image
image | text
string | source_file
string | page_number
int64 |
|---|---|---|---|
ααααα½αα’αααα αα»ααααα·αααΈα‘αΆ
# ααα·ααα·ααααΆ
## ααααα·αααααα
## ααααΆααααΈ α‘α‘
ααααα»αααααααΆαααα
ααααΉαααααΆαααααα»ααααα·αα
ααααααΆα
α’ααΆα α‘α€α¨ αα αΆαα·ααΈ ααααααααααα ααααααα
|
[11] Math - High
| 1
|
|
| | |
|---|---|
| **αααααααααΆααα·αααα** | |
| ααα α’αα»α αααΆαααΈ | αααααααΈ ααΈ αααΌααΈαααα |
| ααα α±α ααΈαααΆ | αααααααΈ α’αα»α αα»αααΈ |
| ααα α
αΆαα αααΆααΆ | ααα ααα»α αα½ |
| ααα ααΌ αααα | |
| **α’αααααΆαα’ααααα** | αααααααΈ ααΆα ααΆααΈα |
| **αα·α
α·ααααα** | ααα ααα ααΆαα· |
| **α’ααααααααα** | ααα α‘α»α αα»ααα | ααα αααα αα½α |
| **α’ααααα
ααΆααααα** | ααα ααα αααΆααΈ |
| **α’αααα―αααα** | ααα α’αα»α ααΉαααααα»α |
| **αααααααααΆααα·αα·ααα** | ααα αα α αα | ααα αα»α ααΈ |
| | ααα αααΌ αα | ααα α αα α
ααααΆ |
ααΆαααα½αααΆαα’αα»ααααΆαα±ααααααα»αααααααΆαααΈ ααααα½αα’αααα αα»ααα αα·αααΈα‘αΆ ααΆααααααΆαααα α£α α α‘ α’αα.αααα. α
α»αααααααΈ α£α αα ααααΌ ααααΆα α’α α α¨ ααΎααααΈααααΎααααΆαααα
ααΆαααΆααΆααα α
**α αΆαααα
ααααααααα
ααα**
αααααΆαα·αααα· Β©
**ααααΉαααααΆαααααα»ααααα·αα
ααααααΆα**
ααααα»αααααΎαααΈα‘α’ ααααΆαα’α α’α α
ααα½αα’α₯ α α α α
αααΆαα
ISBN 9-789-995-000-691
|
[11] Math - High
| 2
|
|
## α’αΆαααααααΆ
ααααα
ααα·ααα·ααααΆααααα·ααααααααααΆααααΈ 11 αα½αααΆααααααΌα ααααα·ααααΆα’αααΈ αααααΈααα·αα’αα»ααΆααα½αααα·ααα·ααααΆ α’αα»ααααα’α·α
αααααΌαααααααααα·αα’αα»ααααααααΆααΈα α’αα»ααααααααΈαααααΆααα αααααααΈααα’αα ααααΌααΆα ααααΈααααααααα»ααααα·α
αα·αααα·α
ααααααα»αααα α αα·ααααααααααΎαααΎαααααα·ααα·ααααΆααααα·ααααααααααΌααααααα·ααα·ααααΆααααα·αααΌαααααΆαααΎααααΈααααααααααα α
ααΆααααα
αααααααα
αααα»αααααα
ααα ααΆαααααααααΌα
ααΆαααααα α
- αααααααΈαα½αα ααΆααααααΆααα’αααΈααααα»ααααα
αααΆααααΆαα
- ααααΉαααΆααααααααΈαα½αα ααααΎαα
ααααΈα§ααΆα ααααααα»αααΈαααΆαααααα
- ααΆαααα αΆααααααα·ααααα·αααααΆαααααααΉαα
ααααααΉααα·ααα
- αα
α
α»ααααααααααααΌαααΈαα½αα ααΆααααααααααααΉαααααααααααΆααα±αααα·αααα
αα
αΆαααΌαα’αααΈαααααΆαααααα½α
- αααααααΈαα½αα ααΆαααα αΆαααα·αααα αΆαααααα½ααααααΆααα±αααα·αααα’αα»ααααααΎααααΈαααααΉαα
ααααααΉα
- αα
α
α»ααααααααααααα
ααα ααΆαααααααααΆααααααααΉααα·αααΆαα
ααααΎαααα αΆαααααααΆααα±αααα·ααααααααααααΆαα α
ααααα
αα·ααααΆαααααΆαααΆαα
αΌααα½αααΈ :
- αααααααΌ α’αααααααΌαααααΆαααΆααααααα½ααααααααααα
- αααααααΌ α’αααααααΌαααααΆαααΆαααααααααααααΌαααααα
αα·ααααΆααα
- αααααααααΆαααΆαααααααααααΆααα½ααααα½ααα·αα·ααααα·ααααααααααα α
ααΎααααΈα±ααααααα
ααα ααΆααααααα’αααααΎα ααΎααααα»αααΉααααα
αΆαααα½αααΆααααΆααα·αααααα·ααααααα’ααααααα’αααΈααααΆαααααααααΌ α’αααααααΌ αα·ααααα·ααα·αααα’αααααααΎααααΆααααααα
ααα·ααα·ααααΆααααα·ααααααααααααααααΈααΈαααΆα α
**αααααααααΆααα·αααα**
|
[11] Math - High
| 3
|
|
# αααααΈα’ααααα
## ααα·ααα·ααααΆααααα·αααααα
| | ααααα |
| :--- | :--- |
| **ααααΌαααΈ 1 : ααααΈααα·αα’αα»ααΆααα½αααα·ααα·ααααΆ** | 1 |
| 1. ααααΌααα½ααααααΈααααααα | 2 |
| 2. ααααΆαααααααα½ααααααΈα | 16 |
| 3. αα·α
αΆαα’αα»ααΆααα½α | 24 |
| **ααααΌαααΈ 2 : α’αα»ααααα’α·α
αααααΌαααααααααα·αα’αα»ααααααααΆααΈα** | 35 |
| 1. α’αα»ααααα’α·α
αααααΌαααααααα | 36 |
| 2. α’αα»ααααααααΆααΈα | 46 |
| **ααααΌαααΈ 3 : αααΈααΆααα·ααα·αααΈααΆαααααΈαααααΆααα** | 57 |
| 1. αααΈααΆααα·ααα·αααΈααΆαααααΈαααααΆααα | 58 |
| **ααααΌαααΈ 4 : αααααααΈααα’ααα** | 73 |
| 1. αααααααΈααα’ααα | 74 |
| **ααααΌαααΈ 5 : ααΈααΈααα·αααααΈαα** | 103 |
| 1. α’αα»ααααααααααΈααα
αααααααΈααΆααα·ααα·αααΈααΆα | 104 |
| **ααααΌαααΈ 6 : ααααΌααΆα** | 117 |
| 1. ααααΌααΆαααΆααααααααα | 118 |
| **ααααΌαααΈ 7 : α
ααα½ααα»ααααα·α
** | 135 |
| 1. α
ααα½ααα»ααααα·α
ααααααααΈαααα·α | 136 |
| 2. α
ααα½ααα»ααααα·α
ααααααααααΈαααααΆααα | 144 |
| 3. ααααααα»αααΈ n αα·αα«αααΈ n ααα
ααα½ααα»ααααα·α
| 164 |
| 4. α’αα»ααααααα
ααα½ααα»ααααα·α
αααα»αααααΈααΆααα | 174 |
| **ααααΌαααΈ 8 : ααα·α
ααααααα»αααα ** | 189 |
| 1. ααα·α
ααααααα»αααα | 190 |
| 2. αααΈααΆααααααΆαααα·ααααΈααΆαααααααααα»αααα | 216 |
| **α
ααααΎαααααΌα** | 235 |
| **αααΆαα»αααα** | 247 |
|
[11] Math - High
| 4
|
|
### ααααΌα α‘ αααααααΈ α‘
# ααααΌα 1
# ααααΈααα·αα’αα»ααΆααα½αααα·ααα·ααααΆ
![pyramids.png: Photo of the great pyramids of Giza]
- **ααααΌααα½ααααααΈααααααα**
- **ααααΆαααααααα½αααααααΈα**
- **αα·α
αΆαα’αα»ααΆααα½αααα·ααα·ααααΆ**
ααΆααα·ααααΆαααααΈααααααΆαα±ααααα·ααα·ααααΆααΆααααααΆαααΆαααΈαα
ααααΎαα‘αΎαααΈαα½ααααααα
αα½ααααα ααΆααααααααΌααα½αααααααΈαα’αΆα
α±αααααααααΆαααΌαααΆααΆαααα’αα»ααααααααΈαααααΆααααα·αα’αα»ααααααααΆααΈααα·αααααΉαααααα»αααααααα’αΆα
α±ααααααααΆααΆαααΌααααααααα e αα·α Ο ααααα α
$$
e = 1 + \frac{1}{1!} + \frac{1}{2!} + \frac{1}{3!} + \frac{1}{4!} + \frac{1}{5!} + \frac{1}{6!} + \frac{1}{7!} + \dots
$$
$$
\frac{\pi}{4} = 1 - \frac{1}{3} + \frac{1}{5} - \frac{1}{7} + \frac{1}{9} - \frac{1}{11} + \frac{1}{13} - \frac{1}{15} + \dots
$$
1
|
[11] Math - High
| 5
|
|
## αααααααΈ 1 ααααΌααα½ααααααΈααααααα
### 1. ααααααααΆααααΌα
α
αααααααααΈααααααα αααα»αα’αΆα
ααααΎααΌαααααααααΌααααααααΈαααααααα α¬αααααΈαααααΈααΆαααααααααΆααααΌαααΆααΆαα‘αΎα αααα»αααααΈαααααααΌαααααΎαα·ααΈαααααααα
ααΆααααααααααααααΈα α
#### ααααα»αααα
- ααααΆααααΌααα½αααααααΈα
- α
ααααααΎαα·αα·ααααααααΆ β αααααΆααααααΌααααααααΈα
- ααααααα½ααΈ n ααΆααααααααΈαααααΆαα 1 αα·αααααΆαα 2
#### α§ααΆα ααα 1
α
ααααααααΌα $S = 1^2 + 2^2 + 3^2 + \dots + n^2$ α
ααα’αΆα
ααααΆααααΌαααααααααααΎααααΆα
$(k+1)^3 - k^3 = 3k^2 + 3k + 1$ αααα±αα k αααααααααΈ 1 ααα n
$2^3 - 1^3 = 3 \times 1^2 + 3 \times 1 + 1$
$3^3 - 2^3 = 3 \times 2^2 + 3 \times 2 + 1$
$4^3 - 3^3 = 3 \times 3^2 + 3 \times 3 + 1$
...
$n^3 - (n-1)^3 = 3 \times (n-1)^2 + 3 \times (n-1) + 1$
$(n+1)^3 - n^3 = 3 \times n^2 + 3 \times n + 1$
ααΌαα’ααααα·αα’ααα ααααΆα
$(n+1)^3 - 1^3 = 3 \times (1^2 + 2^2 + 3^2 + \dots + n^2) + 3 \times (1+2+3+\dots+n) + (1+1+1+\dots+1)$
$= 3 \times S + 3 \times \frac{n(n+1)}{2} + n$ α
ααααΆ S ααααΆα $1^2 + 2^2 + 3^2 + \dots + n^2 = \frac{n(n+1)(2n+1)}{6}$ α
#### α§ααΆα ααα 2
α
ααααααααΌα $S = \frac{1}{1 \cdot 2} + \frac{1}{2 \cdot 3} + \frac{1}{3 \cdot 4} + \dots + \frac{1}{n \cdot (n+1)}$
ααα’αΆα
ααααΆααααΌαααααααααααΎααααΆα
$\frac{1}{k \cdot (k+1)} = \frac{1}{k} - \frac{1}{k+1}$ αααα±αα k αααααααααΈ 1 ααα n
2
|
[11] Math - High
| 6
|
|
### ααααΌα α‘ αααααααΈ α‘
$\frac{1}{1 \cdot 2} = \frac{1}{1} - \frac{1}{2}$
$\frac{1}{2 \cdot 3} = \frac{1}{2} - \frac{1}{3}$
$\frac{1}{3 \cdot 4} = \frac{1}{3} - \frac{1}{4}$
...
$\frac{1}{(n-1) \cdot n} = \frac{1}{n-1} - \frac{1}{n}$
$\frac{1}{n \cdot (n+1)} = \frac{1}{n} - \frac{1}{n+1}$
ααΌαα’ααααα·αα’ααα ααααΆα $S = 1 - \frac{1}{n+1}$ α
α ααα»ααα $\frac{1}{1 \cdot 2} + \frac{1}{2 \cdot 3} + \frac{1}{3 \cdot 4} + \dots + \frac{1}{n \cdot (n+1)} = 1 - \frac{1}{n+1} = \frac{n}{n+1}$ α
#### ααα αΆααααααΌ
ααααΆααααΌα $S = 1^3 + 2^3 + 3^3 + \dots + n^3$ α
**α
ααααΎα**
ααα’αΆα
ααααΆααααΌαααααααααααΎααααΆα
$(k+1)^4 - k^4 = 4k^3 + 6k^2 + 4k + 1$ αααα±αα k αααααααααΈ 1 ααα n
$2^4 - 1^4 = 4 \times 1^3 + 6 \times 1^2 + 4 \times 1 + 1$
$3^4 - 2^4 = 4 \times 2^3 + 6 \times 2^2 + 4 \times 2 + 1$
$4^4 - 3^4 = 4 \times 3^3 + 6 \times 3^2 + 4 \times 3 + 1$
...
$n^4 - (n-1)^4 = 4 \times (n-1)^3 + 6 \times (n-1)^2 + 4 \times (n-1) + 1$
$(n+1)^4 - n^4 = 4 \times n^3 + 6 \times n^2 + 4 \times n + 1$
ααΌαα’ααααα·αα’ααα ααααΆα
$(n+1)^4 - 1^4 = 4 \times (1^3 + 2^3 + 3^3 + \dots + n^3) + 6 \times (1^2 + 2^2 + 3^2 + \dots + n^2)$
$+ 4 \times (1+2+3+\dots+n) + (1+1+1+\dots+1)$
$= 4S + 6 \times (1^2 + 2^2 + 3^2 + \dots + n^2) + 4 \times (1+2+3+\dots+n) + (1+1+1+\dots+1)$
ααααΆ S ααααΆα $1^3 + 2^3 + 3^3 + \dots + n^3 = \frac{n^2(n+1)^2}{4} = [\frac{n(n+1)}{2}]^2$ α
#### ααααα·ααααα·
ααααΆ $S = 12^3 + 13^3 + 14^3 + \dots + 50^3$ α
ααααΆ $S = \frac{1}{1 \cdot 2 \cdot 3} + \frac{1}{2 \cdot 3 \cdot 4} + \frac{1}{3 \cdot 4 \cdot 5} + \dots + \frac{1}{n \cdot (n+1) \cdot (n+2)}$ α
3
|
[11] Math - High
| 7
|
|
### 2. ααααΆααααΌαααΆαααααΆαααααΌ
α
αααααααααΈααααααΆααα½ααΆα
ααα½αααα ααα’αΆα
ααααΆααααΌαααΆααΆαααΆαααααααααααΆαααααΌ α
$1 = 1 = 1^2$
$1+3 = 4 = 2^2$
$1+3+5 = 9 = 3^2$
$1+3+5+7 = 16 = 4^2$
α ααα»ααα $1+3+5+\dots+(2n-1) = n^2$ α
#### ααα αΆααααααΌ
ααααΆααααΌα n αα½ααα
ααα½αααΌ $S = 2+4+6+\dots+2n$ α
**α
ααααΎα**
$2 = 2 = 1 \times 2$
$2+4 = 6 = 2 \times 3$
$2+4+6 = 12 = 3 \times 4$
$2+4+6+8 = 20 = 4 \times 5$
α ααα»ααα $2+4+6+\dots+2n = n \cdot (n+1)$ α
#### ααααα·ααααα·
ααα±ααααααΆαααααΌαααααααΈαααΆαααααα α
$1 = 1$
$1+7 = 8$
$1+7+19 = 27$
$1+7+19+37 = 64$
$1+7+19+37+61 = 125$
ααααΆααααΌααααααααΈαααααααααΆα 10 αα½ α
4
|
[11] Math - High
| 8
|
|
### ααααΌα α‘ αααααααΈ α‘
### 3. αα·αα·ααααααααΆ β αααααΆααααααΌααααααααΈα
#### 3.1 αααααΆα β
αααα»αααΆααααααααααΌααα½αααααααΈα $U_1, U_2, U_3, \dots, U_n$ ααααααΎαα·αα·ααααααααΆ β α’αΆαααΆ ααα·α
αααΆ αααααΆααααΆαααααΌα n αα½αααααααΈα α αααααααααααα $\sum_{k=1}^{n} U_k = U_1 + U_2 + U_3 + \dots + U_n$ α αααα»αααα k αααααααααΈ 1, 2, 3, ... αα αΌαααα n α
**α§ααΆα ααα**
$1+2+3+\dots+n = \sum_{k=1}^{n} k$
$1^2+2^2+3^2+\dots+n^2 = \sum_{k=1}^{n} k^2$ α
#### ααα αΆααααααΌ 1
αααααααααΌαααΆαααααααααααααΎαα·αα·ααααααααΆ β :
α. $1^3+2^3+3^3+\dots+n^3$
α. $2+4+6+8+\dots+100$
α. $\frac{1}{2} - \frac{1}{3} + \frac{1}{4} - \frac{1}{5} + \dots + \frac{1}{50}$
**α
ααααΎα**
α. $1^3+2^3+3^3+\dots+n^3 = \sum_{k=1}^{n} k^3$ α
α. $2+4+6+8+\dots+100 = 2 \times 1 + 2 \times 2 + 2 \times 3 + \dots + 2 \times 50$
ααααΌααααααΈααααααΆααα½ααΌαα
2n α αΎα n αααααααααΈ 1 ααα 50
ααααΆα $2+4+6+8+\dots+100 = \sum_{n=1}^{50} 2n$ α
α. $\frac{1}{2} - \frac{1}{3} + \frac{1}{4} - \frac{1}{5} + \dots + \frac{1}{50}$ ααααΌαααα n αααααααααΈ 2 ααα 50 α
αα½ααΌαα
αααααααΈα $\frac{(-1)^n}{n}$ α
ααααΆα $\frac{1}{2} - \frac{1}{3} + \frac{1}{4} - \frac{1}{5} + \dots + \frac{1}{50} = \sum_{n=2}^{50} \frac{(-1)^n}{n}$ α
5
|
[11] Math - High
| 9
|
|
#### ααα αΆααααααΌ 2
αααααααααααα½ααΆααα’ααααααααΌαααααα·αααααΎαα·αα·ααααααααΆ β
α. $\sum_{k=1}^{6} 2$
α. $\sum_{n=2}^{5} (2n+1)$
α. $\sum_{j=1}^{5} j(j+1)$ α
**α
ααααΎα**
α. $\sum_{k=1}^{6} 2 = 2+2+2+2+2+2$ α
α. $\sum_{n=2}^{5} (2n+1) = 5+7+9+11$ α
α. $\sum_{j=1}^{5} j(j+1) = 1 \times 2 + 2 \times 3 + 3 \times 4 + 4 \times 5 + 5 \times 6$ α
#### ααααα·ααααα·
αααααααααΌαααΆαααααααααααααΎαα·αα·ααααααααΆ β :
α. $2+5+8+\dots+(3n-1)$
α. $3+6+12+\dots+3(2)^{n-1}$
α. $1+\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+\dots+\frac{1}{100}$ α
6
|
[11] Math - High
| 10
|
|
### ααααΌα α‘ αααααααΈ α‘
#### 3.2 ααααΌαααΆαααααααααΌα
ααΆαααααα α
- α. $\sum_{k=1}^{n} c = nc$
- α. $\sum_{k=1}^{n} ca_k = c \sum_{k=1}^{n} a_k$
- α. $\sum_{k=1}^{n} (a_k+b_k) = \sum_{k=1}^{n} a_k + \sum_{k=1}^{n} b_k$
- α. $\sum_{k=1}^{n} (a_k-b_k) = \sum_{k=1}^{n} a_k - \sum_{k=1}^{n} b_k$
- α. $\sum_{k=1}^{n} (a_k+b_k)^2 = \sum_{k=1}^{n} a_k^2 + 2\sum_{k=1}^{n} a_k b_k + \sum_{k=1}^{n} b_k^2$ α
##### αααααΆααααααΆαα
- α. $\sum_{k=1}^{n} c = c+c+c+\dots+c = nc$ α
- α. $\sum_{k=1}^{n} ca_k = ca_1+ca_2+ca_3+\dots+ca_n = c(a_1+a_2+a_3+\dots+a_n) = c \sum_{k=1}^{n} a_k$ α
- α. $\sum_{k=1}^{n} (a_k+b_k) = (a_1+b_1)+(a_2+b_2)+\dots+(a_n+b_n)$
$= (a_1+a_2+\dots+a_n) + (b_1+b_2+\dots+b_n) = \sum_{k=1}^{n} a_k + \sum_{k=1}^{n} b_k$ α
- α. $\sum_{k=1}^{n} (a_k-b_k) = (a_1-b_1)+(a_2-b_2)+(a_3-b_3)+\dots+(a_n-b_n)$
$= (a_1+a_2+a_3+\dots+a_n) - (b_1+b_2+b_3+\dots+b_n)$
$= \sum_{k=1}^{n} a_k - \sum_{k=1}^{n} b_k$ α
- α. $\sum_{k=1}^{n} (a_k+b_k)^2 = \sum_{k=1}^{n} (a_k^2+2a_k b_k+b_k^2) = \sum_{k=1}^{n} a_k^2 + \sum_{k=1}^{n} 2a_k b_k + \sum_{k=1}^{n} b_k^2$
$= \sum_{k=1}^{n} a_k^2 + 2\sum_{k=1}^{n} a_k b_k + \sum_{k=1}^{n} b_k^2$ α
7
|
[11] Math - High
| 11
|
|
#### ααα αΆααααααΌ
ααααΆ
α. $\sum_{k=1}^{15} (4k+3)$
α. $\sum_{k=1}^{20} (k+3)k$ α
**α
ααααΎα**
α. $\sum_{k=1}^{15} (4k+3) = \sum_{k=1}^{15} 4k + \sum_{k=1}^{15} 3 = 4\sum_{k=1}^{15} k + \sum_{k=1}^{15} 3$
$= 4(1+2+3+\dots+15) + 15 \times 3 = 4 \times \frac{(1+15) \times 15}{2} + 45 = 2 \times 16 \times 15 + 45 = 525$ α
α. $\sum_{k=1}^{20} (k+3)k = \sum_{k=1}^{20} (k^2+3k) = \sum_{k=1}^{20} k^2 + 3\sum_{k=1}^{20} k$
$= \frac{(20) \times (20+1) \times (2 \times 20+1)}{6} + 3\frac{(1+20) \times 20}{2}$
$= 10 \times 7 \times 41 + 3 \times 21 \times 10 = 2870 + 630 = 3500$ α
#### ααααα·ααααα·
ααααΆ
- α. $\sum_{k=1}^{20} (3k+1)$
- α. $\sum_{k=1}^{7} k^2$
- α. $\sum_{k=4}^{14} (4k-3)$
- α. $\sum_{k=1}^{n} (3+k)^2$
- α. $\sum_{k=1}^{30} k(k+1)$ α
8
|
[11] Math - High
| 12
|
|
### ααααΌα α‘ αααααααΈ α‘
### 4. ααααααααααα½ααΈ n ααΆααααααα½ααααααΈα
ααα’αΆα
ααααααα½ααΈ n αααααααααΈα $(a_n)$ ααααα»ααααααΆαααααΈαααααααα α¬ααΆαααααΈαααααΈααΆαααααΆαααααααΌα
ααΆαααααα α
#### 4.1 αααααα½ααααΆααααΈ 1 ααααααΈα
##### α§ααΆα ααα 1
ααααααα½ααΈ n αααααααΈα $1, 3, 7, 13, 21, 31, \dots$ α
**α
ααααΎα**
ααααααααααΎαααΆαααααΈαααααα·ααααααΆαααααΈαααααααα α αΎααααα·ααααααΆαααααΈαααααΈααΆααα α αααααααααΌαααααΎαααααααα½αααααααααΆααααααΆαααααααΈα $1, 3, 7, 13, 21, 31, \dots$ ααααΆαααααΆαααααΌααΌα
ααΆ
n : 1 2 3 4 5 6...
αααααΈα $(a_n)$ : 1 3 7 13 21 31...
αααααα½ααααΆααααΈ 1 αααααααΈα : 2 4 6 8 10...
αααααα½αααααααΈαα’αΆα
αα½αα±ααααααααααΆαααααΌ αααα’αΆα
ααααααα½ααΌαα
αααααααΈαααΆαααΌα
ααΆ
$a_2 - a_1 = 3-1 = 2 \times 1$
$a_3 - a_2 = 7-3 = 4 = 2 \times 2$
$a_4 - a_3 = 13-7 = 6 = 2 \times 3$
...
$a_{n-1} - a_{n-2} = 2 \times (n-2)$
$a_n - a_{n-1} = 2 \times (n-1)$ α
ααΌαα’ααααα·αα’αααααΆααα’αα ααααΆα
$a_n - a_1 = 2 \times 1 + 2 \times 2 + 2 \times 3 + \dots + 2 \times (n-2) + 2 \times (n-1)$
$a_n = 2 \times 1 + 2 \times 2 + 2 \times 3 + \dots + 2 \times (n-2) + 2 \times (n-1) + 1$
$= 2 \times [1+2+3+\dots+(n-2)+(n-1)] + 1 = 2 \times \frac{[1+(n-1)](n-1)}{2} + 1$
$= n(n-1)+1 = n^2 - n + 1$ α
ααΌα
ααα αα½ααΈ n αααααααΈα $1, 3, 7, 13, 21, 31, \dots$ αααααααα $a_n = n^2 - n + 1$ α
**ααΆααΌαα
**
ααααΆααααααΈα $(a_n): a_1, a_2, a_3, \dots, a_n$ α αΎα $b_1 = a_2 - a_1$, $b_2 = a_3 - a_2$, $b_3 = a_4 - a_3, \dots, b_{n-1} = a_n - a_{n-1}$ α αααααΈα $(b_n): b_1, b_2, b_3, \dots, b_{n-1}$ α α
ααΆαααααα½ααααΆααααΈ 1 αααααααΈα $(a_n)$ α
9
|
[11] Math - High
| 13
|
|
##### α§ααΆα ααα 2
αααα αΆαααΆ $a_n = a_1 + \sum_{k=1}^{n-1} b_k$ α
αααα $n \ge 2$ α
$a_1 \quad a_2 \quad a_3 \quad a_4 \quad \dots \quad a_{n-1} \quad a_n$
$\quad b_1 \quad b_2 \quad b_3 \quad b_4 \quad \dots \quad b_{n-2} \quad b_{n-1}$
$a_2 - a_1 = b_1$
$a_3 - a_2 = b_2$
$a_4 - a_3 = b_3$
+ ...
$a_{n-1} - a_{n-2} = b_{n-2}$
$a_n - a_{n-1} = b_{n-1}$
$a_n - a_1 = b_1 + b_2 + b_3 + \dots + b_{n-1}$
α
αααα $n \ge 2$ ααααΆα
$a_n = a_1 + (b_1 + b_2 + b_3 + \dots + b_{n-1}) = a_1 + \sum_{k=1}^{n-1} b_k$ α
α
αααα $n=1$ ααααΆα $b_{1-1} = b_0$ αα·αααΆα α
ααΌα
ααα $a_n = a_1 + \sum_{k=1}^{n-1} b_k$ α
αααα $n \ge 2$ α
**ααΆααΌαα
**
ααααΆααααααΈα $(a_n)$ αα·ααααααΈα $(b_n)$ ααααααααααα $b_n = a_{n+1} - a_n$, $n=1, 2, 3, \dots$
αα½ααΈ n αααααααΈα $(a_n)$ αααααααα $a_n = a_1 + \sum_{k=1}^{n-1} b_k$ α
αααα $n \ge 2$ α
#### ααα αΆααααααΌ 1
ααααααα½ααΈ n αααααααΈα $2, 4, 8, 14, 22, 32, \dots$ α
**α
ααααΎα**
ααΆα $a_n$ ααΆαα½ααΈ n αααααααΈααααααα±αα α αααααΈα $(b_n)$ ααΆαααααα½ααααΆααααΈ 1 αααααααΈα $(a_n)$ αααααΆααα½ $b_n$ αααααααα $b_n = a_{n+1} - a_n$
αααααΈα $(b_n): 2, 4, 6, 8, \dots$
αααααΈα $(b_n)$ ααΆαααααΈαααααααααααααΆααα½ααΈ 1 ααααΎααΉα 2 αα·ααααααα½αααααΎααΉα 2 ααα
$b_n = 2 + 2(n-1) = 2n$ α
10
|
[11] Math - High
| 14
|
|
### ααααΌα α‘ αααααααΈ α‘
α
αααα $n \ge 2$ ααααΆα
$a_n = a_1 + \sum_{k=1}^{n-1} b_k = 2 + \sum_{k=1}^{n-1} 2k = 2 + 2 \cdot \frac{1}{2}n(n-1) = 2 + n^2 - n = n^2 - n + 2$
ααα $a_n = n^2 - n + 2$ α
α
αααα $n=1$, $a_1 = 1^2 - 1 + 2 = 2$ αα·α α
ααΌα
ααα αααααΈα $(a_n)$ ααΆααα½ααΈ n αααααααα $a_n = n^2 - n + 2$ α
#### ααα αΆααααααΌ 2
α. ααααααα½ααΈ n αααααααΈα $1, 2, 5, 10, 17, \dots$ α
α. ααααΆααααΌα n αα½ααααΌααααααααΈαααα α
**α
ααααΎα**
α. ααααααα½ααΈ n αααααααΈα $(a_n): 1, 2, 5, 10, 17, \dots$
ααΆα $a_n$ ααΆαα½ααΈ n αααααααΈααααααα±αα $(a_n)$ α αααααΈα $(b_n)$ ααΆαααααα½ααααΆααααΈ 1 αααααααΈα $(a_n)$ αααααΆααα½ $b_n$ αααααααα $b_n = a_{n+1} - a_n$ α αααααΈα $(b_n): 1, 3, 5, 7, \dots$ α αααααΈα $(b_n)$ ααΆαααααΈαααααααααααααΆααα½ααΈ 1 ααααΎααΉα 1 αα·ααααααα½αααααΎααΉα 2 ααα $b_n = 1 + 2(n-1) = 2n-1$ α
α
αααα $n \ge 2$ ααααΆα $a_n = a_1 + \sum_{k=1}^{n-1} b_k = 1 + \sum_{k=1}^{n-1} (2k-1) = 1 + 2\sum_{k=1}^{n-1} k - (n-1)$
$= 1 + 2 \cdot \frac{1}{2}n(n-1) - (n-1) = 1 + n^2 - n - n + 1 = 2 - 2n + n^2$
ααα $a_n = 2 - 2n + n^2$ α
α
αααα $n=1$ ααααΆα $a_1 = 2 - 2 + 1 = 1$ αα·α α ααΌα
ααα $a_n = 2 - 2n + n^2$ α
α. ααααΆααααΌα n αα½ααααΌααααααααΈα $(a_n)$
ααΆα $S_n$ ααΆααααΌααααααααΈα $(a_n)$ ααΊ
$S_n = a_1 + a_2 + a_3 + \dots + a_n = \sum_{k=1}^{n} (2 - 2k + k^2)$
$= 2n - 2\sum_{k=1}^{n} k + \sum_{k=1}^{n} k^2 = 2n - 2 \cdot \frac{1}{2}n(n+1) + \frac{1}{6}n(n+1)(2n+1)$
$= \frac{12n}{6} - \frac{6}{6}n(n+1) + \frac{1}{6}n(n+1)(2n+1) = \frac{1}{6}n(12 - 6n - 6 + 2n^2 + n + 2n + 1)$
$= \frac{1}{6}n(7 - 3n + 2n^2)$
ααΌα
ααα $S_n = \frac{1}{6}n(2n^2 - 3n + 7)$ α
11
|
[11] Math - High
| 15
|
|
#### ααααα·ααααα·
1. α. ααααααα½ααΈ n αααααααΈα $4, 5, 7, 10, 14, \dots$ α
α. ααααΆααααΌα n αα½ααααΌααααααααΈαααα α
2. α. ααααααα½ααΈ n αααααααΈα $3, 5, 8, 12, 17, 23, \dots$ α
α. ααααΆααααΌα n αα½ααααΌααααααααΈαααα α
#### 4.2 αααααα½ααααΆααααΈ 2 ααααααΈα
ααΎαα·αααΆαααααΈα $(b_n)$ αα
αααα·αα’αΆα
ααααααα½ααΈ n αααααααΈα $(a_n)$ ααΆαααα αααααααααΌαααααΎααααααααΆαααα½αααααααΈα $(b_n)$ ααααααα αααααα α
ααΆ αααααα½ααααΆααααΈ 2 αααααααΈα α
##### α§ααΆα ααα
ααααααα½ααΈ n αααααααΈα $(a_n): 1, 2, 6, 15, 31, 56, \dots$ α
**ααααααΈ 1**
n : 1 2 3 4 5 6...
αααααΈα $(a_n)$ : 1 2 6 15 31 56... α
αααααα½ααααΆααααΈ 1: 1 4 9 16 25...
ααΆα $a_n$ ααΆαα½ααΈ n αααααααΈα $(a_n)$ α αααααΈα $(b_n)$ ααΆαααααα½ααααΆααααΈ 1 αααααααΈα $(a_n)$ α
αααααΈα $(b_n): 1, 4, 9, 16, 25, \dots$ ααα $b_n = n^2$ α
α
αααα $n \ge 2$ ααααΆα $a_n = a_1 + \sum_{k=1}^{n-1} k^2 = 1 + \frac{1}{6}(n-1)n(2n-1)$
ααα $a_n = \frac{1}{6}(2n-1)(n-1)n + 1$ α
α
αααα $n=1$, $a_1 = 1$ αα·α α
ααΌα
ααα $a_n = \frac{1}{6}(2n-1)(n-1)n + 1$ α
**ααααααΈ 2**
n: 1 2 3 4 5 6...
αααααΈα $(a_n)$: 1 2 6 15 31 56...
αααααα½ααααΆααααΈ 1: 1 4 9 16 25...
αααααα½ααααΆααααΈ 2: 3 5 7 9
ααΆα $a_n$ ααΆαα½ααΈ n αααααααΈα $(a_n)$ α αααααΈα $(b_n)$ ααΆαααααα½ααααΆααααΈ 1 αααααααΈα $(a_n)$ αααααΆααα½ n αααααααΈαααΊ $b_n = a_{n+1} - a_n$ α ααΆα $c_n$ ααΆαααααα½αααααααΈα $(b_n)$ α α
ααΆ αααααα½ααααΆααααΈ 2 α ααααΆααααααΈα $(c_n): 3, 5, 7, 9, \dots$ α αααααΈα $(c_n)$ ααΆαααααΈαααααααααααααΆααα½ααΈ 1 ααααΎααΉα 3 αα·ααααααα½αααααΎααΉα 2 αα·α $c_n = 3 + 2(n-1) = 2n+1$ α
12
|
[11] Math - High
| 16
|
|
### ααααΌα α‘ αααααααΈ α‘
α
αααα $n \ge 2$ ααααΆα $b_n = b_1 + \sum_{k=1}^{n-1} c_k = b_1 + \sum_{k=1}^{n-1} (2k+1)$
$= 1 + 2\sum_{k=1}^{n-1} k + (n-1) = 1 + 2 \cdot \frac{1}{2}(n-1)n = n^2$ ααα $b_n = n^2$ α
α
αααα $n=1$, $b_1 = 1$ αα·α α ααΌα
ααα $b_n = n^2$ α
α
αααα $n \ge 2$ ααααΆα $a_n = a_1 + \sum_{k=1}^{n-1} b_k = 1 + \sum_{k=1}^{n-1} k^2 = 1 + \frac{1}{6}(n-1)n(2n-1)$
ααα $a_n = \frac{1}{6}(2n-1)(n-1)n + 1$ α
α
αααα $n=1$, $a_1 = 1$ αα·α
ααΌα
ααα $a_n = \frac{1}{6}(2n-1)(n-1)n + 1$ α
**ααΆααΌαα
**
ααααΆααααααΈα $(a_n)$ α
αααααΈα $(b_n)$ αααααααα $b_n = a_{n+1} - a_n$, $n=1, 2, 3, \dots$ α α
ααΆαααααα½ααααΆααααΈ 1 αααααααΈα $(a_n)$ αααααΆααα½ααΈ n αααααααα $a_n = a_1 + \sum_{k=1}^{n-1} b_k$ α
αααα $n \ge 2$ α
αααααΈα $(c_n)$ ααΆαααααα½ααααΆααααΈ 2 αααααααΈα $(a_n)$ ααΊααΆαααααα½ααααΆααααΈ 1 αααααααΈα $(b_n)$ αααααααα $c_n = b_{n+1} - b_n$, $n=1, 2, 3, \dots$ αααααΆααα½ααΈ n αααααααΈα $(b_n)$ αααααααα $b_n = b_1 + \sum_{k=1}^{n-1} c_k$ α
αααα $n \ge 2$ α
#### ααα αΆααααααΌ
α. ααααααα½ααΈ n αααααααΈα $4, 18, 48, 100, 180, 294, \dots$ α
α. ααααΆααααΌα n αα½ααααΌααααααααΈαααα α
**α
ααααΎα**
α. ααααααα½ααΈ n αααααααΈα $4, 18, 48, 100, 180, 294, \dots$
ααΆα $a_n$ ααΆαα½ααΈ n αααααααΈααααααα±αα $(a_n)$
αααααΈα $(b_n)$ ααΆαααααα½ααααΆααααΈ 1 αααααααΈα $(a_n)$ αααααΆααα½ $b_n$ αααααααΈααααααααα $b_n = a_{n+1} - a_n$ α αααααΈα $(b_n): 14, 30, 52, 80, 114, \dots$ α
ααΆααααααΈα $(c_n)$ ααΆαααααα½ααααΆααααΈ 1 αααααααΈα $(b_n)$ αααααΆααα½ $c_n$ ααΆαα½ααΈ n αααααααΈααααααααα $c_n = b_{n+1} - b_n$ ααααΆααααααΈα $(c_n): 16, 22, 28, 34, \dots$ α
13
|
[11] Math - High
| 17
|
|
αααααΈα $(c_n)$ ααΆαααααΈαααααααααααααΆααα½ααΈ 1 ααααΎααΉα 16 αα·ααααααα½αααααΎααΉα 6
$c_n = 16 + 6(n-1) = 16 + 6n - 6 = 6n + 10$ α
α
αααα $n \ge 2$ ααααΆα $b_n = b_1 + \sum_{k=1}^{n-1} c_k = 14 + \sum_{k=1}^{n-1} (6k+10)$
$= 14 + 6\sum_{k=1}^{n-1} k + 10(n-1) = 14 + 6 \cdot \frac{1}{2}n(n-1) + 10(n-1) = 3n^2 + 7n + 4$ α
α
αααα $n=1$ ααααΆα $b_1 = 4+7+3 = 14$ αα·α α ααΌα
ααα $b_n = 3n^2 + 7n + 4$ α
α
αααα $n \ge 2$ ααααΆα $a_n = a_1 + \sum_{k=1}^{n-1} b_k = 4 + \sum_{k=1}^{n-1} (3k^2 + 7k + 4)$
$= 4 + 3\sum_{k=1}^{n-1} k^2 + 7\sum_{k=1}^{n-1} k + 4(n-1)$
$= 4 + 3 \cdot \frac{1}{6}n(n-1)(2n-1) + 7 \cdot \frac{1}{2}(n-1)n + 4n - 4$
$= \frac{1}{2}(n-1)n(2n-1) + \frac{7}{2}(n-1)n + 4n = \frac{1}{2}n(n-1)(2n-1+7) + 4n = n(n-1)(n+3) + 4n = n(n^2+2n-3)+4n = n^3+2n^2-3n+4n = n^3+2n^2+n$
$= \frac{1}{2}(n-1)(7n+2n^2-n)+4n = (n-1)(3n+n^2)+4n = n^3+2n^2+n$ α
ααΌα
ααα $a_n = n^3 + 2n^2 + n$ α
α. ααααΆααααΌα n αα½ααααΌααααααααΈα $(a_n)$
ααΆα $S_n$ ααΆααααΌααααααααΈα $(a_n)$ ααΊ
$S_n = a_1 + a_2 + a_3 + \dots + a_n$
$S_n = \sum_{k=1}^{n} a_k = \sum_{k=1}^{n} (k^3 + 2k^2 + k)$
$= \sum_{k=1}^{n} k^3 + 2\sum_{k=1}^{n} k^2 + \sum_{k=1}^{n} k$
$= \frac{1}{4}n^2(1+n)^2 + 2 \cdot \frac{1}{6}n(n+1)(2n+1) + \frac{1}{2}n(n+1)$
$= \frac{1}{12}n(n+1)[3n(n+1) + 4(2n+1) + 6] = \frac{1}{12}n(n+1)(3n^2+3n+8n+4+6)$
$= \frac{1}{12}n(n+1)(3n^2+11n+10)$ α
α
αααα $n=1$, $a_1 = 4$ αα·α
ααΌα
ααα $S_n = \frac{1}{12}n(n+1)(3n^2+11n+10)$ α
#### ααααα·ααααα·
ααααααα½ααΈ n αααααααΈα $1, 2, 5, 12, 27, 58, 121, \dots$ α
14
|
[11] Math - High
| 18
|
|
### ααα αΆαα
#### ααααΌα α‘ αααααααΈ α‘
1. αααααααααΌαααΆαααααααααααααΎαα·αα·ααααααααΆ β :
- α. $1+2+3+\dots+100$
- α. $1+4+9+16+\dots+484$
- α. $1+8+27+64+\dots+3375$
- α. $1 \times 3 + 2 \times 4 + 3 \times 5 + \dots + 20 \times 22$ α
2. αααααααααααα½ααΆααα’ααααααααΌαααααα·αααααΎαα·αα·ααααααααΆ β :
- α. $\sum_{k=1}^{6} k$
- α. $\sum_{k=1}^{5} k^2$
- α. $\sum_{k=4}^{9} (3k-1)$
- α. $\sum_{k=2}^{7} (-1)^k k$ α
3. ααααΆ
- α. $\sum_{k=1}^{11} k^2$
- α. $\sum_{k=1}^{24} k^2$
- α. $\sum_{k=12}^{24} k^2$ α
4. ααααΆ
- α. $\sum_{k=1}^{24} k^3$
- α. $\sum_{k=1}^{15} k^3$
- α. $\sum_{k=16}^{24} k^3$ α
5. - α. ααααα½ααααααα $\sum_{k=1}^{n} k(k+1)$
- α. αααααααΎα
ααααΎα
- α. ααααΆααααΌα $1 \times 2 + 2 \times 3 + 3 \times 4 + \dots + 28 \times 29$ α
6. - α. ααααα½ααααααα $\sum_{k=1}^{n} k(k+1)(k+2)$
- α. αααααααΎα
ααααΎα
- α. ααααΆααααΌα $1 \times 2 \times 3 + 2 \times 3 \times 4 + 3 \times 4 \times 5 + \dots + 20 \times 21 \times 22$ α
7. - α. ααααααα½ααΈ n αααααααΈα $1, 2, 5, 10, 17, \dots$ α
- α. ααααΆααααΌα n αα½ααααΌααααααααΈαααα α
8. - α. ααααααα½ααΈ n αααααααΈα $1, 5, 14, 30, 55, 91, \dots$ α
- α. ααααααΌα n αα½ααααΌααααααααΈαααα α
9. ααααααα½ααΈ n αααααααΈα $(a_n): p, q, p, q, p, q, \dots$ α
15
|
[11] Math - High
| 19
|
|
## αααααααΈ 2 ααααΆαααααααα½ααααααΈα
### 1. ααααααα½ααΈ n αααααααΎαααααΈααααα½α
ααααΆααα·ααααΆαα½α
ααα αΎαααΌαααααΆαααααααα½αααααααΈαααααααα αα·αααααΈαααααΈααΆααα α
α
αααααααααΈαααααααα $a_{n+1} = a_n + d$, d ααΆαααααα½α α
α
αααααααααΈαααααΈααΆααα $a_{n+1} = a_n \times q$, q ααΆααααααα½α α
ααα
αααααααΉααα·ααααΆα’αααΈααααΆαααααααα½αααααααΈαααααααααααααααααααααα½ααΈ n ααααααΈαααα α
#### ααααα»αααα
- ααααααα½ααΈ n αααααααΎαααΈααααα½α
- ααααααα½ααΈ n ααΆαααααΆαααααααααΆα $a_n$ αα·α $S_n$
- ααααααα½ααΌαα
ααααααΆαααααααα½αααα»ααααααα $a_{n+2} + pa_{n+1} + qa_n = 0$
#### ααα αΆααααααΌ 1
ααααααα½ααΌαα
αααααααΈα $(a_n)$ αααααααΆααααααααααΎα $a_1 = 1$, $a_{n+1} = 1 + 4a_n$ α
**α
ααααΎα**
**ααααααΈ 1**
ααααΆα $a_1 = 1$ ααΈ $a_{n+1} = 1 + 4a_n$ ααα’αΆα
ααααα
$a_2 = 1 + 4a_1 = 1+4$
$a_3 = 1 + 4a_2 = 1 + 4 \times (1+4) = 1+4+4^2$
$a_4 = 1 + 4a_3 = 1 + 4 \times (1+4+4^2) = 1+4+4^2+4^3$
...
ααα $a_n = 1+4+4^2+4^3+\dots+4^{n-1}$
ααααΆα $a_n = 1+4+4^2+4^3+\dots+4^{n-1} = \frac{1 \times (4^n-1)}{4-1} = \frac{4^n-1}{3}$ α
ααΌα
ααα αα½ααΈ n αααααααΈα $(a_n)$ ααΊ $a_n = \frac{4^n-1}{3}$ α
**ααααααΈ 2**
α
αααα $n \ge 2$ ααααΆα $a_{n+1} = 1+4a_n$, $a_n = 1+4a_{n-1}$
16
|
[11] Math - High
| 20
|
|
### ααααΌα α‘ αααααααΈ α’
ααα $a_{n+1} - a_n = 1+4a_n - (1+4a_{n-1}) = 4(a_n - a_{n-1})$ (1)
ααΎααααΆα $b_n = a_{n+1} - a_n$ αααααΈ (1) ααααΆα $b_n = 4b_{n-1} \Rightarrow \frac{b_n}{b_{n-1}} = 4$ ααααααααΈα $(b_n)$ ααΆαααααΈαααααΈααΆααααααααΆαααααααα½αααααΎααΉα 4 αα·αααΆααα½ααΈ 1 αα·ααα½ααΈ n ααΊ
$b_1 = a_2 - a_1 = 1+4 \times 1 - 1 = 4$, $b_n = 4 \times 4^{n-1} = 4^n$ α
αααααΈα $(b_n)$ ααΆαααααααααΈα $(a_n)$ ααΆαααΌααααα α
αααα $n \ge 2$ ααααΆα
$a_n = a_1 + \sum_{k=1}^{n-1} 4^k = 1 + 4 + 4^2 + \dots + 4^{n-1} = \frac{1 \times (4^n-1)}{4-1} = \frac{4^n-1}{3}$
ααΌα
ααα αα½ααΈ n αααααααΈα $(a_n)$ ααΊ $a_n = \frac{4^n-1}{3}$ α
**ααααααΈ 3**
ααααααΎαααααΈααααα½α $(r_n)$ αα½α
ααααΎαααα $a_{n+1} - r_n$ α
ααα±αα $r_n = -\frac{1}{3}$ ααααΆα $a_{n+1} + \frac{1}{3} = 1+4a_n + \frac{1}{3} = 4a_n + \frac{4}{3} = 4(a_n + \frac{1}{3})$
ααΆα $V_{n+1} = a_{n+1} + \frac{1}{3}$ α αΎα $V_n = a_n + \frac{1}{3}$ (1)
ααααΆα $V_{n+1} = 4V_n$ α αΎα $V_n = 4^{n-1} \cdot V_1$ α
ααα’αΆα
ααααΆ $V_1$ ααΆαααααΆαααααα (1) ααΊ $V_1 = a_1 + \frac{1}{3} = 1 + \frac{1}{3} = \frac{4}{3}$
ααααΆα $V_n = 4^{n-1} \cdot \frac{4}{3} = \frac{4^n}{3}$ ααα’αΆα
ααααΆ $a_n$ ααΆαααααΆαααααα (1)
ααΊ $a_n = V_n - \frac{1}{3} = \frac{4^n}{3} - \frac{1}{3} = \frac{4^n-1}{3}$ α
**αααααΆαα**
ααΎααααΈαααααααααααααααααααΈααααα½α $(r_n)$ ααααΆα $r_n = \alpha \cdot n + \beta$ ($\alpha, \beta$ ααα) ααΆαααααΈαααααααααααααΆαα $a_{n+1} = 1+4a_n$ α αααα½αααααΈααααα½ααα
αααα»ααααααΈα $(a_n)$ ααααΆααααΈααΆααααα’αΆα
α±ααααααααααα $\alpha$ αα·α $\beta$ α
$a_{n+1} = 1+4a_n \Rightarrow \alpha \cdot (n+1) + \beta = 1+4 \cdot (\alpha \cdot n + \beta)$ α¬ $3 \cdot \alpha n + 3 \cdot \beta - \alpha + 1 = 0$
ααααΆααααααααα $\begin{cases} 3 \cdot \alpha = 0 \\ 3 \cdot \beta - \alpha + 1 = 0 \end{cases}$ α αΎα $\alpha=0, \beta = -\frac{1}{3}$ α
α ααα»ααα $r_n = -\frac{1}{3}$ ααΌα
ααααααααΈα $(r_n)$ α α
ααΆαααααΈααααα½α α
**ααΆααΌαα
**
ααααααΎαααααΈααααα½α $(r_n)$ ααα $r_n = \alpha \cdot n + \beta$ ($\alpha, \beta$ ααα) ααααααααααααΆααααααΆαααααααααααααΈα α αααα»αααααΈααα αααααα½α $r_n$ αααα»αααααΆαααααααααααααΈα αα½α
ααααΆααααα $\alpha$ αα·α $\beta$ αααααΆαααααααααααΆααααααΈα $(r_n)$ ααααΉαααααΎαααα $a_{n+1} - r_n$ α
17
|
[11] Math - High
| 21
|
|
#### ααα αΆααααααΌ 2
ααα±αααααααΈα $(a_n)$ αααααααα $a_1 = 3$, $a_{n+1} = 2a_n - n + 1$ α ααααααα½ααΈ n α
**α
ααααΎα**
ααΆα $(r_n)$ ααα $r_n = \alpha n + \beta$ ααΆαααΈααααα½α α
ααααΆα $a_{n+1} = 2a_n - n + 1 \Rightarrow \alpha \cdot (n+1) + \beta = 2(\alpha n + \beta) - n + 1$
$n \cdot (\alpha - 1) + \beta - \alpha + 1 = 0$ ααααΆααααααααα $\begin{cases} \alpha - 1 = 0 \\ \beta - \alpha + 1 = 0 \end{cases}$ α αΎα $\alpha = 1, \beta = 0$
ααα $r_n = n$ α ααααΎαααααααΆααααααΈα $a_{n+1} - r_n$ ααααΆα :
$a_{n+1} - n = 2a_n - n + 1 - n = 2a_n - 2n + 1$, $a_{n+1} - (n+1) = 2(a_n - n)$
ααΆα $V_n = a_n - n$ (1) ααα $V_{n+1} = a_{n+1} - (n+1)$ ααααΆα $V_{n+1} = 2 \cdot V_n$
α αΎα $V_n = 2^{n-1} \cdot V_1$ α
ααααααΆ $V_1$ ααΆα $V_n = a_n - n$, $V_1 = a_1 - 1 = 3-1 = 2$, $V_n = 2^{n-1} \cdot 2 = 2^n$
ααα’αΆα
ααααΆ $V_n$ ααΆα (1) ααΊ $V_n = a_n - n$ ααα $a_n = V_n + n = 2^n + n$ α
ααΌα
ααα $a_n = 2^n + n$ α
#### ααααα·ααααα·
ααααααα½ααΈ n αααααααΈα $(a_n)$ αααααααα $a_1 = 5$, $a_n = 2a_{n-1} - n$ α
### 2. ααααΆαααααααααΆα $a_n$ αα·α $S_n$
#### α§ααΆα ααα
ααααΆαααααΌα n αα½ααααΌααααααααΈα $(a_n)$ ααΊ $S_n = n^2 + n$ α ααααααα½ααΈ n αααααααΈα $(a_n)$ α
**α
ααααΎα**
ααααααα½ααΈ n αααααααΈα $(a_n)$
ααααΆαααααΌα n αα½ααααΌααααααααΈα $(a_n)$ ααΊ
$S_n = n^2 + n \Rightarrow S_n = a_1 + a_2 + a_3 + \dots + a_{n-1} + a_n = n^2 + n$
$\Rightarrow S_{n-1} = a_1 + a_2 + a_3 + \dots + a_{n-1} = (n-1)^2 + (n-1)$ α
α
αααα $n \ge 2$ ααααΆα $S_n - S_{n-1} = a_n = n^2 + n - [(n-1)^2 + (n-1)]$
$a_n = n^2 + n - (n-1)^2 - n + 1 = n^2 + n - n^2 + 2n - 1 - n + 1 = 2n$ ααα $a_n = 2n$ α
α
αααα $n=1$ ααααΆα $S_1 = 1^2 + 1 = 2$ αα·α ααααα $(S_1 = a_1 = 2)$ α ααΌα
ααα $a_n = 2n$ α
α§αααΆααΆ $S_n$ ααΊααΆααααΌα n αα½ααααΌααααααααΈα $(a_n)$ ααα $S_n = a_1 + a_2 + \dots + a_{n-1} + a_n$ αα·α $S_{n-1} = a_1 + a_2 + \dots + a_{n-1}$ α
α
αααα $n \ge 2$ ααααΆα $S_n - S_{n-1} = a_n$ α αααααΆαααα $S_1 = a_1$ α
18
|
[11] Math - High
| 22
|
|
### ααααΌα α‘ αααααααΈ α’
**ααΆααΌαα
**
α
αααα $n \ge 2$ ααααΆα $S_n - S_{n-1} = a_n$ αα·α $S_1 = a_1$ α
**αααααΆαα**
$S_n$ αα·α $S_{n+1}$ ααΆαααΆαα
αΆαααΆα
αααΎααααΈαα $a_{n+1}$ ααΊ $a_{n+1} = S_{n+1} - S_n$ α
#### ααα αΆααααααΌ
ααααΆαααααΌα n αα½ααααΌααααααααΈα $(a_n)$ ααΊ $S_n = n^3 + 2n$ α
ααααααα½ααΈ n αααααααΈα $(a_n)$ α
**α
ααααΎα**
ααααααα½ααΈ n αααααααΈα $(a_n)$
ααααΆαααααΌα n αα½ααααΌααααααααΈα $(a_n)$ ααΊ
$S_n = n^3 + 2n \Rightarrow S_{n-1} = (n-1)^3 + 2(n-1)$
$a_n = S_n - S_{n-1} = n^3 + 2n - (n-1)^3 - 2(n-1)$
$= n^3 + 2n - (n^3 - 3n^2 + 3n - 1) - 2n + 2 = n^3 + 2n - n^3 + 3n^2 - 3n + 1 - 2n + 2 = 3n^2 - 3n + 3$ α
ααΌα
ααα αα½ααΈ n αααααααΈα $(a_n)$ ααΊ $a_n = 3n^2 - 3n + 3$ α
#### ααααα·ααααα·
ααααΆα $S_n = 4 - a_n - \frac{1}{2^{n-2}}$ ($n=1, 2, 3, \dots$) ααΆααααΌα n αα½ααααΌααααααααΈα $(a_n)$ α
α. αααααααααΆααααααααααΎααααΆα $a_{n+1}$ αα·α $a_n$ α
α. ααααααα½ααΈ n αααααααΈα $(a_n)$ α
### 3. ααααΆααααααααααΎααααα»ααααααα $a_{n+2} + pa_{n+1} + qa_n = 0$
#### α§ααΆα ααα
ααααααα½ααΈ n αααααααΈα $(a_n)$ ααα $a_1 = 1, a_2 = 3$ ααα $a_{n+2} = 3a_{n+1} - 2a_n$ ($n=1, 2, \dots$) α
ααααααΆαα $a_1, a_2$ ααα’αΆα
ααααΆαα½ $a_3$ αα·ααα½αααααααααΆααααΎααααΈαα·αα·αααααΎαααααΆαααααΌ
$a_3 = 3a_2 - 2a_1 = 9-2 = 7$
$a_4 = 3a_3 - 2a_2 = 21-6 = 15$
$a_5 = 3a_4 - 2a_3 = 45-14 = 31$
$a_6 = 3a_5 - 2a_4 = 93-30 = 63$
ααααΆαααααα½ααααααΈαααα
$a_2 - a_1 = 2$
$a_3 - a_2 = 2^2$
19
|
[11] Math - High
| 23
|
|
$a_4 - a_3 = 2^3$
...
$a_n - a_{n-1} = 2^{n-1}$
$a_n - a_1 = 2 + 2^2 + 2^3 + \dots + 2^{n-1}$ ααα $a_n = 1 + 2 + 2^2 + 2^3 + \dots + 2^{n-1}$
ααΆααααΌααα½αααααααΈαααααΈααΆααααααααΆα 1 ααΆαα½ααΈαα½α αα·αααααααα½αααααΎααΉα 2
ααααΆα $a_n = 2^n - 1$ α
**ααΆααΌαα
**
ααα’αΆα
ααααΆ $a_n$ ααΆααα·ααΈααΆαααααα :
$a_{n+2} = 3a_{n+1} - 2a_n \Leftrightarrow a_{n+2} - 2a_{n+1} = a_{n+1} - 2a_n = a_n - 2a_{n-1}$
$= a_{n-1} - 2a_{n-2} = \dots = a_2 - 2a_1 = 3 - 2 \times 1 = 1$
ααα $a_{n+2} - 2a_{n+1} = a_{n+1} - 2a_n = 1$ (1) α
αααααΆαααα $a_{n+2} = 3a_{n+1} - 2a_n \Leftrightarrow a_{n+2} - a_{n+1} = 2(a_{n+1} - a_n)$
ααΆα $b_n = a_{n+1} - a_n$ ααα $(b_n)$ α¬ $(a_{n+1} - a_n)$ ααΆαααααΈαααααΈααΆααααααααΆαααααααα½αααααΎααΉα 2 αα·ααα½ααΈαα½αααααΎααΉα $b_1 = a_2 - a_1 = 3-1 = 2$ α αΎα $b_n = 2 \cdot (2)^{n-1} = 2^n$ (2)
ααΆα (1) αα·α (2) ααααΆα
$\begin{cases} a_{n+1} - 2a_n = 1 & (1) \\ a_{n+1} - a_n = 2^n & (2) \end{cases}$
$-a_n = 1 - 2^n$
ααΌα
ααα αα½ααΈ n αααααααΈα $(a_n)$ ααΊ $a_n = 2^n - 1$ α
**ααΆααΌαα
αααααααααααΆα $a_{n+2} + pa_{n+1} + qa_n = 0$**
ααΎααααΆαααααααααααααΈα $(a_n)$ ααΆααααααα $a_{n+2} + pa_{n+1} + qa_n = 0$
αααα·α
αΆαααΆααΈαα
ααα½α $\alpha$ αα·α $\beta$ αααααααααααααααα $\alpha + \beta = -p, \alpha\beta = q$ α
αααααααΆα $a_{n+2} - (\alpha + \beta)a_{n+1} + \alpha\beta a_n = 0$ α
ααααΆαααααααααα’αΆα
αααααααΆαααααααΈααααΆαααΌα
ααΆαααααα :
$a_{n+2} - \alpha a_{n+1} = \beta(a_{n+1} - \alpha a_n)$
$a_{n+2} - \beta a_{n+1} = \alpha(a_{n+1} - \beta a_n)$
ααααΆααααααααΆαα 2 ααΆαααΎααα ααΊααΆαααααΈαααααΈααΆαααα’αΆα
α±αααααα $a_n$ ααΆα α
20
|
[11] Math - High
| 24
|
|
### ααααΌα α‘ αααααααΈ α’
ααα’αΆα
αα $\alpha$ αα·α $\beta$ ααααααααααΆααααΈααΆα $x^2 + px + q = 0$ αααααααααααΆααααααΌααα·ααααα»αααΆ α
αααΈααΆα $x^2 + px + q = 0$ α α
ααΆ βαααΈααΆααααααΆαααα $a_{n+2} + pa_{n+1} + qa_n = 0$β α
#### ααα αΆααααααΌ
αααααΈα $(a_n)$ αααααααα $a_1 = 1, a_2 = 13, a_{n+2} = a_{n+1} + 6a_n$ ($n=1, 2, \dots$) α ααααααα½ααΈ n αααααααΈα α
**α
ααααΎα**
ααααααα½ααΈ n αααααααΈα $(a_n)$
$a_{n+2} = a_{n+1} + 6a_n \Leftrightarrow a_{n+2} - a_{n+1} - 6a_n = 0$
αααΈααΆααααααΆαα : $x^2 - x - 6 = 0$ αααΈααΆαααΆαα«α $x=3$ α¬ $x=-2$ α ααααΆα
$\begin{cases} a_{n+2} + 2a_{n+1} = 3(a_{n+1} + 2a_n) & (i) \\ a_{n+2} - 3a_{n+1} = -2(a_{n+1} - 3a_n) & (ii) \end{cases}$
(i) $a_{n+2} + 2a_{n+1} = 3(a_{n+1} + 2a_n)$
ααΆα $b_n = a_{n+1} + 2a_n$ ααααααααΈα $(b_n)$ ααΆαααααΈαααααΈααΆααααααααΆαααααααα½αααααΎααΉα 3 α αΎααα½ααΈ 1 ααΊ $b_1 = a_2 + 2a_1 = 13 + 2 \times 1 = 15$ αα·α $b_n = 15 \cdot (3)^{n-1}$
ααααΆα $a_{n+1} + 2a_n = 15 \cdot (3)^{n-1}$ (1) α
(ii) $a_{n+2} - 3a_{n+1} = -2(a_{n+1} - 3a_n)$
ααΆα $c_n = a_{n+1} - 3a_n$ ααααααααΈα $(c_n)$ ααΆαααααΈαααααΈααΆααααααααΆαααααααα½αααααΎααΉα -2 α αΎααα½ααΈ 1 ααΊ $c_1 = a_2 - 3a_1 = 13 - 3 \times 1 = 10$ αα·α $c_n = 10 \cdot (-2)^{n-1}$
ααααΆα $a_{n+1} - 3a_n = 10 \cdot (-2)^{n-1}$ (2) α
ααΆα (1) αα·α (2) ααααΆα
$\begin{cases} a_{n+1} + 2a_n = 15 \cdot (3)^{n-1} & (1) \\ a_{n+1} - 3a_n = 10 \cdot (-2)^{n-1} & (2) \end{cases}$
$5a_n = 15(3)^{n-1} - 10(-2)^{n-1}$
$a_n = 3 \times 3^{n-1} - 2 \times (-2)^{n-1} = 3^n + (-2)^n$
ααΌα
ααα αα½ααΈ n αααααααΈα $(a_n)$ ααΊ $a_n = 3^n + (-2)^n$ α
#### ααααα·ααααα·
αααααΈα $(a_n)$ ααααααααααααΆαααααα $a_1 = 0, a_2 = 3, a_{n+2} = 4a_{n+1} - 3a_n$ ($n=1, 2, \dots$) α ααααααα½ααΈ n αααααααΈα $(a_n)$ α
21
|
[11] Math - High
| 25
|
|
### ααα αΆαα
1. αααααΈα $(a_n)$ ααααααααααααΆααααααααααΎαααΌα
ααΆαααααα :
- α. $a_1 = 3, a_{n+1} = 2a_n - 4$
- α. $a_1 = 5, a_{n+1} = 3a_n - 2n$ α
ααααααα½ααΈ 4 ααααααΈαααα α
2. αααααΈα $(a_n)$ ααααααααααααΆααααααααααΎα $a_1 = 1, a_2 = 2, a_{n+2} = a_{n+1} + a_n$ α
ααααααα½ααΈ 7 αααααααΈαααα α
3. αααααΈα $(a_n)$ ααααααααααααΆααααααααααΎα $a_1 = 1, a_{n+1} = pa_n + q$ α
ααααΆααααα p αα·α q ααΎααααΉααα½ααΈ 3 ααααΎααΉα 6 αα·ααα½ααΈ 5 ααααΎααΉα 86 α
4. ααααααα½ααΌαα
αααααααΈα $(a_n)$ ααααααααααααΆααααααααααΎαααΌα
ααΆαααααα :
- α. $a_1 = 1, 3a_{n+1} = 2a_n + 3$ ($n=1, 2, \dots$)
- α. $a_1 = 1, 3a_{n+1} = 3a_n + 4$ ($n=1, 2, \dots$)
- α. $a_1 = 1, a_{n+1} = a_n + n$ ($n=1, 2, \dots$) α
5. ααααΆααααααΈα $(a_n)$ αααααααα $a_1 = 4, a_{n+1} = \frac{4a_n - 9}{a_n - 2}$ ($n=1, 2, 3, \dots$) α
- α. ααααΆααααααΆααααΆ $a_n \ne 3$ α
ααααααααα n α
- α. αα $b_n = \frac{1}{a_n - 3}$ αα·αααααααα½ααΌαα
αααααααΈα $(b_n)$ α ααααααα½ααΌαα
αααααααΈα $(a_n)$ α
6. ααααΆα $S_n$ ααΆααααΌα n αα½ααααΌααααααααΈα $(a_n)$ α
ααΎ $S_n$ ααααααααααααα $S_n = 4 - a_n - \frac{1}{2^{n-2}}$ ($n=1, 2, 3, \dots$) αααα
αΌα :
- α. αααααααααΆααααααααααΎααααΆα $a_{n+1}$ αα·α $a_n$ α
- α. ααααααα½ααΈ n αααααααΈα $(a_n)$ α
7. ααααΆα $S_n$ ααΆααααΌα n αα½ααααΌααααααααΈα $(a_n)$ α αΎαααααααααααααα $(a_n): a_1 = 1$, $S_n = a_{n+1} + n^2$ ($n \ge 1$) α ααααααα½ααΈ n ααααααΈα $(a_n)$ α
22
|
[11] Math - High
| 26
|
|
### ααααΌα α‘ αααααααΈ α’
8. ααααΆα $S_n$ ααΆααααΌα n αα½ααααΌααααααααΈα $(a_n)$ α αΎα $S_n$ ααααααααααααα $S_n = \frac{n}{n-1} a_n$ $n \ge 2$ α
- α. αααααΆαααα $a_n$ ($n \ge 3$) α’αα»ααααααΉα n αα·α $a_{n-1}$ α
- α. αααααΆαααα $S_n$ ($n \ge 2$) α’αα»ααααααΉα n αα·α $S_{n-1}$ α
- α. α§αααΆααΆ $a_1 = 1$ αααα½ααΈ n αααααααΈα $S_n$ ααα $n \ge 1$ α
9. ααααΆααααααΈα $(a_n)$ ααααααααααααΆααααααααααΎα
$(a_n): a_1 = 2, a_{n+1} = \frac{a_n}{a_n+3}$ ($n=1, 2, 3, \dots$) α
- α. ααΆα $b_n = \frac{1}{a_n}$ α αααααααααΆααααααααααΎααααΆα $b_n$ αα·α $b_{n+1}$ α
- α. ααααααα½ααΈ n αααααααΈα $(a_n)$ α
23
|
[11] Math - High
| 27
|
|
## αααααααΈ 3 αα·α
αΆαα’αα»ααΆααα½αααα·ααα·ααααΆ
### 1. αααααΆααααααα·α
αΆαα’αα»ααΆααα½αααα·ααα·ααααΆ
ααααααΆααααααααααααΎαα§ααΆα ααααααααααααααΌαα
ααα½ααααααα
ααααΌα 1 αα½ααΆαααα $P(1) = 1 = 1^2$
ααααΌα 2 αα½ααΆαααα $P(2) = 1+3 = 2^2$
ααααΌα 3 αα½ααΆαααα $P(3) = 1+3+5 = 3^2$
ααα’αΆα
ααΆαααΆαααααΌα n αα½ααΆαααα
$P(n) = 1+3+5+\dots+(2n-1) = n^2$ α
#### ααααα»αααα
- ααααααα·α
αΆαα’αα»ααΆααα½αααα·ααα·ααααΆ
- ααΆαααααΎα’αα»ααΆααα½αααα·ααα·ααααΆ
- ααααΉααααΈααααααααΆ α
αααααΆααααΈααΆαααΌαααααααααα ααααααΌααααααΆααααΆααΌααααα $P(n)$ αα·αα
αααααααααα
ααα½αααα n α αααα»αααααΈαααααααααΎααααααααααΆαα $P(n)$ αααα±αα $n=1$ α
ααΎ $n=1$, $1 = 1^2$ α ααα»ααα $P(1)$ αα·α α
α§αααΆααΆ $P(n)$ αα·α α
ααααααααα n ααΆαα½α
ααααααΌαααααΆαααΆ $P(n)$ αα·αααΆαα±ααααΆα $P(n+1)$ αααα·αααα
ααΎ $P(n)$ αα·α ααααΆα $1+3+5+\dots+(2n-1) = n^2$
ααΎααααΈααααΆααααααΆααααΆ $P(n+1)$ αα·ααα»αααααΆααααααΆα $P(n+1) = (n+1)^2$ α
#### ααααααααΆααααααΆαα
ααααΉαααΆ $P(n+1) = 1+3+5+\dots+(2n-1)+[2(n+1)-1] = n^2 + (2n+2-1)$
$= n^2 + 2n + 1 = (n+1)^2$
$P(n+1)$ αα·α α
ααΌα
ααα $P(n)$ αα·αα
ααααααααα $n \in \mathbb{N}$ α
24
|
[11] Math - High
| 28
|
|
### ααααΌα α‘ αααααααΈα£
#### αααααΆααααααα·α
αΆαα’αα»ααΆααα½αααα·ααα·ααααΆ
**αα·α
αΆαα’αα»ααΆααα½α**
$P(n)$ ααΆααααΎαααααΆααααααΉαα
ααα½αααα n
ααΎααααΈααααΆααααααΆααααΆ $P(n)$ αα·αα
ααααααααα $n \in \mathbb{N}$ ααααααΌα :
1. αααααααααΆααααΆ $P(n)$ αα·αα
αααα $n=1$
2. α§αααΆααΆ $P(n)$ αα·αα
ααααααααα n
3. ααααΆααααααΆααααΆ $P(n)$ αα·αααΆαα±ααααΆα $P(n+1)$ αα·α α
#### ααα αΆααααααΌ
ααααΆααααααΆααααααΆα $1 \times 2 + 2 \times 3 + 3 \times 4 + \dots + n \times (n+1) = \frac{1}{3}n(n+1)(n+2)$ αα·αα
ααααααααα $n \in \mathbb{N}$ αααααααΎαα·α
αΆαα’αα»ααΆααα½αααα·ααα·ααααΆ α
**α
ααααΎα**
α
αααα $n=1$
α’αααααΆαααααα $1 \times 2 = 2$
α’αααααΆαααααΆα $\frac{1}{3} \times 1 \times (1+1)(1+2) = 2$ α
αααα’αααααΆαααααα = α’αααααΆαααααΆα
ααΌα
ααα α
αααα $n=1$ ααααΆααα·α α
α§αααΆααΆ ααααΆααα·αα
αααα $n=k$ ααααΆα
$1 \times 2 + 2 \times 3 + 3 \times 4 + \dots + k \times (k+1) = \frac{1}{3}k(k+1)(k+2)$ ($k=1, 2, 3, \dots$) α
αααα αΆαααΆααααΆααα·αα
αααα $n=k$ ααΆαα±ααααΆαααααΆααα·αα
αααα $n=k+1$
$1 \times 2 + 2 \times 3 + 3 \times 4 + \dots + k(k+1) + (k+1)(k+2) = \frac{1}{3}(k+1)(k+2)(k+3)$
ααααΆα $1 \times 2 + 2 \times 3 + 3 \times 4 + \dots + k \times (k+1) = \frac{1}{3}k(k+1)(k+2)$
ααα $(k+1)(k+2)$ ααΎα’αααααΆααααΈααααααΈααΆα ααααΆα
$1 \times 2 + 2 \times 3 + 3 \times 4 + \dots + k \times (k+1) + (k+1)(k+2) = \frac{1}{3}k(k+1)(k+2) + (k+1)(k+2)$
$= (k+1)(k+2)(\frac{k}{3}+1) = (k+1)(k+2)(\frac{k+3}{3})$
$= \frac{1}{3}(k+1)(k+2)(k+3)$ αα·α α
ααΌα
ααα ααααΆααα·αα
ααααααααα $n \in \mathbb{N}$ α
ααΌα
ααα $1 \times 2 + 2 \times 3 + 3 \times 4 + \dots + n \times (n+1) = \frac{1}{3}n(n+1)(n+2)$ ($n=1, 2, 3, \dots$) α
25
|
[11] Math - High
| 29
|
|
#### ααα αΆααααααΌ 2
α
αααα $h>0$ αα·αα
ααα½ααααααααααΆαα· $n \ge 2$ αααα αΆαααΆ $(1+h)^n > 1+nh$ αααααααΎαα·α
αΆαα’αα»ααΆααα½αααα·ααα·ααααΆ α
**α
ααααΎα**
α
αααα $n=2$
ααααΆα $(1+h)^2 = 1+2h+h^2 > 1+2h$ ααααα $h^2 \ge 0$
ααΌα
ααα αα·ααααΆαααααα·α α
αααα $n=2$ α
α§αααΆααΆαα·ααααΆαααααα·α α
αααα $n=k$ ($k \ge 2$) ααα $(1+h)^k > 1+kh$ αα·α
ααααΉααααα αΆαααΆααΆαα·αααα $n=k+1$ ($k \ge 2$) ααΊ $(1+h)^{k+1} > 1+(k+1) \times h$ (1) α
ααααΆα $(1+h)^k > 1+kh$ ααα $h>0$ ααα $1+h>0$
αα»αα’αααααΆααααΈααα (1) αα·α $1+h>0$
ααααΆα $(1+h)^k(1+h) > (1+kh)(1+h)$
$(1+h)^{k+1} > 1+kh+h+kh^2 = 1+(k+1)h+kh^2 > 1+(k+1)h$ ααααα $kh^2 \ge 0$ α
ααΌα
ααα αα·ααααΆα $(1+h)^n > 1+nh$ αα·αα
ααααααααα $n \ge 2$ α
#### ααα αΆααααααΌ 3
ααααΆααααααΆααααΆ $n^3+2n$ ααΆαα α»αα»ααα 3 α
αααααααααα
ααα½αααα n αααααααΎαα·α
αΆαα’αα»ααΆααα½αααα·ααα·ααααΆ α
**α
ααααΎα**
α
αααα $n=0$ ααααΆα 0 ααΆαα α»αα»ααα 3 αα·α α
α§αααΆααΆααααΆααα·αα
ααααα
ααα½αααα $n=k$
ααααΆα $k^3+2k$ ααΆαα α»αα»ααα 3 αα·α
ααΎαααΆα $k^3+2k = 3q$ α
αααααα αΆαααΆααΆαα·αα
αααα $n=k+1$ ααΊ $(k+1)^3+2(k+1)$ ααΆαα α»αα»ααα 3 α
ααααΆα $(k+1)^3+2(k+1) = k^3+3k^2+3k+1+2k+2$
$= k^3+3k^2+5k+3 = (k^3+2k) + 3 \times (k^2+k+1)$
$= 3q + 3(k^2+k+1) = 3(q+k^2+k+1)$
ααΆααααααΆ $(k+1)^3+2(k+1)$ ααΆαα α»αα»ααα 3 α
ααΌα
ααα $n^3+2n$ ααΆαα α»αα»ααα 3 α
αααααααααα
ααα½αααα n α
26
|
[11] Math - High
| 30
|
|
#### ααααα·ααααα·
- α. ααααΆααααααΆαααα·ααααΆα $(1+\frac{1}{2}+\frac{1}{3}+\dots+\frac{1}{n})(1+2+3+\dots+n) \ge n^2$ αααααααΎαα·α
αΆαα’αα»ααΆααα½αααα·ααα·ααααΆ α
- α. ααααΆααααααΆααααΆ $2^{n+1} + 3^{2n-1}$ ααΆαα α»αα»ααα 7 α
ααααααααα $n \in \mathbb{N}$ αααααααΎαα·α
αΆαα’αα»ααΆααα½αααα·ααα·ααααΆ α
- α. ααααΆααααααΆααααααΆα $1^2+2^2+3^2+\dots+n^2 = \frac{n(n+1)(2n+1)}{6}$ αααααααΎαα·α
αΆαα’αα»ααΆααα½αααα·ααα·ααααΆ α
### 2. ααααΉααααΈααααααααΆ
ααααααααααΆαααααααα»αααααααααΆ ααααΆααα α»ααΆ α¬αααααΈαα½α α ααΆαα·ααααααΆαααααΈαααααΈααΆαααα¬αααααΈααααααααα‘αΎα ααααΆααΆαααααΆαααααΌαα½αα
αααΆααααΆαα α αααα»αααααααααααααΉααα·ααααΆααΌααααααααααΆαααααααααα»αααααααααΆα±ααααΆαααΆαααα αα α ααΆααααΌααα·αα·αααααΎααααααΆααααααααα $(x+y)^n$ α
ααααααααααααααα n α
#### α§ααΆα ααα 1
$(x+y)^0 = 1$
$(x+y)^1 = 1x+1y$
$(x+y)^2 = 1x^2+2xy+1y^2$
$(x+y)^3 = 1x^3+3x^2y+3xy^2+1y^3$
$(x+y)^4 = 1x^4+4x^3y+6x^2y^2+4xy^3+1y^4$
$(x+y)^5 = 1x^5+5x^4y+10x^3y^2+10x^2y^3+5xy^4+1y^5$
αααα»ααα½ααααααΈαα½ααααΎαααΆαααααΆαααααΌααΌα
αααααΆ α
ααΈα§ααΆα αααααΆαααΎ αααααααααααα»ααα½ααααααΈαα½αααααααααΆααααααα $(x+y)^n$ ααΎαααΆ α
- αααααΆαααααααΆααΈαα½ααααΆαα α»ααΆαααααΆα $(n+1)$ αα½ α
- αααα»ααααααΆαααααααΆααΈαα½ααααααααα»ααα x α
α»α 1 ααΊααααααΆαααααααααΆαα αα·αααααααα»ααα y ααΎα 1 ααΊααααααΆαααααααααΆααααΎααα’αΆαααΈααααααα
ααααΆα α
- ααααΌαααααααααα»ααααα»ααα½ααΈαα½αααααα α»ααΆααααΎ n α
27
|
[11] Math - High
| 31
|
|
- ααααααα»αα
αΆααααααΎααα·αα
α»ααααααααΊ 1 ααΌα
ααααΆ α αΎαααααΌαααααααααα»αααΈααα½ααΆααααααΆαααααααΆα $(x+y)^n$ ααΆααααααα»ααααααααΆα $(x+y)^{n+1}$... α
| | | | | |
| :---: | :---: | :---: | :---: | :---: |
| | 1 | 3 | 3 | 1 |
| 1 | 4 | 6 | 4 | 1 |
ααααααα»ααα $(x+y)^3$
ααααααα»ααα $(x+y)^4$
α¬ααααααα»ααααα½αααααααααααααααΆαα’αΆαααααααΉαααααααα»ααααα½αααααΆααααΆααα»αααΊ
ααααααα»ααααα½ααΈ $k+1 = \frac{(\text{ααααααα»ααααα½ααΈ } k) \times (\text{ααααααα»ααα } x \text{ αααα»ααα½ααΈ } k)}{k}$ α
#### α§ααΆα ααα 2
$(x+y)^5 = 1x^5 + 5x^4y^1 + 10x^3y^2 + 10x^2y^3 + 5xy^4 + 1y^5$
$\frac{1 \times 5}{1} \quad \frac{5 \times 4}{2} \quad \frac{10 \times 3}{3} \quad \frac{10 \times 2}{4} \quad \frac{5 \times 1}{5}$
ααΈα§ααΆα ααα 1 ααΎααααααΆα
αααααααααααα»ααα·ααααααααααααΆαααααααΆααΆαααααΈαααααααΎαααΆαααααΆαααααΌααΆα
αααΎα α ααααΆα
ααααααααααααα»αααΆαααααααΆααΆαααααΈαααα α
ααΆ ααααΈααααααΆααααΆαα αα·αααΆαααΆααααααΆααααΆαααααααΆααααααααααα»ααααα»ααααααΆαααααααΆ $(x+y)^n$ αααααΆαααααααα»αααΌα
ααααΆαα·ααΆαα’αα»ααααααααΆααααΆααα
ααααααΆααααααΆαααααααΆ $(x+y)^n$ αααααΆαααααααα»ααα α
```
1
1 1
1 2 1
1 3 3 1
1 4 6 4 1
1 5 10 10 5 1
```
α
ααααα―αααααΈα 1, 1, 2, 3, 5, 8, 13, 21, ... α α
ααΆαααααΈα Fibonacci α
#### ααα αΆααααααΌ
ααααΎααααΈααααααΆααααΆαααααααΆααααααα $(x+y)^5$ α
**α
ααααΎα**
αααααααΎαα½ααααααΈααααΆαααααααΈααααααΆααααΆαα ααααΆαααααααα»α
```
1 4 6 4 1
/ \ / \ / \ / \
1 5 10 10 5 1
```
ααααΆα $(x+y)^5 = 1x^5 + 5x^4y + 10x^3y^2 + 10x^2y^3 + 5xy^4 + 1y^5$ α
28
|
[11] Math - High
| 32
|
|
### ααααΌα α‘ αααααααΈα£
#### ααααα·ααααα·
ααααΎααααΈααααααΆααααΆαααααααΆααααααα $(x+y)^6$ α
ααα
αΆαααΆα
ααααααααααΌαα½ααΌαα
αααααααΆα $(x+y)^n$ ααΊ
$(x+y)^n = 1x^n + ?x^{n-1}y^1 + ?x^{n-2}y^2 + \dots + ?x^{n-r}y^r + \dots + 1y^n$
αα½: 1, 2, 3, ..., (r+1), ..., (n+1)
ααααααα»α: $1, \frac{1 \times n}{1}, \frac{n(n-1)}{1 \times 2}, \dots, \frac{n(n-1)(n-2)\dots(n-r+1)}{1 \times 2 \times 3 \times \dots \times r}, \dots, 1$
$= 1x^n + \frac{n}{1}x^{n-1}y^1 + \frac{n(n-1)}{1 \times 2}x^{n-2}y^2 + \dots + \frac{n(n-1)(n-2)\dots(n-r+1)}{1 \times 2 \times 3 \times \dots \times r}x^{n-r}y^r + \dots + 1y^n$ α
ααΌα
ααα
$(x+y)^n = 1x^n + \frac{n}{1}x^{n-1}y^1 + \frac{n(n-1)}{1 \times 2}x^{n-2}y^2 + \dots + \frac{n(n-1)(n-2)\dots(n-r+1)}{1 \times 2 \times 3 \times \dots \times r}x^{n-r}y^r + \dots + 1y^n$ α
ααΈααααΈααααααΆααααΆαα ααααΎαααΆααααααα»αααααααααΆααΆα
ααα½ααααααααααα»αααααΆ α αΎαααααΉαααΆ
$1 = \frac{n!}{n!0!} = C(n,0) = C(n,n)$
$n = \frac{n!}{(n-1)!1!} = C(n,1) = C(n,n-1)$
$\frac{n(n-1)}{1 \cdot 2} = \frac{n!}{(n-2)!2!} = C(n,2) = C(n,n-2)$
$\frac{n(n-1)(n-2)}{1 \times 2 \times 3} = \frac{n!}{(n-3)!3!} = C(n,3) = C(n,n-3)$
...
$\frac{n(n-1)(n-2)\dots(n-r+1)}{1 \times 2 \times 3 \times \dots \times r} = \frac{n!}{(n-r)!r!} = C(n,r) = C(n,n-r)$
α
ααα½αααα $C(n,0), C(n,1), C(n,2), \dots, C(n,n)$ α α
ααΆααααααα»αααααααααΆ $(x+y)^n$ α
ααααΆα
$(x+y)^n = 1x^n + \frac{n}{1}x^{n-1}y^1 + \frac{n(n-1)}{1 \times 2}x^{n-2}y^2 + \dots + \frac{n(n-1)(n-2)\dots(n-r+1)}{1 \times 2 \times 3 \times \dots \times r}x^{n-r}y^r + \dots + 1y^n$
$C(n,0)x^n + C(n,1)x^{n-1}y^1 + C(n,2)x^{n-2}y^2 + \dots + C(n,r)x^{n-r}y^r + \dots + C(n,n)y^n$
α α
ααΆααααΉααααΈααααααααΆ α
> #### ααααΉααααΈααααααααΆ
> $(x+y)^n = \sum_{r=0}^{n} C(n,r)x^{n-r}y^r = C(n,0)x^n + C(n,1)x^{n-1}y^1 + C(n,2)x^{n-2}y^2 + \dots + C(n,n)y^n$ α
29
|
[11] Math - High
| 33
|
|
#### ααα αΆααααααΌ
αααααααΎαα½ααααααΈ 6 ααααααΈααααααΆααααΆααααααΆααααααα»αααααααααΆ
$C(6,0), C(6,1), C(6,2), C(6,3), C(6,4), C(6,5), C(6,6)$ α
**α
ααααΎα**
```
1 5 10 10 5 1
/ \ / \ / \ / \ / \ / \
1 6 15 20 15 6 1
```
$C(6,0) \ C(6,1) \ C(6,2) \ C(6,3) \ C(6,4) \ C(6,5) \ C(6,6)$
#### ααααα·ααααα·
1. αααααααΎααααΉααααΈααααααααΆα
αΌααααααΆαααααααααΆαααααα α
- α. $(1+3x)^4$
- α. $(2x-y)^5$
- α. $(a+\frac{1}{a})^6$
- α. $(1+x)^n$ α
2. ααααααα½ααΈ 4 αααααααΆαααααααΆ $(x+y)^{13}$ α
### ααααααααααα
- αααααΈαααα
ααα½ααα·α ααΊααΆα’αα»ααααααααααααααα $\mathbb{N}$ αα
$\mathbb{R}$ α
- ααααΈαααααααα ααΊααΆαααααΈαααα
ααα½ααα·ααααααΆααα½ααΈαα½αα (αααα
ααΈαα½ααΈ 1) ααααΎααΉααα½αα»ααααααΆααααΌαα
ααα½αααα d αα½α (α α
ααΆαααααα½α $d = U_{n+1} - U_n$) α
- ααΌααααααα½ααΈ n αααααααΈαααααααα $U_n = U_1 + (n-1) \times d$ α
- ααααΌα n αα½ααααααΈαααααααα $S_n = \frac{(U_1+U_n) \times n}{2}$ α
- ααααΈαααααΈααΆααα ααΊααΆαααααΈαα
ααα½ααα·αααααα½ααΈαα½αα (αααα
ααΈαα½ααΈ 1) ααααΎααΉααα½αα»ααααααΆαααα»αααΉαα
ααα½αααα q ααα $q \ne 0$ α α
ααα½αααα q αααα α
ααΆααααααα½α α¬ααααα»ααααααααΈαααααΈααΆααα α
- ααΌααααααα½ααΈ n αααααααΈαααααΈααΆααα $U_n = U_1 \times q^{n-1}$ α
- ααααΌα n αα½ααααΌααααααααΈαααααΈααΆααα $S_n = \frac{U_1(q^n-1)}{q-1}$ ααα $q \ne 1$ α
- ααααΌααααααΈαααααΈααΆααααα·αααααα $S_n = \frac{U_1}{1-q}$ α
αααα $|q|<1$ α
- ααΌαααααααααΌααααααΈααααααααααΆαααααΆααααααα
$\sum_{k=1}^{n} k = \frac{1}{2}n(n+1)$, $\sum_{k=1}^{n} k^2 = \frac{1}{6}n(n+1)(2n+1)$, $\sum_{k=1}^{n} k^3 = [\frac{1}{2}n(n+1)]^2$ α
30
|
[11] Math - High
| 34
|
|
### ααααΌα α‘ αααααααΈα£
- αααααΈα $(b_n)$ ααΆαααααα½ααααΆααααΈ 1 αααααααΈα $(a_n)$ ααα $a_n = a_1 + \sum_{k=1}^{n-1} b_k$, $n \ge 2$ α
- αααααΈα $(c_n)$ ααΆαααααα½ααααΆααααΈ 2 αααααααΈα $(a_n)$ ααα $a_n = a_1 + \sum_{k=1}^{n-1} b_k$, $n \ge 2$ α
- α αΎααααααΈα $(b_n)$ ααΆαααααα½ααααΆααααΈ 1 αααααααΈα $(a_n)$ ααα $b_n = b_1 + \sum_{k=1}^{n-1} c_k$, $n \ge 2$ α
- ααααΆααααααααααΎαααΊααΆα
αααΆαααααααααααα½ $a_{n+1}$ ααΆα’αα»αααααααα½ $a_n$ αααααααΈα $(a_n)$ α
- ααααΆααααααααααΎααααααα $a_{k+1} = pa_k + q$ (p, q ααα) ααααααααα
ααΆ $a_{k+1} - c = p(a_k - c)$ αααααΆααααΈααΆααααααΆαα $c = pc+q$ α
- ααααΆααααααααααΎααααααα $a_{n+1} = a_n + pn$ ααααααααααααΈα $(b_n)$ ααα $b_n = a_{n+1} - a_n = pn$ αα·α $a_n = a_1 + \sum_{k=1}^{n-1} b_k = a_1 + \sum_{k=1}^{n-1} pk$ α
- α
αααα $n \ge 2$ ααααΆα $S_n - S_{n-1} = a_n$ αα·α $S_1 = a_1$ α
- ααααΆααααααααααΎααααααααΈα $(a_n)$ αααααα $a_{n+2} + pa_{n+1} + qa_n = 0$ ααΆααααΈααΆααααααΆαα $x^2 + px + q = 0$ αααα«α $\alpha$ αα·α $\beta$ ααααααααααααα $\alpha+\beta = -p, \alpha\beta = q$ αααααααΆα $a_{n+2} - (\alpha+\beta)a_{n+1} + \alpha\beta a_n = 0$ α’αΆα
αααααααααΆ
$a_{n+2} - \alpha a_{n+1} = \beta(a_{n+1} - \alpha a_n)$
$a_{n+2} - \beta a_{n+1} = \alpha(a_{n+1} - \beta a_n)$ ααααΆααααααααΆαα 2 αααααΊααΆαααααΈαααααΈααΆααα α
- ααααΎ $P(n)$ αα·αα
αααααααααα
ααα½ααααααααααΆαα· n α αΎαααααΎααΆαααααα (i) αα·α (ii) αα·αααΊ :
(i) $P(n)$ αα·αα
αααα $n=1$
(ii) α§αααΆααΆ $P(n)$ αα·αα
αααααααααα
ααα½ααααααααααΆαα· $n=k$ αα½α
αααα αΆαααΆ $P(n+1)$ αα·αα
ααααααααα $n=k+1$ α
> #### ααααΉααααΈααααααααΆ
> $(x+y)^n = \sum_{r=0}^{n} C(n,r)x^{n-r}y^r$
> $= C(n,0)x^n + C(n,1)x^{n-1}y^1 + \dots + C(n,r)x^{n-r}y^r + \dots + C(n,n)y^n$ α
31
|
[11] Math - High
| 35
|
|
### ααα αΆαα
1. ααααΆααααααΆααααΆα
ααααααααα $n \ge 1$; $1^2+2^2+3^2+\dots+n^2 = \frac{n(n+1)(2n+1)}{6}$ αααααααΎαα·α
αΆαα’αα»ααΆααα½αααα·ααα·ααααΆ α
2. αααα αΆαααΆα
αααααααααα
ααα½αααα n
- α. α
ααα½α $4^n+2$ α
ααααΆα
αααΉα 3 α
- α. α
ααα½α $3^{n+3} - 4^{4n+2}$ α
ααααΆα
αααΉα 11 α
3. ααααΆααααΈα $(U_n)$ ααα n ααΆα
ααα½αααααααααααα $U_{n+1} = \sqrt{U_n+2}$ αα·α $U_0 = 0$ α
αααα αΆαααΆααα·α
αΆαα’αα»ααΆααα½αααα·ααα·ααααΆααΆ
- α. α
αααααααααα
ααα½αααα n, $U_n \le 2$ α
- α. α
αααααααααα
ααα½αααα n, $U_n \le U_{n+1}$ α
4. αααα αΆαααΆααα·α
αΆαα’αα»ααΆααα½αααα·ααα·ααααΆααΆ α
αααααααααα
ααα½ααααααααααΆαα· n
$(1+x)^n \ge 1+nx+\frac{n(n-1)}{2}x^2$ α
αααααααααα
ααα½ααα·α $x \ge 0$ α
5. αααααααΎααααΉααααΈααααααααΆ α
αΌααααααΆαααααααααΆαααααα :
- α. $(3x-1)^4$
- α. $(2x+y)^6$
- α. $(a+b)^6$ α
6. ααααααααααΆαααααααααΆαααααααααααααΎ β :
- α. $(x-y)^5$
- α. $(2x+y)^6$
- α. $(a+b)^{12}$ α
32
|
[11] Math - High
| 36
|
|
### ααααΌα α‘ αααααααΈα£
### ααα αΆααααααΌα
1. αααααααΈαα½ααααΌαααααααΈαααααααααααααΉαααΆ $S_{10} = 210$ αα·α $S_{20} = 820$ α
2. ααααΉαααΆααααΌααα½ααΈ 1 αα·ααα½ααΈ 4 αααααααΈααααααααααααΎααΉα 2 αα·αααααΌαααΆααααααααΆααααΎααΉα 20 α ααααΆααααΌαααααΆαααΈαα½ααααΌααααααααΈα α
3. ααααΆα $S_m$ αα·α $S_n$ ααΆααααΌα m αα½ααααΌααα·α n αα½ααααΌααααααααΆαααααααΈαααααααααα½αααα $\frac{S_m}{S_n} = \frac{m^2}{n^2}$; ($n \ne m$) α ααΆααα½ααΈ m ααΊ $u_m$ αα·ααα½ααΈ n ααΊ $u_n$ α αααα αΆαααΆ $\frac{u_m}{u_n} = \frac{2m-1}{2n-1}$ α
4. ααααΆααααααΈαααααΈααΆααα $12, 4, \frac{4}{3}, \dots$ α
- α. ααααΆαα½ααΈ 10
- α. ααΎα
ααα½α $\frac{4}{729}$ ααΆαα½ααΈααα»ααααΆααααααααΈα ?
- α. ααααΆααααΌα 20 αα½ααααΌααααααααΈαααααΈααΆααα α
5. ααα±αα $(U_n)$ ααΆααααΈαααααΈααΆααα ααΎααααΉαααΆ $U_n = 2(3)^{n-1}$ α ααααΆ $S_n$ α
6. ααααΆααααΌααααααΈαααααΈααΆααα $1+2x+3x^2+\dots+(n-1)x^{n-2}+nx^{n-1}$ α
7. ααααΆαα½ααΈ 1 αααααααΈαααααΈααΆαααα’αααααα½αααααΆα $q = \frac{3}{5}$ αα·α $S_\infty = 40$ α
8. ααα±ααααΈα
ααα½αααΆαααααΈαααααΈααΆααα α ααααΆα
ααα½αααΆαααααααΎααααΉαααΆαααα»αααα
ααα½αααΆαααααααααΎααΉα 3375 α αΎαααααΌαααΆααααΎααΉα 93 α
9. ααααΆααααΌα n αα½ααααΌααααααααΈαααΈαα½ααααΆαααααα
- α. αααααΈα $(a_n): 1, \frac{1}{1+2}, \frac{1}{1+2+3}, \dots, \frac{1}{1+2+3+4+\dots+n}$
- α. αααααΈα $(b_n): \frac{2}{(1 \times 3)^2}, \frac{3}{(3 \times 5)^2}, \frac{4}{(5 \times 7)^2}, \dots, \frac{n}{[(2n-1)(2n+1)]^2}$ α
10. - α. ααααααα½ααΈ n αααααααΈα $1, 2, 6, 15, 31, 56, \dots$
- α. ααααΆααααΌα n αα½ααααΌααααααααΈαααα α
11. - α. ααααΆ $\sum_{k=1}^{n} (2k^2-1)$
- α. αααααααΎαααα½α α. ααααΆααααΌα $1+7+17+31+\dots+799$ α
12. αααααααααΌα $1+4+7+10+13+\dots+298$ αααααααΎ β α
33
|
[11] Math - High
| 37
|
|
13. αααααααΎαα·α
αΆαα’αα»ααΆααα½αααα·ααα·ααααΆααααΆααααααΆααααΆ $\sum_{k=1}^{n} 2^{k-1} = 2^n - 1$ α
14. ααααΆααααααΈα $(U_n)$ ααα n ααΆα
ααα½αααααααααααα $U_{n+1} = 2U_n+1$ αα·α $U_0 = 1$ α αΎααααααΈα $(V_n)$ αααααααα $V_n = U_n+1$ α
- α. αααα αΆαααΆαααααΈα $(V_n)$ ααΆαααααΈαααααΈααΆααα α
- α. ααΆααα $U_n$ ααΆα’αα»αααααα n α
- α. αα·ααααΆααΆααααΌααΌααΌααααααααΈα $(U_n)$ α
- α. α
αααααααααα
ααα½αααα n ααααΆααααΌα $S_n = U_0+U_1+U_2+\dots+U_n$ α
15. ααααΆααααααΈα $(U_n)$ αααααααα $U_{n+1} = \frac{U_n+2}{U_n+1}$ αα·α $U_0 = 2$ α
- α. ααααΆ $U_1, U_2, U_3$ α
- α. αααα αΆαααΆα
αααααααααα
ααα½αααα n, $U_{n+1} - \sqrt{2} = \frac{(\sqrt{2}-1)(\sqrt{2}-U_n)}{U_n+1}$ α
- α. αααα αΆαααΆα
αααααααααα
ααα½αααα n, $U_n > 1$ α
- α. ααΆαααΈαααα½α α. αα·α α. ααΆ $|U_{n+1} - \sqrt{2}| \le |\frac{\sqrt{2}-1}{2}| \times |U_n - \sqrt{2}|$ α
- α. αααα αΆαααΆααα·α
αΆαα’αα»ααΆααα½αααα·ααα·ααααΆααΆ $|U_n - \sqrt{2}| \le (\frac{\sqrt{2}-1}{2})^n \times |\sqrt{2}-U_0|$ α
16. ααααΆα $(a_n)$ ααααααααααααΆααααααααααΎα $a_1 = 1, a_{n+1} = \frac{1}{2}a_n + n$ α
- α. ααΆα $b_n = 2^n a_n$ α ααααααα½ααΈ n αααααααΈα $(b_n)$ α
- α. ααααααα½ααΈ n αααααααΈα $(a_n)$ α
17. - α. ααααα $(x^2-2y)^7$ αααααααΎ β
- α. ααααααα½ααΈ 6 αααααααΆα $(x^2-2y)^7$ α
18. αααα αΆαααΆααααΌαααααααααα»ααααα»ααααααΆα $(1+x)^n$ ααΊ $2^n$ α
19. αααα αΆαααΆ $C(n,0)+C(n,2)+C(n,4)+\dots+C(n,n-1) = C(n,1)+C(n,3)+C(n,5)+\dots+C(n,n) = 2^{n-1}$ ααα $C(n,0), C(n,1), C(n,2), \dots, C(n,n)$ ααΆααααααα»ααααα»ααααααΆα $(1+x)^n$ ααα n ααΆα
ααα½ααααααα α
34
|
[11] Math - High
| 38
|
|
### ααααΌα α’ αααααααΈ α‘
# ααααΌα 2
# α’αα»ααααα’α·α
αααααΌαααααααααα·αα’αα»ααααααααΆααΈα
![angkor_wat.png: Photo of Angkor Wat temple complex]
- **α’αα»ααααα’α·α
αααααΌαααααααα**
- **α’αα»ααααααααΆααΈα**
αα
αααα»αααααΌαααα ααα
αΆααααααΎααα·ααααΆααΈαααααααα·ααα’αα»ααααα’α·α
αααααΌαααααααα αααααααΆαααΆαα’αα»αααα αααααααΆααααΈααΆα αα·ααα·αααΈααΆαα’α·α
αααααΌαααααααα αααααΆαααααααΆαααααΆαααααααΆαααα·ααααΆααΈαααααααα· ααααΆα αααΈααΆααα·ααα·αααΈααΆαααααΆααΈα α αααααΆααααΈααα½αααΆααααααααα· α’αα»ααααα’α·α
αααααΌαααααααα αα·αααααΆααΈα αα·αααα’αΆα
ααα
ααααααΉααα·ααααα·ααα
αααααααΆαα
αααααααα αΆαααααΆααααααΉαααΆαααααΆαα ααααΎαααααααααα
α ααααΎααααααΆαααααα αα·ααααααΆαααααααααΆαα
αααααααα αΆααααααααααααα½αααααααααα»αααΈαααΆααααα
αΆααααα α
35
|
[11] Math - High
| 39
|
|
## αααααααΈ 1 α’αα»ααααα’α·α
αααααΌαααααααα
### 1. α’αα»ααααα’α·α
αααααΌαααααααα
#### 1.1 ααααΆαααα’αα»ααααα’α·α
αααααΌαααααααα
αααααααΆαααα’αα»αααααααα»αααααα»ααααα½α
$f(x) = 2^x, g(x) = (1.5)^x$
$h(x) = 1^x, k(x) = (0.5)^x$ α
#### ααααα»αααα
- αααααααΆαααα’αα»ααααα’α·α
αααααΌαααααααα
- αααααααΆααααΈααΆααα·ααα·αααΈααΆαα’α·α
αααααΌαααααααα α
ααΆααΆααααααααααΌαααααΆαα x αα·α y
| x | y = 1.5^x |
|---|---|
| -2 | 0.44 |
| -1 | 0.66 |
| 0 | 1 |
| 1 | 1.5 |
| 2 | 2.25 |
![graph_exponential.png: Graphs of f(x)=2^x, g(x)=(1.5)^x, h(x)=1^x, and k(x)=(0.5)^x]
ααΆαααααΆαααΎαααααααααΎαααΆ αααααααααΆαααααα’αα»ααααα’α·α
αααααΌαααααααα ααΆααααΆα α
ααα»α
αααααΆαααΌα’ααααα (0, 1) ααΆαα·α
αα
α
- ααΎ $x>0$ ααα $(1.5)^x < 2^x$ α ααΌα
ααα αααα»αααΆαααααααΈα‘ ααααΆαααα’αα»αααα $g(x) = (1.5)^x$ αααα·ααα
αααααααααΆαααα’αα»αααα $f(x) = 2^x$ α
- α’αα»αααα $h(x) = 1^x$ ααΆαααααΆαααΆαααααΆααααααααΉαα’ααααα’αΆαααααΈααα·ααα
ααΈααΎα’ααααα’αΆαααααΈαα
ααα½α 1 α―αααΆ α
αααααααααααααα x α
36
|
[11] Math - High
| 40
|
|
### ααααΌα α’ αααααααΈ α‘
- ααΎ $x>0$ ααα $(0.5)^x < 1$ α ααΌα
ααα ααΎ x αααααααααΎαααα‘αΎαα ααα $k(x) = 0.5^x$ ααΆααααααα
α»αααΌα
αα
α α αΎααα·ααα
αα 0 α
- ααΎ $x<0$ ααα $(0.5)^x$ ααΆααααααααα‘αΎααααααΆαααααα α
**ααΆααΌαα
**
α’αα»ααααα’α·α
αααααΌαααααααα $y=a^x$ ααΆαααααΆα :
- ααΎ $a>1$ ααααΆααα $y=a^x$ ααΎαααΈααααααα
ααααΆα ααααΆ $y=a^x$ ααΆα’αα»ααααααΎα
- ααΎ $0<a<1$ ααααΆααα $y=a^x$ α
α»αααΈααααααα
ααααΆα ααααΆ $y=a^x$ ααΆα’αα»ααααα
α»α
- α’αα»αααα $y=a^x$ αα·αααααΆαααΆαα·α
αα
α
ααααααααα $x \in \mathbb{R}$ ααΎ a ααΆα
ααα½ααα·ααα·αααααΆααα·ααα»αααΈ 1
- ααααΆαααα’αα»αααα $y=a^x$ ααΆααα’ααααα’αααααααααα (0,1) ααΆαα·α
αα
α
#### ααα αΆααααααΌ
αααααααΆα α. $y=4^x$ α. $y=(\frac{1}{4})^x$
**α
ααααΎα**
α. ααΆααΆααααααααααΌαααααΆαα x αα·α y
| x | y = 4^x |
|---|---|
| -2 | 0.06 |
| -1 | 0.25 |
| 0 | 1 |
| 1 | 4 |
| 2 | 16 |
![graph_y_eq_4_pow_x.png: Graph of y=4^x]
α. ααΆααΆααααααααααΌαααααΆαα x αα·α y
| x | y = (1/4)^x |
|---|---|
| -2 | 16 |
| -1 | 4 |
| 0 | 1 |
| 1 | 0.25 |
| 2 | 0.06 |
![graph_y_eq_one_fourth_pow_x.png: Graph of y=(1/4)^x]
#### ααααα·ααααα·
αααααααΆαα’αα»αααα α. $y=10^x$ α. $y=(\frac{1}{10})^x$ α
37
|
[11] Math - High
| 41
|
|
#### 1.2 ααααααααααΆαααα’αα»ααααα’α·α
αααααΌαααααααα
##### α§ααΆα ααα
αααααααΆαααα’αα»αααα
α. $y = 2^x + 1$
α. $y = 2^{x+1}$
α. $y = 2^{x-2}$
α. $y = -2^x$
**α
ααααΎα**
α. αααααααΆαααα’αα»αααα $y = 2^x + 1$
ααΆααΆααααααααααΌαααααΆαα x αα·α y
| x | y = 2^x | y = 2^x + 1 |
|---|---|---|
| -2 | 0.25 | 1.25 |
| -1 | 0.5 | 1.50 |
| 0 | 1 | 2 |
| 1 | 2 | 3 |
| 2 | 4 | 5 |
![graph_y_eq_2_pow_x_plus_1.png: Graph of y=2^x and y=2^x+1]
ααΆαααααΆαααααααααααΎαααΆ ααΎααααΈαααααααΆα $y = 2^x + 1$ ααααΌααααααααααΆα $y = 2^x$ αα½α
αααα·α α
ααα½ααα½αα―αααΆα‘αΎαααΎααααα’αααα (oy) ααααΆαααααΆα $y = 2^x + 1$ α
α. αααααααΆαααα’αα»αααα $y = 2^{x+1}$
ααΆααΆααααααααααΌαααααΆαα x αα·α y
| x | y = 2^x | y = 2^{x+1} |
|---|---|---|
| -2 | 0.25 | 0.5 |
| -1 | 0.5 | 1 |
| 0 | 1 | 2 |
| 1 | 2 | 4 |
| 2 | 4 | 8 |
![graph_y_eq_2_pow_x_plus_1_exponent.png: Graph of y=2^x and y=2^(x+1)]
ααΆαααααΆα ααααααααααΎαααΆ ααΎααααΈαααααααΆα $y = 2^{x+1}$ ααααΌααααααααααΆα $y = 2^x$ αα½α
αααα·α α
ααα½ααα½αα―αααΆαα
ααΆααααααααααα’αααα (ox) ααααΆαααααΆα $y = 2^{x+1}$ α
38
|
[11] Math - High
| 42
|
|
### ααααΌα α’ αααααααΈ α‘
α. αααααααΆαααα’αα»αααα $y = 2^{x-2}$
ααΆααΆααααααααααΌαααααΆαα x αα·α y
| x | y = 2^x | y = 2^{x-2} |
|---|---|---|
| -2 | 0.25 | 0.06 |
| -1 | 0.5 | 0.12 |
| 0 | 1 | 0.25 |
| 1 | 2 | 0.5 |
| 2 | 4 | 1 |
![graph_y_eq_2_pow_x_minus_2.png: Graph of y=2^x and y=2^(x-2)]
ααΆαααααΆα ααααααααααΎαααΆ ααΎααααΈαααααααΆα $y = 2^{x-2}$ ααααΌααααααααααΆα $y = 2^x$ αα½α
αααα·αα
ααα½αααΈαα―αααΆαα
ααΆαααααΆαααααα’αααα (ox) ααααΆαααααΆα $y = 2^{x-2}$ α
α. αααααααΆαααα’αα»αααα $y = -2^x$
ααΆααΆααααααααααΌαααααΆαα x αα·α y
| x | y = 2^x | y = -2^x |
|---|---|---|
| -2 | 0.25 | -0.25 |
| -1 | 0.5 | -0.5 |
| 0 | 1 | -1 |
| 1 | 2 | -2 |
| 2 | 4 | -4 |
![graph_y_eq_neg_2_pow_x.png: Graph of y=2^x and y=-2^x]
ααΆαααααΆα ααααααααααΎαααΆ ααααΌααααααααααΆα $y = 2^x$ αα½α
ααΌα ααααΆααααα»αααααΆ αααααΉαα’αααα (ox) ααααΆαααααΆα $y = -2^x$ α
39
|
[11] Math - High
| 43
|
|
**ααΆααΌαα
**
- ααΎααααΈαααααααΆαααα’αα»αααα $y = a^x + q$ ααααααΌααααααααΆα $y = a^x$ αα½α
ααααΎααααααα·αααααα’αααα (oy) α
ααα½α q α―αααΆα‘αΎαααΎααΎ $q>0$ α αΎαα
ααα½α q α―αααΆα
α»ααααααααΎ $q<0$ α
- ααΎααααΈαααααααΆαααα’αα»αααα $y = a^{x-p}$ ααααααΌααααααααΆα $y = a^x$ αα½α
ααααΎααααααα·αααααα’αααα (ox) α
ααα½α p α―αααΆαα
ααΆαααααΆαααΎ $p>0$ α αΎα p α―αααΆαα
ααΆααααααααΎ $p<0$ α
- ααΎααααΈαααααααΆαααα’αα»αααα $y = -a^x$ ααααααΌααααααααΆα $y = a^x$ αα½α
ααΌαααααΆααααα»ααααααΉαα’αααα (ox) ααααΆαααααΆα $y = -a^x$ α
#### ααααα·ααααα·
αααααααΆαα’αα»αααα
- α. $y = 3^x - 3$
- α. $y = 3^{x+1}$
- α. $y = 1.5^{x-2}$
- α. $y = 1.5^{x+3}$
- α. $y = -5^x$ α
### 2. αααΈααΆααα·ααα·αααΈααΆαα’α·α
αααααΌαααααααα
#### 2.1 αααΈααΆαα’α·α
αααααΌαααααααα
αααΈααΆαα’α·α
αααααΌαααααααα ααΆαααΈααΆααααααΆαα’ααααΆα ααΆαα·ααααααα α
##### α§ααΆα ααα
ααΎ $2^x = 2^4$ ααα $x=4$ α
**ααΆααΌαα
**
ααΎ a ααΆα
ααα½ααα·αααααΆααα»αααΈ 1 ααα $a^x = a^y$ ααααΆα $x=y$ α
40
|
[11] Math - High
| 44
|
|
### ααααΌα α’ αααααααΈ α‘
#### ααα αΆααααααΌ
αααααααΆααααΈααΆα
- α. $2^{3x+5} = 128$
- α. $5^{x-3} = \frac{1}{25}$
- α. $(\frac{1}{9})^x = 81^{x+4}$
- α. $49^x = 7^{x^2-15}$
- α. $36^{2x} = 216^{x-1}$
- α
. $10^{x-1} = 100^{2x-3}$ α
**α
ααααΎα**
- α. $2^{3x+5} = 128 \Rightarrow 2^{3x+5} = 2^7 \Rightarrow 3x+5=7 \Rightarrow x = \frac{2}{3}$ α
- α. $5^{x-3} = \frac{1}{25} \Rightarrow 5^{x-3} = 5^{-2} \Rightarrow x-3=-2 \Rightarrow x=1$ α
- α. $(\frac{1}{9})^x = 81^{x+4} \Rightarrow 9^{-x} = 9^{2(x+4)} \Rightarrow -x = 2x+8 \Rightarrow x = -\frac{8}{3}$ α
- α. $49^x = 7^{x^2-15} \Rightarrow 7^{2x} = 7^{x^2-15} \Rightarrow 2x = x^2-15 \Rightarrow x=5, x=-3$ α
- α. $36^{2x} = 216^{x-1} \Rightarrow 6^{2(2x)} = 6^{3(x-1)} \Rightarrow 4x = 3x-3 \Rightarrow x=-3$ α
- α
. $10^{x-1} = 100^{2x-3} \Rightarrow 10^{x-1} = 10^{2(2x-3)} \Rightarrow x-1 = 4x-6 \Rightarrow x = \frac{5}{3}$ α
#### ααααα·ααααα·
- α. $5^x \cdot 2^{\frac{2x-1}{x+1}} = 50$
- α. $27^x + 12^x = 2 \cdot 8^x$
- α. $3^{4x+8} - 4 \cdot 3^{2x+5} + 27 = 0$
- α. $3^{2x^2-6x+3} + 6^{x^2-3x+1} = 2^{2x^2-6x+3}$ α
#### 2.2 αα·αααΈααΆαα’α·α
αααααΌαααααααα
αα·αααΈααΆαα’α·α
αααααΌααααααααααΆααααααα $a^x > a^y$
- ααΎ $a>1$ αα·αααΈααΆα $a^x > a^y$ ααααΌα $x>y$ α αΎααα·αααΈααΆα $a^x < a^y$ ααααΌα $x<y$
- ααΎ $0<a<1$ αα·αααΈααΆα $a^x > a^y$ ααααΌα $x<y$ α αΎααα·αααΈααΆα $a^x < a^y$ ααααΌα $x>y$ α
##### α§ααΆα ααα
αααααααΆααα·αααΈααΆα $2^{3x+1} < \frac{1}{32}$ α
$2^{3x+1} < \frac{1}{32}$ αα·αααΈααΆαααΎα
41
|
[11] Math - High
| 45
|
|
$2^{3x+1} < 2^{-5}$ ααααα $\frac{1}{32} = \frac{1}{2^5}$ α¬ $2^{-5}$ α α ααα»ααα α’αααααΆααααΈαααΆααααααΌα
ααααΆ α
$3x+1 < -5$ αααααααααα·αααΈααΆαα’α·α
αααααΌαααααααα
$3x < -6$ αα 1 ααΈα’αααααΆααααΈα ααααΆα $x < -2$ α
ααα’αααααΆααααΈαααΉα 3 α
**αααααααααΆαα**
α±ααααααα x ααΌα
ααΆα -2 ααΌα
ααΆ $x=-3$
$2^{3x+1} < \frac{1}{32}$
$2^{3(-3)+1} < \frac{1}{32}$ αααα½α x ααα -3 ααααΆα $2^{-8} < \frac{1}{32}$ ααααα½α
$\frac{1}{256} < \frac{1}{32}$ αα·α (ααααΎαααααα $a^{-n} = \frac{1}{a^n}$)
#### ααα αΆααααααΌ
αααααααΆααα·αααΈααΆα
- α. $25^{2x} \le 5\sqrt{5}$
- α. $(\frac{2}{3})^{-3x} > \frac{16}{81}$
- α. $5^x > -7$
- α. $(0.1)^x > 10$
- α. $27^x \cdot 3^{1-x} < \frac{1}{3}$
- α
. $2^{9x-x^3} < 1$
- α. $(\frac{1}{3})^x + 3(\frac{1}{3})^{x+1} > 12$ α
**α
ααααΎα**
- α. $25^{2x} \le 5\sqrt{5} \Rightarrow 25^{2x} \le 5(5^{\frac{1}{2}}) \Rightarrow 5^{4x} \le 5^{\frac{3}{2}} \Rightarrow 4x \le \frac{3}{2} \Rightarrow x \le \frac{3}{8}$ α
- α. $(\frac{2}{3})^{-3x} > \frac{16}{81} \Rightarrow (\frac{2}{3})^{-3x} > (\frac{2}{3})^4 \Rightarrow -3x < 4 \Rightarrow x > -\frac{4}{3}$ α
- α. $5^x > -7$
$5^x$ αα·αααααΆαααΆαα·α
αα
$-7$ α’αα·αααααΆαααΆαα·α
αα
ααΌα
ααα αα·αααΈααΆαααΆαα«ααααααααααα $x \in \mathbb{R}$ α
- α. $(0.1)^x > 10 \Rightarrow 10^{-x} > 10 \Rightarrow -x > 1 \Rightarrow x < -1$ α
- α. $27^x \cdot 3^{1-x} < \frac{1}{3} \Rightarrow 3^{3x} \cdot 3^{1-x} < 3^{-1} \Rightarrow 3^{2x+1} < 3^{-1} \Rightarrow 2x+1 < -1 \Rightarrow 2x < -2 \Rightarrow x < -1$ α
- α
. $2^{9x-x^3} < 1 \Rightarrow 2^{9x-x^3} < 2^0 \Rightarrow 9x-x^3 < 0 \Rightarrow (9-x^2)x < 0 \Rightarrow -3 < x < 0$ α¬ $x > 3$ α
42
|
[11] Math - High
| 46
|
|
### ααααΌα α’ αααααααΈ α‘
- α. $(\frac{1}{3})^x + 3(\frac{1}{3})^{x+1} > 12 \Rightarrow [(\frac{1}{3})^x]^2 + (\frac{1}{3})^x - 12 > 0$, ($x \ne 0$) α
ααΆα $t = (\frac{1}{3})^x, t>0$ ααΎαααΆα $t^2+t-12>0$
$t<-4, t>3$
ααα $t>0$ ααΆαα±ααα«ααααααα·αααΈααΆαααΊ $t>3$
α¬ $(\frac{1}{3})^x > 3 = (\frac{1}{3})^{-1}$
ααα $\frac{1}{3} < 1$ ααα $\frac{1}{x} < -1$
$-1 < x < 0$
ααΌα
ααα αα·αααΈααΆαααΆαα
ααααΎα $-1 < x < 0$ α
#### ααααα·ααααα·
αααααααΆααα·αααΈααΆα
- α. $(7)^{3x+1} > 49$
- α. $(\frac{1}{5})^x < \sqrt[3]{0.04}$
- α. $3^x < \frac{1}{9\sqrt{3}}$
- α. $3^x \le -3$
- α. $(0.2)^x > 25$
- α
. $(0.1)^{4x^2-2x-2} < (0.1)^{2x-3}$ α
43
|
[11] Math - High
| 47
|
|
### ααα αΆαα
1. αααααααΆαααα’αα»ααααααΆαααααααααα»αααααα»ααααα½α :
- α. $f(x) = 2^x; g(x) = 5^x; h(x) = 10^x$
- α. $f(x) = (\frac{1}{2})^x; g(x) = (\frac{1}{5})^x; h(x) = (\frac{1}{10})^x$ α
2. α
αΌαααααααα a ααΎααααααααα $f(x) = a^x$ ααΆααααΆαα
ααα»α
ααΈαα½ααααΌα
ααΆαααααα α
- α. A(3, 216)
- α. B(5, 32)
- α. C(3, 512)
- α. D(4, 256)
- α. E(-2, 64)
- α
. F(-3, $\frac{1}{216}$)
- α. G(3, 343)
- α. H($\frac{1}{3}$, 3) α
3. αααα αΆαααΆ ααΎ $f(x) = a^x$ ααα $f(x)f(y) = f(x+y)$ α
4. - α. ααΎ $(x_1, y_1)$ αα·α $(x_2, y_2)$ ααΆα
ααα»α
ααΈααα
ααΎααααααα $f(x) = a^x$ αααα
ααα»α
ααΆαα ααΈα $(x_1+x_2, y_1y_2)$ αα·α $(x_1-x_2, \frac{y_1}{y_2})$ ααΆα
ααα»α
αα
ααΎααααααα α
- α. ααΎ $(x_1, y_1)$ ααΆα
ααα»α
ααΈααα
ααΎααααααα $f(x) = a^x$ αααα
ααα»α
ααΆααααΈα $(2x_1, y_1^2)$ αα·α $(-x_1, \frac{1}{y_1})$ ααΆα
ααα»α
αα
ααΎααααααα $f(x) = a^x$ α
5. - α. αααααααΆαααα’αα»αααα $f(x) = 2^x$ α
- α. αααααααΆαααα’αα»ααααααΈαα½αααααα»αααααα»ααααα½αααΆαα½αααααΆαααα’αα»αααα $f(x) = 2^x$
- i). $y = f(x)-1$
- ii). $y = f(x-1)$
- iii). $y = f(x+1)$
- iv). $y = f(0.5x)$
- v). $y = f(2x)$
- vi). $y = f(-x)$ α
44
|
[11] Math - High
| 48
|
|
### ααααΌα α’ αααααααΈ α‘
6. ααΎ $a>0$ α α
αΌαααααααα a αα·α x αααααααΎα±ααααααΆα αα·ααα·ααααΆαααΆαααααααααααααααΆαα
- α. $a^x = 1$
- α. $a^x > 1$
- α. $0 < a^x < 1$ α
7. αααααααΆαααα’αα»αααα
- α. $f(x) = 2^{|x|}$
- α. $f(x) = x(2^x)$
- α. $f(x) = x^x$ α
8. αααααααΆαααα’αα»ααααααΆαααααα
- α. $y = 2^{x-1}$
- α. $y = 2^{|x-1|}$
- α. $y = 2^x + 2^{-x}$
- α. $y = 2^{-x^2}$
- α. $y = 3^{-|x+1|^2}$
- α
. $y = 2^{|x^2-8|}$ α
9. αααααααΆααααΈααΆα
- α. $3^{x^2+4x} = \frac{1}{27}$
- α. $3^{5x} \cdot 9^{x^2} = 27$
- α. $4^{3x^2+2x+1} = 16$ α
45
|
[11] Math - High
| 49
|
|
## αααααααΈ 2 α’αα»ααααααααΆααΈα
### 1. ααααΆαααα’αα»ααααααααΆααΈα
α’αα»ααααααααΆααΈα ααΆα’αα»ααααα
αααΆαααα’αα»ααααα’α·α
αααααΌαααααααα α α ααα»αααααααΆαααααααΆαααα»αααααΆαααααΉααααααΆαα $y=x$ α
#### ααααα»αααα
- αααααααΆαααα’αα»ααααααααΆααΈα
- αααααααΆααααΈααΆααα·ααα·αααΈααΆαααααΆααΈα α
**α. ααΎααα a > 1**
αααααααΆαααα’αα»αααα $y=4^x$ αα·α $y = \log_4 x$ α
ααΆααΆααααααααααΌαααααΆαα x αα·α y
| y = 4^x | y = log_4 x |
|---|---|---|---|
| x | y | x | y |
| -2 | 0.06 | 0.25 | -1 |
| -1 | 0.25 | 0.50 | -0.50 |
| 0 | 1 | 1 | 0 |
| 1 | 4 | 2 | 0.50 |
| 2 | 16 | 4 | 1 |
![graph_log_4_x.png: Graph of y=4^x and y=log_4(x) showing reflection across y=x]
ααΆαααααΆα ααΎαααααααααΎαααΆ
- ααααα x ααΎα ααααΆααααααααααΌαααααΆαα $y = \log_4 x$ ααΎα
- ααααΆαααα’αα»αααα $y = \log_4 x$ ααΆααα’αααα (ox) αααααα
ααα»α
(1,0) ααΆαα·α
αα
- ααααΆαααα’αα»αααα $y = \log_4 x$ αα·α $y = 4^x$ αααα»αααααΆαααααΉααααααΆαα $y=x$ α
**ααΆααΌαα
**
ααΎ $a>1$ αααα’αα»αααα $y = \log_a x$ ααΆα’αα»ααααααΎα α
46
|
[11] Math - High
| 50
|
|
### ααααΌα α’ αααααααΈ α’
**α. ααΎααα 0 < a < 1**
αααααααΆαααα’αα»αααα $y = (\frac{1}{4})^x$ αα·α $y = \log_{\frac{1}{4}} x$
ααΆααΆααααααααααΌαααααΆαα x αα·α y
| y = (1/4)^x | y = log_(1/4) x |
|---|---|---|---|
| x | y | x | y |
| -2 | 16 | 0.25 | 1 |
| -1 | 4 | 0.50 | 0.50 |
| 0 | 1 | 1 | 0 |
| 1 | 0.25 | 2 | -0.50 |
| 2 | 0.06 | 4 | -1 |
![graph_log_one_fourth_x.png: Graph of y=(1/4)^x and y=log_(1/4)(x) showing reflection across y=x]
ααΆαααααΆα ααΎαααααααααΎαααΆ
- ααααα x ααΎα ααααΆααααααααααΌαααααΆαα $y = \log_{\frac{1}{4}} x$ α
α»α
- ααααΆαααα’αα»αααα $y = \log_{\frac{1}{4}} x$ ααΆααα’αααα (ox) αααααα
ααα»α
(1,0)
- ααααΆαααα’αα»αααα $y = \log_{\frac{1}{4}} x$ αα·α $y = (\frac{1}{4})^x$ αααα»αααααΆαααααΉααααααΆαα $y=x$ α
**ααΆααΌαα
**
ααΎ $0 < a < 1$ αααα’αα»αααα $y = \log_a x$ ααΆα’αα»ααααα
α»α α
**ααΆααααα·ααααΆα**
α’αα»αααα $y = \log_a x$ ααΆαααααΆα α
- ααΎ $a>1$ ααααΆαααΎαααΈααααααα
ααααΆα ααααΆ $y = \log_a x$ ααΆα’αα»ααααααΎα
- ααΎ $0<a<1$ ααααΆαα
α»αααΈααααααα
ααααΆα ααααΆ $y = \log_a x$ ααΆα’αα»ααααα
α»α
- α’αα»αααα $y = \log_a x$ ααΆαααααΆαααΆααααΆαα
ααα»α
αααααΆαααΌα’ααααα (1,0) ααΆαα·α
αα
- α’αα»αααα $y = \log_a x$ ααΆααααααααα $x>0$ ααΆαα·α
αα
- α’αα»αααα $y = \log_a x$ αα·α $y = a^x$ ααΆα’αα»ααααα
αααΆαααααΆ α αΎαααΆαααααΆααααα»αααααΆαααααΉααααααΆαα $y=x$ α
47
|
[11] Math - High
| 51
|
|
#### ααα αΆααααααΌ
αααααααΆαααα’αα»αααα
α. $y = \log_{10} x$
α. $y = \log_{\frac{1}{10}} x$
**α
ααααΎα**
α. ααΆααΆααααααααααΌαααααΆαα x αα·α y
| x | y = log_10 x |
|---|---|
| 1/100 | -2 |
| 1/10 | -1 |
| 1 | 0 |
| 10 | 1 |
| 100 | 2 |
![graph_log_10_x.png: Graph of y=log_10(x)]
ααΎαααααααααΎαααΆ α’αα»αααα $y = \log_{10} x$ ααΆαααααΆαααΎα α
α. ααΆααΆααααααααααΌαααααΆαα x αα·α y
| x | y = log_(1/10) x |
|---|---|
| 1/100 | 2 |
| 1/10 | 1 |
| 1 | 0 |
| 10 | -1 |
| 100 | -2 |
![graph_log_one_tenth_x.png: Graph of y=log_(1/10)(x)]
ααΎαααααααααΎαααΆ α’αα»αααα $y = \log_{\frac{1}{10}} x$ ααΆαααααΆαα
α»α α
#### ααααα·ααααα·
αααααααΆαααα’αα»αααα
α. $y = \log_7 x$
α. $y = \log_{\frac{1}{7}} x$ α
48
|
[11] Math - High
| 52
|
|
### ααααΌα α’ αααααααΈ α’
### 2. ααααααααααΆαααα’αα»ααααααααΆααΈα
αααααααΆαααα’αα»αααα
α. $y = -2 + \log_3 x$
α. $y = \log_3(x-2)$
α. $y = -\log_3 x$ α
**α
ααααΎα**
α. αααααααΆαααα’αα»αααα $y = -2 + \log_3 x$
ααΆααΆααααααααααΌαααααΆαα x αα·α y
| x | log_3 x | y = -2 + log_3 x |
|---|---|---|
| 1/3 | -1 | -3 |
| 1 | 0 | -2 |
| 3 | 1 | -1 |
| 9 | 2 | 0 |
![graph_log_3_x_minus_2.png: Graph of y=log_3(x) and y=-2+log_3(x)]
ααΎααααΈαααααααΆαααα’αα»αααα $y = -2 + \log_3 x$ ααααΌααααααααααΆαααα’αα»αααα $y = \log_3 x$ αα½α
αααα·αα
ααα½α 2 α―αααΆα
α»ααααααααααα’αααα (oy) ααααΆαααααΆαααα’αα»αααα $y = -2 + \log_3 x$ α
α. αααααααΆαααα’αα»αααα $y = \log_3(x-2)$
ααΆααΆααααααααααΌαααααΆαα x αα·α y
| x | y = log_3(x-2) |
|---|---|
| 7/3 | -1 |
| 3 | 0 |
| 5 | 1 |
| 11 | 2 |
![graph_log_3_x_minus_2_arg.png: Graph of y=log_3(x) and y=log_3(x-2)]
ααΎααααΈαααααααΆαααα’αα»αααα $y = \log_3(x-2)$ ααααΌααααααααααΆαααα’αα»αααα $y = \log_3 x$ αα½α
αααα·αα
ααα½α 2 α―αααΆ αα
ααΆαααααΆαααααα’αααα (ox) ααααΆαααααΆαααα’αα»αααα $y = \log_3(x-2)$ α
49
|
[11] Math - High
| 53
|
|
α. αααααααΆαααα’αα»αααα $y = -\log_3 x$
ααΆααΆααααααααααΌαααααΆαα x αα·α y
| x | y = -log_3 x |
|---|---|
| 1/9 | 2 |
| 1/3 | 1 |
| 1 | 0 |
| 3 | -1 |
| 9 | -2 |
![graph_neg_log_3_x.png: Graph of y=log_3(x) and y=-log_3(x)]
ααΎααααΈαααααααΆαααα’αα»αααα $y = -\log_3 x$ ααααΌααααααααααΆαααα’αα»αααα $y = \log_3 x$ αα½α
ααΌαααααΆααααα»αααααααΆαααααΉαα’αααα (ox) ααααΆαααααΆαααα’αα»αααα $y = -\log_3 x$ α
**ααΆααΌαα
:**
- ααΎααααΈαααααααΆαααα’αα»αααα $y = \log_a x + q$ ααααααΌααααααααΆα $y = \log_a x$ αα½α
ααααΎααααααα·αααααα’αααα (oy) α
ααα½α q α―αααΆα‘αΎαααΎααΎ $q>0$ α αΎαα
ααα½α q α―αααΆ α
α»ααααααααΎ $q<0$ α
- ααΎααααΈαααααααΆαααα’αα»αααα $y = \log_a(x-p)$ ααααααΌααααααααΆα $y = \log_a x$ αα½α
ααααΎααααααα·αααααα’αααα (ox) α
ααα½α p α―αααΆαα
ααΆαααααΆαααΎ $p>0$ α αΎα p α―αααΆαα
ααΆααααααααΎ $p<0$ α
- ααΎααααΈαααααααΆαααα’αα»αααα $y = -\log_a x$ ααααααΌααααααααΆα $y = \log_a x$ αα½α
ααΌαααααΆααααα»ααααααΉαα’αααα (ox) ααααΆαααααΆα $y = -\log_a x$ α
50
|
[11] Math - High
| 54
|
|
### ααααΌα α’ αααααααΈ α’
#### ααααα·ααααα·
αααααααΆαααα’αα»αααα
- α. $y = \log_7(x+3)$
- α. $y = \log_7 x + 3$
- α. $y = -\log_7 x$
- α. $y = \log_2(x-1)^2$
- α. $y = 2 - \log_2 x^2$ α
### 3. ααΌαααααααααΌαααα
ααααΉααααΈααααΆαααααα α’αΆα
α±ααα’ααααα·ααααΆα’αα»ααααααααΆααΈα ααααΎαααααΆααααααΌααααααααααΆααΈααα
ααΆ ααααααααα’αΆα
ααααΆααΆα α
ααΎ a, b αα·α x ααΆα
ααα½ααα·ααα·αααααΆα α αΎα $a \ne 1, b \ne 1$ ααααΆα $\log_a x = \frac{\log_b x}{\log_b a}$ α
**αααααΆααααααΆαα**
$\log_a x = \frac{\log_b x}{\log_b a}$ ?
ααΆα $M = \log_a x$
$a^M = x$ αααααααΆααααααα’α·α
αααααΌαααααααα
ααααΆα $\log_b a^M = \log_b x$ ααΎαααααΆααΈαααα b ααΎα’αααααΆααααΈα
$M \log_b a = \log_b x$ ααΆαααααααααααααΆααΈα
$M = \frac{\log_b x}{\log_b a}$ αααααααΆααα M
$\log_a x = \frac{\log_b x}{\log_b a}$ αααα½α M ααα $\log_a x$ α
#### ααα αΆααααααΌ
ααααΆ
- α. $\log_9 27$
- α. $\log_{27} \frac{1}{3}$
- α. $\log_7 27$
- α. $\log_5 125$ α
**α
ααααΎα**
- α. $\log_9 27 = \frac{\log_3 27}{\log_3 9} = \frac{\log_3 3^3}{\log_3 3^2} = \frac{3}{2}$ α
- α. $\log_{27} \frac{1}{3} = \log_{3^3} 3^{-1} = -\frac{1}{3}$ α
51
|
[11] Math - High
| 55
|
|
- α. $\log_7 27 = \frac{\log_{10} 27}{\log_{10} 7} \approx 1.6937$ α
- α. $\log_5 125 = \log_5 5^3 = 3$ α
#### ααααα·ααααα·
ααααΆ
- α. $\log_5 625$
- α. $\log_5 346$
- α. $\log_6 4870$ α
### 4. αααΈααΆααα·ααα·αααΈααΆαααααΆααΈα
#### 4.1 αααΈααΆαααααΆααΈα
ααΎ $a>0, a \ne 1$ ααααααΈααΆα $\log_a x = \log_a y$ ααααΆα $x=y$ α
##### α§ααΆα ααα 1
ααΎ $\log_4 x = \log_4 7$ ααα $x=7$ α
##### α§ααΆα ααα 2
αααααααΆααααΈααΆα $\log_4 x = \frac{5}{2}$ α
$\log_4 x = \frac{5}{2}$, $\log_4 x = \log_4 4^{\frac{5}{2}}$ ααααΎααΌααααα $\log_a a = 1$
$x = 4^{\frac{5}{2}} = (2^2)^{\frac{5}{2}}$ ααααα $4=2^2$
$x = 2^5$ α¬ $x=32$ α
##### α§ααΆα ααα 3
αααααααΆααααΈααΆα $6(\log_8 8 + \log_8 x) = 13$ α
$6(\frac{1}{\log_x 8} + \log_8 x) = 13$ ααααΎααΌαααααααααΌαααα $\log_b 8 = \frac{1}{\log_8 b}$
$6 + 6(\log_8 x)^2 = 13 \log_8 x$ ααααα
$6(\log_8 x)^2 - 13 \log_8 x + 6 = 0$
$(3\log_8 x - 2)(2\log_8 x - 3) = 0$
$\log_8 x = \frac{2}{3}, \log_8 x = \frac{3}{2}$
$x = 8^{\frac{2}{3}}, x = 8^{\frac{3}{2}}$
$x = 4, x = 16\sqrt{2}$ α
##### α§ααΆα ααα 4
αααααααΆααααΈααΆα $\log_9 x + \log_x 9 = \frac{5}{2}$
$\log_9 x + \frac{1}{\log_9 x} = \frac{5}{2} \Rightarrow (\log_9 x)^2 + 1 = \frac{5}{2} \log_9 x$
$2(\log_9 x)^2 - 5\log_9 x + 2 = 0$
52
|
[11] Math - High
| 56
|
|
### ααααΌα α’ αααααααΈ α’
$(2\log_9 x - 1)(\log_9 x - 2) = 0$ ααααΌα $\log_9 x = \frac{1}{2}, \log_9 x = 2$
ααΎ $\log_9 x = \frac{1}{2}$ α¬ $x = 9^{\frac{1}{2}} \Rightarrow x=3$
$\log_9 x = 2$ α¬ $x = 9^2 \Rightarrow x=81$ α
#### ααααα·ααααα·
αααααααΆααααΈααΆα
- α. $\log_9 x = \frac{3}{2}$
- α. $\log_x \frac{1}{10} = -3$
- α. $\log_x 9 = 2$ α
#### 4.2 αα·αααΈααΆαααααΆααΈα
- ααΎ $a>1$ ααααα·αααΈααΆα $\log_a x > \log_a y$ ααααΌα $x>y$ α αΎα $\log_a x < \log_a y$ ααααΌα $x<y$ α
- ααΎ $0<a<1$ ααααα·αααΈααΆα $\log_a x > \log_a y$ ααααΌα $x<y$ α αΎα $\log_a x < \log_a y$ ααααΌα $x>y$ α
##### α§ααΆα ααα
ααΎ $\log_3 x > \log_3 7$ ααα $x>7$ α
ααΎ $\log_{\frac{1}{2}} x > \log_{\frac{1}{2}} 7$ ααα $x<7$ α
#### ααα αΆααααααΌ
αααααααΆααα·αααΈααΆαααΆαααααα αα½α
αααααααααΆαα
- α. $\log_3(3x-5) > \log_3(x+7)$
- α. $\log_{\frac{1}{3}}(2x-1) \le 2$ α
**α
ααααΎα**
α. $\log_3(3x-5) > \log_3(x+7)$
$3x-5 > x+7$ αααααααααα·αααΈααΆαααααα’αα»ααααααααΆααΈα
$2x > 12$ ααααααααΌααααααα α¬ααααΎα’αααααΆααααΈααααα·αααΈααΆα
$x > 6$ α
ααα’αααααΆααααΈαααΉα 2 α
**αααααααααΆαα**
$\log(3x-5)$ ααΆααααααΎ $3x-5>0$ α¬ $x > \frac{5}{3}$
$\log(x+7)$ ααΆααααααΎ $x+7>0$ α¬ $x > -7$
ααΌα
ααα αα·αααΈααΆαααΆααααα»αα
ααααΎα $x>6$ α¬ $x \in (6, +\infty)$ α
53
|
[11] Math - High
| 57
|
|
α. $\log_{\frac{1}{3}}(2x-1) \le 2$
$\log_{\frac{1}{3}}(2x-1) \le \log_{\frac{1}{3}}(\frac{1}{3})^2$ ααααα $2 = \log_{\frac{1}{3}}(\frac{1}{3})^2$ ααΆαααααααααααΆααΈαααααα
$2x-1 \ge \frac{1}{9}$ ααααΌααα·ααα
ααααααααααααΎααΉα $\frac{1}{3} < 1$
$x \ge \frac{5}{8}$ α
**αααααααααΆαα**
$\log_{\frac{1}{3}}(2x-1)$ ααΆααααααΎ $2x-1>0$ α¬ $x > \frac{1}{2}$
ααΌα
ααα αα·αααΈααΆαααΆααααα»αα
ααααΎα $x \ge \frac{5}{8}$ α¬ $x \in [\frac{5}{8}, +\infty)$ α
#### ααααα·ααααα·
αααααααΆααα·αααΈααΆα
- α. $\log_5(x^2-6) > \log_5 x$
- α. $\log_{\frac{1}{5}} x < 0$
- α. $\log_x 27 \ge 3$ α
54
|
[11] Math - High
| 58
|
|
### ααα αΆαα
1. αααααα’αα»ααααα
αααΆαααα’αα»ααααααΆαααααα :
- α. $f(x) = 10^x$
- α. $g(x) = 3^x$
- α. $h(x) = 7^x$
- α. $f(x) = (\frac{1}{2})^x$
- α. $g(x) = (\frac{1}{5})^x$
- α
. $h(x) = (\frac{1}{10})^x$ α
2. αααααα’αα»ααααα
αααΆαααα’αα»ααααααΆαααααα :
- α. $f(x) = \log x$
- α. $g(x) = \log_3 x$
- α. $h(x) = \log_5 x$
- α. $f(x) = \log_{\frac{1}{3}} x$
- α. $g(x) = \log_{\frac{1}{4}} x$
- α
. $h(x) = \log_{2.1} x$ α
3. - α. αααααααΆαααα’αα»ααααα’α·α
αααααΌαααααααα $f(x) = 5^x$
- α. αααααααΆαααα’αα»ααααα
αααΆαααααα’αα»αααα $f(x) = 5^x$ αααα»αααααα»ααααα½α
- α. ααααααααΈααΆαα’αα»ααααα
αααΆαααααα’αα»ααααααΆαααΎ α
4. αααααααΆαααα’αα»ααααααΆαααααα :
- α. $f(x) = \log_6 x$
- α. $g(x) = \log_{\frac{1}{6}} x$
- α. $h(x) = \log_{0.8} x$ α
5. αααα αΆαααΆ ααΎ $f(x) = \log_a x$ ααα $f(xy) = f(x) + f(y)$ α
6. - α. αααα αΆαααΆ ααΎ $(x_1, y_1)$ αα·α $(x_2, y_2)$ ααΆα
ααα»α
ααΈααα
ααΎααααααα $y = \log_a x$ αααα
ααα»α
$(\frac{x_1}{x_2}, y_1-y_2)$ αααααα·ααα
ααΎααααααα $y = \log_a x$ α
- α. αααα αΆαααΆ ααΎ $(x_1, y_1)$ ααΆα
ααα»α
αα
ααΎααααααα $y = \log_a x$ αααα
ααα»α
$(x_1^2, 2y_1)$ αα·αα
ααα»α
$(\frac{1}{x_1}, -y_1)$ αααααα·ααα
ααΎααααααα $y = \log_a x$ α
55
|
[11] Math - High
| 59
|
|
7. ααα±ααα’αα»αααα $f(x) = a^x$ αα·αα’αα»ααααα
αααΆα $f^{-1}(x) = \log_a x$ ααα $a>0$ α ααααααα a ααΎααααΈα±αααααααααααα’αα»αααα $f(x)$ αα·α $f^{-1}(x)$ ααΆααααααΆ α
8. ααα±αα $f(x) = x - \log_2 x$ α αΎα $g(x) = 2^x$ α
ααααΆ
- α. $f(g(x))$
- α. $g(f(x))$ α
9. αααααααΆααααΈααΆααα·ααααααααααΆαα
- α. $\log_2(2x+4) - \log_2(x-1) = 3$
- α. $\log_2 x + \log_4 x = 5$
- α. $\log_5 x + \log_{10} \sqrt{x} = 5$
- α. $\log(x+10) + \frac{1}{2}\log x^2 = 2 - \log 4$ α
10. ααααααα m ααΎααααΈα±αααα·αααΈααΆα $1 + \log_5(x^2+1) \ge \log_5(mx^2+4x+m)$ αααααααααΆααα
ααααααααα x α
11. ααααααα a ααΎααααΈα±αααα·αααΈααΆα $\log_{\frac{1}{a+1}}(x^2+2) \ge 1$ ααΆααααα»αα«αα
ααααααααα x α
56
|
[11] Math - High
| 60
|
|
### ααααΌαα£ αααααααΈ α‘
# ααααΌα 3
# αααΈααΆααα·ααα·αααΈααΆαααααΈαααααΆααα
![pendulum_clock.png: Photo of an octagonal pendulum wall clock]
- **αααΈααΆααα·ααα·αααΈααΆαααααΈαααααΆααα**
αα
αααα»αααα·ααα·ααααΆααααα·αααΌαααααΆα ααΎαααΆααα·ααααΆααΈαααΈααΆααα·ααα·αααΈααΆαααααΈαααααΆαααααΆαα ααΌα
ααΆ $\cos x = a, \sin x = a, \tan x = t, \cos x > a, \sin x < a, \dots$ α
αααααααα ααΎαααΉααα·ααααΆααΈαααΈααΆααα·ααα·αααΈααΆαααααΈαααααΆαααααααααααααααααα αααααΆααααααααααΈααααα
ααΆααα»α α
57
|
[11] Math - High
| 61
|
|
## αααααααΈ 1 αααΈααΆααα·ααα·αααΈααΆαααααΈαααααΆααα
ααΎαααΆααα·ααααΆαα½α
ααα αΎαααΈαααΈααΆααα·ααα·αααΈααΆαααααΈαααααΆαααααΆαα αα
αααα»αααααα
ααα·ααα·ααααΆααααα·αααΌαααααΆα α αααα»αααααααααααΎαααΉααα·ααααΆααΈαααΈααΆααα·ααα·αααΈααΆαααααΈαααααΆααα αααααΆαααααααααααααααα α
#### ααααα»αααα
- αααααααΆααααΈααΆαααααΈαααααΆααα
- αααααααΆααα·αααΈααΆαααααΈαααααΆααα α
### 1. αααΈααΆαααααΈαααααΆααα
#### 1.1 αααΈααΆαααΊααααααΈ 1 αααααΉα sinx αα·α cosx
ααΆαααΈααΆαααααΈαααααΆαααααααααααααΈααααα½αα αΎαααΆαααΆα : $a \cos x + b \sin x = c$ α
αααααααααΆααααΈααΆααααααΆαααΈααααα :
**ααααααΈ 1**
- ααααααααΈααΆααααααΆααΆα $\cos(x-\theta) = \frac{c}{r}$ ααα $r = \sqrt{a^2+b^2}, (r \ge 0)$ αα·α $\cos \theta = \frac{a}{r}, \sin \theta = \frac{b}{r}$ α ααα $\cos \theta = \frac{a}{r}$ ααΆαα±αα $a = r \cos \theta$, $\sin \theta = \frac{b}{r}$ ααΆαα²αα $b = r \sin \theta$ α
- αααΈααΆα $a \cos x + b \sin x = r \cos \theta \cos x + r \sin \theta \sin x = c$
$r(\cos x \cos \theta + \sin x \sin \theta) = c, r \cos(x-\theta) = c$ α¬ $\cos(x-\theta) = \frac{c}{r}$ α
- αααααααΆααααΈααΆα $\cos(x-\theta) = \frac{c}{r}$ ααΆαααΌααααα $\cos x = \cos \alpha$ α
##### α§ααΆα ααα 1
αααααααΆααααΈααΆα $\sin x + \sqrt{3} \cos x = 0$ (1) α
ααααΆα $r = \sqrt{1+3} = 2, \cos \theta = \frac{a}{r} = \frac{\sqrt{3}}{2}, \sin \theta = \frac{b}{r} = \frac{1}{2}$ ααΆαα±αα $\theta = \frac{\pi}{6}$ α
αααΈααΆα $\sin x + \sqrt{3} \cos x = 2 \cos \frac{\pi}{3} \sin x + 2 \sin \frac{\pi}{3} \cos x = 0$
$2(\sin x \cos \frac{\pi}{3} + \cos x \sin \frac{\pi}{3}) = 0$
$2 \sin(x+\frac{\pi}{3}) = 0$ α¬ $\sin(x+\frac{\pi}{3}) = 0$
58
|
[11] Math - High
| 62
|
|
### ααααΌαα£ αααααααΈ α‘
αααΈααΆα (1) ααΆαα
ααααΎα : $\sin(x+\frac{\pi}{3}) = 0$ ααΆαα±αα $x+\frac{\pi}{3} = k\pi$ α
ααΌα
ααα $x = -\frac{\pi}{3} + k\pi, k \in \mathbb{Z}$ α α
ααα½αα
α»αααααΌα
ααααΎαααΆαααΈα α
**ααααααΈ 2**
ααααΆ $\sin x$ αα·α $\cos x$ ααΆα’αα»αααααα $t = \tan \frac{x}{2}$ ααα $x \ne \pi + 2k\pi, (k \in \mathbb{Z})$ α
αααααα½α $\cos x = \frac{1-t^2}{1+t^2}$ αα·α $\sin x = \frac{2t}{1+t^2}$ ααααααΈααΆα $a \cos x + b \sin x = c$ αα
ααΆαααΈααΆαααΊααααααΈ 2 αααααΆα t ααΆα’ααααΆα : $(c+a)t^2 - 2bt + c-a = 0$ α
αααααααΆααααΈααΆαααΊααααααΈ 2 ααα«ααα½α
αααααααΆααααΈααΆαααΆαααΌααααα $\tan \frac{x}{2} = \tan \alpha$ α
**αααααΆαα**
αα·ααΈαααα’αΆα
ααΆααα«αααααα $x = \pi + 2k\pi$ α ααΌα
ααα ααααΌαααααααΌααα·αα·αααααΎαααΆααΎ $x = \pi + 2k\pi$ ααΆα«ααααααααΈααΆα $a \cos x + b \sin x = c$ α¬αα ?
##### α§ααΆα ααα 2
αααααααΆααααΈααΆα $\sin x + 7 \cos x = 5$ α
ααΆα $t = \tan \frac{x}{2}$ ($x \ne \pi + 2k\pi$) αααΈααΆαααααα $\frac{2t}{1+t^2} + 7(\frac{1-t^2}{1+t^2}) = 5$
$2t + 7 - 7t^2 = 5 + 5t^2$ α¬ $6t^2 - t - 1 = 0$ α
αααΈααΆαααΆαα«α $t_1 = \frac{1}{2}, t_2 = -\frac{1}{3}$ α
- ααΎ $t_1 = \frac{1}{2}$ ααααΆα $\tan \frac{x}{2} = \frac{1}{2}$ αααααΆα $\alpha_1 \in \mathbb{R}$ ααα $\tan \alpha_1 = \frac{1}{2}$
ααΆαα±αα $\frac{x}{2} = \alpha_1 + k\pi$ α¬ $x = 2\alpha_1 + 2k\pi, k \in \mathbb{Z}$ α
- ααΎ $t_2 = -\frac{1}{3}$ ααααΆα $\tan \frac{x}{2} = -\frac{1}{3}$ αααααΆα $\alpha_2 \in \mathbb{R}$ ααα $\tan \alpha_2 = -\frac{1}{3}$
ααααΆα $\tan \frac{x}{2} = \tan \alpha_2$ ααΆαα±αα $\frac{x}{2} = \alpha_2 + k\pi$ α¬ $x = 2\alpha_2 + 2k\pi, k \in \mathbb{Z}$ α
ααΌα
ααα αααΈααΆαααΆαα
ααααΎα : $x = 2\alpha_1 + 2k\pi, x = 2\alpha_2 + 2k\pi, k \in \mathbb{Z}$ α
#### ααα αΆααααααΌ
αααααααΆααααΈααΆα $\sqrt{3} \sin 2x + \cos 2x = \sqrt{2}$ (1) α
**α
ααααΎα**
ααααΆα $\cos 2x + \sqrt{3} \sin 2x = \sqrt{2}$ α αΎα $r = \sqrt{1+3} = 2, \cos \theta = \frac{1}{2}, \sin \theta = \frac{\sqrt{3}}{2}$
αααΈααΆαααααα $2(\frac{1}{2} \cos 2x + \frac{\sqrt{3}}{2} \sin 2x) = \sqrt{2}$ α¬ $\cos(2x - \frac{\pi}{3}) = \frac{\sqrt{2}}{2} = \cos \frac{\pi}{4}$
αααΈααΆα (1) ααΆαα
ααααΎα :
$2x - \frac{\pi}{3} = \frac{\pi}{4} + 2k\pi, x = \frac{7\pi}{24} + k\pi, k \in \mathbb{Z}$
$2x - \frac{\pi}{3} = -\frac{\pi}{4} + 2k\pi, x = -\frac{\pi}{24} + k\pi, k \in \mathbb{Z}$
α
ααα½αα
α»αααααΌα
ααααΎαααΆα 4 α
59
|
[11] Math - High
| 63
|
|
#### ααααα·ααααα·
αααααααΆααααΈααΆα
- α. $2 \sin x - 3 \cos x = 3$
- α. $\cos 2x - \sin 2x = -1$
- α. $\cos x + \sqrt{3} \sin x = 1$
- α. $\cos x - \sqrt{3} \sin x = 3$ α
#### 1.2 αααΈααΆαααΊααααααΈ 2 αααααΉαα’αα»ααααααααΈαααααΆαααααα
ααα½ααα·α x
αααΈααΆαααΆαααααααΆαααΆα : $a \cos^2 x + b \cos x + c = 0, a \sin^2 x + b \sin x + c = 0$
$a \tan^2 x + b \tan x + c = 0$ αα·α $a \cot^2 x + b \cot x + c = 0$ α
ααΎααααΈαααααααΆααααΈααΆααααααα ααααααΌα :
- ααααααααααΈααΆαα±αααα
ααΆαααΈααΆαααΊααααααΈ 2 ααΌα
ααΆαααΎαα·α α
- ααΆαα’αα»ααααααααΈαααααΆαααααα
ααα½ααα·α x ααα X α¬ Y α¬ t... α
- ααΆαααααααααααα
ααΆαα’αα»ααααααααΈαααααΆααααααα±αα α
- αααααααΆααααΈααΆαααΊααααααΈ 2 αααααΆαα’ααααΆα X α¬ Y α¬ t... α
- ααα«ααααααΆααααααα½αααΆα’αα»ααααααααΈαααααΆααααα·α αα½α
αααααααααΆααααααααααααΆαααΎ α αΎααααααααΆαααΆαααΌααααααααΈααΆαααααΈαααααΆααα α
##### α§ααΆα ααα 1
αααααααΆααααΈααΆα $4 - \cos 2x - 7 \sin x = 0$ α
ααα $\cos 2x = 1 - 2 \sin^2 x$ αααΈααΆαα’αΆα
ααααα $2 \sin^2 x - 7 \sin x + 3 = 0$ (1)
ααΆα $t = \sin x$ αααααΆααααααααα $-1 \le t \le 1$ α
αααΈααΆα (1) α’αΆα
ααααα $2t^2 - 7t + 3 = 0, \Delta = 25, t_1 = \frac{1}{2}, t_2 = 3$ αα·ααα
αααααααααΆααααααααααααΆαααΎ
ααΌα
ααα $t_1 = \sin x = \frac{1}{2} = \sin \frac{\pi}{6}$ ααΎαααΆα $x_1 = \frac{\pi}{6} + 2k\pi, x_2 = \frac{5\pi}{6} + 2k\pi, k \in \mathbb{Z}$ α
##### α§ααΆα ααα 2
αααααααΆααααΈααΆα $\tan \frac{x}{2} - (1-\sqrt{3})\tan \frac{x}{2} - \sqrt{3} = 0$ α
ααααΆα $\tan \frac{x}{2} = t$ αααΈααΆα (1) ααααα $t^2 - (1-\sqrt{3})t - \sqrt{3} = 0$ αααααααΆααααΈααΆαααα
ααααΆα $t_1 = -1, t_2 = \sqrt{3}$ α $t_1 = \tan \frac{x}{2} = -1 = \tan(-\frac{\pi}{4}), t_2 = \tan \frac{x}{2} = \sqrt{3} = \tan \frac{\pi}{3}$ α
60
|
[11] Math - High
| 64
|
|
### ααααΌαα£ αααααααΈ α‘
ααΌα
ααα $\tan \frac{x}{2} = \tan(-\frac{\pi}{4}), \frac{x}{2} = -\frac{\pi}{4} + k\pi, x = -\frac{\pi}{2} + 2k\pi$
$\tan \frac{x}{2} = \tan \frac{\pi}{3}, \frac{x}{2} = \frac{\pi}{3} + k\pi, x = \frac{2\pi}{3} + 2k\pi, k \in \mathbb{Z}$ α
ααΌα
ααα αααΈααΆαααΆαα
ααααΎα : $x = -\frac{\pi}{2} + 2k\pi$ αα·α $x = \frac{2\pi}{3} + 2k\pi, k \in \mathbb{Z}$ α
#### ααα αΆααααααΌ
αααααααΆααααΈααΆα $\frac{\cos x(2 \sin x + 3\sqrt{2}) - 2 \cos^2 x - 1}{1+\sin 2x} = 1$ α
**α
ααααΎα**
αααααααα $1+\sin 2x \ne 0, \sin 2x \ne -1, 2x \ne -\frac{\pi}{2} + 2k\pi, x \ne -\frac{\pi}{4} + k\pi, k \in \mathbb{Z}$
αααΈααΆαααααα $\cos x(2 \sin x + 3\sqrt{2}) - 2 \cos^2 x - 1 = 1 + \sin 2x$
$2 \sin x \cos x + 3\sqrt{2} \cos x - 2 \cos^2 x - 1 = 1 + \sin 2x, 2 \cos^2 x - 3\sqrt{2} \cos x + 2 = 0$
ααΆα $t = \cos x$ αααΈααΆααα
ααΆ $2t^2 - 3\sqrt{2}t + 2 = 0$ α
αααααααΆααααΈααΆαααα ααααΆα $t_1 = \sqrt{2}, t_2 = \frac{1}{\sqrt{2}}$ α
$t_1 = \cos x = \sqrt{2}$ (αα·ααα) α $t_2 = \cos x = \frac{1}{\sqrt{2}} = \cos \frac{\pi}{4}$ α
ααΌα
ααα $\cos x = \cos \frac{\pi}{4}$ ααΆαα±αα $x = \pm \frac{\pi}{4} + 2k\pi, k \in \mathbb{Z}$ α
ααΆαααααααααααΆαααΎαααΈααΆαααΆαα«α $x = \frac{\pi}{4} + 2k\pi, k \in \mathbb{Z}$ α
#### ααααα·ααααα·
αααααααΆααααΈααΆα
- α. $2 \cos^2 x - 3\sqrt{2} \cos x + 2 = 0$
- α. $\frac{1}{\sin^2 x} = \cot x + 3$
- α. $5 \tan^2 y + 5 \tan y = 2(1+\tan^2 y)$
- α. $8 \sin^2 x - 6 \sin x = 5$ α
61
|
[11] Math - High
| 65
|
|
### 2. αααΈααΆααααααΆαααΆα $a \sin^2 x + b \sin x \cos x + c \cos^2 x = d$ ααα $a, b, c \ne 0$
ααΎααααΈαααααααΆααααΈααΆαααα ααααααΌαα
ααα’αααααΆααααΈααααααΈααΆαααΉα $\cos^2 x \ne 0, x \ne \frac{\pi}{2} + k\pi$ α
ααααΆααααΈααΆα $a \tan^2 x + b \tan x + c = 0$ αα½α
αααααααΆαααΆααααΈααΆαααΊααααααΈ 2 ααΌα
ααΆαααΎ α
#### α§ααΆα ααα 1
αααααααΆααααΈααΆα $3 \sin^2 x + 3 \sin x \cos x - 6 \cos^2 x = 0$ α
α
ααα’αααααΆααααΈααααααΈααΆαααΉα $\cos^2 x$ αααΈααΆαααααα $3 \tan^2 x + 3 \tan x - 6 = 0$ α
ααΆα $t = \tan x$ ααααΆα $3t^2 + 3t - 6 = 0$ αααΈααΆααααααΆαα«αααΈαααΊ : $t_1 = 1, t_2 = -2$ α
$t_1 = \tan x = 1$ ααααΆα $\tan x = \tan \frac{\pi}{4}, x = \frac{\pi}{4} + k\pi; (k \in \mathbb{Z})$ α $t_2 = -2$ ααΆα $\tan \alpha = -2$
$\tan x = \tan \alpha, x = \alpha + k\pi, k \in \mathbb{Z}$ α αααΈααΆαααΆαα
ααααΎα $x = \frac{\pi}{4} + k\pi$ αα·α $x = \alpha + k\pi, k \in \mathbb{Z}$ α
#### α§ααΆα ααα 2
αααααααΆααααΈααΆα $\frac{\sin^4 \frac{x}{2} + \cos^4 \frac{x}{2}}{1-\sin x} - \tan^2 x \sin x = \frac{1+\sin x}{2} + \tan^2 x$ α
αααααααα $\sin x \ne 1, \cos x \ne 0, x \ne \frac{\pi}{2} + k\pi$
$\frac{(\sin^2 \frac{x}{2} + \cos^2 \frac{x}{2})^2 - 2 \sin^2 \frac{x}{2} \cos^2 \frac{x}{2}}{1-\sin x} = \frac{1+\sin x}{2} + \tan^2 x(1+\sin x)$
$\frac{1 - \frac{1}{2} \sin^2 x}{1-\sin x} = (1+\sin x)(\frac{1}{2} + \tan^2 x)$
$\frac{1 - \frac{1}{2} \sin^2 x}{1-\sin x} = (1+\sin x)(\frac{\cos^2 x + 2 \sin^2 x}{2 \cos^2 x})$
$1 - \frac{1}{2} \sin^2 x = (1-\sin^2 x)(\frac{\cos^2 x + \sin^2 x + \sin^2 x}{2(1-\sin^2 x)})$
$2 - \sin^2 x = 1 + \sin^2 x$
$\sin^2 x = \frac{1}{2}, \sin x = \pm \frac{\sqrt{2}}{2}, x = \pm \frac{\pi}{4} + k\pi (k \in \mathbb{Z})$ ααΆα
ααααΎααααααααΈααΆα α
#### ααα αΆααααααΌ 1
αααααααΆααααΈααΆα $3 \sin^2 x + 2 \sin x \cos x = 2$ α
**α
ααααΎα**
αααΈααΆαααααα $3 \sin^2 x + 2 \sin x \cos x = 2(\sin^2 x + \cos^2 x)$
$\sin^2 x + 2 \sin x \cos x - 2 \cos^2 x = 0$ α
ααα’αααααΆααααΈααααααΈααΆαααΉα $\cos^2 x \ne 0$ α
ααααΆα $\tan^2 x + 2 \tan x - 2 = 0, \tan x = -1 \pm \sqrt{3}$ α ααΆα $\tan \alpha = -1 \pm \sqrt{3}$
$\tan x = \tan \alpha, x = \alpha + k\pi, k \in \mathbb{Z}$ α
62
|
[11] Math - High
| 66
|
|
### ααααΌαα£ αααααααΈ α‘
#### ααα αΆααααααΌ 2
ααααααα m αααααΆαα±αααααΈααΆα $1+m \cos x = m^2 - \cos^2 x$ ααΆαα«α α
**α
ααααΎα**
ααΆα $\cos x = t$ ααα $-1 \le t \le 1$ α
ααααΆα $t^2 + mt + 1 - m^2 = 0$ α ααΆα $f(t) = t^2 + mt + 1 - m^2$
αααΈααΆαααΆαα«ααα½αααΎα
ααααα $[-1, 1]$ ααΆαααΆ $f(-1) \times f(1) \le 0$ α
ααααΆα $(1-m+1-m^2)(1+m+1-m^2) \le 0$
$(2-m-m^2)(2+m-m^2) \le 0$
$(m^2+m-2)(m^2-m-2) \le 0, -2 \le m \le -1$ α¬ $1 \le m \le 2$ (1)
αααΈααΆαααΆαα«αααΈαααΎ $[-1, 1]$ ααΆαααΆ
$\begin{cases} \Delta \ge 0 \\ f(-1) > 0 \\ f(1) > 0 \\ -1 < -\frac{S}{2} < 1 \end{cases} \Rightarrow \begin{cases} m^2-4(1-m^2) \ge 0 \\ 1-m+1-m^2 > 0 \\ 1+m+1-m^2 > 0 \\ -1 < -\frac{m}{2} < 1 \end{cases} \Rightarrow \begin{cases} m \le -\frac{2}{\sqrt{5}}, \frac{2}{\sqrt{5}} \le m \\ -2 < m < 1 \\ -1 < m < 2 \\ -2 < m < 2 \end{cases}$
$-1 < m \le -\frac{2\sqrt{5}}{5}, \frac{2\sqrt{5}}{5} \le m < 1$ (2)
ααΆα (1) αα·α (2) ααΆαα±αααααΈααΆαααΆαα«αααΆαααΆ $-2 \le m \le -\frac{2\sqrt{5}}{5}$ α¬ $\frac{2\sqrt{5}}{5} \le m \le 2$ α
#### ααααα·ααααα· 1
αααααααΆααααΈααΆαααΆαααααα :
- α. $7 \cos^2 x + 6\sqrt{3} \sin x \cos x - \sin^2 x + 2 = 0$
- α. $2 \cos^2 x + \cot^2 x = \frac{\sin^3 x + 1}{\sin^2 x}$
- α. $\tan^3 x + \tan^2 x - 3 \tan x = 3$
- α. $\sin^2 x + 3 \cos^2 x - 2 \sin 2x = 0$ α
#### ααααα·ααααα· 2
ααααααα m ααΎααααΈα±αααααΈααΆα $\sin 4x = m \tan x$ ααΆαα«α $x \ne k\pi$ α
63
|
[11] Math - High
| 67
|
|
### 3. αααααααααααΈααΆαααααΈαααααΆααα
#### α§ααΆα ααα 1
αααααααΆααααααααααααΈααΆα $\begin{cases} x+y = \frac{\pi}{3} \\ \sin x + \sin y = 1 \end{cases}$
ααααΎααΌααααααααααααΈααααΌααα
ααΆαααα»α ααααΆααααΈααΆα
$\sin x + \sin y = 2 \sin(\frac{x+y}{2}) \cos(\frac{x-y}{2}) = 1$ ααα $x+y = \frac{\pi}{3}$ ααααΆα $2 \sin \frac{\pi}{6} \cos(\frac{x-y}{2}) = 1$ α
ααααΆα $\sin \frac{\pi}{6} = \frac{1}{2}$ ααΆαα±αα $\cos(\frac{x-y}{2}) = 1 = \cos 0$ ααααΆα $\frac{x-y}{2} = 2k\pi$ α¬ $x-y = 4k\pi$
ααααΆααααααααα $\begin{cases} x+y = \frac{\pi}{3} \\ x-y = 4k\pi \end{cases} \Rightarrow \begin{cases} x = \frac{\pi}{6} + 2k\pi \\ y = \frac{\pi}{6} - 2k\pi \end{cases} (k \in \mathbb{Z})$ α
ααΌα
ααααααααααααααΈααΆαααΆαααΌα
ααααΎα $(x = \frac{\pi}{6} + 2k\pi, y = \frac{\pi}{6} - 2k\pi), (k \in \mathbb{Z})$ α
#### α§ααΆα ααα 2
αααααααΆααααααααααααΈααΆα $\begin{cases} x+y = \frac{\pi}{2} & (1) \\ \tan x + \tan y = \sqrt{3} + \frac{\sqrt{3}}{3} & (2) \end{cases}$
αααααααα $x \ne \frac{\pi}{2} + k\pi, y \ne \frac{\pi}{2} + k\pi, k \in \mathbb{Z}$ α ααα $\tan y = \cot x$ ααΆααα»αααααα
ααααΆα $\begin{cases} x+y = \frac{\pi}{2} & (1) \\ \tan x + \tan y = \sqrt{3} + \frac{\sqrt{3}}{3} & (2) \end{cases} \Rightarrow \begin{cases} x+y = \frac{\pi}{2} \\ \tan^2 x - (\sqrt{3} + \frac{\sqrt{3}}{3})\tan x + 1 = 0 \end{cases}$
αααΈααΆα $\tan^2 x - (\sqrt{3} + \frac{\sqrt{3}}{3})\tan x + 1 = 0$ ααΆαα«αααΈαααΊ $t_1 = \sqrt{3}, t_2 = \frac{\sqrt{3}}{3}$ α
ααΌα
αααααααΆα $\begin{cases} x+y = \frac{\pi}{2} \\ \tan x = \sqrt{3} = \tan \frac{\pi}{3} \end{cases}$ α¬ $\begin{cases} x+y = \frac{\pi}{2} \\ \tan x = \frac{\sqrt{3}}{3} = \tan \frac{\pi}{6} \end{cases}$
$\begin{cases} x = \frac{\pi}{3} + k\pi \\ y = \frac{\pi}{6} - k\pi \end{cases}$ α¬ $\begin{cases} x = \frac{\pi}{6} + k\pi \\ y = \frac{\pi}{3} - k\pi \end{cases}$ ($k \in \mathbb{Z}$) ααΆα
ααααΎααααααααααααααααΈααΆα α
#### ααα αΆααααααΌ
ααααααα m αααααΆαα±αααααααααααααΈααΆαααΆααααααααΆαα«α
$\begin{cases} x-y = m & (1) \\ 2(\cos 2x + \cos 2y) - 1 - 4\cos^2 m = 0 & (2) \end{cases}$
64
|
[11] Math - High
| 68
|
|
### ααααΌαα£ αααααααΈ α‘
**α
ααααΎα**
ααααΆααααΈααΆα (2) ααααα : $4 \cos(x+y) \cos(y-x) - 1 - 4\cos^2 m = 0$ ααΆα (1)
αααα½αα
αΌα
ααααΆα : $4 \cos^2 m - 4 \cos(x+y) \cos m + 1 = 0, [2 \cos m - \cos(x+y)]^2 + \sin^2(x+y) = 0$
$\begin{cases} \sin(x+y) = 0 \\ \cos(x+y) = 2 \cos m \end{cases} \Rightarrow \begin{cases} x+y = 2k\pi \\ \cos m = \frac{1}{2} \end{cases}$ α¬ $\begin{cases} x-y = \pi + 2k\pi \\ \cos m = -\frac{1}{2} \end{cases}$
- α
αααα $\cos m = \frac{1}{2}, m = \pm \frac{\pi}{3} + 2k\pi$ ααα $\cos(x+y) = 1$ ααααΆα
$\begin{cases} x+y = 2k\pi \\ x-y = \pm \frac{\pi}{3} + 2k\pi \end{cases} \Rightarrow \begin{cases} x = \frac{\pi}{6} + k\pi \\ y = -\frac{\pi}{6} + k\pi, k \in \mathbb{Z} \end{cases}$
- α
αααα $\cos m = -\frac{1}{2}, m = \pm \frac{2\pi}{3} + 2k\pi$ ααα $\cos(x+y) = -1$
ααααΆααααααααα : $\begin{cases} x+y = \pi + 2k\pi \\ x-y = m \end{cases} \Rightarrow \begin{cases} x+y = (2k+1)\pi \\ x-y = m \end{cases}, \begin{cases} m = \frac{m}{2} + (2k+1)\frac{\pi}{2} \\ y = -\frac{m}{2} + (2k+1)\frac{\pi}{2} \end{cases}$
ααΌα
ααα
- $m = \pm \frac{\pi}{3} + 2k\pi$ ααααααααααΆαα«α $\begin{cases} x = \frac{m}{2} + k\pi \\ y = -\frac{m}{2} + k\pi \end{cases}$
- $m = \pm \frac{2\pi}{3} + 2k\pi$ ααααααααααΆαα«α $\begin{cases} x = \frac{m}{2} + (2k+1)\frac{\pi}{2} \\ y = -\frac{m}{2} + (2k+1)\frac{\pi}{2}, k \in \mathbb{Z} \end{cases}$
#### ααααα·ααααα· 1
αααααααΆααααααααααααΈααΆα
- α. $\begin{cases} \cos^2 x + \cos^2 y = \frac{1}{4} \\ x+y = \frac{5\pi}{6} \end{cases}$
- α. $\begin{cases} \sin x \cos y = \frac{1}{4} \\ 3 \tan x = \tan y \end{cases}$
#### ααααα·ααααα· 2
ααα±αααααααααααααΈααΆα $\begin{cases} \sin x \cos 2y = m^2+1 \\ \cos x \sin 2y = m \end{cases}$ α
- α. αααααΆααααααα m αααααΆαα±αααααααααααααΈααΆαααΆαα«α α
- α. αααααααΆααααααααααααΈααΆαααΆαααααα m αααααααΆαααα α
65
|
[11] Math - High
| 69
|
|
### 4. αα·αααΈααΆαααααΈαααααΆααα
#### 4.1 αα·αααΈααΆααααααΆαααΆα $a \cos x + b \sin x > c$
##### α§ααΆα ααα 1
αααααααΆααα·ααΈαααΆα $\sin x - \cos x > 0$ α
ααααΆα $\sqrt{2}(\frac{\sqrt{2}}{2} \sin x - \frac{\sqrt{2}}{2} \cos x) > 0, \sqrt{2}(\sin x \cos \frac{\pi}{4} - \sin \frac{\pi}{4} \cos x) > 0, \sqrt{2} \sin(x-\frac{\pi}{4}) > 0$ α
$\sin(x-\frac{\pi}{4}) > 0$ ααααΌαααΉαα
α»αααααΌαα
ααΎαααααααααααααααα·ααααα½α
![unit_circle_inequality.png: Unit circle showing the solution for sin(angle)>0]
ααΌα
ααααα·αααΈααΆαααΆαα
ααααΎα $2k\pi < x - \frac{\pi}{4} < \pi + 2k\pi$
α¬ $\frac{\pi}{4} + 2k\pi < x < \frac{5\pi}{4} + 2k\pi (k \in \mathbb{Z})$ α
**αααααΆαα**
ααΎαα·αααΈααΆααααααΆαααααααααΆαααααααααααΆαα¬αα·ααΈαααΆαααΊααααααΈ 2 ααααααΌααααααααΆααααααα»αα«αααααα·ααααΆαααααΆ α
##### α§ααΆα ααα 2
αααααααΆααα·αααΈααΆα $\sin x + \frac{1}{\sin x} \ge \frac{5}{2}$ α
ααααΆα $\sin x + \frac{1}{\sin x} - \frac{5}{2} \ge 0, \sin x + \frac{1}{\sin x} - \frac{5}{2} \ge 0$
$\frac{2 \sin^2 x - 5 \sin x + 2}{2 \sin x} \ge 0$
$\frac{(\sin x - 2)(2 \sin x - 1)}{2 \sin x} \ge 0$ ααα $\sin x - 2 < 0$ α
ααααααααα $x \in \mathbb{R}$
ααααΆα $\frac{2 \sin x - 1}{2 \sin x} \le 0$ (1)
αα·αααΈααΆαααΆαα
ααααΎααα»αααααΆαα $\sin x \ne 0$ α¬ $x \ne k\pi$ α
ααΆα $\sin x = X$ ααα $-1 \le X \le 1$ αα·αααΈααΆα (1) αα
ααΆ $\frac{2X-1}{2X} \le 0$ αα·ααααΆαααααΆααααΆα
| X | -1 | | 0 | | 1/2 | | 1 |
|---|---|---|---|---|---|---|---|
| 2X-1 | | - | | - | 0 | + | |
| X | | - | 0 | + | | + | |
| (2X-1)/X | | + | \| | - | 0 | + | |
![unit_circle_solution.png: Unit circle showing the solution interval for sin(x)]
66
|
[11] Math - High
| 70
|
|
### ααααΌαα£ αααααααΈ α‘
ααααΆα $0 \le X \le \frac{1}{2}, 0 < X \le \frac{1}{2}, 0 < \sin x \le \frac{1}{2}$ ααααΌαααΉα $2k\pi < x \le \frac{\pi}{6} + 2k\pi$
α¬ $\frac{5\pi}{6} + 2k\pi \le x < \pi + 2k\pi, k \in \mathbb{Z}$ α
α»αααααΌα
ααααΎααα
ααΎααααΌααααΈααΆααα (ααααααα) α
#### ααα αΆααααααΌ 1
αααααααΆααα·αααΈααΆα $2 \cos 2x + \sin^2 x \cos x + \sin x \cos^2 x > 2(\sin x + \cos x)$ α
**α
ααααΎα**
ααααΆα $2 \cos 2x + \sin^2 x \cos x + \sin x \cos^2 x > 2(\sin x + \cos x)$
$2(\cos^2 x - \sin^2 x) + \sin x \cos x(\sin x + \cos x) - 2(\sin x + \cos x) > 0$
$(\cos x + \sin x)[2(\cos x - \sin x) + \sin x \cos x - 2] > 0$
ααΆα $f(x) = (\cos x + \sin x)[2(\cos x - \sin x) + \sin x \cos x - 2]$
ααα $f(x)$ ααΆα’αα»αααααααααΆααα½α $2\pi$ αααααα’αΆα
αα·ααααΆαα·αααΈααΆααααα»αα
ααααα $(0, 2\pi)$ α
ααααΆα $f(x) = 0, \cos x + \sin x = 0$ (1) α¬ $2(\cos x - \sin x) + \sin x \cos x - 2 = 0$ (2)
(1) : $\tan x = -1, (x = \frac{3\pi}{4}, x = \frac{7\pi}{4})$ α
(2) ααΆα $t = \cos x - \sin x = \sqrt{2} \cos(x+\frac{\pi}{4})$ ααα $-\sqrt{2} \le t \le \sqrt{2}$
$t^2 = 1 - 2 \sin x \cos x, \sin x \cos x = \frac{1-t^2}{2}$
(2) α’αΆα
ααααα $2t + \frac{1-t^2}{2} - 2 = 0$ α¬ $t^2 - 4t + 3 = 0$
αααααααΆααααΈααΆαααα ααααΆα $t=1; t=3$ (αα·ααα)
$t=1, \sqrt{2} \cos(x+\frac{\pi}{4}) = 1, \cos(x+\frac{\pi}{4}) = \frac{\sqrt{2}}{2} = \cos \frac{\pi}{4}$ α ααααΆα $x=0, x = \frac{3\pi}{2}$ α
ααααΆααααααΆααΌα
ααΆαααααα
| x | 0 | | $\frac{3\pi}{4}$ | | $\frac{3\pi}{2}$ | | $\frac{7\pi}{4}$ | | $2\pi$ |
|---|---|---|---|---|---|---|---|---|---|
| f(x) | | 0 | + | 0 | | + | 0 | |
ααΌα
ααα αα·αααΈααΆαααΆααααα»αα
ααααΎα $\frac{3\pi}{4} + 2k\pi < x < \frac{3\pi}{2} + 2k\pi$ α¬
$\frac{7\pi}{4} + 2k\pi < x < 2\pi + 2k\pi; (k \in \mathbb{Z})$ α
67
|
[11] Math - High
| 71
|
|
#### ααα αΆααααααΌ 2
αααα αΆαααΆαααα»αααααΈααα ABC ααααΆα $\cos A + \cos B + \cos C \le \frac{3}{2}$
**α
ααααΎα**
ααααΆα $\frac{3}{2} - (\cos A + \cos B + \cos A) = \frac{1}{2}[3-2(\cos A + \cos B + \cos C)]$
$= \frac{1}{2}[1-4 \cos \frac{A+B}{2} \cos \frac{A-B}{2} + 2 - 2 \cos C]$
$= \frac{1}{2}[1-4 \cos \frac{A+B}{2} \cos \frac{A-B}{2} + 2(1-\cos C)]$
$= \frac{1}{2}[1-4 \cos \frac{A+B}{2} \cos \frac{A-B}{2} + 4 \sin^2 \frac{C}{2}]$
$= \frac{1}{2}[1-4 \sin \frac{C}{2} \cos \frac{A-B}{2} + 4 \sin^2 \frac{C}{2}]$
ααααα $\cos \frac{A+B}{2} = \sin \frac{C}{2}$
$= \frac{1}{2}[(2 \sin \frac{C}{2} - \cos \frac{A-B}{2})^2 + 1 - \cos^2 \frac{A-B}{2}]$
$= \frac{1}{2}[(2 \sin \frac{C}{2} - \cos \frac{A-B}{2})^2 + \sin^2 \frac{A-B}{2}] \ge 0$ α
ααΌα
ααα $\frac{3}{2} \ge \cos A + \cos B + \cos C$ α¬ $\cos A + \cos B + \cos C \le \frac{3}{2}$ α
#### ααααα·ααααα· 1
αααααααΆααα·αααΈααΆαααΆαααααα :
- α. $\sin x + \cos x > \cos \frac{\pi}{6}$
- α. $\frac{2 \cos x - 5 \sin x}{\cos x} > 0$
- α. $\frac{\cos x}{1-3 \cos x} < \frac{1-\cos x}{1-9 \cos^2 x}$
- α. $2 \sin^2(x+\frac{\pi}{4}) + \sqrt{3} \cos 2x > 0$ α
#### ααααα·ααααα· 2
αααα αΆαααΆαααα»αααααΈααα ABC ααααΆα $\sin^2 A + \sin^2 B + \sin^2 C \le \frac{9}{4}$ α
68
|
[11] Math - High
| 72
|
|
### ααααΌαα£ αααααααΈ α‘
## ααααααααααα
ααΎααααΈαααααααΆααααΈααΆα $a \cos x + b \sin x = c$ ααααΆαααΈααααα :
1. ααααααααααΈααΆααααααΆααΆα $\cos(x-\theta) = \frac{c}{r}$ ααα $r = \sqrt{a^2+b^2}, (r \ge 0)$ αα·α $\cos \theta = \frac{a}{r}$, $\sin \theta = \frac{b}{r}$ αα½α
αααααααΆααααΈααΆα $\cos(x-\theta) = \frac{c}{r}$ ααΆαααΌααααα $\cos x = \cos a$ α
2. ααααΆ $\sin x$ αα·α $\cos x$ ααΆα’αα»αααααα $\tan \frac{x}{2} (x \ne \pi + 2k\pi, k \in \mathbb{Z})$ α
αααααα½α $\cos x = \frac{1-t^2}{1+t^2}$ αα·α $\sin x = \frac{2t}{1+t^2}$ ααααααΈααΆα $a \cos x + b \sin x = c$ αα
ααΆ
αααΈααΆαααΊααααααΈ 2 αααααΆα t ααΆα’ααααΆα : $(c+a)t^2 - 2bt + c-a = 0$ α ααα«ααααΈααΆαααΊααααααΈ2
αα½α
αααααααΆααααΈααΆαααΆαααΌααααα $\tan \frac{x}{2} = \tan a$ α
ααΎααααΈαααααααΆααααΈααΆα : $a \cos^2 x + b \cos x + c = 0, a \sin^2 x + b \sin x + c = 0$
$a \tan^2 x + b \tan x + c = 0$ αα·α $a \cot^2 x + b \cot x + c = 0$ ααααααΌα :
- ααααααααααΈααΆαα±αααα
ααΆαααΈααΆαααΊααααααΈ2 ααΌα
ααΆαααΎαα·α
- ααΆαα’αα»ααααααααΈαααααΆαααααα
ααα½ααα·α x ααα X α¬ Y α¬ t...
- ααΆαααααααααααα
ααΆαα’αα»ααααααααΈαααααΆααααααα±αα
- αααααααΆααααΈααΆαααΊααααααΈ 2 αααααΆαα’ααααΆα X α¬ Y α¬ t...
- ααα«ααααααΆααααααα½αααΆα’αα»ααααααααΈαααααΆααααα·α αα½α
αααααααααΆααααααααααααΆαααΎ
α αΎααααααααΆαααΆαααΌααααααααΈααΆαααααΈαααααΆααα α
ααΎααααΈαααααααΆααα·αααΈααΆααααααΆααααααα $a \cos x + b \sin x < c$ α¬ $a \cos x + b \sin x > c,...$ ααααααΌα :
- αααααα’αααααΈ 1 αααα·αααΈααΆαα±ααααΆαααΆα $\cos(x-\theta)$ α¬ $\sin(x-\theta)$
- αααααααΆααα·αααΈααΆα $\cos(x-\theta) > \cos a$ α¬ $\sin(x-\theta) < \sin a,...$
ααΆααααααααααααΆααα·αααΈααΆα $\cos x > a$ α¬ $\sin x < a$ α
ααΎαα·αααΈααΆααααααΆαααααααααΆαααααααααααΆα α¬αα·αααΈααΆαααΊααααααΈ 2 ααααααΌααααααααΆααα
αααα»αα«αααααα·ααααΆαααααΆ α
69
|
[11] Math - High
| 73
|
|
## = ααα αΆαα
1. αααααααΆααααΈααΆαααΆαααααα α
- α. $\cos x + \sqrt{3} \sin x = \cos 3x$
- α. $\sin 3x + 2 \cos x - 2 = 0$
- α. $\sin 2x + \tan x = 2$
- α. $\sin 5x + \cos 5x = \sqrt{2} \cos 13x$
- α. $6 \sin x - 2 \cos^3 x = 5 \sin 2x \cos x$
- α
. $\sqrt{5} \cos x - \cos 2x + 2 \sin x = 0$ α
2. αααααααΆααααΈααΆαααΆαααααα α
- α. $1 + \cot^2 2x = \frac{1-\cos 2x}{\sin^2 2x}$
- α. $\cos^4 x + \sin^4 x = \frac{7}{16}$
- α. $(1-\tan x)(1+\sin 2x) = 1 + \tan x$
- α. $3 \sin 3x - \sqrt{3} \cos 9x = 1 + 4 \sin^3 3x$
- α. $1 + 3 \cos x + \cos 2x = \cos 3x + 2 \sin x \sin 2x$
- α
. $\cos^4 x + \sin^6 x = \cos^2 2x$ α
3. ααα±αααααΈααΆα $\cos 2x - (2m+1)\cos x + m + 1 = 0$ (1) α
- α. αααααααΆααααΈααΆα (1) ααΆαααΆ $m = \frac{3}{2}$ α
- α. ααααααα m αααααααΎα±αααααΈααΆαααΆαα«α x αα
α
ααααα $\frac{\pi}{2} < x < \frac{3\pi}{2}$ α
4. αααααααΆααααααααααααΈααΆαααααΈαααααΆαααααΆαααααα α
- α. $\begin{cases} \sin(x+y) = \frac{1}{2} \\ \cos(x-y) = \frac{\sqrt{2}}{2} \end{cases}$
- α. $\begin{cases} \sin x + \sin y = \sqrt{2} \\ \cos x + \sin y = \sqrt{2} \end{cases}$
- α. $\begin{cases} \sin(x+y) = \cos(x-y) \\ \tan x - \tan y = 1 \end{cases}$ α
70
|
[11] Math - High
| 74
|
|
### ααααΌαα£ αααααααΈ α‘
5. αααααααΆααα·αααΈααΆαααΆαααααα α
- α. $\sin^2(\frac{x}{2} - \frac{\pi}{4}) < \cos^2 \frac{2x}{2}$
- α. $6 \sin^2 x - \sin x \cos x - \cos^2 x > 2$
- α. $\frac{\cos x}{1+2 \cos x} > \frac{1-\cos x}{1-2 \cos x}$
- α. $\frac{1-\sin x}{1-3 \sin x} < \frac{1+\sin x}{1-9 \sin^2 x}$ α
6. αααα αΆαααΆ
- α. $\sin^4 \alpha + \cos^4 \alpha \ge \frac{1}{2}$
- α. $\sin^6 \alpha + \cos^6 \alpha \ge \frac{1}{4}$
- α. $\sin^8 \alpha + \cos^8 \alpha \ge \frac{1}{8}$ α
7. αααα αΆαααΆαααα»α $\triangle ABC$ ααααΆα : $\sin \frac{A}{2} \cdot \sin \frac{B}{2} \cdot \sin \frac{C}{2} \le \frac{1}{8}$ α
8. αααα αΆαααΆ $\triangle ABC$ αααααααααααααααα α $\tan A \tan B \tan^2 \frac{C}{2} = 1$ ααΆααααΈααααααααααΆα α
## ααα αΆααααααΌα
1. αααααααΆααααΈααΆαααΆαααααα α
- α. $2 \cos 3x + \sqrt{3} \sin x + \cos x = 0$
- α. $\sin^6 x + \cos^8 x = 2(\sin^8 x + \cos^8 x)$
- α. $\frac{\sin 5x}{5 \sin x} = 1$
- α. $\sin^2 x + \sin^2 3x = \cos^2 2x + \cos^2 4x$
- α. $\cos^3 x + \frac{3}{4} \sin 2x - 2 \cos x = 0$
- α
. $\sin x + \sin 2x + \sin 3x + \sin 4x + \sin 5x + \sin 6x = 0$ α
71
|
[11] Math - High
| 75
|
|
2. αααααααΆααα·αααΈααΆαααΆαααααα α
- α. $2 \cos^2 x - \cos x + 1 \le 0$ αααα»α $[0, \pi]$
- α. $\frac{2 \sin^2 x - \sin x - 1}{\sin x} > 0$ αααα»α $[0, \pi]$
- α. $\frac{\sin x - \cos x + 1}{\sin x + \cos x - 1} > 0$ α
3.
- α. αααααΆααααααα $(x+\frac{1}{x})(x-8)(x-1)$ α
- α. αααααααΆααααΈααΆα $2 \sin^3 2x - 17 \sin^2 x + 7 \sin x + 8 = 0$ α
4. ααα±αααααΈααΆα $\cos 2x - (2m+1)\cos x + m + 1 = 0$ (1) α
- α. αααααααΆααααΈααΆα (1) ααΆαααΆ $m=1$
- α. ααααααα m αααααααΎα±αααααΈααΆαααΆαα«ααα
α
ααααα $[0, \pi]$ α
5. ααααΆααα·αααΈααΆα (E) : $2 \sin^2 x - 5 \sin x + 2 > 0$ α
ααααΆα $X = \sin x$ αα·αααΈααΆαααααα $2X^2 - 5X + 2 > 0$ α
- α. ααΆααααΆαααα»ααααααΆαααααα $2x^2 - 5x + 2$ α
- α. αααα αΆαααΆαα·αααΈααΆα (E) αααααααΆααΆα $2(\sin x - 2)(\sin x - \frac{1}{2}) > 0$ α
- α. αα·ααααΆαααααΆ $(\sin x - 2)(\sin x - \frac{1}{2})$ αα
ααΎα
ααααα $[0, 2\pi]$ α
- α. αααααα»αα
ααααΎααααα·αααΈααΆα (E) α
6. αααααααΆααααααααααααΈααΆαααααΈαααααΆαααααΆαααααα α
- α. $\begin{cases} \sin x + \sin y = \frac{3}{2} \\ \sin^2 x + \sin^2 y = \frac{5}{4} \end{cases}$
- α. $\begin{cases} \cos^3 x - \cos x + \sin y = 0 \\ \sin^3 x - \sin y + \cos x = 0 \end{cases}$
- α. $\begin{cases} \frac{1-\tan x}{1+\tan x} = \tan y \\ x-y = \frac{\pi}{6} \end{cases}$ α
7. ααααΆα $\triangle ABC$ αααααΆααα»α αα·ααααα»αααααααααααααα $\frac{1+\cos A}{1+\cos B} = \frac{2a+c}{2a-c}$ α
αααα αΆαααΆ $\triangle ABC$ ααΆααααΈαααααααΆα α
8. αα»α A, B, C αα $\triangle ABC$ αα½αααΆα $\frac{A+C}{2} = B$ α
αααααααΆαααα»αααααααΈααααααααΎααααΉαααΆ $\sin A + \sin B + \sin C = \frac{3+\sqrt{3}}{2}$ α
72
|
[11] Math - High
| 76
|
|
### ααααΌα α€ αααααααΈ α‘
# ααααΌα 4 αααααααΈααα’ααα
TAKAKAZU SEKI
## β αααααααΈααα’ααα
α’ααααααααΆαα±ααααΆααααααααα·αααΆααααΈααα»ααα ααΊ Arthur Cayley (1875-1921) α ααΆααα·α αααααααΈαααααΆαααΎαα‘αΎααα»ααααΆααααΈα α αΎαααΆαααααααα
αααα»αα
ααααΎαααΌαα
αααααααααααααΈααΆαααΈααα’ααα α ααΎαααααΎααααααααααΈαααααααΌαααΆααααααααα Leibniz αα
ααααΆα 1678 αααααααααααα
αααα»ααααααααααα»α TAKAKAZU SEKI ααΆααααααα
ααααΆαααααααααααΎαααα αΆααααααα»αααααΆα 1680 α
αα
α
α»αααααααααΈ 19 Cayley, Jame Syluester and Ferdinand Frobenius ααΆααα·ααααΆααααΉααααΈαααΆααααΈα αα·ααααααααΈααααααΆααααΈααααα
α αααααΆαααΆαααΈαα
ααααΎααααα»αααΈαααα·αααΈααα’ααα αααΆααααΈαααααΌαααΆααααααα
αααααααΆααααα»ααααααααΈααα’ααα α¬α’αα»ααααααΈααα’ααα α αΎαααααααΆαααΆαααααααα·ααααΆαααααα»αααααΉααααΈααΈαααα·αααΈααα’ααα α
73
|
[11] Math - High
| 77
|
|
### αααααααΈ 1 αααααααΈααα’ααα
## 1. αααααααΈααα’ααα
### 1.1 αααααΆα
ααΎαααααΆααααααα½α
ααα αΎαααΌααααααα
ααα»α
αα½αα
ααα½ααα
αααα»αααααα α
**α§ααΆα ααα** ααΎαααααααα
ααα»α
$M(x, y)$ αα
ααΆ $M'(x', y')$ ααΆαααααα H αααααΆαααα
α·α O αα·αααααα 4 αααααααΆααααΈααΆα
$x' = 4x$
$y' = 4y$
αααΈααΆααααααααα
αΆααααααααααααααααα α
ααΆαααΈααΆαααΊααααααΈ 1 α¬α α
ααΆαααΈααΆαααΈααα’ααα α
αααα»αααααΈααα ααααΆαααααα
αΆααααΆαααααααΈααα’ααα α
**ααΆααΌαα
** αααααααΈααα’αααααααααααα
ααα»α
$M(x, y)$ αα
$M'(x', y')$
αααααααααααΈααΆα
$x' = ax + cy$
$y' = bx + dy$
ααΎααααΈααΆααααα½ααα·ααααΆαα
αα»α ααα’αΆα
ααααααΆααααααααΈααα’αααα±ααααΆαααααααααΆαααΆααααΈα
$\begin{bmatrix} x' \\ y' \end{bmatrix} = \begin{bmatrix} a & c \\ b & d \end{bmatrix} \begin{bmatrix} x \\ y \end{bmatrix}$ α
αααα»αααααΈαααααααΆ $A = \begin{bmatrix} a & c \\ b & d \end{bmatrix}$ ααΆαααΆααααΈααααααααααΈααα’ααα α
α
ααα»α
$\begin{bmatrix} x' \\ y' \end{bmatrix}$ ααΆααΌαααΆααα $\begin{bmatrix} x \\ y \end{bmatrix}$ ααΆααααααααΈααα’ααα A α
**ααααα»αααα**
- ααααααΆααααααααΈααα’αααααΆαααΆααααΈα
- αααααααΌαααΆαααΆααααααααΈααα’ααα
- αααααααΌαααΆαααΆααααααααΈααα’ααααααααΆαα
- αααααααΌαααΆαααΆααααααααΈααα’ααααααααΆαααααα»αααααΈαααααΆααα α
74
|
[11] Math - High
| 78
|
|
### ααααΌα α€ αααααααΈ α‘
ααα’αΆα
ααααΆ $x', y'$ ααΆαααααααα»ααααΆααααΈα $\begin{bmatrix} a & c \\ b & d \end{bmatrix}$ αα·α $\begin{bmatrix} x \\ y \end{bmatrix}$ αα½α
ααααΉααα½αααΆααααΈαααα’αααααΈ 1 αα·ααα½αααΆααααΈαααα’αααααΈ 2 α
**αααααΆαα**
ααΎααααΈααααα½ααααα»αααααα ααα’αΆα
ααααΎα’αααα A ααααααααααααΆαααααααΆααααΆααΆαααΆααααΈααα αα·αααΆαααααααΈααα’ααααα α
#### α§ααΆα ααα 1
ααΌαααΆαααα
ααα»α
$(2, \frac{1}{2})$ ααΆαααααα $A = \begin{bmatrix} -\frac{1}{2} & 1 \\ 4 & 3 \end{bmatrix}$ ααΊ
$\begin{bmatrix} x' \\ y' \end{bmatrix} = \begin{bmatrix} -\frac{1}{2} & 1 \\ 4 & 3 \end{bmatrix} \begin{bmatrix} 2 \\ \frac{1}{2} \end{bmatrix} = \begin{bmatrix} -\frac{1}{2} \times 2 + 1 \times \frac{1}{2} \\ 4 \times 2 + 3 \times \frac{1}{2} \end{bmatrix} = \begin{bmatrix} -\frac{1}{2} \\ \frac{19}{2} \end{bmatrix}$
ααΌα
ααα ααΌαααΆααα $(2, \frac{1}{2})$ ααΊ $(-\frac{1}{2}, \frac{19}{2})$ α
α
αααααααααααΈααα’ααααααααΆααααΆααααΈα $I = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}$
ααΌαααΆαααΆααΊ
$\begin{bmatrix} x' \\ y' \end{bmatrix} = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix} \begin{bmatrix} x \\ y \end{bmatrix} = \begin{bmatrix} x+0 \\ 0+y \end{bmatrix} = \begin{bmatrix} x \\ y \end{bmatrix}$ α
ααααααααααΎαααΆ ααΌαααΆααα $\begin{bmatrix} x' \\ y' \end{bmatrix}$ αααα½αααΎααΆαααΎα $\begin{bmatrix} x \\ y \end{bmatrix}$ α
ααααΆ I ααΆαααααααΈααα’ααααααα α ααΆααααααΆα
ααα»α
ααααααααααα I αα
ααααΆα
ααα»α
ααααααα α
#### α§ααΆα ααα 2
ααΌαααΆαααα
ααα»α
$(-\frac{1}{2}, 3)$ ααΆα $I = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}$ ααΊ
$\begin{bmatrix} x' \\ y' \end{bmatrix} = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix} \begin{bmatrix} -\frac{1}{2} \\ 3 \end{bmatrix} = \begin{bmatrix} -\frac{1}{2} \\ 3 \end{bmatrix}$ α
ααΌα
ααα $x' = -\frac{1}{2}, y' = 3$ α
75
|
[11] Math - High
| 79
|
|
#### ααα αΆααααααΌ 1
αααα αΆαααΆ ααΎαααΆααααΈα A αααααα
ααα»α
$M \ne 0$ αα
ααααΆα
ααα»α
M αααα ααααααΆααααΈα $A = I$ α
**α
ααααΎα**
ααΆα $A = \begin{bmatrix} a & c \\ b & d \end{bmatrix}$ ααΆαααΆααααΈααααααααα $M(x, y)$ αα
$M(x, y)$ αααα
ααΎαααΆα $\begin{bmatrix} x \\ y \end{bmatrix} = \begin{bmatrix} a & c \\ b & d \end{bmatrix} \begin{bmatrix} x \\ y \end{bmatrix}$ α¬ $\begin{bmatrix} x \\ y \end{bmatrix} = \begin{bmatrix} ax+cy \\ bx+dy \end{bmatrix}$
ααΎαααΆα
$x = ax+cy$
$y = bx+dy$
$x(1-a) - cy = 0$ (1)
$y(1-d) - bx = 0$ (2) α
ααα $M \ne 0$ ααα $x \ne 0, y \ne 0$ α αΎααααΈααΆα (1) αα·α (2) αααααααααΆαα
αα»αααααΆ $\begin{cases} 1-a=0 \\ c=0 \end{cases}$ αα·α $\begin{cases} 1-d=0 \\ b=0 \end{cases}$ ααΆαα±αα $\begin{cases} a=1; b=0 \\ c=0; d=1 \end{cases}$
ααΎαααΆα $A = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}$ α
#### ααα αΆααααααΌ 2
ααααααααΆααααΈα A ααααααααααΈα
ααα»α
(1, 2) αα
α
ααα»α
(8, 1) αα·ααααααααΈα
ααα»α
(-1, 1) αα
(1, 2) α
**α
ααααΎα**
ααΆα $A = \begin{bmatrix} a & c \\ b & d \end{bmatrix}$ ααΆαααΆααααΈααααααααΈα
ααα»α
(1, 2) αα
α
ααα»α
(8, 1) αααααΎαααΆα
$\begin{bmatrix} 8 \\ 1 \end{bmatrix} = \begin{bmatrix} a & c \\ b & d \end{bmatrix} \begin{bmatrix} 1 \\ 2 \end{bmatrix}$
ααΎαααΆααααΈααΆα
$a+2c = 8$ (1)
$b+2d = 1$ (2)
$\begin{bmatrix} 1 \\ 2 \end{bmatrix} = \begin{bmatrix} a & c \\ b & d \end{bmatrix} \begin{bmatrix} -1 \\ 1 \end{bmatrix}$
ααΎαααΆααααΈααΆα
$-a+c = 1$ (3)
$-b+d = 2$ (4)
αααααααΆααααααααααααΈααΆα (1) αα·α (3) ααΎαααΆα $a=2, c=3$ α
76
|
[11] Math - High
| 80
|
|
### ααααΌα α€ αααααααΈ α‘
αααααααΆααααααααααααΈααΆα (2) αα·α (4) ααΎαααΆα $b = -1, d = 1$ α
ααΌα
ααα αααΆααααΈααααααααΌαααααΊ $A = \begin{bmatrix} 2 & 3 \\ -1 & 1 \end{bmatrix}$ α
#### ααααα·ααααα·
1. αααααααα
ααα»α
$M(x, y)$ αα
$M'(x', y')$ ααΆααααΆααααΈα A ααα
$x' = x \cos \alpha - y \sin \alpha, y' = x \sin \alpha + y \cos \alpha$ α ααααααααΆααααΈα A α
2. ααααα $\alpha$ ααΎααααΈα±ααα
ααα»α
M' αααα½αααΎ M α
### 1.2 α
ααα»α
α₯ααααααααα½α
#### α§ααΆα ααα
αααααααΌαααΆαααα
ααα»α
$P(3, 2)$ αα·α $Q(5, 2)$ ααααααααααααααααααΈααα’αα $A = \begin{bmatrix} 3 & -5 \\ 2 & -4 \end{bmatrix}$ α
ααΆααααααΆααααΌαααΆααα P αααααααα :
$\begin{bmatrix} x' \\ y' \end{bmatrix} = \begin{bmatrix} 3 & -5 \\ 2 & -4 \end{bmatrix} \begin{bmatrix} 3 \\ 2 \end{bmatrix} = \begin{bmatrix} 3 \times 3 - 5 \times 2 \\ 2 \times 3 - 4 \times 2 \end{bmatrix} = \begin{bmatrix} -1 \\ -2 \end{bmatrix}$ ααΌα
ααα $P'(-1, -2)$ α
ααΌαααΆααα Q αααααααα :
$\begin{bmatrix} x' \\ y' \end{bmatrix} = \begin{bmatrix} 3 & -5 \\ 2 & -4 \end{bmatrix} \begin{bmatrix} 5 \\ 2 \end{bmatrix} = \begin{bmatrix} 3 \times 5 - 5 \times 2 \\ 2 \times 5 - 4 \times 2 \end{bmatrix} = \begin{bmatrix} 5 \\ 2 \end{bmatrix}$ ααΌα
ααα $Q'(5, 2)$ α
ααΆαααααα A ααΌαααΆα $P(3, 2)$ ααΊ $P'(-1, -2)$ α αΎα $Q(5, 2)$ ααΊ $Q'(5, 2)$ ααα
ααααααααααΎαααΆ α
ααα»α
Q αα·αααααΆααααααΌαααΈααΆαααα
ααΆα A ααααα αααααααΌαααΆαααααααΆ
Q' αα
ααααΆα
ααα»α
Q αααα α
αααα»αααααΈααα ααααΆ Q ααΆα
ααα»α
α₯ααααααααα½αααΆαααααα A α
**ααΆααΌαα
** Q ααΆα
ααα»α
α₯ααααααααα½αααΆαααααα A ααΆαααΆ $A(Q) = Q$ α
- α
αααααααααα
αΆαα $H(0, k)$ αααααααα $x' = kx, y' = ky$ α
- ααα O ααΊααΆα
ααα»α
α₯ααααααααα½α α ααααα $O(0, 0)$ αααααααΌαααΆα $O(0, 0)$ αααα α
- α
ααααααααααααααΆααααΆααααΈαα―αααΆ I ααΊ $I = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}$ α
77
|
[11] Math - High
| 81
|
|
αααααα
ααα»α
αααααααααααααααααΆα I αα
ααααΆα
ααα»α
αααα½αα―ααααα α
ααααα $\begin{bmatrix} x' \\ y' \end{bmatrix} = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix} \begin{bmatrix} x \\ y \end{bmatrix} = \begin{bmatrix} x \\ y \end{bmatrix}$ α
#### ααα αΆααααααΌ 1
αααααααα
ααα»α
$M(x, y)$ αα
$M'(x', y')$ αααααααα
$\begin{bmatrix} x' \\ y' \end{bmatrix} = \begin{bmatrix} 2 & -1 \\ 3 & 4 \end{bmatrix} \begin{bmatrix} x \\ y \end{bmatrix} + \begin{bmatrix} 3 \\ -3 \end{bmatrix}$ α
αααααα
ααα»α
α₯ααααααααα½α α
**α
ααααΎα**
ααΎ $M(x, y)$ ααΆα
ααα»α
α₯ααααααααα½α αα»αααααΆαα $x' = x, y' = y$
$\begin{bmatrix} x \\ y \end{bmatrix} = \begin{bmatrix} 2 & -1 \\ 3 & 4 \end{bmatrix} \begin{bmatrix} x \\ y \end{bmatrix} + \begin{bmatrix} 3 \\ -3 \end{bmatrix}$
$\begin{bmatrix} x \\ y \end{bmatrix} = \begin{bmatrix} 2x-y+3 \\ 3x+4y-3 \end{bmatrix}$
ααΎαααΆααααααααα $\begin{cases} 2x-y+3 = x \\ 3x+4y-3 = y \end{cases}$ α¬ $\begin{cases} x-y = -3 \\ 3x+3y = 3 \end{cases}$
αααααααΆααααααααα ααΎαααΆα $x = -1, y = 2$ α
ααΌα
ααα α
ααα»α
α₯ααααααααα½αααΊ $(-1, 2)$ α
#### ααα αΆααααααΌ 2
αααα αΆαααΆαααααΆαα $y = 2x-1$ αααααααΆααααΆααααΈα $\begin{bmatrix} 3 & -1 \\ 4 & -1 \end{bmatrix}$ ααΆαααΌαααΆααααα½αα―ααααα α
**α
ααααΎα**
αα $M(x, y)$ ααααααα
$M'(x', y')$ ααΆααααΆααααΈα $\begin{bmatrix} 3 & -1 \\ 4 & -1 \end{bmatrix}$ αααααΎαααΆα
$\begin{bmatrix} x' \\ y' \end{bmatrix} = \begin{bmatrix} 3 & -1 \\ 4 & -1 \end{bmatrix} \begin{bmatrix} x \\ y \end{bmatrix}$ α
ααΎ M ααΆα
ααα»α
αα½ααααααααΆαα $y = 2x-1$ ααα M ααΆαααΌα’ααααα $(a, 2a-1)$ α
ααΌα
ααα $\begin{bmatrix} x' \\ y' \end{bmatrix} = \begin{bmatrix} 3 & -1 \\ 4 & -1 \end{bmatrix} \begin{bmatrix} a \\ 2a-1 \end{bmatrix} = \begin{bmatrix} 3a-2a+1 \\ 4a-2a+1 \end{bmatrix} = \begin{bmatrix} a+1 \\ 2a+1 \end{bmatrix}$ α
αααααΎαααΆα $x' = a+1, y' = 2a+1$ α
αααααααΆαα a ααΎαααΆα $y' = 2x' - 1$ αααααΆααΌαααΆααααααααΆαα $y = 2x-1$ α
αααααΆααααΎα αα·αααΌαααΆαααΆααΆααααΈααΆαααΌα
ααααΆααΊ $y = 2x-1$ α
78
|
[11] Math - High
| 82
|
|
### ααααΌα α€ αααααααΈ α‘
#### ααααα·ααααα·
1. αααααααα
ααα»α
$M(x, y)$ αα
$M'(x', y')$ αααααααα
- α. $x' = 2y-3, y' = x+1$
- α. $\begin{bmatrix} x' \\ y' \end{bmatrix} = \begin{bmatrix} 3 & 3 \\ -2 & -1 \end{bmatrix} \begin{bmatrix} x \\ y \end{bmatrix} + \begin{bmatrix} -4 \\ 2 \end{bmatrix}$ α
αααααα
ααα»α
α₯ααααααααα½α α
2. αααα αΆαααΆαααααΆαα $y = 2x-3$ αααααααΆααααΆααααΈα $\begin{bmatrix} 2 & 1 \\ 3 & -1 \end{bmatrix}$ ααΆαααΌαααΆααααα½αα―ααααα α
3. ααα
ααα»α
α₯ααααααααα½αααΆαααααα $\begin{bmatrix} 2 & 1 \\ 2 & 3 \end{bmatrix}$ α
4. ααα
ααα»α
α₯ααααααααα½αααΆαααααα $\begin{bmatrix} 3 & 2 \\ 3 & 4 \end{bmatrix}$ α
### 1.3 αααααααΈααα’ααα
αααΆα
#### α§ααΆα ααα 1
αααααααα
ααα»α
$M(x, y)$ αα
$M'(x', y')$ ααΆααααααα
αΆαααααααΆααααΆααααΈα
$H = \begin{bmatrix} k & 0 \\ 0 & k \end{bmatrix}$ ααα $k \ne 0$ α
ααααΆα $\begin{bmatrix} x' \\ y' \end{bmatrix} = \begin{bmatrix} k & 0 \\ 0 & k \end{bmatrix} \begin{bmatrix} x \\ y \end{bmatrix} = \begin{bmatrix} kx \\ ky \end{bmatrix}$ ααΆαα±ααααααΆα $x' = kx, y' = ky$
ααααΆ M' ααΆααΌαααΆααα M ααΆαααααα H α
α₯α‘αΌαααα αααα M ααΆααΌαααΆααα M' αα·α α αΎααααα»αααααΈαααααααααΌαααααΆ x, y ααΆα’αα»αααααα x' αα·α y' α
ααααΆα $x = \frac{1}{k}x'$ αα·α $y = \frac{1}{k}y'$ ααααααΆααααΈααΆααααααΆαααΆααααΈα ααΊ
$\begin{bmatrix} x \\ y \end{bmatrix} = \begin{bmatrix} \frac{1}{k} & 0 \\ 0 & \frac{1}{k} \end{bmatrix} \begin{bmatrix} x' \\ y' \end{bmatrix}$ αααααααΆ M ααΆααΌαααΆααα M' ααΆααααααα
αΆαα H α
ααααααααααααα
αΆαααα H ααα $H^{-1}$ αααααΆααααΆααααΈα $\begin{bmatrix} \frac{1}{k} & 0 \\ 0 & \frac{1}{k} \end{bmatrix}$ α
79
|
[11] Math - High
| 83
|
|
ααΌα
ααα $H^{-1} = \begin{bmatrix} \frac{1}{k} & 0 \\ 0 & \frac{1}{k} \end{bmatrix}$ ααΆαααΆααααΈαα
αααΆααα $H = \begin{bmatrix} k & 0 \\ 0 & k \end{bmatrix}$ α
**ααΆααΌαα
**
ααααααα $M(x, y)$ αα
$M'(x', y')$ ααΆαααααα $A = \begin{bmatrix} a & c \\ b & d \end{bmatrix}$ ααα
$M' = A(M)$ α¬ $\begin{bmatrix} x' \\ y' \end{bmatrix} = \begin{bmatrix} a & c \\ b & d \end{bmatrix} \begin{bmatrix} x \\ y \end{bmatrix}$ ααΆαα±αα $x' = ax+cy, y' = bx+dy$ α
ααΎααααΈααααααααααα
αααΆα $A^{-1}$ αααααααα $M'(x', y')$ αα
$M(x, y)$ αα·αααα ααααααΌα
ααααΆ $(x, y)$ ααΆα’αα»αααααα $(x', y')$ α
$\begin{cases} ax+cy = x' \\ bx+dy = y' \end{cases}$ α¬ $\begin{cases} adx+cdy = dx' \\ -bcx-cdy = -cy' \end{cases}$ ααααΌα
$(ad-bc)x = dx' - cy'$ αααα»αααααΈ $ad-bc \ne 0$ ααΆαα±ααααααΆα
$x = \frac{1}{ad-bc}(dx'-cy'), y = \frac{1}{ad-bc}(-bx'+ay')$
ααααααΆαααΆαααΆααααΈα ααααΆα
$\begin{bmatrix} x \\ y \end{bmatrix} = \frac{1}{ad-bc} \begin{bmatrix} d & -c \\ -b & a \end{bmatrix} \begin{bmatrix} x' \\ y' \end{bmatrix}$ α¬ $M = A^{-1}(M')$ α
ααααΆ $A^{-1}$ ααΆαααααα
αααΆααααααΆααααΆααααΈαα
αααΆα $A^{-1} = \frac{1}{ad-bc} \begin{bmatrix} d & -c \\ -b & a \end{bmatrix}$ ααα $ad-bc \ne 0$ α
| α ααα»ααα αααΆααααΈαα
αααΆααα $A = \begin{bmatrix} a & c \\ b & d \end{bmatrix}$ ααΊ $A^{-1} = \frac{1}{ad-bc} \begin{bmatrix} d & -c \\ -b & a \end{bmatrix}$ ααα $ad-bc \ne 0$ α |
|---|
#### ααα αΆααααααΌ 1
L ααΆαααααΆαααααααΆααααΈααΆα $x+3y-3=0$ α
ααααααΆααααΈαα L ααΆααααΆααααΈα $\begin{bmatrix} 1 & -1 \\ 2 & 4 \end{bmatrix}$ α
**α
ααααΎα**
ααΆα L' ααΆααααΆααααΈαα $L: x+3y-3=0$ ααΆααααΆααααΈα $\begin{bmatrix} 1 & -1 \\ 2 & 4 \end{bmatrix}$ α
80
|
[11] Math - High
| 84
|
|
### ααααΌα α€ αααααααΈ α‘
ααΎααααΈααααα L' ααααααΌαααα
ααα»α
A', B' αααααΆααΌαααΆααααααααΆααααα
ααα»α
A, B αααααααΆαα L α αΎαααααΆααααΈαα L ααΊααΆαααααΆαααααααΆααααΆαα
ααα»α
A', B' α
αααα $A(0, 1)$ αα·α $B(3, 0)$ ααΆα
ααα»α
αα L αααααααΆα
ααΌαααΆααα A' αααααααα $\begin{bmatrix} 1 & -1 \\ 2 & 4 \end{bmatrix} \begin{bmatrix} 0 \\ 1 \end{bmatrix} = \begin{bmatrix} -1 \\ 4 \end{bmatrix}$
ααΆαα±αα $A'(-1, 4)$ α
ααΌαααΆααα B' ααΊ $\begin{bmatrix} 1 & -1 \\ 2 & 4 \end{bmatrix} \begin{bmatrix} 3 \\ 0 \end{bmatrix} = \begin{bmatrix} 3 \\ 6 \end{bmatrix}$
ααΆαα±αα $B'(3, 6)$ α
ααΌα
ααα αααααΆαα L' ααΆααααΆαααΈαα
ααα»α
A' αα·α B' αααααααααααΈααΆα : $L': x-2y+9=0$ α
![graph_of_line_transformation.png: Graph showing line L and its transformation L']
**ααΆααΌαα
** ααΎααααΈαααααααΌαααΆααααααααΆαα L ααααααΌαααααΆ x, y ααΆα’αα»αααααα x', y' αα½α
αααα
αααα½ααααα»ααααΈααΆα $L: x+3y-3=0$ α
ααΎ $A = \begin{bmatrix} 1 & -1 \\ 2 & 4 \end{bmatrix}$ ααα $A^{-1} = \frac{1}{6} \begin{bmatrix} 4 & 1 \\ -2 & 1 \end{bmatrix}$
$\begin{bmatrix} x \\ y \end{bmatrix} = A^{-1} \begin{bmatrix} x' \\ y' \end{bmatrix} = \frac{1}{6} \begin{bmatrix} 4 & 1 \\ -2 & 1 \end{bmatrix} \begin{bmatrix} x' \\ y' \end{bmatrix} = \frac{1}{6} \begin{bmatrix} 4x'+y' \\ -2x'+y' \end{bmatrix}$
ααΎαααΆα $x = \frac{4x'+y'}{6}, y = \frac{-2x'+y'}{6}$
αααα½ααααα»ααααΈααΆααααααΆαα L: $\frac{4x'+y'}{6} + 3(\frac{-2x'+y'}{6}) - 3 = 0$ α¬ $x'-2y'+9=0$ α
ααΌα
ααα αααααΆαα $L: x+3y-3=0$ ααααΆααααΈαααααΆαα $L': x-2y+9=0$ α
#### ααα αΆααααααΌ 2
αααααααΌαααΆααααααααα $C: x^2+y^2=4$ ααΆα $A = \begin{bmatrix} 3 & 0 \\ 0 & 3 \end{bmatrix}$ α
81
|
[11] Math - High
| 85
|
End of preview. Expand
in Data Studio
README.md exists but content is empty.
- Downloads last month
- 3