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شائع علم المثلثات >

sinh^2(x)-5cosh(x)+7=0

الحلّ

sinh^{2} (x) - 5 cosh (x) + 7 = 0

الحلّ

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

درجات

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

خطوات الحلّ

sinh^{2} (x) - 5 cosh (x) + 7 = 0

Rewrite using trig identities

sinh^{2} (x) - 5 cosh (x) + 7 = 0

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

:Use the Hyperbolic identity

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

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

:Use the Hyperbolic identity

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

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

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

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

فعّل قانون القوى

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

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

:فعّل قانون القوى

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

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

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

e^{x} = u

أعد كتابة المعادلة، بحيث أنّ

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

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

حلّ:

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

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

بسّط

\frac{( u ^{2} - 1 ) ^{2}}{4 u ^{2}} - \frac{5 ( u ^{2} + 1 )}{2 u} + 7 = 0

اضرب بالمضاعف المشترك الأصغر

\frac{( u ^{2} - 1 ) ^{2}}{4 u ^{2}} - \frac{5 ( u ^{2} + 1 )}{2 u} + 7 = 0

Find Least Common Multiplier of

4 u^{2}, 2 u : 4 u^{2}

4 u^{2}, 2 u

Lowest Common Multiplier (LCM)

4, 2

المضاعف المشترك الأصغر لـ:

4

4, 2

المضاعف المشترك الأصغر

4

تحليل لعوامل أوّليّة لـ:

2 \cdot 2

4

4 = 2 \cdot 2, 2

ينقسم على

4

= 2 \cdot 2

2

تحليل لعوامل أوّليّة لـ:

2

2

هو عدد أوّليّ لذلك لا يمكن تحليله لعوامل أوّليّة

2

= 2

2

أو

4

احسب عدد مركّب من عوامل أوّليّة تظهر في

= 2 \cdot 2

2 \cdot 2 = 4 :

اضرب الأعداد

= 4

Compute an expression comprised of factors that appear either in

4 u^{2}

2 u

= 4 u^{2}

4 u^{2} =

اضرب بالمضاعف المشترك الأصغر

\frac{( u ^{2} - 1 ) ^{2}}{4 u ^{2}} \cdot 4 u^{2} - \frac{5 ( u ^{2} + 1 )}{2 u} \cdot 4 u^{2} + 7 \cdot 4 u^{2} = 0 \cdot 4 u^{2}

بسّط

\frac{( u ^{2} - 1 ) ^{2}}{4 u ^{2}} \cdot 4 u^{2} - \frac{5 ( u ^{2} + 1 )}{2 u} \cdot 4 u^{2} + 7 \cdot 4 u^{2} = 0 \cdot 4 u^{2}

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

بسّط:

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

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

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

:اضرب كسور

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

4 :

إلغ العوامل المشتركة

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

u^{2} :

إلغ العوامل المشتركة

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

- \frac{5 ( u ^{2} + 1 )}{2 u} \cdot 4 u^{2}

بسّط:

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

- \frac{5 ( u ^{2} + 1 )}{2 u} \cdot 4 u^{2}

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

:اضرب كسور

= - \frac{5 ( u ^{2} + 1 ) \cdot 4 u ^{2}}{2 u}

5 \cdot 4 = 20 :

اضرب الأعداد

= - \frac{20 u ^{2} ( u ^{2} + 1 )}{2 u}

\frac{20}{2} = 10 :

اقسم الأعداد

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

u :

إلغ العوامل المشتركة

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

7 \cdot 4 u^{2}

بسّط:

28 u^{2}

7 \cdot 4 u^{2}

7 \cdot 4 = 28 :

اضرب الأعداد

= 28 u^{2}

0 \cdot 4 u^{2}

بسّط:

0

0 \cdot 4 u^{2}

0 \cdot a = 0

فعّل القانون

= 0

(u^{2} - 1)^{2} - 10 u (u^{2} + 1) + 28 u^{2} = 0

(u^{2} - 1)^{2} - 10 u (u^{2} + 1) + 28 u^{2} = 0

(u^{2} - 1)^{2} - 10 u (u^{2} + 1) + 28 u^{2} = 0

(u^{2} - 1)^{2} - 10 u (u^{2} + 1) + 28 u^{2} = 0

حلّ:

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

(u^{2} - 1)^{2} - 10 u (u^{2} + 1) + 28 u^{2} = 0

(u^{2} - 1)^{2} - 10 u (u^{2} + 1) + 28 u^{2}

وسّع:

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

(u^{2} - 1)^{2} - 10 u (u^{2} + 1) + 28 u^{2}

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

(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

= u^{4} - 2 u^{2} + 1 - 10 u (u^{2} + 1) + 28 u^{2}

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

وسٌع:

- 10 u^{3} - 10 u

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

a (b + c) = ab + a c

: افتح أقواس بالاستعانة بـ

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

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

فعّل قوانين سالب-موجب

+ (- a) = - a

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

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

بسّط:

- 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

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

:فعّل قانون القوى

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

= 10 u^{2 + 1}

2 + 1 = 3 :

اجمع الأعداد

= 10 u^{3}

10 \cdot 1 \cdot u = 10 u

10 \cdot 1 \cdot u

10 \cdot 1 = 10 :

اضرب الأعداد

= 10 u

= - 10 u^{3} - 10 u

= - 10 u^{3} - 10 u

= u^{4} - 2 u^{2} + 1 - 10 u^{3} - 10 u + 28 u^{2}

u^{4} - 2 u^{2} + 1 - 10 u^{3} - 10 u + 28 u^{2}

بسّط:

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

u^{4} - 2 u^{2} + 1 - 10 u^{3} - 10 u + 28 u^{2}

جمّع التعابير المتشابهة

= u^{4} - 10 u^{3} - 2 u^{2} + 28 u^{2} - 10 u + 1

- 2 u^{2} + 28 u^{2} = 26 u^{2} :

اجمع العناصر المتشابهة

= 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

بطريقة نيوتون ريبسون

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

جدّ حلًا لـ:

u \approx 0.17157 \dots

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

تعريف تقريب نيوتن-ريبسون

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

f^{^{'}} (u)

جد:

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

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

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

:استعمل قانون الجمع

= \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})

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

:استعمل قانون الأسس

= 4 u^{4 - 1}

بسّط

= 4 u^{3}

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

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

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

:استخرج الثابت

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

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

:استعمل قانون الأسس

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

بسّط

= 30 u^{2}

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

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

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

:استخرج الثابت

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

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

:استعمل قانون الأسس

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

بسّط

= 52 u

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

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

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

:استخرج الثابت

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

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

:استعمل المشتقة الأساسية

= 10 \cdot 1

بسّط

= 10

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

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

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

:مشتقة الثابت

= 0

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

بسّط

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

u_{0} = 0

استبدل

Δ u_{n + 1} < 0.000001

حتّى

u_{n + 1}

احسب

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

فعّل القسمة الطويلة:

\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

بطريقة نيوتون ريبسون

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

جدّ حلًا لـ:

u \approx 0.26794 \dots

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

تعريف تقريب نيوتن-ريبسون

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

f^{^{'}} (u)

جد:

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)

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

:استعمل قانون الجمع

= \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})

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

:استعمل قانون الأسس

= 3 u^{3 - 1}

بسّط

= 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})

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

:استخرج الثابت

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

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

:استعمل قانون الأسس

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

بسّط

= 19.65685 \dots u

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

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

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

:استخرج الثابت

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

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

:استعمل المشتقة الأساسية

= 24.31370 \dots \cdot 1

بسّط

= 24.31370 \dots

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

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

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

:مشتقة الثابت

= 0

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

بسّط

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

u_{0} = 0

استبدل

Δ u_{n + 1} < 0.000001

حتّى

u_{n + 1}

احسب

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

فعّل القسمة الطويلة:

\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

بطريقة نيوتون ريبسون

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

جدّ حلًا لـ:

u \approx 3.73205 \dots

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

تعريف تقريب نيوتن-ريبسون

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

f^{^{'}} (u)

جد:

2 u - 9.56047 \dots

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

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

:استعمل قانون الجمع

= \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})

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

:استعمل قانون الأسس

= 2 u^{2 - 1}

بسّط

= 2 u

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

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

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

:استخرج الثابت

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

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

:استعمل المشتقة الأساسية

= 9.56047 \dots \cdot 1

بسّط

= 9.56047 \dots

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

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

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

:مشتقة الثابت

= 0

= 2 u - 9.56047 \dots + 0

بسّط

= 2 u - 9.56047 \dots

u_{0} = 2

استبدل

Δ u_{n + 1} < 0.000001

حتّى

u_{n + 1}

احسب

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

فعّل القسمة الطويلة:

\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

The solutions are

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

افحص الإجبات

جد نقاط غير معرّفة:

u = 0

وقم بمساواتها لصفر

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

خذ المقامات في

u = 0

النقاط التالية غير معرّفة

u = 0

ضمّ النقاط غير المعرّفة مع الحلول

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

Substitute back

u = e^{x},

solve for

x

e^{x} = 0.17157 \dots

حلّ:

x = ln (0.17157 \dots)

e^{x} = 0.17157 \dots

فعّل قانون القوى

e^{x} = 0.17157 \dots

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

إذا

, f (x) = g (x)

إذا تحقّق أنّ

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

ln (e^{a}) = a

:فعّل قانون اللوغارتمات

ln (e^{x}) = x

x = ln (0.17157 \dots)

x = ln (0.17157 \dots)

e^{x} = 0.26794 \dots

حلّ:

x = ln (0.26794 \dots)

e^{x} = 0.26794 \dots

فعّل قانون القوى

e^{x} = 0.26794 \dots

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

إذا

, f (x) = g (x)

إذا تحقّق أنّ

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

ln (e^{a}) = a

:فعّل قانون اللوغارتمات

ln (e^{x}) = x

x = ln (0.26794 \dots)

x = ln (0.26794 \dots)

e^{x} = 3.73205 \dots

حلّ:

x = ln (3.73205 \dots)

e^{x} = 3.73205 \dots

فعّل قانون القوى

e^{x} = 3.73205 \dots

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

إذا

, f (x) = g (x)

إذا تحقّق أنّ

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

ln (e^{a}) = a

:فعّل قانون اللوغارتمات

ln (e^{x}) = x

x = ln (3.73205 \dots)

x = ln (3.73205 \dots)

e^{x} = 5.82842 \dots

حلّ:

x = ln (5.82842 \dots)

e^{x} = 5.82842 \dots

فعّل قانون القوى

e^{x} = 5.82842 \dots

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

إذا

, f (x) = g (x)

إذا تحقّق أنّ

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

ln (e^{a}) = a

:فعّل قانون اللوغارتمات

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)

رسم

أمثلة شائعة

4+4sin(x)=0

4 + 4 sin (x) = 0

tan(A)= 67/94

tan (A) = \frac{67}{94}

sin(8x)-2cos(4x)=0

sin (8 x) - 2 cos (4 x) = 0

1.812=316*cos(1.496*x)+1.496

1.812 = 316 \cdot cos (1.496 \cdot x) + 1.496

solvefor t,cos(wt+d)=0

so l v e f or t, cos (wt + d) = 0