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شائع علم المثلثات >

6cosh^2(x)+4sinh(x)=7

الحلّ

6 cosh^{2} (x) + 4 sinh (x) = 7

الحلّ

x = ln (1.21230 \dots), x = ln (0.45880 \dots)

درجات

x = 11.03066 \dots^{\circ}, x = - 44.64062 \dots^{\circ}

خطوات الحلّ

6 cosh^{2} (x) + 4 sinh (x) = 7

Rewrite using trig identities

6 cosh^{2} (x) + 4 sinh (x) = 7

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

:Use the Hyperbolic identity

6 cosh^{2} (x) + 4 \cdot \frac{e ^{x} - e ^{- x}}{2} = 7

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

:Use the Hyperbolic identity

6 (\frac{e ^{x} + e ^{- x}}{2})^{2} + 4 \cdot \frac{e ^{x} - e ^{- x}}{2} = 7

6 (\frac{e ^{x} + e ^{- x}}{2})^{2} + 4 \cdot \frac{e ^{x} - e ^{- x}}{2} = 7

6 (\frac{e ^{x} + e ^{- x}}{2})^{2} + 4 \cdot \frac{e ^{x} - e ^{- x}}{2} = 7 : x = ln (1.21230 \dots), x = ln (0.45880 \dots)

6 (\frac{e ^{x} + e ^{- x}}{2})^{2} + 4 \cdot \frac{e ^{x} - e ^{- x}}{2} = 7

فعّل قانون القوى

6 (\frac{e ^{x} + e ^{- x}}{2})^{2} + 4 \cdot \frac{e ^{x} - e ^{- x}}{2} = 7

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

:فعّل قانون القوى

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

6 (\frac{e ^{x} + ( e ^{x} ) ^{- 1}}{2})^{2} + 4 \cdot \frac{e ^{x} - ( e ^{x} ) ^{- 1}}{2} = 7

6 (\frac{e ^{x} + ( e ^{x} ) ^{- 1}}{2})^{2} + 4 \cdot \frac{e ^{x} - ( e ^{x} ) ^{- 1}}{2} = 7

e^{x} = u

أعد كتابة المعادلة، بحيث أنّ

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

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

حلّ:

u \approx 1.21230 \dots, u \approx 0.45880 \dots, u \approx - 0.82487 \dots, u \approx - 2.17956 \dots

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

بسّط

\frac{3 ( u ^{2} + 1 ) ^{2}}{2 u ^{2}} + \frac{2 ( u ^{2} - 1 )}{u} = 7

اضرب بالمضاعف المشترك الأصغر

\frac{3 ( u ^{2} + 1 ) ^{2}}{2 u ^{2}} + \frac{2 ( u ^{2} - 1 )}{u} = 7

Find Least Common Multiplier of

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

2 u^{2}, u

Lowest Common Multiplier (LCM)

Compute an expression comprised of factors that appear either in

2 u^{2}

u

= 2 u^{2}

2 u^{2} =

اضرب بالمضاعف المشترك الأصغر

\frac{3 ( u ^{2} + 1 ) ^{2}}{2 u ^{2}} \cdot 2 u^{2} + \frac{2 ( u ^{2} - 1 )}{u} \cdot 2 u^{2} = 7 \cdot 2 u^{2}

بسّط

\frac{3 ( u ^{2} + 1 ) ^{2}}{2 u ^{2}} \cdot 2 u^{2} + \frac{2 ( u ^{2} - 1 )}{u} \cdot 2 u^{2} = 7 \cdot 2 u^{2}

\frac{3 ( u ^{2} + 1 ) ^{2}}{2 u ^{2}} \cdot 2 u^{2}

بسّط:

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

\frac{3 ( u ^{2} + 1 ) ^{2}}{2 u ^{2}} \cdot 2 u^{2}

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

:اضرب كسور

= \frac{3 ( u ^{2} + 1 ) ^{2} \cdot 2 u ^{2}}{2 u ^{2}}

2 :

إلغ العوامل المشتركة

= \frac{3 ( u ^{2} + 1 ) ^{2} u ^{2}}{u ^{2}}

u^{2} :

إلغ العوامل المشتركة

= 3 (u^{2} + 1)^{2}

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

بسّط:

4 u (u^{2} - 1)

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

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

:اضرب كسور

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

2 \cdot 2 = 4 :

اضرب الأعداد

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

u :

إلغ العوامل المشتركة

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

7 \cdot 2 u^{2}

بسّط:

14 u^{2}

7 \cdot 2 u^{2}

7 \cdot 2 = 14 :

اضرب الأعداد

= 14 u^{2}

3 (u^{2} + 1)^{2} + 4 u (u^{2} - 1) = 14 u^{2}

3 (u^{2} + 1)^{2} + 4 u (u^{2} - 1) = 14 u^{2}

3 (u^{2} + 1)^{2} + 4 u (u^{2} - 1) = 14 u^{2}

3 (u^{2} + 1)^{2} + 4 u (u^{2} - 1) = 14 u^{2}

حلّ:

u \approx 1.21230 \dots, u \approx 0.45880 \dots, u \approx - 0.82487 \dots, u \approx - 2.17956 \dots

3 (u^{2} + 1)^{2} + 4 u (u^{2} - 1) = 14 u^{2}

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

وسّع:

3 u^{4} + 6 u^{2} + 3 + 4 u^{3} - 4 u

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

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

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

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

:فعّل صيغة الضرب المختصر

a = u^{2}, b = 1

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

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

بسّط:

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

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

1^{a} = 1

فعّل القانون

1^{2} = 1

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

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

(u^{2})^{2}

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

:فعّل قانون القوى

= u^{2 \cdot 2}

2 \cdot 2 = 4 :

اضرب الأعداد

= u^{4}

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

2 u^{2} \cdot 1

2 \cdot 1 = 2 :

اضرب الأعداد

= 2 u^{2}

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

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

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

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

وسٌع:

3 u^{4} + 6 u^{2} + 3

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

فعّل قانون ضرب الأقواس

= 3 u^{4} + 3 \cdot 2 u^{2} + 3 \cdot 1

3 u^{4} + 3 \cdot 2 u^{2} + 3 \cdot 1

بسّط:

3 u^{4} + 6 u^{2} + 3

3 u^{4} + 3 \cdot 2 u^{2} + 3 \cdot 1

3 \cdot 2 = 6 :

اضرب الأعداد

= 3 u^{4} + 6 u^{2} + 3 \cdot 1

3 \cdot 1 = 3 :

اضرب الأعداد

= 3 u^{4} + 6 u^{2} + 3

= 3 u^{4} + 6 u^{2} + 3

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

4 u (u^{2} - 1)

وسٌع:

4 u^{3} - 4 u

4 u (u^{2} - 1)

a (b - c) = ab - a c

: افتح أقواس بالاستعانة بـ

a = 4 u, b = u^{2}, c = 1

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

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

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

بسّط:

4 u^{3} - 4 u

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

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

4 u^{2} u

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

:فعّل قانون القوى

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

= 4 u^{2 + 1}

2 + 1 = 3 :

اجمع الأعداد

= 4 u^{3}

4 \cdot 1 \cdot u = 4 u

4 \cdot 1 \cdot u

4 \cdot 1 = 4 :

اضرب الأعداد

= 4 u

= 4 u^{3} - 4 u

= 4 u^{3} - 4 u

= 3 u^{4} + 6 u^{2} + 3 + 4 u^{3} - 4 u

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

انقل

14 u^{2}

إلى الجانب الأيسر

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

من الطرفين

14 u^{2}

اطرح

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

بسّط

3 u^{4} + 4 u^{3} - 8 u^{2} - 4 u + 3 = 0

3 u^{4} + 4 u^{3} - 8 u^{2} - 4 u + 3 = 0

بطريقة نيوتون ريبسون

3 u^{4} + 4 u^{3} - 8 u^{2} - 4 u + 3 = 0

جدّ حلًا لـ:

u \approx 1.21230 \dots

3 u^{4} + 4 u^{3} - 8 u^{2} - 4 u + 3 = 0

تعريف تقريب نيوتن-ريبسون

f (u) = 3 u^{4} + 4 u^{3} - 8 u^{2} - 4 u + 3

f^{^{'}} (u)

جد:

12 u^{3} + 12 u^{2} - 16 u - 4

\frac{d}{d u} (3 u^{4} + 4 u^{3} - 8 u^{2} - 4 u + 3)

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

:استعمل قانون الجمع

= \frac{d}{d u} (3 u^{4}) + \frac{d}{d u} (4 u^{3}) - \frac{d}{d u} (8 u^{2}) - \frac{d}{d u} (4 u) + \frac{d}{d u} (3)

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

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

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

:استخرج الثابت

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

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

:استعمل قانون الأسس

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

بسّط

= 12 u^{3}

\frac{d}{d u} (4 u^{3}) = 12 u^{2}

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

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

:استخرج الثابت

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

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

:استعمل قانون الأسس

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

بسّط

= 12 u^{2}

\frac{d}{d u} (8 u^{2}) = 16 u

\frac{d}{d u} (8 u^{2})

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

:استخرج الثابت

= 8 \frac{d}{d u} (u^{2})

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

:استعمل قانون الأسس

= 8 \cdot 2 u^{2 - 1}

بسّط

= 16 u

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

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

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

:استخرج الثابت

= 4 \frac{d u}{d u}

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

:استعمل المشتقة الأساسية

= 4 \cdot 1

بسّط

= 4

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

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

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

:مشتقة الثابت

= 0

= 12 u^{3} + 12 u^{2} - 16 u - 4 + 0

بسّط

= 12 u^{3} + 12 u^{2} - 16 u - 4

u_{0} = 1

استبدل

Δ u_{n + 1} < 0.000001

حتّى

u_{n + 1}

احسب

u_{1} = 1.5 : Δ u_{1} = 0.5

f (u_{0}) = 3 \cdot 1^{4} + 4 \cdot 1^{3} - 8 \cdot 1^{2} - 4 \cdot 1 + 3 = - 2

f^{^{'}} (u_{0}) = 12 \cdot 1^{3} + 12 \cdot 1^{2} - 16 \cdot 1 - 4 = 4

u_{1} = 1.5

Δ u_{1} = ∣ 1.5 - 1 ∣ = 0.5

Δ u_{1} = 0.5

u_{2} = 1.30537 \dots : Δ u_{2} = 0.19462 \dots

f (u_{1}) = 3 \cdot 1. 5^{4} + 4 \cdot 1. 5^{3} - 8 \cdot 1. 5^{2} - 4 \cdot 1.5 + 3 = 7.6875

f^{^{'}} (u_{1}) = 12 \cdot 1. 5^{3} + 12 \cdot 1. 5^{2} - 16 \cdot 1.5 - 4 = 39.5

u_{2} = 1.30537 \dots

Δ u_{2} = ∣ 1.30537 \dots - 1.5 ∣ = 0.19462 \dots

Δ u_{2} = 0.19462 \dots

u_{3} = 1.22652 \dots : Δ u_{3} = 0.07885 \dots

f (u_{2}) = 3 \cdot 1.30537 \dots^{4} + 4 \cdot 1.30537 \dots^{3} - 8 \cdot 1.30537 \dots^{2} - 4 \cdot 1.30537 \dots + 3 = 1.75491 \dots

f^{^{'}} (u_{2}) = 12 \cdot 1.30537 \dots^{3} + 12 \cdot 1.30537 \dots^{2} - 16 \cdot 1.30537 \dots - 4 = 22.25477 \dots

u_{3} = 1.22652 \dots

Δ u_{3} = ∣ 1.22652 \dots - 1.30537 \dots ∣ = 0.07885 \dots

Δ u_{3} = 0.07885 \dots

u_{4} = 1.21271 \dots : Δ u_{4} = 0.01381 \dots

f (u_{3}) = 3 \cdot 1.22652 \dots^{4} + 4 \cdot 1.22652 \dots^{3} - 8 \cdot 1.22652 \dots^{2} - 4 \cdot 1.22652 \dots + 3 = 0.22886 \dots

f^{^{'}} (u_{3}) = 12 \cdot 1.22652 \dots^{3} + 12 \cdot 1.22652 \dots^{2} - 16 \cdot 1.22652 \dots - 4 = 16.56956 \dots

u_{4} = 1.21271 \dots

Δ u_{4} = ∣ 1.21271 \dots - 1.22652 \dots ∣ = 0.01381 \dots

Δ u_{4} = 0.01381 \dots

u_{5} = 1.21230 \dots : Δ u_{5} = 0.00040 \dots

f (u_{4}) = 3 \cdot 1.21271 \dots^{4} + 4 \cdot 1.21271 \dots^{3} - 8 \cdot 1.21271 \dots^{2} - 4 \cdot 1.21271 \dots + 3 = 0.00639 \dots

f^{^{'}} (u_{4}) = 12 \cdot 1.21271 \dots^{3} + 12 \cdot 1.21271 \dots^{2} - 16 \cdot 1.21271 \dots - 4 = 15.64663 \dots

u_{5} = 1.21230 \dots

Δ u_{5} = ∣ 1.21230 \dots - 1.21271 \dots ∣ = 0.00040 \dots

Δ u_{5} = 0.00040 \dots

u_{6} = 1.21230 \dots : Δ u_{6} = 3.5348 E - 7

f (u_{5}) = 3 \cdot 1.21230 \dots^{4} + 4 \cdot 1.21230 \dots^{3} - 8 \cdot 1.21230 \dots^{2} - 4 \cdot 1.21230 \dots + 3 = 5.52123 E - 6

f^{^{'}} (u_{5}) = 12 \cdot 1.21230 \dots^{3} + 12 \cdot 1.21230 \dots^{2} - 16 \cdot 1.21230 \dots - 4 = 15.61963 \dots

u_{6} = 1.21230 \dots

Δ u_{6} = ∣ 1.21230 \dots - 1.21230 \dots ∣ = 3.5348 E - 7

Δ u_{6} = 3.5348 E - 7

u \approx 1.21230 \dots

فعّل القسمة الطويلة:

\frac{3 u ^{4} + 4 u ^{3} - 8 u ^{2} - 4 u + 3}{u - 1.21230 \dots} = 3 u^{3} + 7.63690 \dots u^{2} + 1.25824 \dots u - 2.47462 \dots

3 u^{3} + 7.63690 \dots u^{2} + 1.25824 \dots u - 2.47462 \dots \approx 0

بطريقة نيوتون ريبسون

3 u^{3} + 7.63690 \dots u^{2} + 1.25824 \dots u - 2.47462 \dots = 0

جدّ حلًا لـ:

u \approx 0.45880 \dots

3 u^{3} + 7.63690 \dots u^{2} + 1.25824 \dots u - 2.47462 \dots = 0

تعريف تقريب نيوتن-ريبسون

f (u) = 3 u^{3} + 7.63690 \dots u^{2} + 1.25824 \dots u - 2.47462 \dots

f^{^{'}} (u)

جد:

9 u^{2} + 15.27381 \dots u + 1.25824 \dots

\frac{d}{d u} (3 u^{3} + 7.63690 \dots u^{2} + 1.25824 \dots u - 2.47462 \dots)

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

:استعمل قانون الجمع

= \frac{d}{d u} (3 u^{3}) + \frac{d}{d u} (7.63690 \dots u^{2}) + \frac{d}{d u} (1.25824 \dots u) - \frac{d}{d u} (2.47462 \dots)

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

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

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

:استخرج الثابت

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

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

:استعمل قانون الأسس

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

بسّط

= 9 u^{2}

\frac{d}{d u} (7.63690 \dots u^{2}) = 15.27381 \dots u

\frac{d}{d u} (7.63690 \dots u^{2})

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

:استخرج الثابت

= 7.63690 \dots \frac{d}{d u} (u^{2})

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

:استعمل قانون الأسس

= 7.63690 \dots \cdot 2 u^{2 - 1}

بسّط

= 15.27381 \dots u

\frac{d}{d u} (1.25824 \dots u) = 1.25824 \dots

\frac{d}{d u} (1.25824 \dots u)

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

:استخرج الثابت

= 1.25824 \dots \frac{d u}{d u}

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

:استعمل المشتقة الأساسية

= 1.25824 \dots \cdot 1

بسّط

= 1.25824 \dots

\frac{d}{d u} (2.47462 \dots) = 0

\frac{d}{d u} (2.47462 \dots)

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

:مشتقة الثابت

= 0

= 9 u^{2} + 15.27381 \dots u + 1.25824 \dots - 0

بسّط

= 9 u^{2} + 15.27381 \dots u + 1.25824 \dots

u_{0} = 2

استبدل

Δ u_{n + 1} < 0.000001

حتّى

u_{n + 1}

احسب

u_{1} = 1.19491 \dots : Δ u_{1} = 0.80508 \dots

f (u_{0}) = 3 \cdot 2^{3} + 7.63690 \dots \cdot 2^{2} + 1.25824 \dots \cdot 2 - 2.47462 \dots = 54.58948 \dots

f^{^{'}} (u_{0}) = 9 \cdot 2^{2} + 15.27381 \dots \cdot 2 + 1.25824 \dots = 67.80587 \dots

u_{1} = 1.19491 \dots

Δ u_{1} = ∣ 1.19491 \dots - 2 ∣ = 0.80508 \dots

Δ u_{1} = 0.80508 \dots

u_{2} = 0.72978 \dots : Δ u_{2} = 0.46512 \dots

f (u_{1}) = 3 \cdot 1.19491 \dots^{3} + 7.63690 \dots \cdot 1.19491 \dots^{2} + 1.25824 \dots \cdot 1.19491 \dots - 2.47462 \dots = 15.05138 \dots

f^{^{'}} (u_{1}) = 9 \cdot 1.19491 \dots^{2} + 15.27381 \dots \cdot 1.19491 \dots + 1.25824 \dots = 32.35955 \dots

u_{2} = 0.72978 \dots

Δ u_{2} = ∣ 0.72978 \dots - 1.19491 \dots ∣ = 0.46512 \dots

Δ u_{2} = 0.46512 \dots

u_{3} = 0.51598 \dots : Δ u_{3} = 0.21379 \dots

f (u_{2}) = 3 \cdot 0.72978 \dots^{3} + 7.63690 \dots \cdot 0.72978 \dots^{2} + 1.25824 \dots \cdot 0.72978 \dots - 2.47462 \dots = 3.67695 \dots

f^{^{'}} (u_{2}) = 9 \cdot 0.72978 \dots^{2} + 15.27381 \dots \cdot 0.72978 \dots + 1.25824 \dots = 17.19813 \dots

u_{3} = 0.51598 \dots

Δ u_{3} = ∣ 0.51598 \dots - 0.72978 \dots ∣ = 0.21379 \dots

Δ u_{3} = 0.21379 \dots

u_{4} = 0.46223 \dots : Δ u_{4} = 0.05374 \dots

f (u_{3}) = 3 \cdot 0.51598 \dots^{3} + 7.63690 \dots \cdot 0.51598 \dots^{2} + 1.25824 \dots \cdot 0.51598 \dots - 2.47462 \dots = 0.61999 \dots

f^{^{'}} (u_{3}) = 9 \cdot 0.51598 \dots^{2} + 15.27381 \dots \cdot 0.51598 \dots + 1.25824 \dots = 11.53548 \dots

u_{4} = 0.46223 \dots

Δ u_{4} = ∣ 0.46223 \dots - 0.51598 \dots ∣ = 0.05374 \dots

Δ u_{4} = 0.05374 \dots

u_{5} = 0.45882 \dots : Δ u_{5} = 0.00341 \dots

f (u_{4}) = 3 \cdot 0.46223 \dots^{3} + 7.63690 \dots \cdot 0.46223 \dots^{2} + 1.25824 \dots \cdot 0.46223 \dots - 2.47462 \dots = 0.03500 \dots

f^{^{'}} (u_{4}) = 9 \cdot 0.46223 \dots^{2} + 15.27381 \dots \cdot 0.46223 \dots + 1.25824 \dots = 10.24137 \dots

u_{5} = 0.45882 \dots

Δ u_{5} = ∣ 0.45882 \dots - 0.46223 \dots ∣ = 0.00341 \dots

Δ u_{5} = 0.00341 \dots

u_{6} = 0.45880 \dots : Δ u_{6} = 0.00001 \dots

f (u_{5}) = 3 \cdot 0.45882 \dots^{3} + 7.63690 \dots \cdot 0.45882 \dots^{2} + 1.25824 \dots \cdot 0.45882 \dots - 2.47462 \dots = 0.00013 \dots

f^{^{'}} (u_{5}) = 9 \cdot 0.45882 \dots^{2} + 15.27381 \dots \cdot 0.45882 \dots + 1.25824 \dots = 10.16082 \dots

u_{6} = 0.45880 \dots

Δ u_{6} = ∣ 0.45880 \dots - 0.45882 \dots ∣ = 0.00001 \dots

Δ u_{6} = 0.00001 \dots

u_{7} = 0.45880 \dots : Δ u_{7} = 2.12808 E - 10

f (u_{6}) = 3 \cdot 0.45880 \dots^{3} + 7.63690 \dots \cdot 0.45880 \dots^{2} + 1.25824 \dots \cdot 0.45880 \dots - 2.47462 \dots = 2.16224 E - 9

f^{^{'}} (u_{6}) = 9 \cdot 0.45880 \dots^{2} + 15.27381 \dots \cdot 0.45880 \dots + 1.25824 \dots = 10.16050 \dots

u_{7} = 0.45880 \dots

Δ u_{7} = ∣ 0.45880 \dots - 0.45880 \dots ∣ = 2.12808 E - 10

Δ u_{7} = 2.12808 E - 10

u \approx 0.45880 \dots

فعّل القسمة الطويلة:

\frac{3 u ^{3} + 7.63690 \dots u ^{2} + 1.25824 \dots u - 2.47462 \dots}{u - 0.45880 \dots} = 3 u^{2} + 9.01332 \dots u + 5.39361 \dots

3 u^{2} + 9.01332 \dots u + 5.39361 \dots \approx 0

بطريقة نيوتون ريبسون

3 u^{2} + 9.01332 \dots u + 5.39361 \dots = 0

جدّ حلًا لـ:

u \approx - 0.82487 \dots

3 u^{2} + 9.01332 \dots u + 5.39361 \dots = 0

تعريف تقريب نيوتن-ريبسون

f (u) = 3 u^{2} + 9.01332 \dots u + 5.39361 \dots

f^{^{'}} (u)

جد:

6 u + 9.01332 \dots

\frac{d}{d u} (3 u^{2} + 9.01332 \dots u + 5.39361 \dots)

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

:استعمل قانون الجمع

= \frac{d}{d u} (3 u^{2}) + \frac{d}{d u} (9.01332 \dots u) + \frac{d}{d u} (5.39361 \dots)

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

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

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

:استخرج الثابت

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

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

:استعمل قانون الأسس

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

بسّط

= 6 u

\frac{d}{d u} (9.01332 \dots u) = 9.01332 \dots

\frac{d}{d u} (9.01332 \dots u)

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

:استخرج الثابت

= 9.01332 \dots \frac{d u}{d u}

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

:استعمل المشتقة الأساسية

= 9.01332 \dots \cdot 1

بسّط

= 9.01332 \dots

\frac{d}{d u} (5.39361 \dots) = 0

\frac{d}{d u} (5.39361 \dots)

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

:مشتقة الثابت

= 0

= 6 u + 9.01332 \dots + 0

بسّط

= 6 u + 9.01332 \dots

u_{0} = - 1

استبدل

Δ u_{n + 1} < 0.000001

حتّى

u_{n + 1}

احسب

u_{1} = - 0.79434 \dots : Δ u_{1} = 0.20565 \dots

f (u_{0}) = 3 (- 1)^{2} + 9.01332 \dots (- 1) + 5.39361 \dots = - 0.61970 \dots

f^{^{'}} (u_{0}) = 6 (- 1) + 9.01332 \dots = 3.01332 \dots

u_{1} = - 0.79434 \dots

Δ u_{1} = ∣ - 0.79434 \dots - (- 1) ∣ = 0.20565 \dots

Δ u_{1} = 0.20565 \dots

u_{2} = - 0.82421 \dots : Δ u_{2} = 0.02987 \dots

f (u_{1}) = 3 (- 0.79434 \dots)^{2} + 9.01332 \dots (- 0.79434 \dots) + 5.39361 \dots = 0.12688 \dots

f^{^{'}} (u_{1}) = 6 (- 0.79434 \dots) + 9.01332 \dots = 4.24726 \dots

u_{2} = - 0.82421 \dots

Δ u_{2} = ∣ - 0.82421 \dots - (- 0.79434 \dots) ∣ = 0.02987 \dots

Δ u_{2} = 0.02987 \dots

u_{3} = - 0.82487 \dots : Δ u_{3} = 0.00065 \dots

f (u_{2}) = 3 (- 0.82421 \dots)^{2} + 9.01332 \dots (- 0.82421 \dots) + 5.39361 \dots = 0.00267 \dots

f^{^{'}} (u_{2}) = 6 (- 0.82421 \dots) + 9.01332 \dots = 4.06801 \dots

u_{3} = - 0.82487 \dots

Δ u_{3} = ∣ - 0.82487 \dots - (- 0.82421 \dots) ∣ = 0.00065 \dots

Δ u_{3} = 0.00065 \dots

u_{4} = - 0.82487 \dots : Δ u_{4} = 3.19753 E - 7

f (u_{3}) = 3 (- 0.82487 \dots)^{2} + 9.01332 \dots (- 0.82487 \dots) + 5.39361 \dots = 1.2995 E - 6

f^{^{'}} (u_{3}) = 6 (- 0.82487 \dots) + 9.01332 \dots = 4.06407 \dots

u_{4} = - 0.82487 \dots

Δ u_{4} = ∣ - 0.82487 \dots - (- 0.82487 \dots) ∣ = 3.19753 E - 7

Δ u_{4} = 3.19753 E - 7

u \approx - 0.82487 \dots

فعّل القسمة الطويلة:

\frac{3 u ^{2} + 9.01332 \dots u + 5.39361 \dots}{u + 0.82487 \dots} = 3 u + 6.53869 \dots

3 u + 6.53869 \dots \approx 0

u \approx - 2.17956 \dots

The solutions are

u \approx 1.21230 \dots, u \approx 0.45880 \dots, u \approx - 0.82487 \dots, u \approx - 2.17956 \dots

u \approx 1.21230 \dots, u \approx 0.45880 \dots, u \approx - 0.82487 \dots, u \approx - 2.17956 \dots

افحص الإجبات

جد نقاط غير معرّفة:

u = 0

وقم بمساواتها لصفر

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

خذ المقامات في

u = 0

النقاط التالية غير معرّفة

u = 0

ضمّ النقاط غير المعرّفة مع الحلول

u \approx 1.21230 \dots, u \approx 0.45880 \dots, u \approx - 0.82487 \dots, u \approx - 2.17956 \dots

u \approx 1.21230 \dots, u \approx 0.45880 \dots, u \approx - 0.82487 \dots, u \approx - 2.17956 \dots

Substitute back

u = e^{x},

solve for

x

e^{x} = 1.21230 \dots

حلّ:

x = ln (1.21230 \dots)

e^{x} = 1.21230 \dots

فعّل قانون القوى

e^{x} = 1.21230 \dots

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

إذا

, f (x) = g (x)

إذا تحقّق أنّ

ln (e^{x}) = ln (1.21230 \dots)

ln (e^{a}) = a

:فعّل قانون اللوغارتمات

ln (e^{x}) = x

x = ln (1.21230 \dots)

x = ln (1.21230 \dots)

e^{x} = 0.45880 \dots

حلّ:

x = ln (0.45880 \dots)

e^{x} = 0.45880 \dots

فعّل قانون القوى

e^{x} = 0.45880 \dots

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

إذا

, f (x) = g (x)

إذا تحقّق أنّ

ln (e^{x}) = ln (0.45880 \dots)

ln (e^{a}) = a

:فعّل قانون اللوغارتمات

ln (e^{x}) = x

x = ln (0.45880 \dots)

x = ln (0.45880 \dots)

e^{x} = - 0.82487 \dots

حلّ:

x \in R

لا يوجد حلّ لـ

e^{x} = - 0.82487 \dots

x \in R

لا يمكن أن يكون سالبًا أو صفرًا لـ

a^{f (x)}

x \in R لا يوجد حلّ لـ

e^{x} = - 2.17956 \dots

حلّ:

x \in R

لا يوجد حلّ لـ

e^{x} = - 2.17956 \dots

x \in R

لا يمكن أن يكون سالبًا أو صفرًا لـ

a^{f (x)}

x \in R لا يوجد حلّ لـ

x = ln (1.21230 \dots), x = ln (0.45880 \dots)

x = ln (1.21230 \dots), x = ln (0.45880 \dots)

رسم

أمثلة شائعة

5sqrt(3)tan(x)+3=8sqrt(3)tan(x)

53 tan (x) + 3 = 83 tan (x)

cos^2(x)=-0.5

cos^{2} (x) = - 0.5

cos(x)=(1.5)/(4.272)

cos (x) = \frac{1.5}{4.272}

pi/(12)=arcsin(x/2)

\frac{π}{12} = arcsin (\frac{x}{2})

6sin^2(x)=0

6 sin^{2} (x) = 0