 ¨§à¢ÕÂ¹¡ÃÒ¿áÊ´§¤ÇÒÁÊÑÁ¾Ñ¹ ̧ÃÐËÇ‹Ò§ 𝐸𝐸𝑘𝑘¡Ñº 𝑣𝑣 áÅÐ 𝐸𝐸𝑘𝑘 ¡Ñº 𝑣𝑣2


ÊÁ¡ÒÃ 𝑛𝑛 = 2𝜋𝜋�𝑚𝑚𝑘𝑘 ¨§à¢ÕÂ¹¡ÃÒ¿áÊ´§¤ÇÒÁÊÑÁ¾Ñ¹ ̧ÃÐËÇ‹Ò§ 𝑛𝑛 ¡Ñº 𝑚𝑚 áÅÐ 𝑛𝑛 ¡Ñº √𝑚𝑚




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ÊÁ¡ÒÃ 𝑃𝑃𝑃𝑃 = 𝑛𝑛𝑛𝑛𝑛𝑛 ¨§à¢ÕÂ¹¡ÃÒ¿áÊ´§¤ÇÒÁÊÑÁ¾Ñ¹ ̧ÃÐËÇ‹Ò§ 𝑃𝑃¡Ñº 𝑃𝑃

áºº½ƒ¡·Õè 10

1. ¨Ò¡¡ÃÒ¿µ‹Íä»¹Õé ¨§ËÒ¤‹Ò slope ¢Í§¡ÃÒ¿

2. ¨Ò¡¡ÃÒ¿·Õè¡íÒË¹´ãËŒ ¨§ËÒ¤‹Ò¤ÇÒÁªÑ¹¢Í§¡ÃÒ¿ 1. 1

4

2. 12

3. 1 4. 2




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3. ¨§ËÒ slope ¨Ò¡ÊÁ¡ÒÃµ‹Íä»¹Õé 1. 2𝑥𝑥 − 4𝑥𝑥 = 1

2. 3𝑥𝑥 + 2𝑥𝑥 = 4

4. ¨Ò¡¤ÇÒÁÊÑÁ¾Ñ¹ ̧¢Í§»ÃÔÁÒ³ 2 »ÃÔÁÒ³ à¢ÕÂ¹à»š¹ÊÁ¡ÒÃä´ŒÇ‹Ò 2x + 3y = 6 àÁ×èÍ¹íÒä»à¢ÕÂ¹¡ÃÒ¿ÃÐºº¾Ô¡Ñ´©Ò¡¨Ðä´Œ¡ÃÒ¿ÁÕ¤‹Ò¤ÇÒÁªÑ¹à·‹ÒäÃ

1. −23

2. 23

3. −32

4. 32

5. Í¹ØÀÒ¤Ë¹Öè§ÁÕÁÇÅ 𝑚𝑚 ¡íÒÅÑ§à¤Å×èÍ¹·Õè´ŒÇÂÍÑµÃÒàÃçÇ 𝑣𝑣 ¾ºÇ‹ÒÍ¹ØÀÒ¤¹Õé¨ÐÁÕ¾ÅÑ§§Ò¹¨Å¹ (E𝑘𝑘) â´Â E𝑘𝑘 ÁÕ¤ÇÒÁÊÑÁ¾Ñ¹ ̧¡Ñº 𝑣𝑣 ´Ñ§ÊÁ¡ÒÃ E𝑘𝑘 =

12𝑚𝑚𝑣𝑣2 àÁ×èÍ ÁÇÅ 𝑚𝑚 ¤§·Õè ¡ÃÒ¿¤ÇÒÁÊÑÁ¾Ñ¹ ̧ÃÐËÇ‹Ò§ E𝑘𝑘 ¡Ñº 𝑣𝑣

¤ÇÃÁÕÅÑ¡É³ÐÍÂ‹Ò§äÃ




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6. ¡ÒÃÈÖ¡ÉÒ¡ÒÃá¡Ç‹§¢Í§ÅÙ¡µØŒÁ¹ÒÌÔ¡Ò ¾ºÇ‹Ò¤Òº¡ÒÃá¡Ç‹§¢Í§ÅÙ¡µØŒÁ (𝑛𝑛) ¨Ðà»ÅÕèÂ¹á»Å§â´Â¤ÇÒÁÊÑÁ¾Ñ¹ ̧

¡Ñº¤ÇÒÁÂÒÇ¢Í§ÊÒÂÅÙ¡µØŒÁ (𝑙𝑙) µÒÁÊÁ¡ÒÃ 𝑛𝑛 = 2𝜋𝜋�𝑙𝑙𝑔𝑔 ¨§ËÒ¡ÃÒ¿¤ÇÒÁÊÑÁ¾Ñ¹ ̧ÃÐËÇ‹Ò§ 𝑛𝑛 ¡Ñº 𝑙𝑙 ¤ÇÃ

ÁÕÅÑ¡É³ÐµÒÁ¢ŒÍã´

7. àÁ×èÍÍÍ¡áÃ§ (𝐹𝐹) ¡ÃÐ·íÒ¡ÑºÁÇÅ (𝑚𝑚) ¢¹Ò´µ‹Ò§æ «Öè§ÇÒ§ÍÂÙ‹º¹¾×é¹·ÕèäÁ‹ÁÕ¤ÇÒÁ½„´·íÒãËŒÁÇÅà¤Å×èÍ¹·Õè´ŒÇÂ¤ÇÒÁàÃ‹§ (𝑎𝑎)¢¹Ò´µ‹Ò§æ ¡Ñ¹ ä´Œ¤ÇÒÁÊÑÁ¾Ñ¹ ̧ÃÐËÇ‹Ò§áÃ§ ÁÇÅ áÅÐ¤ÇÒÁàÃ‹§´Ñ§ÊÁ¡ÒÃ 𝐹𝐹 = 𝑚𝑚𝑎𝑎 ¡ÃÒ¿áÊ´§¤ÇÒÁÊÑÁ¾Ñ¹ ̧ÃÐËÇ‹Ò§ 𝑎𝑎 ¡Ñº 1

𝑚𝑚 ¨Ðà»š¹´Ñ§ÃÙ»ã´

8. ã¹¡ÒÃ·´ÅÍ§àÃ×èÍ§¡ÃÐ¨¡àÇŒÒà¾×èÍËÒ¤ÇÒÁÊÑÁ¾Ñ¹ ̧ÃÐËÇ‹Ò§ÃÐÂÐÇÑµ¶Ø (𝑠𝑠) ÃÐÂÐÀÒ¾ (𝑠𝑠´) áÅÐ¤ÇÒÁÂÒÇâ¿¡ÑÊ (𝑓𝑓) ä´Œ¤ÇÒÁÊÑÁ¾Ñ¹ ̧ 1

𝑓𝑓= 1

𝑠𝑠+ 1

𝑠𝑠´ ¡ÃÒ¿ã¹¢ŒÍã´áÊ´§¤ÇÒÁÊÑÁ¾Ñ¹ ̧ä´Œ¶Ù¡µŒÍ§




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àÃ¢Ò¤³ÔµáÅÐµÃÕâ¡³ÁÔµÔ 3.1 ¡ÒÃËÒ¤ÇÒÁÂÒÇÃÐËÇ‹Ò§¨Ǿ ÊÍ§¨Ǿ ã¹ÃÐºº¾Ô¡Ñ́ ©Ò¡

3.2 ¢ŒÍÁÙÅÃÙ»·Ã§àÃ¢Ò¤³Ôµ

3.3 ·ÄÉ®Õº·¾Õ·Òâ¡ÃÑÊ (Pythagoras Theorem)




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3.4 ¿§̃¡ªÑ¹µÃÕâ¡³ÁÔµÔ

𝑠𝑠𝑠𝑠𝑛𝑛𝑠𝑠 =¢ŒÒÁ

©Ò¡=

𝑐𝑐𝑐𝑐𝑠𝑠𝑠𝑠 =ªỐ

©Ò¡=

𝑡𝑡𝑎𝑎𝑛𝑛𝑠𝑠 =¢ŒÒÁ

ªỐ=

ÃÙ»ÊÒÁàËÅÕèÂÁ·Õè¹‹ÒÊ¹ã¨

µÒÃÒ§áÊ´§¤‹Ò sin ,cos , tam ¢Í§ÁØÁ·Õè¹‹ÒÊ¹ã¨

ÁØÁ 0 30 37 45 53 60 90

Sin 0 12

35

√22

45

√32

1

Cos 1 √32

45

√22

35

12

0

tan 0 1√3

34

1 43

√3 ∞

áºº½ƒ¡·Õè 11 ¨§ËÒ¤ÇÒÁÂÒÇ¢Í§´ŒÒ¹·Õè a b áÅÐ c




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New ËŒÍ§ Á.4/ àÅ¢·Õè Math For Physics 1. ¡c¬¶ ¬q · 2016. 5. 11. ·...

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