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受欢迎的三角函数 >

cosh(2x)+sinh^2(x)-13sinh(x)=-3

解答

cosh (2 x) + sinh^{2} (x) - 13 sinh (x) = - 3

解答

x = ln (1.38742 \dots), x = ln (8.12310 \dots)

+1

度数

x = 18.76151 \dots^{\circ}, x = 120.01818 \dots^{\circ}

求解步骤

cosh (2 x) + sinh^{2} (x) - 13 sinh (x) = - 3

使用三角恒等式改写

cosh (2 x) + sinh^{2} (x) - 13 sinh (x) = - 3

使用双曲函数恒等式:

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

cosh (2 x) + (\frac{e ^{x} - e ^{- x}}{2})^{2} - 13 \cdot \frac{e ^{x} - e ^{- x}}{2} = - 3

使用双曲函数恒等式:

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

\frac{e ^{2 x} + e ^{- 2 x}}{2} + (\frac{e ^{x} - e ^{- x}}{2})^{2} - 13 \cdot \frac{e ^{x} - e ^{- x}}{2} = - 3

\frac{e ^{2 x} + e ^{- 2 x}}{2} + (\frac{e ^{x} - e ^{- x}}{2})^{2} - 13 \cdot \frac{e ^{x} - e ^{- x}}{2} = - 3

\frac{e ^{2 x} + e ^{- 2 x}}{2} + (\frac{e ^{x} - e ^{- x}}{2})^{2} - 13 \cdot \frac{e ^{x} - e ^{- x}}{2} = - 3 : x = ln (1.38742 \dots), x = ln (8.12310 \dots)

\frac{e ^{2 x} + e ^{- 2 x}}{2} + (\frac{e ^{x} - e ^{- x}}{2})^{2} - 13 \cdot \frac{e ^{x} - e ^{- x}}{2} = - 3

在两边乘以

2

\frac{e ^{2 x} + e ^{- 2 x}}{2} \cdot 2 + (\frac{e ^{x} - e ^{- x}}{2})^{2} \cdot 2 - 13 \cdot \frac{e ^{x} - e ^{- x}}{2} \cdot 2 = - 3 \cdot 2

化简

\frac{e ^{2 x} + e ^{- 2 x}}{2} \cdot 2 + (\frac{e ^{x} - e ^{- x}}{2})^{2} \cdot 2 - 13 \cdot \frac{e ^{x} - e ^{- x}}{2} \cdot 2 : e^{2 x} + e^{- 2 x} + \frac{( e ^{x} - e ^{- x} ) ^{2}}{2} - 13 (e^{x} - e^{- x})

\frac{e ^{2 x} + e ^{- 2 x}}{2} \cdot 2 + (\frac{e ^{x} - e ^{- x}}{2})^{2} \cdot 2 - 13 \cdot \frac{e ^{x} - e ^{- x}}{2} \cdot 2

\frac{e ^{2 x} + e ^{- 2 x}}{2} \cdot 2 = e^{2 x} + e^{- 2 x}

\frac{e ^{2 x} + e ^{- 2 x}}{2} \cdot 2

分式相乘:

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

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

约分:

2

= e^{2 x} + e^{- 2 x}

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

(\frac{e ^{x} - e ^{- x}}{2})^{2} \cdot 2

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

(\frac{e ^{x} - e ^{- x}}{2})^{2}

使用指数法则:

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

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

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

分式相乘:

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

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

约分:

2

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

13 \cdot \frac{e ^{x} - e ^{- x}}{2} \cdot 2 = 13 (e^{x} - e^{- x})

13 \cdot \frac{e ^{x} - e ^{- x}}{2} \cdot 2

分式相乘:

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

= \frac{( e ^{x} - e ^{- x} ) \cdot 13 \cdot 2}{2}

约分:

2

= (e^{x} - e^{- x}) \cdot 13

= e^{2 x} + e^{- 2 x} + \frac{( e ^{x} - e ^{- x} ) ^{2}}{2} - 13 (e^{x} - e^{- x})

e^{2 x} + e^{- 2 x} + \frac{( e ^{x} - e ^{- x} ) ^{2}}{2} - 13 (e^{x} - e^{- x}) = - 6

使用指数运算法则

e^{2 x} + e^{- 2 x} + \frac{( e ^{x} - e ^{- x} ) ^{2}}{2} - 13 (e^{x} - e^{- x}) = - 6

使用指数法则:

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

e^{2 x} = (e^{x})^{2}, e^{- 2 x} = (e^{x})^{- 2}, e^{- x} = (e^{x})^{- 1}

(e^{x})^{2} + (e^{x})^{- 2} + \frac{( e ^{x} - ( e ^{x} ) ^{- 1} ) ^{2}}{2} - 13 (e^{x} - (e^{x})^{- 1}) = - 6

(e^{x})^{2} + (e^{x})^{- 2} + \frac{( e ^{x} - ( e ^{x} ) ^{- 1} ) ^{2}}{2} - 13 (e^{x} - (e^{x})^{- 1}) = - 6

用

e^{x} = u

改写方程式

(u)^{2} + (u)^{- 2} + \frac{( u - ( u ) ^{- 1} ) ^{2}}{2} - 13 (u - (u)^{- 1}) = - 6

解

u^{2} + u^{- 2} + \frac{( u - u ^{- 1} ) ^{2}}{2} - 13 (u - u^{- 1}) = - 6 : u \approx - 0.12310 \dots, u \approx - 0.72075 \dots, u \approx 1.38742 \dots, u \approx 8.12310 \dots

u^{2} + u^{- 2} + \frac{( u - u ^{- 1} ) ^{2}}{2} - 13 (u - u^{- 1}) = - 6

整理后得

u^{2} + \frac{1}{u ^{2}} + \frac{( u ^{2} - 1 ) ^{2}}{2 u ^{2}} - 13 (u - \frac{1}{u}) = - 6

乘以最小公倍数

u^{2} + \frac{1}{u ^{2}} + \frac{( u ^{2} - 1 ) ^{2}}{2 u ^{2}} - 13 (u - \frac{1}{u}) = - 6

找到

u^{2}, 2 u^{2}

的最小公倍数:

2 u^{2}

u^{2}, 2 u^{2}

最小公倍数 (LCM)

计算出由出现在

u^{2}

或

2 u^{2}

中的因子组成的表达式

= 2 u^{2}

乘以最小公倍数=

2 u^{2}

u^{2} \cdot 2 u^{2} + \frac{1}{u ^{2}} \cdot 2 u^{2} + \frac{( u ^{2} - 1 ) ^{2}}{2 u ^{2}} \cdot 2 u^{2} - 13 (u - \frac{1}{u}) \cdot 2 u^{2} = - 6 \cdot 2 u^{2}

化简

u^{2} \cdot 2 u^{2} + \frac{1}{u ^{2}} \cdot 2 u^{2} + \frac{( u ^{2} - 1 ) ^{2}}{2 u ^{2}} \cdot 2 u^{2} - 13 (u - \frac{1}{u}) \cdot 2 u^{2} = - 6 \cdot 2 u^{2}

化简

u^{2} \cdot 2 u^{2} : 2 u^{4}

u^{2} \cdot 2 u^{2}

使用指数法则:

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

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

= 2 u^{2 + 2}

数字相加:

2 + 2 = 4

= 2 u^{4}

化简

\frac{1}{u ^{2}} \cdot 2 u^{2} : 2

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

分式相乘:

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

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

约分:

u^{2}

= 1 \cdot 2

数字相乘:

1 \cdot 2 = 2

= 2

化简

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

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

分式相乘:

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

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

约分:

2

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

约分:

u^{2}

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

化简

- 13 (u - \frac{1}{u}) \cdot 2 u^{2} : - 26 u^{2} (u - \frac{1}{u})

- 13 (u - \frac{1}{u}) \cdot 2 u^{2}

数字相乘:

13 \cdot 2 = 26

= - 26 u^{2} (u - \frac{1}{u})

化简

- 6 \cdot 2 u^{2} : - 12 u^{2}

- 6 \cdot 2 u^{2}

数字相乘:

6 \cdot 2 = 12

= - 12 u^{2}

2 u^{4} + 2 + (u^{2} - 1)^{2} - 26 u^{2} (u - \frac{1}{u}) = - 12 u^{2}

2 u^{4} + 2 + (u^{2} - 1)^{2} - 26 u^{2} (u - \frac{1}{u}) = - 12 u^{2}

展开

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

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

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

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

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

乘开

- 26 u^{2} (u - \frac{1}{u}) : - 26 u^{3} + 26 u

- 26 u^{2} (u - \frac{1}{u})

使用分配律:

a (b - c) = ab - a c

a = - 26 u^{2}, b = u, c = \frac{1}{u}

= - 26 u^{2} u - (- 26 u^{2}) \frac{1}{u}

使用加减运算法则

- (- a) = a

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

化简

- 26 u^{2} u + 26 \cdot \frac{1}{u} u^{2} : - 26 u^{3} + 26 u

- 26 u^{2} u + 26 \cdot \frac{1}{u} u^{2}

26 u^{2} u = 26 u^{3}

26 u^{2} u

使用指数法则:

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

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

= 26 u^{2 + 1}

数字相加:

2 + 1 = 3

= 26 u^{3}

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

26 \cdot \frac{1}{u} u^{2}

分式相乘:

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

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

数字相乘:

1 \cdot 26 = 26

= \frac{26 u ^{2}}{u}

约分:

u

= 26 u

= - 26 u^{3} + 26 u

= - 26 u^{3} + 26 u

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

化简

2 u^{4} + 2 + u^{4} - 2 u^{2} + 1 - 26 u^{3} + 26 u : 3 u^{4} - 26 u^{3} - 2 u^{2} + 26 u + 3

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

对同类项分组

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

同类项相加:

2 u^{4} + u^{4} = 3 u^{4}

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

数字相加:

2 + 1 = 3

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

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

3 u^{4} - 26 u^{3} - 2 u^{2} + 26 u + 3 = - 12 u^{2}

3 u^{4} - 26 u^{3} - 2 u^{2} + 26 u + 3 = - 12 u^{2}

解

3 u^{4} - 26 u^{3} - 2 u^{2} + 26 u + 3 = - 12 u^{2} : u \approx - 0.12310 \dots, u \approx - 0.72075 \dots, u \approx 1.38742 \dots, u \approx 8.12310 \dots

3 u^{4} - 26 u^{3} - 2 u^{2} + 26 u + 3 = - 12 u^{2}

将

12 u^{2}

para o lado esquerdo

3 u^{4} - 26 u^{3} - 2 u^{2} + 26 u + 3 = - 12 u^{2}

两边加上

12 u^{2}

3 u^{4} - 26 u^{3} - 2 u^{2} + 26 u + 3 + 12 u^{2} = - 12 u^{2} + 12 u^{2}

化简

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

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

使用牛顿-拉弗森方法找到

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

的一个解:

u \approx - 0.12310 \dots

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

牛顿-拉弗森近似法定义

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

找到

f^{^{'}} (u) : 12 u^{3} - 78 u^{2} + 20 u + 26

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

使用微分加减法定则:

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

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

\frac{d}{d u} (3 u^{4}) = 12 u^{3}

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

将常数提出:

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

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

使用幂法则:

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

= 3 \cdot 4 u^{4 - 1}

化简

= 12 u^{3}

\frac{d}{d u} (26 u^{3}) = 78 u^{2}

\frac{d}{d u} (26 u^{3})

将常数提出:

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

= 26 \frac{d}{d u} (u^{3})

使用幂法则:

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

= 26 \cdot 3 u^{3 - 1}

化简

= 78 u^{2}

\frac{d}{d u} (10 u^{2}) = 20 u

\frac{d}{d u} (10 u^{2})

将常数提出:

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

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

使用幂法则:

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

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

化简

= 20 u

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

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

将常数提出:

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

= 26 \frac{d u}{d u}

使用常见微分定则:

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

= 26 \cdot 1

化简

= 26

\frac{d}{d u} (3) = 0

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

常数微分:

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

= 0

= 12 u^{3} - 78 u^{2} + 20 u + 26 + 0

化简

= 12 u^{3} - 78 u^{2} + 20 u + 26

令

u_{0} = 0

计算

u_{n + 1}

至

Δ u_{n + 1} < 0.000001

u_{1} = - 0.11538 \dots : Δ u_{1} = 0.11538 \dots

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

f^{^{'}} (u_{0}) = 12 \cdot 0^{3} - 78 \cdot 0^{2} + 20 \cdot 0 + 26 = 26

u_{1} = - 0.11538 \dots

Δ u_{1} = ∣ - 0.11538 \dots - 0 ∣ = 0.11538 \dots

Δ u_{1} = 0.11538 \dots

u_{2} = - 0.12305 \dots : Δ u_{2} = 0.00766 \dots

f (u_{1}) = 3 (- 0.11538 \dots)^{4} - 26 (- 0.11538 \dots)^{3} + 10 (- 0.11538 \dots)^{2} + 26 (- 0.11538 \dots) + 3 = 0.17360 \dots

f^{^{'}} (u_{1}) = 12 (- 0.11538 \dots)^{3} - 78 (- 0.11538 \dots)^{2} + 20 (- 0.11538 \dots) + 26 = 22.63541 \dots

u_{2} = - 0.12305 \dots

Δ u_{2} = ∣ - 0.12305 \dots - (- 0.11538 \dots) ∣ = 0.00766 \dots

Δ u_{2} = 0.00766 \dots

u_{3} = - 0.12310 \dots : Δ u_{3} = 0.00005 \dots

f (u_{2}) = 3 (- 0.12305 \dots)^{4} - 26 (- 0.12305 \dots)^{3} + 10 (- 0.12305 \dots)^{2} + 26 (- 0.12305 \dots) + 3 = 0.00114 \dots

f^{^{'}} (u_{2}) = 12 (- 0.12305 \dots)^{3} - 78 (- 0.12305 \dots)^{2} + 20 (- 0.12305 \dots) + 26 = 22.33544 \dots

u_{3} = - 0.12310 \dots

Δ u_{3} = ∣ - 0.12310 \dots - (- 0.12305 \dots) ∣ = 0.00005 \dots

Δ u_{3} = 0.00005 \dots

u_{4} = - 0.12310 \dots : Δ u_{4} = 2.33489 E - 9

f (u_{3}) = 3 (- 0.12310 \dots)^{4} - 26 (- 0.12310 \dots)^{3} + 10 (- 0.12310 \dots)^{2} + 26 (- 0.12310 \dots) + 3 = 5.21461 E - 8

f^{^{'}} (u_{3}) = 12 (- 0.12310 \dots)^{3} - 78 (- 0.12310 \dots)^{2} + 20 (- 0.12310 \dots) + 26 = 22.33340 \dots

u_{4} = - 0.12310 \dots

Δ u_{4} = ∣ - 0.12310 \dots - (- 0.12310 \dots) ∣ = 2.33489 E - 9

Δ u_{4} = 2.33489 E - 9

u \approx - 0.12310 \dots

使用长除法

Eq u a t i o n 0 : \frac{3 u ^{4} - 26 u ^{3} + 10 u ^{2} + 26 u + 3}{u + 0.12310 \dots} = 3 u^{3} - 26.36931 \dots u^{2} + 13.24621 \dots u + 24.36931 \dots

3 u^{3} - 26.36931 \dots u^{2} + 13.24621 \dots u + 24.36931 \dots \approx 0

使用牛顿-拉弗森方法找到

3 u^{3} - 26.36931 \dots u^{2} + 13.24621 \dots u + 24.36931 \dots = 0

的一个解:

u \approx - 0.72075 \dots

3 u^{3} - 26.36931 \dots u^{2} + 13.24621 \dots u + 24.36931 \dots = 0

牛顿-拉弗森近似法定义

f (u) = 3 u^{3} - 26.36931 \dots u^{2} + 13.24621 \dots u + 24.36931 \dots

找到

f^{^{'}} (u) : 9 u^{2} - 52.73863 \dots u + 13.24621 \dots

\frac{d}{d u} (3 u^{3} - 26.36931 \dots u^{2} + 13.24621 \dots u + 24.36931 \dots)

使用微分加减法定则:

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

= \frac{d}{d u} (3 u^{3}) - \frac{d}{d u} (26.36931 \dots u^{2}) + \frac{d}{d u} (13.24621 \dots u) + \frac{d}{d u} (24.36931 \dots)

\frac{d}{d u} (3 u^{3}) = 9 u^{2}

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

将常数提出:

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

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

使用幂法则:

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

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

化简

= 9 u^{2}

\frac{d}{d u} (26.36931 \dots u^{2}) = 52.73863 \dots u

\frac{d}{d u} (26.36931 \dots u^{2})

将常数提出:

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

= 26.36931 \dots \frac{d}{d u} (u^{2})

使用幂法则:

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

= 26.36931 \dots \cdot 2 u^{2 - 1}

化简

= 52.73863 \dots u

\frac{d}{d u} (13.24621 \dots u) = 13.24621 \dots

\frac{d}{d u} (13.24621 \dots u)

将常数提出:

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

= 13.24621 \dots \frac{d u}{d u}

使用常见微分定则:

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

= 13.24621 \dots \cdot 1

化简

= 13.24621 \dots

\frac{d}{d u} (24.36931 \dots) = 0

\frac{d}{d u} (24.36931 \dots)

常数微分:

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

= 0

= 9 u^{2} - 52.73863 \dots u + 13.24621 \dots + 0

化简

= 9 u^{2} - 52.73863 \dots u + 13.24621 \dots

令

u_{0} = - 2

计算

u_{n + 1}

至

Δ u_{n + 1} < 0.000001

u_{1} = - 1.14944 \dots : Δ u_{1} = 0.85055 \dots

f (u_{0}) = 3 (- 2)^{3} - 26.36931 \dots (- 2)^{2} + 13.24621 \dots (- 2) + 24.36931 \dots = - 131.60037 \dots

f^{^{'}} (u_{0}) = 9 (- 2)^{2} - 52.73863 \dots (- 2) + 13.24621 \dots = 154.72347 \dots

u_{1} = - 1.14944 \dots

Δ u_{1} = ∣ - 1.14944 \dots - (- 2) ∣ = 0.85055 \dots

Δ u_{1} = 0.85055 \dots

u_{2} = - 0.79668 \dots : Δ u_{2} = 0.35276 \dots

f (u_{1}) = 3 (- 1.14944 \dots)^{3} - 26.36931 \dots (- 1.14944 \dots)^{2} + 13.24621 \dots (- 1.14944 \dots) + 24.36931 \dots = - 30.25251 \dots

f^{^{'}} (u_{1}) = 9 (- 1.14944 \dots)^{2} - 52.73863 \dots (- 1.14944 \dots) + 13.24621 \dots = 85.75760 \dots

u_{2} = - 0.79668 \dots

Δ u_{2} = ∣ - 0.79668 \dots - (- 1.14944 \dots) ∣ = 0.35276 \dots

Δ u_{2} = 0.35276 \dots

u_{3} = - 0.72390 \dots : Δ u_{3} = 0.07277 \dots

f (u_{2}) = 3 (- 0.79668 \dots)^{3} - 26.36931 \dots (- 0.79668 \dots)^{2} + 13.24621 \dots (- 0.79668 \dots) + 24.36931 \dots = - 4.43722 \dots

f^{^{'}} (u_{2}) = 9 (- 0.79668 \dots)^{2} - 52.73863 \dots (- 0.79668 \dots) + 13.24621 \dots = 60.97432 \dots

u_{3} = - 0.72390 \dots

Δ u_{3} = ∣ - 0.72390 \dots - (- 0.79668 \dots) ∣ = 0.07277 \dots

Δ u_{3} = 0.07277 \dots

u_{4} = - 0.72076 \dots : Δ u_{4} = 0.00314 \dots

f (u_{3}) = 3 (- 0.72390 \dots)^{3} - 26.36931 \dots (- 0.72390 \dots)^{2} + 13.24621 \dots (- 0.72390 \dots) + 24.36931 \dots = - 0.17646 \dots

f^{^{'}} (u_{3}) = 9 (- 0.72390 \dots)^{2} - 52.73863 \dots (- 0.72390 \dots) + 13.24621 \dots = 56.14052 \dots

u_{4} = - 0.72076 \dots

Δ u_{4} = ∣ - 0.72076 \dots - (- 0.72390 \dots) ∣ = 0.00314 \dots

Δ u_{4} = 0.00314 \dots

u_{5} = - 0.72075 \dots : Δ u_{5} = 5.80676 E - 6

f (u_{4}) = 3 (- 0.72076 \dots)^{3} - 26.36931 \dots (- 0.72076 \dots)^{2} + 13.24621 \dots (- 0.72076 \dots) + 24.36931 \dots = - 0.00032 \dots

f^{^{'}} (u_{4}) = 9 (- 0.72076 \dots)^{2} - 52.73863 \dots (- 0.72076 \dots) + 13.24621 \dots = 55.93389 \dots

u_{5} = - 0.72075 \dots

Δ u_{5} = ∣ - 0.72075 \dots - (- 0.72076 \dots) ∣ = 5.80676 E - 6

Δ u_{5} = 5.80676 E - 6

u_{6} = - 0.72075 \dots : Δ u_{6} = 1.98067 E - 11

f (u_{5}) = 3 (- 0.72075 \dots)^{3} - 26.36931 \dots (- 0.72075 \dots)^{2} + 13.24621 \dots (- 0.72075 \dots) + 24.36931 \dots = - 1.10786 E - 9

f^{^{'}} (u_{5}) = 9 (- 0.72075 \dots)^{2} - 52.73863 \dots (- 0.72075 \dots) + 13.24621 \dots = 55.93351 \dots

u_{6} = - 0.72075 \dots

Δ u_{6} = ∣ - 0.72075 \dots - (- 0.72075 \dots) ∣ = 1.98067 E - 11

Δ u_{6} = 1.98067 E - 11

u \approx - 0.72075 \dots

使用长除法

Eq u a t i o n 0 : \frac{3 u ^{3} - 26.36931 \dots u ^{2} + 13.24621 \dots u + 24.36931 \dots}{u + 0.72075 \dots} = 3 u^{2} - 28.53159 \dots u + 33.81062 \dots

3 u^{2} - 28.53159 \dots u + 33.81062 \dots \approx 0

使用牛顿-拉弗森方法找到

3 u^{2} - 28.53159 \dots u + 33.81062 \dots = 0

的一个解:

u \approx 1.38742 \dots

3 u^{2} - 28.53159 \dots u + 33.81062 \dots = 0

牛顿-拉弗森近似法定义

f (u) = 3 u^{2} - 28.53159 \dots u + 33.81062 \dots

找到

f^{^{'}} (u) : 6 u - 28.53159 \dots

\frac{d}{d u} (3 u^{2} - 28.53159 \dots u + 33.81062 \dots)

使用微分加减法定则:

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

= \frac{d}{d u} (3 u^{2}) - \frac{d}{d u} (28.53159 \dots u) + \frac{d}{d u} (33.81062 \dots)

\frac{d}{d u} (3 u^{2}) = 6 u

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

将常数提出:

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

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

使用幂法则:

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

= 3 \cdot 2 u^{2 - 1}

化简

= 6 u

\frac{d}{d u} (28.53159 \dots u) = 28.53159 \dots

\frac{d}{d u} (28.53159 \dots u)

将常数提出:

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

= 28.53159 \dots \frac{d u}{d u}

使用常见微分定则:

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

= 28.53159 \dots \cdot 1

化简

= 28.53159 \dots

\frac{d}{d u} (33.81062 \dots) = 0

\frac{d}{d u} (33.81062 \dots)

常数微分:

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

= 0

= 6 u - 28.53159 \dots + 0

化简

= 6 u - 28.53159 \dots

令

u_{0} = 1

计算

u_{n + 1}

至

Δ u_{n + 1} < 0.000001

u_{1} = 1.36744 \dots : Δ u_{1} = 0.36744 \dots

f (u_{0}) = 3 \cdot 1^{2} - 28.53159 \dots \cdot 1 + 33.81062 \dots = 8.27902 \dots

f^{^{'}} (u_{0}) = 6 \cdot 1 - 28.53159 \dots = - 22.53159 \dots

u_{1} = 1.36744 \dots

Δ u_{1} = ∣ 1.36744 \dots - 1 ∣ = 0.36744 \dots

Δ u_{1} = 0.36744 \dots

u_{2} = 1.38736 \dots : Δ u_{2} = 0.01992 \dots

f (u_{1}) = 3 \cdot 1.36744 \dots^{2} - 28.53159 \dots \cdot 1.36744 \dots + 33.81062 \dots = 0.40503 \dots

f^{^{'}} (u_{1}) = 6 \cdot 1.36744 \dots - 28.53159 \dots = - 20.32694 \dots

u_{2} = 1.38736 \dots

Δ u_{2} = ∣ 1.38736 \dots - 1.36744 \dots ∣ = 0.01992 \dots

Δ u_{2} = 0.01992 \dots

u_{3} = 1.38742 \dots : Δ u_{3} = 0.00005 \dots

f (u_{2}) = 3 \cdot 1.38736 \dots^{2} - 28.53159 \dots \cdot 1.38736 \dots + 33.81062 \dots = 0.00119 \dots

f^{^{'}} (u_{2}) = 6 \cdot 1.38736 \dots - 28.53159 \dots = - 20.20739 \dots

u_{3} = 1.38742 \dots

Δ u_{3} = ∣ 1.38742 \dots - 1.38736 \dots ∣ = 0.00005 \dots

Δ u_{3} = 0.00005 \dots

u_{4} = 1.38742 \dots : Δ u_{4} = 5.15864 E - 10

f (u_{3}) = 3 \cdot 1.38742 \dots^{2} - 28.53159 \dots \cdot 1.38742 \dots + 33.81062 \dots = 1.04241 E - 8

f^{^{'}} (u_{3}) = 6 \cdot 1.38742 \dots - 28.53159 \dots = - 20.20703 \dots

u_{4} = 1.38742 \dots

Δ u_{4} = ∣ 1.38742 \dots - 1.38742 \dots ∣ = 5.15864 E - 10

Δ u_{4} = 5.15864 E - 10

u \approx 1.38742 \dots

使用长除法

Eq u a t i o n 0 : \frac{3 u ^{2} - 28.53159 \dots u + 33.81062 \dots}{u - 1.38742 \dots} = 3 u - 24.36931 \dots

3 u - 24.36931 \dots \approx 0

u \approx 8.12310 \dots

解为

u \approx - 0.12310 \dots, u \approx - 0.72075 \dots, u \approx 1.38742 \dots, u \approx 8.12310 \dots

u \approx - 0.12310 \dots, u \approx - 0.72075 \dots, u \approx 1.38742 \dots, u \approx 8.12310 \dots

验证解

找到无定义的点（奇点）:

u = 0

取

u^{2} + u^{- 2} + \frac{( u - u ^{- 1} ) ^{2}}{2} - 13 (u - u^{- 1})

的分母，令其等于零

解

u^{2} = 0 : u = 0

u^{2} = 0

使用法则

x^{n} = 0 \Rightarrow x = 0

u = 0

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 - 0.12310 \dots, u \approx - 0.72075 \dots, u \approx 1.38742 \dots, u \approx 8.12310 \dots

u \approx - 0.12310 \dots, u \approx - 0.72075 \dots, u \approx 1.38742 \dots, u \approx 8.12310 \dots

代回

u = e^{x}

，求解

x

解

e^{x} = - 0.12310 \dots : x \in R

无解

e^{x} = - 0.12310 \dots

a^{f (x)}

对于

x

不能为零或负值

\in R

x \in R 无解

解

e^{x} = - 0.72075 \dots : x \in R

无解

e^{x} = - 0.72075 \dots

a^{f (x)}

对于

x

不能为零或负值

\in R

x \in R 无解

解

e^{x} = 1.38742 \dots : x = ln (1.38742 \dots)

e^{x} = 1.38742 \dots

使用指数运算法则

e^{x} = 1.38742 \dots

若

f (x) = g (x)

，则

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

ln (e^{x}) = ln (1.38742 \dots)

使用对数计算法则:

ln (e^{a}) = a

ln (e^{x}) = x

x = ln (1.38742 \dots)

x = ln (1.38742 \dots)

解

e^{x} = 8.12310 \dots : x = ln (8.12310 \dots)

e^{x} = 8.12310 \dots

使用指数运算法则

e^{x} = 8.12310 \dots

若

f (x) = g (x)

，则

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

ln (e^{x}) = ln (8.12310 \dots)

使用对数计算法则:

ln (e^{a}) = a

ln (e^{x}) = x

x = ln (8.12310 \dots)

x = ln (8.12310 \dots)

x = ln (1.38742 \dots), x = ln (8.12310 \dots)

x = ln (1.38742 \dots), x = ln (8.12310 \dots)

作图

流行的例子

(1-tan^2(A))/(1+tan^2(A))=1

tan(a)=sqrt(15/7),sin(a)

2cos^2(θ)+sin(θ)=2