เอกสารประกอบการสอน คณิตศาสตร์...
TRANSCRIPT
Page 1 of 93
Mr.Jaranawit Kongkaew : Mathermatics II
เอกสารประกอบการสอน คณตศาสตร 2
โดย
นายจรณวชณ กองแกว
แผนกคณตศาสตร – วทยาศาสตร คณะชางอตสาหกรรม
โรงเรยนพายพเทคโนโลยและบรหารธรกจ ปการศกษา 2554
@ลขสทธโรงเรยนพายพเทคโนโลยและบรหารธรกจ
Page 2 of 93
Mr.Jaranawit Kongkaew : Mathermatics II
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
Page 3 of 93
Mr.Jaranawit Kongkaew : Mathermatics II
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
Page 4 of 93
Mr.Jaranawit Kongkaew : Mathermatics II
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
Page 5 of 93
Mr.Jaranawit Kongkaew : Mathermatics II
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
Page 6 of 93
Mr.Jaranawit Kongkaew : Mathermatics II
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
Page 7 of 93
Mr.Jaranawit Kongkaew : Mathermatics II
………….. 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 = ……………………………………………………………………
Page 8 of 93
Mr.Jaranawit Kongkaew : Mathermatics II
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
= ……………………………………………………………………
Page 9 of 93
Mr.Jaranawit Kongkaew : Mathermatics II
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
ใบงาน 2 รายวชา คณตศาสตร 2
Page 10 of 93
Mr.Jaranawit Kongkaew : Mathermatics II
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 = ……………………………………………………………………
= ……………………………………………………………………
Page 11 of 93
Mr.Jaranawit Kongkaew : Mathermatics II
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 = ……………………………………………………………………
= ……………………………………………………………………
= ……………………………………………………………………
= ……………………………………………………………………
Page 12 of 93
Mr.Jaranawit Kongkaew : Mathermatics II
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 = ………………………………………………………………
= ………………………………………………………………
Page 13 of 93
Mr.Jaranawit Kongkaew : Mathermatics II
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
ใบงาน 3 รายวชา คณตศาสตร 2
Page 14 of 93
Mr.Jaranawit Kongkaew : Mathermatics II
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 = …………………………………….
Page 15 of 93
Mr.Jaranawit Kongkaew : Mathermatics II
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 = …………………………………….
Page 16 of 93
Mr.Jaranawit Kongkaew : Mathermatics II
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 ................................................................................................................................. ................................................................................................................................. ................................................................................................................................. ................................................................................................................................. .................................................................................................................................
Page 17 of 93
Mr.Jaranawit Kongkaew : Mathermatics II
1. จดประสงค 1.1 มความเขาใจเกยวกบการแกสมการเลขยกก าลง 1.2 สามารถค านวณการแกสมการเลขยกก าลงได 1.3 สามารถค านวณหาคาตวแปรในระบบสมการเลขยกก าลงได
2. เนอหาโดยสงเขป
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
ใบงาน 4 รายวชา คณตศาสตร 2
Page 18 of 93
Mr.Jaranawit Kongkaew : Mathermatics II
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
.................................................................................................................................
.................................................................................................................................
.................................................................................................................................
.................................................................................................................................
.................................................................................................................................
Page 19 of 93
Mr.Jaranawit Kongkaew : Mathermatics II
5.) 43x - 1 = x16
1
.................................................................................................................................
.................................................................................................................................
.................................................................................................................................
.................................................................................................................................
6.) 22x – 2x + 1+ 1 = 0 ................................................................................................................................. ................................................................................................................................. ................................................................................................................................. .................................................................................................................................
7.) 32x – (4)3x + 1+ 27 = 0 ................................................................................................................................. ................................................................................................................................. ................................................................................................................................. .................................................................................................................................
8.) 4x = 161
.................................................................................................................................
.................................................................................................................................
.................................................................................................................................
.................................................................................................................................
9.) 3x + 2 = 81 ................................................................................................................................. ................................................................................................................................. ................................................................................................................................. .................................................................................................................................
10.) 25x + 2 = x
1251
.................................................................................................................................
.................................................................................................................................
................................................................................................................................
.................................................................................................................................
.................................................................................................................................
Page 20 of 93
Mr.Jaranawit Kongkaew : Mathermatics II
11.) 23 - x = 8x + 2 ................................................................................................................................. ................................................................................................................................. ................................................................................................................................. ................................................................................................................................. .................................................................................................................................
12.) 32x - 3x + 2 = 0 ................................................................................................................................. ................................................................................................................................. ................................................................................................................................. ................................................................................................................................. .................................................................................................................................
13.) 32x - 3x + 2 = 0 ................................................................................................................................. ................................................................................................................................. ................................................................................................................................. ................................................................................................................................. .................................................................................................................................
14.) 33
33
xx
.................................................................................................................................
.................................................................................................................................
.................................................................................................................................
.................................................................................................................................
.................................................................................................................................
.................................................................................................................................
.................................................................................................................................
15.) 23
14
x
.................................................................................................................................
.................................................................................................................................
.................................................................................................................................
.................................................................................................................................
.................................................................................................................................
.................................................................................................................................
Page 21 of 93
Mr.Jaranawit Kongkaew : Mathermatics II
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
ใบงาน 5 รายวชา คณตศาสตร 2
Log1.7 x
Loge x
Log10 x
Page 22 of 93
Mr.Jaranawit Kongkaew : Mathermatics II
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
Page 23 of 93
Mr.Jaranawit Kongkaew : Mathermatics II
3. แบบฝกหด I. จงหาคาฟงกชนลอการทม ตอไปน
1) log 100 + log 10 + log 1
.................................................................................................................................
.................................................................................................................................
.................................................................................................................................
2) ( log 1000) ( log 105 )
.................................................................................................................................
.................................................................................................................................
.................................................................................................................................
3) log 2 32 + log 5 25 + log 3 81
.................................................................................................................................
.................................................................................................................................
.................................................................................................................................
.................................................................................................................................
4) log 3 log 2 log 3 log 2 512
.................................................................................................................................
.................................................................................................................................
.................................................................................................................................
.................................................................................................................................
.................................................................................................................................
5) log3 3 33
.................................................................................................................................
.................................................................................................................................
.................................................................................................................................
.................................................................................................................................
Page 24 of 93
Mr.Jaranawit Kongkaew : Mathermatics II
6) log2 16
.................................................................................................................................
.................................................................................................................................
.................................................................................................................................
.................................................................................................................................
7) 81log
3
1
.................................................................................................................................
.................................................................................................................................
.................................................................................................................................
.................................................................................................................................
8)
81
1log
9
1
.................................................................................................................................
.................................................................................................................................
.................................................................................................................................
.................................................................................................................................
.................................................................................................................................
9) log7 343
.................................................................................................................................
.................................................................................................................................
.................................................................................................................................
.................................................................................................................................
10) log16 2
.................................................................................................................................
.................................................................................................................................
.................................................................................................................................
.................................................................................................................................
Page 25 of 93
Mr.Jaranawit Kongkaew : Mathermatics II
11) log8 32
.................................................................................................................................
.................................................................................................................................
.................................................................................................................................
.................................................................................................................................
12) log12 4 + log12 3
.................................................................................................................................
.................................................................................................................................
.................................................................................................................................
.................................................................................................................................
.................................................................................................................................
13) 3log2 33
.................................................................................................................................
.................................................................................................................................
.................................................................................................................................
.................................................................................................................................
.................................................................................................................................
14) )251
log(01.0
.................................................................................................................................
.................................................................................................................................
.................................................................................................................................
.................................................................................................................................
.................................................................................................................................
15) ก าหนดให 3010.02log , 4771.03log , 6990.05log และ 8451.07log จงหาคาของ 125.0log420log
.................................................................................................................................
.................................................................................................................................
.............................................................................................................................. ...
.................................................................................................................................
.................................................................................................................................
.................................................................................................................................
.................................................................................................................................
Page 26 of 93
Mr.Jaranawit Kongkaew : Mathermatics II
1. จดประสงค 1.1 มความเขาใจเกยวกบการแกสมการฟงกชนลอการทม 1.2 สามารถอธบายการแกสมการฟงกชนลอการทมได 1.3 สามารถค านวณหาคาสมการฟงกชนลอการทมได
2. เนอหาโดยสงเขป
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
ใบงาน 6 รายวชา คณตศาสตร 2
Page 27 of 93
Mr.Jaranawit Kongkaew : Mathermatics II
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
Page 28 of 93
Mr.Jaranawit Kongkaew : Mathermatics II
3. แบบฝกหด I. จงแกสมการลอการทมตอไปน
1) log 3 )8(x log 3 x = 2
.................................................................................................................................
.................................................................................................................................
.................................................................................................................................
.................................................................................................................................
.................................................................................................................................
.................................................................................................................................
2) log 5 x log 5 )4( x = 1
.................................................................................................................................
.................................................................................................................................
.................................................................................................................................
.................................................................................................................................
.................................................................................................................................
.................................................................................................................................
3) log 49 16
1x
.................................................................................................................................
.................................................................................................................................
.................................................................................................................................
.................................................................................................................................
.................................................................................................................................
.................................................................................................................................
4) log 2 log 2 log 2 x = 0
.................................................................................................................................
.................................................................................................................................
.................................................................................................................................
.................................................................................................................................
.................................................................................................................................
.................................................................................................................................
Page 29 of 93
Mr.Jaranawit Kongkaew : Mathermatics II
5) ln 2ln xe = xe 4lnln
.................................................................................................................................
.................................................................................................................................
.................................................................................................................................
.................................................................................................................................
.................................................................................................................................
.................................................................................................................................
6) log x log )8( x = log )3(x log )4( x
.................................................................................................................................
.............................................................................................................................. ...
.................................................................................................................................
.......................................................................................................................... .......
.................................................................................................................................
...................................................................................................................... ...........
7) ก าหนดให log 2 = 0.3010 จงแกสมการ 02.0)2.0( x
.................................................................................................................................
.................................................................................................................................
.................................................................................................................................
.................................................................................................................................
8) log 8 x + log 8 )2( x = log 8 )32( x
.................................................................................................................................
.................................................................................................................................
.................................................................................................................................
.................................................................................................................................
.................................................................................................................................
9) ln (2 ln x + 3 ) = ln 3
............................................................................................................................ .....
.................................................................................................................................
.................................................................................................................................
.................................................................................................................................
.................................................................................................................................
Page 30 of 93
Mr.Jaranawit Kongkaew : Mathermatics II
10) ln 102xe = )22ln( xxe
.................................................................................................................................
.................................................................................................................................
.................................................................................................................................
.................................................................................................................................
.................................................................................................................................
11) x2log3 = 5log1 3
.................................................................................................................................
.................................................................................................................................
.................................................................................................................................
.................................................................................................................................
.................................................................................................................................
12) )13(log6 x = 2log10log 66
.................................................................................................................................
.................................................................................................................................
.................................................................................................................................
.................................................................................................................................
.................................................................................................................................
13) 5loglog 2x = x2log7log
.................................................................................................................................
.................................................................................................................................
.................................................................................................................................
.................................................................................................................................
.................................................................................................................................
14) zyx zy 27logloglog = 5log3
.................................................................................................................................
.................................................................................................................................
.................................................................................................................................
.................................................................................................................................
.................................................................................................................................
Page 31 of 93
Mr.Jaranawit Kongkaew : Mathermatics II
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
ใบงาน 7 รายวชา คณตศาสตร 2
Page 32 of 93
Mr.Jaranawit Kongkaew : Mathermatics II
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
Page 33 of 93
Mr.Jaranawit Kongkaew : Mathermatics II
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
= ……………………………………………………………………
Page 34 of 93
Mr.Jaranawit Kongkaew : Mathermatics II
III. จงหาสมประสทธทวนาม
1)
36 = ……………………………………………………………………
= ……………………………………………………………………
= ……………………………………………………………………
2)
59 = ……………………………………………………………………
= ……………………………………………………………………
= ……………………………………………………………………
3)
05 = ……………………………………………………………………
= ……………………………………………………………………
= ……………………………………………………………………
4)
58 = ……………………………………………………………………
= ……………………………………………………………………
= ……………………………………………………………………
5)
47 = ……………………………………………………………………
= ……………………………………………………………………
= ……………………………………………………………………
6)
312 = ……………………………………………………………………
= ……………………………………………………………………
= ……………………………………………………………………
Page 35 of 93
Mr.Jaranawit Kongkaew : Mathermatics II
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
= ……………………………………………………………………
= ……………………………………………………………………
= ……………………………………………………………………
Page 36 of 93
Mr.Jaranawit Kongkaew : Mathermatics II
1. จดประสงค
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
ใบงาน 8 รายวชา คณตศาสตร 2
Page 37 of 93
Mr.Jaranawit Kongkaew : Mathermatics II
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
Page 38 of 93
Mr.Jaranawit Kongkaew : Mathermatics II
3. แบบฝกหด I. จงกระจายทวนามตอไปน โดยใชสามเหลยมปาสกาล
1) 2ba = …………………………………………………………………… = …………………………………………………………………… = ……………………………………………………………………
2) 3ba = …………………………………………………………………… = …………………………………………………………………… = ……………………………………………………………………
3) 4ba = …………………………………………………………………… = …………………………………………………………………… = ……………………………………………………………………
4) 6ba = …………………………………………………………………… = …………………………………………………………………… = ……………………………………………………………………
5) 52 y3x = …………………………………………………………………… = …………………………………………………………………… = …………………………………………………………………… = ……………………………………………………………………
6) 52 yx = …………………………………………………………………… = …………………………………………………………………… = …………………………………………………………………… = ……………………………………………………………………
7) 52x = …………………………………………………………………… = …………………………………………………………………… = …………………………………………………………………… = ……………………………………………………………………
8) 432 yx = …………………………………………………………………… = …………………………………………………………………… = …………………………………………………………………… = ……………………………………………………………………
Page 39 of 93
Mr.Jaranawit Kongkaew : Mathermatics II
= …………………………………………………………………… 9) 3x21 = ……………………………………………………………………
= …………………………………………………………………… = …………………………………………………………………… = ……………………………………………………………………
10) 24x = …………………………………………………………………… = …………………………………………………………………… = …………………………………………………………………… = ……………………………………………………………………
II. จงกระจายคาตอไปน โดยใชทฤษฏทวนาม
1) 5ba = …………………………………………………………………… = …………………………………………………………………… = …………………………………………………………………… = ……………………………………………………………………
2) 52 yx = …………………………………………………………………… = …………………………………………………………………… = …………………………………………………………………… = ……………………………………………………………………
3) 52 3yx = …………………………………………………………………… = …………………………………………………………………… = …………………………………………………………………… = ……………………………………………………………………
4) 52x = …………………………………………………………………… = …………………………………………………………………… = …………………………………………………………………… = ……………………………………………………………………
5) 331 x = …………………………………………………………………… = …………………………………………………………………… = …………………………………………………………………… = ……………………………………………………………………
Page 40 of 93
Mr.Jaranawit Kongkaew : Mathermatics II
III. จงหาหาคาตอไปน โดยใชทฤษฏทวนาม
1) จงหาพจนท 5 ของ 103y2x ................................................................................................................................. ................................................................................................................................. ................................................................................................................................. ................................................................................................................................. ................................................................................................................................. ................................................................................................................................. .................................................................................................................................
2) จงหาพจนท 6 ของ 102y3x ................................................................................................................................. ................................................................................................................................. ................................................................................................................................. ................................................................................................................................. ................................................................................................................................. ................................................................................................................................. .................................................................................................................................
3) จงหาพจนท 8 ของ 153x ................................................................................................................................. ................................................................................................................................. ................................................................................................................................. ................................................................................................................................. ................................................................................................................................. ................................................................................................................................. .................................................................................................................................
3) จงหาพจนท 5 ของ 532 x ................................................................................................................................. ................................................................................................................................. ................................................................................................................................. ................................................................................................................................. ................................................................................................................................. ................................................................................................................................ .................................................................................................................................
Page 41 of 93
Mr.Jaranawit Kongkaew : Mathermatics II
1. จดประสงค 1.1 มความเขาใจเกยวกบการบวก ลบ เมทรกซ 1.2 สามารถอธบายการบวก ลบ เมทรกซได 1.3 สามารถค านวณการบวก ลบ เมทรกซได
2. เนอหาโดยสงเขป
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
ใบงาน 9 รายวชา คณตศาสตร 2
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)
Page 42 of 93
Mr.Jaranawit Kongkaew : Mathermatics II
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
Page 43 of 93
Mr.Jaranawit Kongkaew : Mathermatics II
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
.................................................................................................................................
.................................................................................................................................
.................................................................................................................................
.................................................................................................................................
.................................................................................................................................
.................................................................................................................................
.................................................................................................................................
.................................................................................................................................
.................................................................................................................................
.................................................................................................................................
.................................................................................................................................
2. ก าหนดให A =
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) ................................................................................................................................. ................................................................................................................................. ................................................................................................................................. ................................................................................................................................. ................................................................................................................................. ................................................................................................................................. ................................................................................................................................. ................................................................................................................................. ................................................................................................................................. ................................................................................................................................. ................................................................................................................................. .................................................................................................................................
Page 44 of 93
Mr.Jaranawit Kongkaew : Mathermatics II
3. ก าหนดให A =
323
421
245
, B =
214
503
410
จงหาคาของ 21
A - 4B
.................................................................................................................................
.................................................................................................................................
.................................................................................................................................
.................................................................................................................................
.................................................................................................................................
.................................................................................................................................
.................................................................................................................................
.................................................................................................................................
4. ก าหนดให A =
24
32
03
, B =
12
30
12
จงหาคาของ -3A + 21
B
.................................................................................................................................
.................................................................................................................................
.................................................................................................................................
.................................................................................................................................
.................................................................................................................................
.................................................................................................................................
.................................................................................................................................
.................................................................................................................................
5. ให
wzyx
wzyx 2 =
41
53 จงหาคาของ x + y + z + w
.................................................................................................................................
.................................................................................................................................
.................................................................................................................................
.................................................................................................................................
.................................................................................................................................
.................................................................................................................................
.................................................................................................................................
.................................................................................................................................
Page 45 of 93
Mr.Jaranawit Kongkaew : Mathermatics II
1. จดประสงค 1.1 มความเขาใจเกยวกบหลกการคณเมทรกซ 1.2 สามารถอธบายวธการคณเมทรกซได 1.3 สามารถค านวณหาคาเมทรกซโดยวธการคณเมทรกซได
2. เนอหาโดยสงเขป
การคณเมทรกซ ดวย สเกลาร ก าหนด k เปนสเกลาร ใด ๆ แลว
kA =
gc
hb
da
k =
kgkc
khkb
kdka
Ex. ก าหนดให A =
24
32
03
จงหา 2A
2A = 2
24
32
03
=
2242
)3222
0232
(
=
48
64
06
เรอง การคณเมทรกซ
ใบงาน 10 รายวชา คณตศาสตร 2
Page 46 of 93
Mr.Jaranawit Kongkaew : Mathermatics II
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
Page 47 of 93
Mr.Jaranawit Kongkaew : Mathermatics II
3. แบบฝกหด I. จงหาคาของเมทรกซ ตอไปน
1. ก าหนดให A =
3
2
1
, B = 321 และ C =
42
31
จงหาคาของ 1) AB 2) 2AB 3) 2A 2C ................................................................................................................................. ................................................................................................................................. ................................................................................................................................. ................................................................................................................................. ................................................................................................................................. ................................................................................................................................. ................................................................................................................................. ................................................................................................................................. ................................................................................................................................. ................................................................................................................................. ................................................................................................................................. ................................................................................................................................. ................................................................................................................................. ................................................................................................................................. .................................................................................................................................
2. ก าหนดให A =
75
14 , B =
574
242 จงหาคาของ AB และ BA
.................................................................................................................................
.................................................................................................................................
.................................................................................................................................
.................................................................................................................................
.................................................................................................................................
.................................................................................................................................
.................................................................................................................................
.................................................................................................................................
.................................................................................................................................
.................................................................................................................................
Page 48 of 93
Mr.Jaranawit Kongkaew : Mathermatics II
3. ก าหนดให A =
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) ................................................................................................................................. ................................................................................................................................. ................................................................................................................................. ................................................................................................................................. ................................................................................................................................. ................................................................................................................................. ................................................................................................................................. ................................................................................................................................. ................................................................................................................................. ................................................................................................................................. ................................................................................................................................. ................................................................................................................................. ................................................................................................................................. ................................................................................................................................. ................................................................................................................................. ................................................................................................................................. ................................................................................................................................. ................................................................................................................................. ................................................................................................................................. ................................................................................................................................. ................................................................................................................................. ................................................................................................................................. ................................................................................................................................. ................................................................................................................................. ................................................................................................................................. ................................................................................................................................. .................................................................................................................................
Page 49 of 93
Mr.Jaranawit Kongkaew : Mathermatics II
1. จดประสงค
1.1 มความเขาใจเกยวกบไมเนอรและโคแฟกเตอร
1.2 สามารถอธบายวธการหลกการของไมเนอรและโคแฟกเตอรได
1.3 สามารถค านวณหาคาไมเนอรและโคแฟกเตอรได
2. เนอหาโดยสงเขป
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
ใบงาน 11 รายวชา คณตศาสตร 2
Page 50 of 93
Mr.Jaranawit Kongkaew : Mathermatics II
Ex. ก าหนดให A =
323
421
245
จงหา 22
M22 =
33
25 = (5×3) – (3×2) = 15 – 6 = 9
3. แบบฝกหด
I. จงหาคาไมเนอร ของเมทรกซ ตอไปน
1) ก าหนด A =
25
31จงหาคาของ 22
.................................................................................................................................
.................................................................................................................................
.................................................................................................................................
2) ก าหนด A =
1210
2152
3423
3211
จงหาคาของ 12 , 13 , 22 , 31
.................................................................................................................................
.................................................................................................................................
.................................................................................................................................
.................................................................................................................................
.................................................................................................................................
.................................................................................................................................
.................................................................................................................................
.................................................................................................................................
.................................................................................................................................
.................................................................................................................................
.................................................................................................................................
.................................................................................................................................
.................................................................................................................................
.................................................................................................................................
.................................................................................................................................
Page 51 of 93
Mr.Jaranawit Kongkaew : Mathermatics II
3) ก าหนด A =
22
12จงหาคาของ 12 , 21
.................................................................................................................................
.................................................................................................................................
.................................................................................................................................
.................................................................................................................................
.................................................................................................................................
.................................................................................................................................
.................................................................................................................................
II. จงหาคาโคแฟกเตอร ของเมทรกซ ตอไปน
1) ก าหนด A =
734
225
123
จงหาคาของ C12 , C 21
.................................................................................................................................
.................................................................................................................................
.................................................................................................................................
.................................................................................................................................
.................................................................................................................................
.................................................................................................................................
.................................................................................................................................
.................................................................................................................................
3) ก าหนด A =
570
684
312
จงหาคาของ C12 , C 23 , C 32
.................................................................................................................................
.................................................................................................................................
.................................................................................................................................
.................................................................................................................................
.................................................................................................................................
.................................................................................................................................
.................................................................................................................................
.................................................................................................................................
Page 52 of 93
Mr.Jaranawit Kongkaew : Mathermatics II
4) ให A =
1210
2152
3423
3211
จงหาคาของ C12 , C13 , C 22 , C 32 ,
(C33+C11) ................................................................................................................................. ................................................................................................................................. ................................................................................................................................. ................................................................................................................................. ................................................................................................................................. ................................................................................................................................. ................................................................................................................................. ................................................................................................................................. ................................................................................................................................. ................................................................................................................................. ................................................................................................................................. ................................................................................................................................. ................................................................................................................................. ................................................................................................................................. ................................................................................................................................. ................................................................................................................................. ................................................................................................................................. ................................................................................................................................. ................................................................................................................................. ................................................................................................................................. ................................................................................................................................. ................................................................................................................................. ................................................................................................................................. ................................................................................................................................. ................................................................................................................................. ................................................................................................................................. ................................................................................................................................. ................................................................................................................................. .................................................................................................................................
Page 53 of 93
Mr.Jaranawit Kongkaew : Mathermatics II
1. จดประสงค 1.1 มความเขาใจเกยวกบดเทอรมแนนต 1.2 สามารถอธบายวธการหลกการของดเทอรมแนนตได 1.3 สามารถค านวณหาคาดเทอรมแนนตได
2. เนอหาโดยสงเขป
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
ใบงาน 12 รายวชา คณตศาสตร 2
Page 54 of 93
Mr.Jaranawit Kongkaew : Mathermatics II
Ex. ก าหนดให A =
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
Page 55 of 93
Mr.Jaranawit Kongkaew : Mathermatics II
3. แบบฝกหด I. จงหาคาดเทอรมแนนต ตอไปน
1) A =
323
421
245
2) B =
570
684
312
.................................................................................................................................
.................................................................................................................................
.................................................................................................................................
.................................................................................................................................
.................................................................................................................................
.................................................................................................................................
.................................................................................................................................
.................................................................................................................................
.................................................................................................................................
3) K =
1210
2152
3423
3211
4) G =
12121
2152
344321
211
.................................................................................................................................
.................................................................................................................................
.................................................................................................................................
.................................................................................................................................
.................................................................................................................................
.................................................................................................................................
.................................................................................................................................
.................................................................................................................................
.................................................................................................................................
.................................................................................................................................
.................................................................................................................................
Page 56 of 93
Mr.Jaranawit Kongkaew : Mathermatics II
5) ให H =
3211
115321
48321
2121
และ P =
62
43 จงหา det H – det P
.................................................................................................................................
.................................................................................................................................
.................................................................................................................................
.................................................................................................................................
.................................................................................................................................
.................................................................................................................................
.................................................................................................................................
.................................................................................................................................
.................................................................................................................................
.................................................................................................................................
5) ก าหนดให H =
574
043
0020
และ D =
920
710
5120
ถา HD = B จงหา det B
.................................................................................................................................
.................................................................................................................................
.................................................................................................................................
.................................................................................................................................
.................................................................................................................................
.................................................................................................................................
.................................................................................................................................
.................................................................................................................................
.................................................................................................................................
.................................................................................................................................
.................................................................................................................................
.................................................................................................................................
.................................................................................................................................
Page 57 of 93
Mr.Jaranawit Kongkaew : Mathermatics II
1. จดประสงค 1.1 มความเขาใจเกยวกบอนเวอรสของเมทรกซ 1.2 สามารถอธบายการหาอนเวอรสของเมทรกซได 1.3 สามารถค านวณหาคาอนเวอรสของเมทรกซได
2. เนอหาโดยสงเขป
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
ใบงาน 13 รายวชา คณตศาสตร 2
Page 58 of 93
Mr.Jaranawit Kongkaew : Mathermatics II
Ex. ก าหนดให A =
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
Ex. ก าหนดให A =
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
Page 59 of 93
Mr.Jaranawit Kongkaew : Mathermatics II
=
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
.................................................................................................................................
.................................................................................................................................
.................................................................................................................................
.................................................................................................................................
.................................................................................................................................
.................................................................................................................................
.................................................................................................................................
3) A =
23
36 4) A =
38
12
.................................................................................................................................
.................................................................................................................................
.................................................................................................................................
.................................................................................................................................
.................................................................................................................................
.................................................................................................................................
.................................................................................................................................
Page 60 of 93
Mr.Jaranawit Kongkaew : Mathermatics II
5) A =
151
743
412
6) A =
131
543
012
.................................................................................................................................
.................................................................................................................................
.................................................................................................................................
.................................................................................................................................
.................................................................................................................................
.................................................................................................................................
.................................................................................................................................
.................................................................................................................................
.................................................................................................................................
.................................................................................................................................
.................................................................................................................................
7) A =
1210
2152
3423
3211
8) A =
3 8210
2142
3723
4201
.................................................................................................................................
.................................................................................................................................
.................................................................................................................................
.................................................................................................................................
.................................................................................................................................
.................................................................................................................................
.................................................................................................................................
.................................................................................................................................
.................................................................................................................................
.................................................................................................................................
.................................................................................................................................
.................................................................................................................................
.................................................................................................................................
Page 61 of 93
Mr.Jaranawit Kongkaew : Mathermatics II
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
ใบงาน 14 รายวชา คณตศาสตร 2
Page 62 of 93
Mr.Jaranawit Kongkaew : Mathermatics II
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
Page 63 of 93
Mr.Jaranawit Kongkaew : Mathermatics II
D2 =
333
222
111
cda
cda
cda
D3 =
333
222
111
dba
dba
dba
แลวค าตอบของสมการ คอ x = D
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
แลวค าตอบของสมการ คอ x = D
D1 = 17-17-
= 1
y = D
D2 = 17-51
= -3
Page 64 of 93
Mr.Jaranawit Kongkaew : Mathermatics II
3. แบบฝกหด
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
.................................................................................................................................
.................................................................................................................................
.................................................................................................................................
.................................................................................................................................
.................................................................................................................................
.................................................................................................................................
.................................................................................................................................
.................................................................................................................................
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
.................................................................................................................................
.................................................................................................................................
.................................................................................................................................
.................................................................................................................................
.................................................................................................................................
.................................................................................................................................
.................................................................................................................................
.................................................................................................................................
Page 65 of 93
Mr.Jaranawit Kongkaew : Mathermatics II
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
.................................................................................................................................
.................................................................................................................................
.................................................................................................................................
.................................................................................................................................
.................................................................................................................................
.................................................................................................................................
.................................................................................................................................
9) x + 2y + z = 0 10) x + y - z = 6 3x + y = -11 x - y + z = -4 2x + z = -7 x + y + z = 12
.................................................................................................................................
.................................................................................................................................
.................................................................................................................................
.................................................................................................................................
.................................................................................................................................
.................................................................................................................................
.................................................................................................................................
.................................................................................................................................
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
.................................................................................................................................
.................................................................................................................................
.................................................................................................................................
.................................................................................................................................
.................................................................................................................................
.................................................................................................................................
.................................................................................................................................
.................................................................................................................................
Page 66 of 93
Mr.Jaranawit Kongkaew : Mathermatics II
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
ใบงาน 15 รายวชา คณตศาสตร 2
Page 67 of 93
Mr.Jaranawit Kongkaew : Mathermatics II
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
Page 68 of 93
Mr.Jaranawit Kongkaew : Mathermatics II
3. แบบฝกหด
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
.................................................................................................................................
.................................................................................................................................
.................................................................................................................................
.................................................................................................................................
.................................................................................................................................
.................................................................................................................................
.................................................................................................................................
.................................................................................................................................
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
.................................................................................................................................
.................................................................................................................................
.................................................................................................................................
.................................................................................................................................
.................................................................................................................................
.................................................................................................................................
.................................................................................................................................
.................................................................................................................................
Page 69 of 93
Mr.Jaranawit Kongkaew : Mathermatics II
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
.................................................................................................................................
.................................................................................................................................
.................................................................................................................................
.................................................................................................................................
.................................................................................................................................
.................................................................................................................................
.................................................................................................................................
9) x + 2y + z = 0 10) x + y - z = 6 3x + y = -11 x - y + z = -4 2x + z = -7 x + y + z = 12
.................................................................................................................................
.................................................................................................................................
.................................................................................................................................
.................................................................................................................................
.................................................................................................................................
.................................................................................................................................
.................................................................................................................................
.................................................................................................................................
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
.................................................................................................................................
.................................................................................................................................
.................................................................................................................................
.................................................................................................................................
.................................................................................................................................
.................................................................................................................................
.................................................................................................................................
.................................................................................................................................
Page 70 of 93
Mr.Jaranawit Kongkaew : Mathermatics II
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
ใบงาน 16 รายวชา คณตศาสตร 2
y
x
Page 71 of 93
Mr.Jaranawit Kongkaew : Mathermatics II
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
3. แบบฝกหด
I. จงเปลยนมมองศา ใหอยในรปของหนวยเรเดยน 1) 160o 2) 40o
.................................................................................................................................
.................................................................................................................................
.................................................................................................................................
.................................................................................................................................
.................................................................................................................................
3) 60o 4) 206o ................................................................................................................................. ................................................................................................... .............................. ................................................................................................................................. ................................................................................................................................. .................................................................................................................................
Page 72 of 93
Mr.Jaranawit Kongkaew : Mathermatics II
5) 144o 6) 336o ................................................................................................................................. ................................................................................................................................. ................................................................................................................................. ................................................................................................................................. .................................................................................................................................
II. จงเปลยนมมเรเดยน ใหอยในรปของหนวยองศา
1) 8π
2) 3
5π
.................................................................................................................................
.................................................................................................................................
.................................................................................................................................
............................................................................................................................... ..
.................................................................................................................................
........................................................................................................................... ......
3) 4π
4) 5
2π
.................................................................................................................................
.................................................................................................................................
.................................................................................................................................
.................................................................................................................................
.................................................................................................................................
.................................................................................................................................
.................................................................................................................................
5) 115π
6) 6
5π
.................................................................................................................................
.................................................................................................................................
.................................................................................................................................
.................................................................................................................................
.................................................................................................................................
.................................................................................................................................
.................................................................................................................................
Page 73 of 93
Mr.Jaranawit Kongkaew : Mathermatics II
III. จงเปลยนมมตอไปน
1) จงเปลยนมม 48ππ
เรเดยน ใหอยในรปของหนวยองศา
.................................................................................................................................
.................................................................................................................................
.................................................................................................................................
.................................................................................................................................
.................................................................................................................................
.................................................................................................................................
2) จงเปลยนมม
43
25
ππ เรเดยน ใหอยในรปของหนวยองศา
.................................................................................................................................
.................................................................................................................................
.................................................................................................................................
.................................................................................................................................
.................................................................................................................................
.................................................................................................................................
.................................................................................................................................
3) จงเปลยนมม 9,680 ฟลปดา ใหอยในรปของหนวยเรเดยน ................................................................................................................................. ................................................................................................................................. ................................................................................................................................. ................................................................................................................................. ................................................................................................................................. ................................................................................................................................. .................................................................................................................................
4) จงเปลยนมม 690,350 ฟลปดา ใหอยในรปของหนวยองศา ................................................................................................................................. ................................................................................................................................. ................................................................................................................................. ................................................................................................................................. ................................................................................................................................. ................................................................................................................................. .................................................................................................................................
Page 74 of 93
Mr.Jaranawit Kongkaew : Mathermatics II
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
ใบงาน 17 รายวชา คณตศาสตร 2
c a
b
B
C A
c2 = a2 + b2
Page 75 of 93
Mr.Jaranawit Kongkaew : Mathermatics II
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
Page 76 of 93
Mr.Jaranawit Kongkaew : Mathermatics II
3. แบบฝกหด
I. จงหาคาตอไปน 1) sin 45 o sec 60 o + cot 60 o cos 30 o
.................................................................................................................................
.................................................................................................................................
.................................................................................................................................
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
.................................................................................................................................
.................................................................................................................................
.................................................................................................................................
.................................................................................................................................
5) oo
oo
30 cosec30 cot
30sin 45 cot+ cot 45 o – 2sin 30 o
.................................................................................................................................
.................................................................................................................................
.................................................................................................................................
.................................................................................................................................
6)
oo
2oo
30 sec60 cot
60sin 45tan - cot 60 o – 2cosec 30 o
.................................................................................................................................
.................................................................................................................................
.................................................................................................................................
.................................................................................................................................
Page 77 of 93
Mr.Jaranawit Kongkaew : Mathermatics II
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
Page 78 of 93
Mr.Jaranawit Kongkaew : Mathermatics II
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
.................................................................................................................................
.................................................................................................................................
.................................................................................................................................
.................................................................................................................................
.................................................................................................................................
2) tan A = 125
, c = 24 หนวย จงหา a
.................................................................................................................................
.................................................................................................................................
.................................................................................................................................
.................................................................................................................................
.................................................................................................................................
3) a = 12 หนวย , c = 5 หนวย จงหา sin C ................................................................................................................................. ................................................................................................................................. ................................................................................................................................. ................................................................................................................................. .................................................................................................................................
4) cos C = 32
, a = 10 หนวย จงหา b
.................................................................................................................................
.................................................................................................................................
.................................................................................................................................
.................................................................................................................................
.................................................................................................................................
5) a = 4 หนวย , c = 6 หนวย จงหา tan A ................................................................................................................................. ................................................................................................................................. ................................................................................................................................. ................................................................................................................................ .................................................................................................................................
Page 79 of 93
Mr.Jaranawit Kongkaew : Mathermatics II
1. จดประสงค 1.1 มความเขาใจเกยวกบฟงกชนตรโกณมตของมมรรอบจดศนยกลาง 1.2 สามารถหาคาฟงกชนตรโกณมตของมมรรอบจดศนยกลางได 1.3 สามารถค านวณหาคา sin cos tan cot และฟงกชนตรโกณมตอน ๆ ได
2. เนอหาโดยสงเขป
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
ใบงาน 18 รายวชา คณตศาสตร 2
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 (+)
Page 80 of 93
Mr.Jaranawit Kongkaew : Mathermatics II
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
3. แบบฝกหด
I. จงหาคาตอไปน 1) 2sin2 30 o – 4cos2 60 o + 3tan2 45o
.................................................................................................................................
.................................................................................................................................
.................................................................................................................................
2) o30 coseco60 sec3o45 4cot
o602tano30 23cos -o45 2sin
.................................................................................................................................
.................................................................................................................................
.................................................................................................................................
3) tan2 3
+ 2tan2 4
.................................................................................................................................
.................................................................................................................................
.................................................................................................................................
4) 2cosec2 4
- 3sec2 6
.................................................................................................................................
.................................................................................................................................
.................................................................................................................................
Page 81 of 93
Mr.Jaranawit Kongkaew : Mathermatics II
5) oo
oo
30 cosec30 cot
30sin 45 cot+ cot
6 – 2sin 30 o
.................................................................................................................................
.................................................................................................................................
.................................................................................................................................
.................................................................................................................................
6)
oo
2oo
30 sec330 cot
150sin 135tan - cot 60 o – 2cosec 270 o
.................................................................................................................................
.................................................................................................................................
.................................................................................................................................
.................................................................................................................................
7) cot2 4
+ cos2 3
- sin2 3
- 43
cot2 3
.................................................................................................................................
.................................................................................................................................
.................................................................................................................................
.................................................................................................................................
8) cos (-300 o ) – cot (-690 o ) ................................................................................................................................. ................................................................................................................................. ................................................................................................................................. .................................................................................................................................
9) sin 150 o + cot (-600 o ) ................................................................................................................................. ................................................................................................................................. ................................................................................................................................. .................................................................................................................................
10) cos 585 o + tan
32
.................................................................................................................................
.................................................................................................................................
.................................................................................................................................
.................................................................................................................................
Page 82 of 93
Mr.Jaranawit Kongkaew : Mathermatics II
11) ถา A cos60 sec45 cot
30 cosec60 cos60tan oo
ooo จงหาคา 6 tan2 A + 8 sin 2A cos A
.................................................................................................................................
.................................................................................................................................
.................................................................................................................................
.................................................................................................................................
.................................................................................................................................
12) ถา cos A = 0.8 แลวคาของ tan A มคาเทาใด ................................................................................................................................. ................................................................................................................................. ................................................................................................................................. ................................................................................................................................. .................................................................................................................................
13) ก าหนดให 13 cos A = 12 คาของ Ctan
1A cos
1 มคาเทาใด
.................................................................................................................................
.................................................................................................................................
.................................................................................................................................
.................................................................................................................................
.................................................................................................................................
14) 2 sin 30 o – 6 cot + 3 tan 45o ................................................................................................................................. ................................................................................................................................. .................................................................................................................................
15) o30 coseco60 sec32 cot4
o60tan o30 cos 3 -sin
co
.................................................................................................................................
.................................................................................................................................
.................................................................................................................................
16) cot2 3
+ 2cosec 4
.................................................................................................................................
.................................................................................................................................
.................................................................................................................................
Page 83 of 93
Mr.Jaranawit Kongkaew : Mathermatics II
1. จดประสงค
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
ใบงาน 19 รายวชา คณตศาสตร 2
Page 84 of 93
Mr.Jaranawit Kongkaew : Mathermatics II
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
Page 85 of 93
Mr.Jaranawit Kongkaew : Mathermatics II
3. แบบฝกหด
I. จงพสจนเอกลกษณ ตอไปน 1) sec4 A – 1 = 2 tan2 A + tan4 A
.................................................................................................................................
.................................................................................................................................
.................................................................................................................................
.................................................................................................................................
.................................................................................................................................
.................................................................................................................................
.................................................................................................................................
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 ................................................................................................................................. ................................................................................................................................. ................................................................................................................................. ................................................................................................................................. ................................................................................................................................. ................................................................................................................................. .................................................................................................................................
Page 86 of 93
Mr.Jaranawit Kongkaew : Mathermatics II
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
.................................................................................................................................
.................................................................................................................................
.................................................................................................................................
.................................................................................................................................
.................................................................................................................................
.................................................................................................................................
.................................................................................................................................
8) A secA cos
A cosecAsin
= 1
.................................................................................................................................
.................................................................................................................................
.................................................................................................................................
.................................................................................................................................
.................................................................................................................................
.............................................................................................................................. ...
.................................................................................................................................
Page 87 of 93
Mr.Jaranawit Kongkaew : Mathermatics II
9) A cosAsin AcosAsin
2Asin 21
133
.................................................................................................................................
.................................................................................................................................
.................................................................................................................................
.................................................................................................................................
.................................................................................................................................
.................................................................................................................................
.................................................................................................................................
10) (sec A – tan A)(sec A + tan A)2 = A cos
Asin 1
.................................................................................................................................
.................................................................................................................................
.................................................................................................................................
.................................................................................................................................
.................................................................................................................................
.................................................................................................................................
.................................................................................................................................
11) cos4 A – sin4 A = cos 2A ................................................................................................................................. ................................................................................................................................. ................................................................................................................................. ................................................................................................................................. ................................................................................................................................. ................................................................................................................................. .................................................................................................................................
12) (1 – cos2 A)(1 + cot2 A) = 1 ................................................................................................................................. ................................................................................................................................. ................................................................................................................................. ................................................................................................................................. ................................................................................................................................. ................................................................................................................................. .................................................................................................................................
Page 88 of 93
Mr.Jaranawit Kongkaew : Mathermatics II
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
ใบงาน 20 รายวชา คณตศาสตร 2
โดย C(h, k) เปนจดศนยกลาง r เปนรศม P(x, y) เปนจดใด ๆ บนเสนรอบ
วงกลม
r
C(h, k)
P(x, y)
Page 89 of 93
Mr.Jaranawit Kongkaew : Mathermatics II
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
6. เลตสเรกตม (L.R) ยาว a
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
6. เลตสเรกตม (L.R) ยาว a
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
Page 90 of 93
Mr.Jaranawit Kongkaew : Mathermatics II
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)
Page 91 of 93
Mr.Jaranawit Kongkaew : Mathermatics II
3. แบบฝกหด
I. จงตอบค าถามตอไปน 1) ก าหนดใหสมการวงกลมมจดศนยกลางอยท (1, 2) และมรศมยาว √ หนวย จงหาสมการวงกลม
...................................................................................................................................
...................................................................................................................................
...................................................................................................................................
...................................................................................................................................
...................................................................................................................................
...................................................................................................................................
2) จงหาจดศนยกลางและรศมของวงกลม x2 + y2 – 4x + 6y + 4 = 0 ................................................................................................................................... ................................................................................................................................... ................................................................................................................................... ................................................................................................................................... ................................................................................................................................... ...................................................................................................................................
3) จงหาจดยอด จดโฟกส สมการไดเรกทรกซ ความยาวของเสนเลตสเรกตมของสมการพาราโบลา y2 = 16x ................................................................................................................................... ................................................................................................................................... ................................................................................................................................... ................................................................................................................................... ................................................................................................................................... ................................................................................................................................... ...................................................................................................................................
4) จงหาสมการพาราโบลาทมจดยอดอยท (3, 3) และมสมการไดเรกทรกซคอ y = 1 ................................................................................................................................... ................................................................................................................................... ................................................................................................................................... ................................................................................................................................... ................................................................................................................................... ................................................................................................................................... ...................................................................................................................................
Page 92 of 93
Mr.Jaranawit Kongkaew : Mathermatics II
5) จงหาสมการวงรทมจดศนยกลางอยทจดก าเนดมจดยอดท (4,0)และจดโฟกส (3,0) ................................................................................................................................... ................................................................................................................................... ................................................................................................................................... ................................................................................................................................... ................................................................................................................................... ...................................................................................................................................
6) จงหาจดศนยกลาง จดโฟกส จดยอด ความยาวของเลตสเรกตมของสมการวงร
19
y4
x 22
...................................................................................................................................
...................................................................................................................................
...................................................................................................................................
...................................................................................................................................
...................................................................................................................................
...................................................................................................................................
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 ................................................................................................................................... ................................................................................................................................... ................................................................................................................................... ................................................................................................................................... ................................................................................................................................... ................................................................................................................................... ................................................................................................................................... ...................................................................................................................................