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受欢迎的三角函数 >

6tanh^2(x)+5sech(x)-7=0

解答

6 tanh^{2} (x) + 5 sech (x) - 7 = 0

解答

x = ln (0.17157 \dots), x = ln (0.26794 \dots), x = ln (3.73205 \dots), x = ln (5.82842 \dots)

+1

度数

x = - 100.99797 \dots^{\circ}, x = - 75.45612 \dots^{\circ}, x = 75.45612 \dots^{\circ}, x = 100.99797 \dots^{\circ}

求解步骤

6 tanh^{2} (x) + 5 sech (x) - 7 = 0

使用三角恒等式改写

6 tanh^{2} (x) + 5 sech (x) - 7 = 0

使用双曲函数恒等式:

tanh (x) = \frac{e ^{x} - e ^{- x}}{e ^{x} + e ^{- x}}

6 (\frac{e ^{x} - e ^{- x}}{e ^{x} + e ^{- x}})^{2} + 5 sech (x) - 7 = 0

使用双曲函数恒等式:

sech (x) = \frac{2}{e ^{x} + e ^{- x}}

6 (\frac{e ^{x} - e ^{- x}}{e ^{x} + e ^{- x}})^{2} + 5 \cdot \frac{2}{e ^{x} + e ^{- x}} - 7 = 0

6 (\frac{e ^{x} - e ^{- x}}{e ^{x} + e ^{- x}})^{2} + 5 \cdot \frac{2}{e ^{x} + e ^{- x}} - 7 = 0

6 (\frac{e ^{x} - e ^{- x}}{e ^{x} + e ^{- x}})^{2} + 5 \cdot \frac{2}{e ^{x} + e ^{- x}} - 7 = 0 : x = ln (0.17157 \dots), x = ln (0.26794 \dots), x = ln (3.73205 \dots), x = ln (5.82842 \dots)

6 (\frac{e ^{x} - e ^{- x}}{e ^{x} + e ^{- x}})^{2} + 5 \cdot \frac{2}{e ^{x} + e ^{- x}} - 7 = 0

使用指数运算法则

6 (\frac{e ^{x} - e ^{- x}}{e ^{x} + e ^{- x}})^{2} + 5 \cdot \frac{2}{e ^{x} + e ^{- x}} - 7 = 0

使用指数法则:

a^{b c} = (a^{b})^{c}

e^{- x} = (e^{x})^{- 1}

6 (\frac{e ^{x} - ( e ^{x} ) ^{- 1}}{e ^{x} + ( e ^{x} ) ^{- 1}})^{2} + 5 \cdot \frac{2}{e ^{x} + ( e ^{x} ) ^{- 1}} - 7 = 0

6 (\frac{e ^{x} - ( e ^{x} ) ^{- 1}}{e ^{x} + ( e ^{x} ) ^{- 1}})^{2} + 5 \cdot \frac{2}{e ^{x} + ( e ^{x} ) ^{- 1}} - 7 = 0

用

e^{x} = u

改写方程式

6 (\frac{u - ( u ) ^{- 1}}{u + ( u ) ^{- 1}})^{2} + 5 \cdot \frac{2}{u + ( u ) ^{- 1}} - 7 = 0

解

6 (\frac{u - u ^{- 1}}{u + u ^{- 1}})^{2} + 5 \cdot \frac{2}{u + u ^{- 1}} - 7 = 0 : u \approx 0.17157 \dots, u \approx 0.26794 \dots, u \approx 3.73205 \dots, u \approx 5.82842 \dots

6 (\frac{u - u ^{- 1}}{u + u ^{- 1}})^{2} + 5 \cdot \frac{2}{u + u ^{- 1}} - 7 = 0

整理后得

\frac{6 ( u ^{2} - 1 ) ^{2}}{( u ^{2} + 1 ) ^{2}} + \frac{10 u}{u ^{2} + 1} - 7 = 0

乘以最小公倍数

\frac{6 ( u ^{2} - 1 ) ^{2}}{( u ^{2} + 1 ) ^{2}} + \frac{10 u}{u ^{2} + 1} - 7 = 0

找到

(u^{2} + 1)^{2}, u^{2} + 1

的最小公倍数:

(u^{2} + 1)^{2}

(u^{2} + 1)^{2}, u^{2} + 1

最小公倍数 (LCM)

计算出由出现在

(u^{2} + 1)^{2}

或

u^{2} + 1

中的因子组成的表达式

= (u^{2} + 1)^{2}

乘以最小公倍数=

(u^{2} + 1)^{2}

\frac{6 ( u ^{2} - 1 ) ^{2}}{( u ^{2} + 1 ) ^{2}} (u^{2} + 1)^{2} + \frac{10 u}{u ^{2} + 1} (u^{2} + 1)^{2} - 7 (u^{2} + 1)^{2} = 0 \cdot (u^{2} + 1)^{2}

化简

\frac{6 ( u ^{2} - 1 ) ^{2}}{( u ^{2} + 1 ) ^{2}} (u^{2} + 1)^{2} + \frac{10 u}{u ^{2} + 1} (u^{2} + 1)^{2} - 7 (u^{2} + 1)^{2} = 0 \cdot (u^{2} + 1)^{2}

化简

\frac{6 ( u ^{2} - 1 ) ^{2}}{( u ^{2} + 1 ) ^{2}} (u^{2} + 1)^{2} : 6 (u^{2} - 1)^{2}

\frac{6 ( u ^{2} - 1 ) ^{2}}{( u ^{2} + 1 ) ^{2}} (u^{2} + 1)^{2}

分式相乘:

a \cdot \frac{b}{c} = \frac{a \cdot b}{c}

= \frac{6 ( u ^{2} - 1 ) ^{2} ( u ^{2} + 1 ) ^{2}}{( u ^{2} + 1 ) ^{2}}

约分:

(u^{2} + 1)^{2}

= 6 (u^{2} - 1)^{2}

化简

\frac{10 u}{u ^{2} + 1} (u^{2} + 1)^{2} : 10 u (u^{2} + 1)

\frac{10 u}{u ^{2} + 1} (u^{2} + 1)^{2}

分式相乘:

a \cdot \frac{b}{c} = \frac{a \cdot b}{c}

= \frac{10 u ( u ^{2} + 1 ) ^{2}}{u ^{2} + 1}

约分:

u^{2} + 1

= 10 u (u^{2} + 1)

化简

0 \cdot (u^{2} + 1)^{2} : 0

0 \cdot (u^{2} + 1)^{2}

使用法则

0 \cdot a = 0

= 0

6 (u^{2} - 1)^{2} + 10 u (u^{2} + 1) - 7 (u^{2} + 1)^{2} = 0

6 (u^{2} - 1)^{2} + 10 u (u^{2} + 1) - 7 (u^{2} + 1)^{2} = 0

6 (u^{2} - 1)^{2} + 10 u (u^{2} + 1) - 7 (u^{2} + 1)^{2} = 0

解

6 (u^{2} - 1)^{2} + 10 u (u^{2} + 1) - 7 (u^{2} + 1)^{2} = 0 : u \approx 0.17157 \dots, u \approx 0.26794 \dots, u \approx 3.73205 \dots, u \approx 5.82842 \dots

6 (u^{2} - 1)^{2} + 10 u (u^{2} + 1) - 7 (u^{2} + 1)^{2} = 0

展开

6 (u^{2} - 1)^{2} + 10 u (u^{2} + 1) - 7 (u^{2} + 1)^{2} : - u^{4} + 10 u^{3} - 26 u^{2} + 10 u - 1

6 (u^{2} - 1)^{2} + 10 u (u^{2} + 1) - 7 (u^{2} + 1)^{2}

(u^{2} - 1)^{2} = u^{4} - 2 u^{2} + 1

(u^{2} - 1)^{2}

使用完全平方公式:

(a - b)^{2} = a^{2} - 2 ab + b^{2}

a = u^{2}, b = 1

= (u^{2})^{2} - 2 u^{2} \cdot 1 + 1^{2}

化简

(u^{2})^{2} - 2 u^{2} \cdot 1 + 1^{2} : u^{4} - 2 u^{2} + 1

(u^{2})^{2} - 2 u^{2} \cdot 1 + 1^{2}

使用法则

1^{a} = 1

1^{2} = 1

= (u^{2})^{2} - 2 \cdot 1 \cdot u^{2} + 1

(u^{2})^{2} = u^{4}

(u^{2})^{2}

使用指数法则:

(a^{b})^{c} = a^{b c}

= u^{2 \cdot 2}

数字相乘:

2 \cdot 2 = 4

= u^{4}

2 u^{2} \cdot 1 = 2 u^{2}

2 u^{2} \cdot 1

数字相乘:

2 \cdot 1 = 2

= 2 u^{2}

= u^{4} - 2 u^{2} + 1

= u^{4} - 2 u^{2} + 1

= 6 (u^{4} - 2 u^{2} + 1) + 10 u (u^{2} + 1) - 7 (u^{2} + 1)^{2}

(u^{2} + 1)^{2} = u^{4} + 2 u^{2} + 1

(u^{2} + 1)^{2}

使用完全平方公式:

(a + b)^{2} = a^{2} + 2 ab + b^{2}

a = u^{2}, b = 1

= (u^{2})^{2} + 2 u^{2} \cdot 1 + 1^{2}

化简

(u^{2})^{2} + 2 u^{2} \cdot 1 + 1^{2} : u^{4} + 2 u^{2} + 1

(u^{2})^{2} + 2 u^{2} \cdot 1 + 1^{2}

使用法则

1^{a} = 1

1^{2} = 1

= (u^{2})^{2} + 2 \cdot 1 \cdot u^{2} + 1

(u^{2})^{2} = u^{4}

(u^{2})^{2}

使用指数法则:

(a^{b})^{c} = a^{b c}

= u^{2 \cdot 2}

数字相乘:

2 \cdot 2 = 4

= u^{4}

2 u^{2} \cdot 1 = 2 u^{2}

2 u^{2} \cdot 1

数字相乘:

2 \cdot 1 = 2

= 2 u^{2}

= u^{4} + 2 u^{2} + 1

= u^{4} + 2 u^{2} + 1

= 6 (u^{4} - 2 u^{2} + 1) + 10 u (u^{2} + 1) - 7 (u^{4} + 2 u^{2} + 1)

乘开

6 (u^{4} - 2 u^{2} + 1) : 6 u^{4} - 12 u^{2} + 6

6 (u^{4} - 2 u^{2} + 1)

打开括号

= 6 u^{4} + 6 (- 2 u^{2}) + 6 \cdot 1

使用加减运算法则

+ (- a) = - a

= 6 u^{4} - 6 \cdot 2 u^{2} + 6 \cdot 1

化简

6 u^{4} - 6 \cdot 2 u^{2} + 6 \cdot 1 : 6 u^{4} - 12 u^{2} + 6

6 u^{4} - 6 \cdot 2 u^{2} + 6 \cdot 1

数字相乘:

6 \cdot 2 = 12

= 6 u^{4} - 12 u^{2} + 6 \cdot 1

数字相乘:

6 \cdot 1 = 6

= 6 u^{4} - 12 u^{2} + 6

= 6 u^{4} - 12 u^{2} + 6

= 6 u^{4} - 12 u^{2} + 6 + 10 u (u^{2} + 1) - 7 (u^{4} + 2 u^{2} + 1)

乘开

10 u (u^{2} + 1) : 10 u^{3} + 10 u

10 u (u^{2} + 1)

使用分配律:

a (b + c) = ab + a c

a = 10 u, b = u^{2}, c = 1

= 10 u u^{2} + 10 u \cdot 1

= 10 u^{2} u + 10 \cdot 1 \cdot u

化简

10 u^{2} u + 10 \cdot 1 \cdot u : 10 u^{3} + 10 u

10 u^{2} u + 10 \cdot 1 \cdot u

10 u^{2} u = 10 u^{3}

10 u^{2} u

使用指数法则:

a^{b} \cdot a^{c} = a^{b + c}

u^{2} u = u^{2 + 1}

= 10 u^{2 + 1}

数字相加:

2 + 1 = 3

= 10 u^{3}

10 \cdot 1 \cdot u = 10 u

10 \cdot 1 \cdot u

数字相乘:

10 \cdot 1 = 10

= 10 u

= 10 u^{3} + 10 u

= 10 u^{3} + 10 u

= 6 u^{4} - 12 u^{2} + 6 + 10 u^{3} + 10 u - 7 (u^{4} + 2 u^{2} + 1)

乘开

- 7 (u^{4} + 2 u^{2} + 1) : - 7 u^{4} - 14 u^{2} - 7

- 7 (u^{4} + 2 u^{2} + 1)

打开括号

= (- 7) u^{4} + (- 7) \cdot 2 u^{2} + (- 7) \cdot 1

使用加减运算法则

+ (- a) = - a

= - 7 u^{4} - 7 \cdot 2 u^{2} - 7 \cdot 1

化简

- 7 u^{4} - 7 \cdot 2 u^{2} - 7 \cdot 1 : - 7 u^{4} - 14 u^{2} - 7

- 7 u^{4} - 7 \cdot 2 u^{2} - 7 \cdot 1

数字相乘:

7 \cdot 2 = 14

= - 7 u^{4} - 14 u^{2} - 7 \cdot 1

数字相乘:

7 \cdot 1 = 7

= - 7 u^{4} - 14 u^{2} - 7

= - 7 u^{4} - 14 u^{2} - 7

= 6 u^{4} - 12 u^{2} + 6 + 10 u^{3} + 10 u - 7 u^{4} - 14 u^{2} - 7

化简

6 u^{4} - 12 u^{2} + 6 + 10 u^{3} + 10 u - 7 u^{4} - 14 u^{2} - 7 : - u^{4} + 10 u^{3} - 26 u^{2} + 10 u - 1

6 u^{4} - 12 u^{2} + 6 + 10 u^{3} + 10 u - 7 u^{4} - 14 u^{2} - 7

对同类项分组

= 6 u^{4} - 7 u^{4} + 10 u^{3} - 12 u^{2} - 14 u^{2} + 10 u + 6 - 7

同类项相加:

- 12 u^{2} - 14 u^{2} = - 26 u^{2}

= 6 u^{4} - 7 u^{4} + 10 u^{3} - 26 u^{2} + 10 u + 6 - 7

同类项相加:

6 u^{4} - 7 u^{4} = - u^{4}

= - u^{4} + 10 u^{3} - 26 u^{2} + 10 u + 6 - 7

数字相加/相减:

6 - 7 = - 1

= - u^{4} + 10 u^{3} - 26 u^{2} + 10 u - 1

= - u^{4} + 10 u^{3} - 26 u^{2} + 10 u - 1

- u^{4} + 10 u^{3} - 26 u^{2} + 10 u - 1 = 0

使用牛顿-拉弗森方法找到

- u^{4} + 10 u^{3} - 26 u^{2} + 10 u - 1 = 0

的一个解:

u \approx 0.17157 \dots

- u^{4} + 10 u^{3} - 26 u^{2} + 10 u - 1 = 0

牛顿-拉弗森近似法定义

f (u) = - u^{4} + 10 u^{3} - 26 u^{2} + 10 u - 1

找到

f^{^{'}} (u) : - 4 u^{3} + 30 u^{2} - 52 u + 10

\frac{d}{d u} (- u^{4} + 10 u^{3} - 26 u^{2} + 10 u - 1)

使用微分加减法定则:

(f \pm g)^{'} = f^{'} \pm g^{'}

= - \frac{d}{d u} (u^{4}) + \frac{d}{d u} (10 u^{3}) - \frac{d}{d u} (26 u^{2}) + \frac{d}{d u} (10 u) - \frac{d}{d u} (1)

\frac{d}{d u} (u^{4}) = 4 u^{3}

\frac{d}{d u} (u^{4})

使用幂法则:

\frac{d}{d x} (x^{a}) = a \cdot x^{a - 1}

= 4 u^{4 - 1}

化简

= 4 u^{3}

\frac{d}{d u} (10 u^{3}) = 30 u^{2}

\frac{d}{d u} (10 u^{3})

将常数提出:

(a \cdot f)^{'} = a \cdot f^{'}

= 10 \frac{d}{d u} (u^{3})

使用幂法则:

\frac{d}{d x} (x^{a}) = a \cdot x^{a - 1}

= 10 \cdot 3 u^{3 - 1}

化简

= 30 u^{2}

\frac{d}{d u} (26 u^{2}) = 52 u

\frac{d}{d u} (26 u^{2})

将常数提出:

(a \cdot f)^{'} = a \cdot f^{'}

= 26 \frac{d}{d u} (u^{2})

使用幂法则:

\frac{d}{d x} (x^{a}) = a \cdot x^{a - 1}

= 26 \cdot 2 u^{2 - 1}

化简

= 52 u

\frac{d}{d u} (10 u) = 10

\frac{d}{d u} (10 u)

将常数提出:

(a \cdot f)^{'} = a \cdot f^{'}

= 10 \frac{d u}{d u}

使用常见微分定则:

\frac{d u}{d u} = 1

= 10 \cdot 1

化简

= 10

\frac{d}{d u} (1) = 0

\frac{d}{d u} (1)

常数微分:

\frac{d}{d x} (a) = 0

= 0

= - 4 u^{3} + 30 u^{2} - 52 u + 10 - 0

化简

= - 4 u^{3} + 30 u^{2} - 52 u + 10

令

u_{0} = 0

计算

u_{n + 1}

至

Δ u_{n + 1} < 0.000001

u_{1} = 0.1 : Δ u_{1} = 0.1

f (u_{0}) = - 0^{4} + 10 \cdot 0^{3} - 26 \cdot 0^{2} + 10 \cdot 0 - 1 = - 1

f^{^{'}} (u_{0}) = - 4 \cdot 0^{3} + 30 \cdot 0^{2} - 52 \cdot 0 + 10 = 10

u_{1} = 0.1

Δ u_{1} = ∣ 0.1 - 0 ∣ = 0.1

Δ u_{1} = 0.1

u_{2} = 0.14907 \dots : Δ u_{2} = 0.04907 \dots

f (u_{1}) = - 0. 1^{4} + 10 \cdot 0. 1^{3} - 26 \cdot 0. 1^{2} + 10 \cdot 0.1 - 1 = - 0.2501

f^{^{'}} (u_{1}) = - 4 \cdot 0. 1^{3} + 30 \cdot 0. 1^{2} - 52 \cdot 0.1 + 10 = 5.096

u_{2} = 0.14907 \dots

Δ u_{2} = ∣ 0.14907 \dots - 0.1 ∣ = 0.04907 \dots

Δ u_{2} = 0.04907 \dots

u_{3} = 0.16783 \dots : Δ u_{3} = 0.01875 \dots

f (u_{2}) = - 0.14907 \dots^{4} + 10 \cdot 0.14907 \dots^{3} - 26 \cdot 0.14907 \dots^{2} + 10 \cdot 0.14907 \dots - 1 = - 0.05441 \dots

f^{^{'}} (u_{2}) = - 4 \cdot 0.14907 \dots^{3} + 30 \cdot 0.14907 \dots^{2} - 52 \cdot 0.14907 \dots + 10 = 2.90143 \dots

u_{3} = 0.16783 \dots

Δ u_{3} = ∣ 0.16783 \dots - 0.14907 \dots ∣ = 0.01875 \dots

Δ u_{3} = 0.01875 \dots

u_{4} = 0.17143 \dots : Δ u_{4} = 0.00360 \dots

f (u_{3}) = - 0.16783 \dots^{4} + 10 \cdot 0.16783 \dots^{3} - 26 \cdot 0.16783 \dots^{2} + 10 \cdot 0.16783 \dots - 1 = - 0.00755 \dots

f^{^{'}} (u_{3}) = - 4 \cdot 0.16783 \dots^{3} + 30 \cdot 0.16783 \dots^{2} - 52 \cdot 0.16783 \dots + 10 = 2.09886 \dots

u_{4} = 0.17143 \dots

Δ u_{4} = ∣ 0.17143 \dots - 0.16783 \dots ∣ = 0.00360 \dots

Δ u_{4} = 0.00360 \dots

u_{5} = 0.17157 \dots : Δ u_{5} = 0.00014 \dots

f (u_{4}) = - 0.17143 \dots^{4} + 10 \cdot 0.17143 \dots^{3} - 26 \cdot 0.17143 \dots^{2} + 10 \cdot 0.17143 \dots - 1 = - 0.00027 \dots

f^{^{'}} (u_{4}) = - 4 \cdot 0.17143 \dots^{3} + 30 \cdot 0.17143 \dots^{2} - 52 \cdot 0.17143 \dots + 10 = 1.94704 \dots

u_{5} = 0.17157 \dots

Δ u_{5} = ∣ 0.17157 \dots - 0.17143 \dots ∣ = 0.00014 \dots

Δ u_{5} = 0.00014 \dots

u_{6} = 0.17157 \dots : Δ u_{6} = 2.13816 E - 7

f (u_{5}) = - 0.17157 \dots^{4} + 10 \cdot 0.17157 \dots^{3} - 26 \cdot 0.17157 \dots^{2} + 10 \cdot 0.17157 \dots - 1 = - 4.15046 E - 7

f^{^{'}} (u_{5}) = - 4 \cdot 0.17157 \dots^{3} + 30 \cdot 0.17157 \dots^{2} - 52 \cdot 0.17157 \dots + 10 = 1.94113 \dots

u_{6} = 0.17157 \dots

Δ u_{6} = ∣ 0.17157 \dots - 0.17157 \dots ∣ = 2.13816 E - 7

Δ u_{6} = 2.13816 E - 7

u \approx 0.17157 \dots

使用长除法

Eq u a t i o n 0 : \frac{- u ^{4} + 10 u ^{3} - 26 u ^{2} + 10 u - 1}{u - 0.17157 \dots} = - u^{3} + 9.82842 \dots u^{2} - 24.31370 \dots u + 5.82842 \dots

- u^{3} + 9.82842 \dots u^{2} - 24.31370 \dots u + 5.82842 \dots \approx 0

使用牛顿-拉弗森方法找到

- u^{3} + 9.82842 \dots u^{2} - 24.31370 \dots u + 5.82842 \dots = 0

的一个解:

u \approx 0.26794 \dots

- u^{3} + 9.82842 \dots u^{2} - 24.31370 \dots u + 5.82842 \dots = 0

牛顿-拉弗森近似法定义

f (u) = - u^{3} + 9.82842 \dots u^{2} - 24.31370 \dots u + 5.82842 \dots

找到

f^{^{'}} (u) : - 3 u^{2} + 19.65685 \dots u - 24.31370 \dots

\frac{d}{d u} (- u^{3} + 9.82842 \dots u^{2} - 24.31370 \dots u + 5.82842 \dots)

使用微分加减法定则:

(f \pm g)^{'} = f^{'} \pm g^{'}

= - \frac{d}{d u} (u^{3}) + \frac{d}{d u} (9.82842 \dots u^{2}) - \frac{d}{d u} (24.31370 \dots u) + \frac{d}{d u} (5.82842 \dots)

\frac{d}{d u} (u^{3}) = 3 u^{2}

\frac{d}{d u} (u^{3})

使用幂法则:

\frac{d}{d x} (x^{a}) = a \cdot x^{a - 1}

= 3 u^{3 - 1}

化简

= 3 u^{2}

\frac{d}{d u} (9.82842 \dots u^{2}) = 19.65685 \dots u

\frac{d}{d u} (9.82842 \dots u^{2})

将常数提出:

(a \cdot f)^{'} = a \cdot f^{'}

= 9.82842 \dots \frac{d}{d u} (u^{2})

使用幂法则:

\frac{d}{d x} (x^{a}) = a \cdot x^{a - 1}

= 9.82842 \dots \cdot 2 u^{2 - 1}

化简

= 19.65685 \dots u

\frac{d}{d u} (24.31370 \dots u) = 24.31370 \dots

\frac{d}{d u} (24.31370 \dots u)

将常数提出:

(a \cdot f)^{'} = a \cdot f^{'}

= 24.31370 \dots \frac{d u}{d u}

使用常见微分定则:

\frac{d u}{d u} = 1

= 24.31370 \dots \cdot 1

化简

= 24.31370 \dots

\frac{d}{d u} (5.82842 \dots) = 0

\frac{d}{d u} (5.82842 \dots)

常数微分:

\frac{d}{d x} (a) = 0

= 0

= - 3 u^{2} + 19.65685 \dots u - 24.31370 \dots + 0

化简

= - 3 u^{2} + 19.65685 \dots u - 24.31370 \dots

令

u_{0} = 0

计算

u_{n + 1}

至

Δ u_{n + 1} < 0.000001

u_{1} = 0.23971 \dots : Δ u_{1} = 0.23971 \dots

f (u_{0}) = - 0^{3} + 9.82842 \dots \cdot 0^{2} - 24.31370 \dots \cdot 0 + 5.82842 \dots = 5.82842 \dots

f^{^{'}} (u_{0}) = - 3 \cdot 0^{2} + 19.65685 \dots \cdot 0 - 24.31370 \dots = - 24.31370 \dots

u_{1} = 0.23971 \dots

Δ u_{1} = ∣ 0.23971 \dots - 0 ∣ = 0.23971 \dots

Δ u_{1} = 0.23971 \dots

u_{2} = 0.26758 \dots : Δ u_{2} = 0.02786 \dots

f (u_{1}) = - 0.23971 \dots^{3} + 9.82842 \dots \cdot 0.23971 \dots^{2} - 24.31370 \dots \cdot 0.23971 \dots + 5.82842 \dots = 0.55101 \dots

f^{^{'}} (u_{1}) = - 3 \cdot 0.23971 \dots^{2} + 19.65685 \dots \cdot 0.23971 \dots - 24.31370 \dots = - 19.77400 \dots

u_{2} = 0.26758 \dots

Δ u_{2} = ∣ 0.26758 \dots - 0.23971 \dots ∣ = 0.02786 \dots

Δ u_{2} = 0.02786 \dots

u_{3} = 0.26794 \dots : Δ u_{3} = 0.00036 \dots

f (u_{2}) = - 0.26758 \dots^{3} + 9.82842 \dots \cdot 0.26758 \dots^{2} - 24.31370 \dots \cdot 0.26758 \dots + 5.82842 \dots = 0.00705 \dots

f^{^{'}} (u_{2}) = - 3 \cdot 0.26758 \dots^{2} + 19.65685 \dots \cdot 0.26758 \dots - 24.31370 \dots = - 19.26866 \dots

u_{3} = 0.26794 \dots

Δ u_{3} = ∣ 0.26794 \dots - 0.26758 \dots ∣ = 0.00036 \dots

Δ u_{3} = 0.00036 \dots

u_{4} = 0.26794 \dots : Δ u_{4} = 6.27517 E - 8

f (u_{3}) = - 0.26794 \dots^{3} + 9.82842 \dots \cdot 0.26794 \dots^{2} - 24.31370 \dots \cdot 0.26794 \dots + 5.82842 \dots = 1.20873 E - 6

f^{^{'}} (u_{3}) = - 3 \cdot 0.26794 \dots^{2} + 19.65685 \dots \cdot 0.26794 \dots - 24.31370 \dots = - 19.26206 \dots

u_{4} = 0.26794 \dots

Δ u_{4} = ∣ 0.26794 \dots - 0.26794 \dots ∣ = 6.27517 E - 8

Δ u_{4} = 6.27517 E - 8

u \approx 0.26794 \dots

使用长除法

Eq u a t i o n 0 : \frac{- u ^{3} + 9.82842 \dots u ^{2} - 24.31370 \dots u + 5.82842 \dots}{u - 0.26794 \dots} = - u^{2} + 9.56047 \dots u - 21.75198 \dots

- u^{2} + 9.56047 \dots u - 21.75198 \dots \approx 0

使用牛顿-拉弗森方法找到

- u^{2} + 9.56047 \dots u - 21.75198 \dots = 0

的一个解:

u \approx 3.73205 \dots

- u^{2} + 9.56047 \dots u - 21.75198 \dots = 0

牛顿-拉弗森近似法定义

f (u) = - u^{2} + 9.56047 \dots u - 21.75198 \dots

找到

f^{^{'}} (u) : - 2 u + 9.56047 \dots

\frac{d}{d u} (- u^{2} + 9.56047 \dots u - 21.75198 \dots)

使用微分加减法定则:

(f \pm g)^{'} = f^{'} \pm g^{'}

= - \frac{d}{d u} (u^{2}) + \frac{d}{d u} (9.56047 \dots u) - \frac{d}{d u} (21.75198 \dots)

\frac{d}{d u} (u^{2}) = 2 u

\frac{d}{d u} (u^{2})

使用幂法则:

\frac{d}{d x} (x^{a}) = a \cdot x^{a - 1}

= 2 u^{2 - 1}

化简

= 2 u

\frac{d}{d u} (9.56047 \dots u) = 9.56047 \dots

\frac{d}{d u} (9.56047 \dots u)

将常数提出:

(a \cdot f)^{'} = a \cdot f^{'}

= 9.56047 \dots \frac{d u}{d u}

使用常见微分定则:

\frac{d u}{d u} = 1

= 9.56047 \dots \cdot 1

化简

= 9.56047 \dots

\frac{d}{d u} (21.75198 \dots) = 0

\frac{d}{d u} (21.75198 \dots)

常数微分:

\frac{d}{d x} (a) = 0

= 0

= - 2 u + 9.56047 \dots - 0

化简

= - 2 u + 9.56047 \dots

令

u_{0} = 2

计算

u_{n + 1}

至

Δ u_{n + 1} < 0.000001

u_{1} = 3.19252 \dots : Δ u_{1} = 1.19252 \dots

f (u_{0}) = - 2^{2} + 9.56047 \dots \cdot 2 - 21.75198 \dots = - 6.63103 \dots

f^{^{'}} (u_{0}) = - 2 \cdot 2 + 9.56047 \dots = 5.56047 \dots

u_{1} = 3.19252 \dots

Δ u_{1} = ∣ 3.19252 \dots - 2 ∣ = 1.19252 \dots

Δ u_{1} = 1.19252 \dots

u_{2} = 3.64038 \dots : Δ u_{2} = 0.44785 \dots

f (u_{1}) = - 3.19252 \dots^{2} + 9.56047 \dots \cdot 3.19252 \dots - 21.75198 \dots = - 1.42212 \dots

f^{^{'}} (u_{1}) = - 2 \cdot 3.19252 \dots + 9.56047 \dots = 3.17542 \dots

u_{2} = 3.64038 \dots

Δ u_{2} = ∣ 3.64038 \dots - 3.19252 \dots ∣ = 0.44785 \dots

Δ u_{2} = 0.44785 \dots

u_{3} = 3.72836 \dots : Δ u_{3} = 0.08798 \dots

f (u_{2}) = - 3.64038 \dots^{2} + 9.56047 \dots \cdot 3.64038 \dots - 21.75198 \dots = - 0.20057 \dots

f^{^{'}} (u_{2}) = - 2 \cdot 3.64038 \dots + 9.56047 \dots = 2.27971 \dots

u_{3} = 3.72836 \dots

Δ u_{3} = ∣ 3.72836 \dots - 3.64038 \dots ∣ = 0.08798 \dots

Δ u_{3} = 0.08798 \dots

u_{4} = 3.73204 \dots : Δ u_{4} = 0.00367 \dots

f (u_{3}) = - 3.72836 \dots^{2} + 9.56047 \dots \cdot 3.72836 \dots - 21.75198 \dots = - 0.00774 \dots

f^{^{'}} (u_{3}) = - 2 \cdot 3.72836 \dots + 9.56047 \dots = 2.10374 \dots

u_{4} = 3.73204 \dots

Δ u_{4} = ∣ 3.73204 \dots - 3.72836 \dots ∣ = 0.00367 \dots

Δ u_{4} = 0.00367 \dots

u_{5} = 3.73205 \dots : Δ u_{5} = 6.45822 E - 6

f (u_{4}) = - 3.73204 \dots^{2} + 9.56047 \dots \cdot 3.73204 \dots - 21.75198 \dots = - 0.00001 \dots

f^{^{'}} (u_{4}) = - 2 \cdot 3.73204 \dots + 9.56047 \dots = 2.09638 \dots

u_{5} = 3.73205 \dots

Δ u_{5} = ∣ 3.73205 \dots - 3.73204 \dots ∣ = 6.45822 E - 6

Δ u_{5} = 6.45822 E - 6

u_{6} = 3.73205 \dots : Δ u_{6} = 1.98957 E - 11

f (u_{5}) = - 3.73205 \dots^{2} + 9.56047 \dots \cdot 3.73205 \dots - 21.75198 \dots = - 4.17089 E - 11

f^{^{'}} (u_{5}) = - 2 \cdot 3.73205 \dots + 9.56047 \dots = 2.09637 \dots

u_{6} = 3.73205 \dots

Δ u_{6} = ∣ 3.73205 \dots - 3.73205 \dots ∣ = 1.98957 E - 11

Δ u_{6} = 1.98957 E - 11

u \approx 3.73205 \dots

使用长除法

Eq u a t i o n 0 : \frac{- u ^{2} + 9.56047 \dots u - 21.75198 \dots}{u - 3.73205 \dots} = - u + 5.82842 \dots

- u + 5.82842 \dots \approx 0

u \approx 5.82842 \dots

解为

u \approx 0.17157 \dots, u \approx 0.26794 \dots, u \approx 3.73205 \dots, u \approx 5.82842 \dots

u \approx 0.17157 \dots, u \approx 0.26794 \dots, u \approx 3.73205 \dots, u \approx 5.82842 \dots

验证解

找到无定义的点（奇点）:

u = 0

取

6 (\frac{u - u ^{- 1}}{u + u ^{- 1}})^{2} + 5 \frac{2}{u + u ^{- 1}} - 7

的分母，令其等于零

u = 0

以下点无定义

u = 0

将不在定义域的点与解相综合:

u \approx 0.17157 \dots, u \approx 0.26794 \dots, u \approx 3.73205 \dots, u \approx 5.82842 \dots

u \approx 0.17157 \dots, u \approx 0.26794 \dots, u \approx 3.73205 \dots, u \approx 5.82842 \dots

代回

u = e^{x}

，求解

x

解

e^{x} = 0.17157 \dots : x = ln (0.17157 \dots)

e^{x} = 0.17157 \dots

使用指数运算法则

e^{x} = 0.17157 \dots

若

f (x) = g (x)

，则

ln (f (x)) = ln (g (x))

ln (e^{x}) = ln (0.17157 \dots)

使用对数计算法则:

ln (e^{a}) = a

ln (e^{x}) = x

x = ln (0.17157 \dots)

x = ln (0.17157 \dots)

解

e^{x} = 0.26794 \dots : x = ln (0.26794 \dots)

e^{x} = 0.26794 \dots

使用指数运算法则

e^{x} = 0.26794 \dots

若

f (x) = g (x)

，则

ln (f (x)) = ln (g (x))

ln (e^{x}) = ln (0.26794 \dots)

使用对数计算法则:

ln (e^{a}) = a

ln (e^{x}) = x

x = ln (0.26794 \dots)

x = ln (0.26794 \dots)

解

e^{x} = 3.73205 \dots : x = ln (3.73205 \dots)

e^{x} = 3.73205 \dots

使用指数运算法则

e^{x} = 3.73205 \dots

若

f (x) = g (x)

，则

ln (f (x)) = ln (g (x))

ln (e^{x}) = ln (3.73205 \dots)

使用对数计算法则:

ln (e^{a}) = a

ln (e^{x}) = x

x = ln (3.73205 \dots)

x = ln (3.73205 \dots)

解

e^{x} = 5.82842 \dots : x = ln (5.82842 \dots)

e^{x} = 5.82842 \dots

使用指数运算法则

e^{x} = 5.82842 \dots

若

f (x) = g (x)

，则

ln (f (x)) = ln (g (x))

ln (e^{x}) = ln (5.82842 \dots)

使用对数计算法则:

ln (e^{a}) = a

ln (e^{x}) = x

x = ln (5.82842 \dots)

x = ln (5.82842 \dots)

x = ln (0.17157 \dots), x = ln (0.26794 \dots), x = ln (3.73205 \dots), x = ln (5.82842 \dots)

x = ln (0.17157 \dots), x = ln (0.26794 \dots), x = ln (3.73205 \dots), x = ln (5.82842 \dots)

作图

流行的例子

2sin(x)+sqrt(2)=0,0<= x<= 2pi

2 sin (x) + 2 = 0, 0 \leq x \leq 2 π

2cos^2(x)+3cos(x)=2,0<= x<= 2pi

2 cos^{2} (x) + 3 cos (x) = 2, 0 \leq x \leq 2 π

cos (θ) = \frac{7}{11}

sin(θ/2)=sin(θ)

sin (\frac{θ}{2}) = sin (θ)

2 tan (x) = 1