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sin^{22}(x)=4sin^2(x)cos^2(x)

Solution

sin^{22} (x) = 4 sin^{2} (x) cos^{2} (x)

Solution

x = 2 πn, x = π + 2 πn, x = 1.25989 \dots + 2 πn, x = π - 1.25989 \dots + 2 πn, x = - 1.25989 \dots + 2 πn, x = π + 1.25989 \dots + 2 πn

Degrees

x = 0^{\circ} + 36 0^{\circ} n, x = 18 0^{\circ} + 36 0^{\circ} n, x = 72.18663 \dots^{\circ} + 36 0^{\circ} n, x = 107.81336 \dots^{\circ} + 36 0^{\circ} n, x = - 72.18663 \dots^{\circ} + 36 0^{\circ} n, x = 252.18663 \dots^{\circ} + 36 0^{\circ} n

Solution steps

sin^{22} (x) = 4 sin^{2} (x) cos^{2} (x)

Subtract

4 sin^{2} (x) cos^{2} (x)

from both sides

sin^{22} (x) - 4 sin^{2} (x) cos^{2} (x) = 0

Rewrite using trig identities

sin^{22} (x) - 4 cos^{2} (x) sin^{2} (x)

Use the Pythagorean identity:

cos^{2} (x) + sin^{2} (x) = 1

cos^{2} (x) = 1 - sin^{2} (x)

= sin^{22} (x) - 4 (1 - sin^{2} (x)) sin^{2} (x)

sin^{22} (x) - (1 - sin^{2} (x)) \cdot 4 sin^{2} (x) = 0

Solve by substitution

sin^{22} (x) - (1 - sin^{2} (x)) \cdot 4 sin^{2} (x) = 0

Let:

sin (x) = u

u^{22} - (1 - u^{2}) \cdot 4 u^{2} = 0

u^{22} - (1 - u^{2}) \cdot 4 u^{2} = 0 : u = 0, u = 0.90641 \dots, u = - 0.90641 \dots

u^{22} - (1 - u^{2}) \cdot 4 u^{2} = 0

Expand

u^{22} - (1 - u^{2}) \cdot 4 u^{2} : u^{22} - 4 u^{2} + 4 u^{4}

u^{22} - (1 - u^{2}) \cdot 4 u^{2}

= u^{22} - 4 u^{2} (1 - u^{2})

Expand

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

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

Apply the distributive law:

a (b - c) = ab - a c

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

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

Apply minus-plus rules

- (- a) = a

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

Simplify

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

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

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

4 \cdot 1 \cdot u^{2}

Multiply the numbers:

4 \cdot 1 = 4

= 4 u^{2}

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

4 u^{2} u^{2}

Apply exponent rule:

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

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

= 4 u^{2 + 2}

Add the numbers:

2 + 2 = 4

= 4 u^{4}

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

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

= u^{22} - 4 u^{2} + 4 u^{4}

u^{22} - 4 u^{2} + 4 u^{4} = 0

Write in the standard form

a_{n} x^{n} + \dots + a_{1} x + a_{0} = 0

u^{22} + 4 u^{4} - 4 u^{2} = 0

Rewrite the equation with

v = u^{2}, v^{2} = u^{4}

and

v^{11} = u^{22}

v^{11} + 4 v^{2} - 4 v = 0

Solve

v^{11} + 4 v^{2} - 4 v = 0 : v = 0, v \approx 0.90641 \dots, v \approx - 1.24548 \dots

v^{11} + 4 v^{2} - 4 v = 0

Factor

v^{11} + 4 v^{2} - 4 v : v (v^{10} + 4 v - 4)

v^{11} + 4 v^{2} - 4 v

Apply exponent rule:

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

v^{2} = vv

= v^{10} v + 4 vv - 4 v

Factor out common term

v

= v (v^{10} + 4 v - 4)

v (v^{10} + 4 v - 4) = 0

Using the Zero Factor Principle:

ab = 0

then

a = 0

b = 0

v = 0 or v^{10} + 4 v - 4 = 0

Solve

v^{10} + 4 v - 4 = 0 : v \approx 0.90641 \dots, v \approx - 1.24548 \dots

v^{10} + 4 v - 4 = 0

Find one solution for

v^{10} + 4 v - 4 = 0

using Newton-Raphson:

v \approx 0.90641 \dots

v^{10} + 4 v - 4 = 0

Newton-Raphson Approximation Definition

f (v) = v^{10} + 4 v - 4

Find

f^{^{'}} (v) : 10 v^{9} + 4

\frac{d}{d v} (v^{10} + 4 v - 4)

Apply the Sum/Difference Rule:

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

= \frac{d}{d v} (v^{10}) + \frac{d}{d v} (4 v) - \frac{d}{d v} (4)

\frac{d}{d v} (v^{10}) = 10 v^{9}

\frac{d}{d v} (v^{10})

Apply the Power Rule:

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

= 10 v^{10 - 1}

Simplify

= 10 v^{9}

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

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

Take the constant out:

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

= 4 \frac{d v}{d v}

Apply the common derivative:

\frac{d v}{d v} = 1

= 4 \cdot 1

Simplify

= 4

\frac{d}{d v} (4) = 0

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

Derivative of a constant:

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

= 0

= 10 v^{9} + 4 - 0

Simplify

= 10 v^{9} + 4

Let

v_{0} = 1

Compute

v_{n + 1}

until

Δ v_{n + 1} < 0.000001

v_{1} = 0.92857 \dots : Δ v_{1} = 0.07142 \dots

f (v_{0}) = 1^{10} + 4 \cdot 1 - 4 = 1

f^{^{'}} (v_{0}) = 10 \cdot 1^{9} + 4 = 14

v_{1} = 0.92857 \dots

Δ v_{1} = ∣ 0.92857 \dots - 1 ∣ = 0.07142 \dots

Δ v_{1} = 0.07142 \dots

v_{2} = 0.90766 \dots : Δ v_{2} = 0.02090 \dots

f (v_{1}) = 0.92857 \dots^{10} + 4 \cdot 0.92857 \dots - 4 = 0.19088 \dots

f^{^{'}} (v_{1}) = 10 \cdot 0.92857 \dots^{9} + 4 = 9.13260 \dots

v_{2} = 0.90766 \dots

Δ v_{2} = ∣ 0.90766 \dots - 0.92857 \dots ∣ = 0.02090 \dots

Δ v_{2} = 0.02090 \dots

v_{3} = 0.90641 \dots : Δ v_{3} = 0.00125 \dots

f (v_{2}) = 0.90766 \dots^{10} + 4 \cdot 0.90766 \dots - 4 = 0.01023 \dots

f^{^{'}} (v_{2}) = 10 \cdot 0.90766 \dots^{9} + 4 = 8.18168 \dots

v_{3} = 0.90641 \dots

Δ v_{3} = ∣ 0.90641 \dots - 0.90766 \dots ∣ = 0.00125 \dots

Δ v_{3} = 0.00125 \dots

v_{4} = 0.90641 \dots : Δ v_{4} = 3.97918 E - 6

f (v_{3}) = 0.90641 \dots^{10} + 4 \cdot 0.90641 \dots - 4 = 0.00003 \dots

f^{^{'}} (v_{3}) = 10 \cdot 0.90641 \dots^{9} + 4 = 8.13008 \dots

v_{4} = 0.90641 \dots

Δ v_{4} = ∣ 0.90641 \dots - 0.90641 \dots ∣ = 3.97918 E - 6

Δ v_{4} = 3.97918 E - 6

v_{5} = 0.90641 \dots : Δ v_{5} = 3.99335 E - 11

f (v_{4}) = 0.90641 \dots^{10} + 4 \cdot 0.90641 \dots - 4 = 3.24656 E - 10

f^{^{'}} (v_{4}) = 10 \cdot 0.90641 \dots^{9} + 4 = 8.12992 \dots

v_{5} = 0.90641 \dots

Δ v_{5} = ∣ 0.90641 \dots - 0.90641 \dots ∣ = 3.99335 E - 11

Δ v_{5} = 3.99335 E - 11

v \approx 0.90641 \dots

Apply long division:

\frac{v ^{10} + 4 v - 4}{v - 0.90641 \dots} = v^{9} + 0.90641 \dots v^{8} + 0.82158 \dots v^{7} + 0.74469 \dots v^{6} + 0.67500 \dots v^{5} + 0.61183 \dots v^{4} + 0.55457 \dots v^{3} + 0.50267 \dots v^{2} + 0.45563 \dots v + 4.41299 \dots

v^{9} + 0.90641 \dots v^{8} + 0.82158 \dots v^{7} + 0.74469 \dots v^{6} + 0.67500 \dots v^{5} + 0.61183 \dots v^{4} + 0.55457 \dots v^{3} + 0.50267 \dots v^{2} + 0.45563 \dots v + 4.41299 \dots \approx 0

Find one solution for

v^{9} + 0.90641 \dots v^{8} + 0.82158 \dots v^{7} + 0.74469 \dots v^{6} + 0.67500 \dots v^{5} + 0.61183 \dots v^{4} + 0.55457 \dots v^{3} + 0.50267 \dots v^{2} + 0.45563 \dots v + 4.41299 \dots = 0

using Newton-Raphson:

v \approx - 1.24548 \dots

v^{9} + 0.90641 \dots v^{8} + 0.82158 \dots v^{7} + 0.74469 \dots v^{6} + 0.67500 \dots v^{5} + 0.61183 \dots v^{4} + 0.55457 \dots v^{3} + 0.50267 \dots v^{2} + 0.45563 \dots v + 4.41299 \dots = 0

Newton-Raphson Approximation Definition

f (v) = v^{9} + 0.90641 \dots v^{8} + 0.82158 \dots v^{7} + 0.74469 \dots v^{6} + 0.67500 \dots v^{5} + 0.61183 \dots v^{4} + 0.55457 \dots v^{3} + 0.50267 \dots v^{2} + 0.45563 \dots v + 4.41299 \dots

Find

f^{^{'}} (v) : 9 v^{8} + 7.25131 \dots v^{7} + 5.75111 \dots v^{6} + 4.46819 \dots v^{5} + 3.37502 \dots v^{4} + 2.44733 \dots v^{3} + 1.66372 \dots v^{2} + 1.00535 \dots v + 0.45563 \dots

\frac{d}{d v} (v^{9} + 0.90641 \dots v^{8} + 0.82158 \dots v^{7} + 0.74469 \dots v^{6} + 0.67500 \dots v^{5} + 0.61183 \dots v^{4} + 0.55457 \dots v^{3} + 0.50267 \dots v^{2} + 0.45563 \dots v + 4.41299 \dots)

Apply the Sum/Difference Rule:

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

= \frac{d}{d v} (v^{9}) + \frac{d}{d v} (0.90641 \dots v^{8}) + \frac{d}{d v} (0.82158 \dots v^{7}) + \frac{d}{d v} (0.74469 \dots v^{6}) + \frac{d}{d v} (0.67500 \dots v^{5}) + \frac{d}{d v} (0.61183 \dots v^{4}) + \frac{d}{d v} (0.55457 \dots v^{3}) + \frac{d}{d v} (0.50267 \dots v^{2}) + \frac{d}{d v} (0.45563 \dots v) + \frac{d}{d v} (4.41299 \dots)

\frac{d}{d v} (v^{9}) = 9 v^{8}

\frac{d}{d v} (v^{9})

Apply the Power Rule:

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

= 9 v^{9 - 1}

Simplify

= 9 v^{8}

\frac{d}{d v} (0.90641 \dots v^{8}) = 7.25131 \dots v^{7}

\frac{d}{d v} (0.90641 \dots v^{8})

Take the constant out:

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

= 0.90641 \dots \frac{d}{d v} (v^{8})

Apply the Power Rule:

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

= 0.90641 \dots \cdot 8 v^{8 - 1}

Simplify

= 7.25131 \dots v^{7}

\frac{d}{d v} (0.82158 \dots v^{7}) = 5.75111 \dots v^{6}

\frac{d}{d v} (0.82158 \dots v^{7})

Take the constant out:

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

= 0.82158 \dots \frac{d}{d v} (v^{7})

Apply the Power Rule:

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

= 0.82158 \dots \cdot 7 v^{7 - 1}

Simplify

= 5.75111 \dots v^{6}

\frac{d}{d v} (0.74469 \dots v^{6}) = 4.46819 \dots v^{5}

\frac{d}{d v} (0.74469 \dots v^{6})

Take the constant out:

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

= 0.74469 \dots \frac{d}{d v} (v^{6})

Apply the Power Rule:

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

= 0.74469 \dots \cdot 6 v^{6 - 1}

Simplify

= 4.46819 \dots v^{5}

\frac{d}{d v} (0.67500 \dots v^{5}) = 3.37502 \dots v^{4}

\frac{d}{d v} (0.67500 \dots v^{5})

Take the constant out:

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

= 0.67500 \dots \frac{d}{d v} (v^{5})

Apply the Power Rule:

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

= 0.67500 \dots \cdot 5 v^{5 - 1}

Simplify

= 3.37502 \dots v^{4}

\frac{d}{d v} (0.61183 \dots v^{4}) = 2.44733 \dots v^{3}

\frac{d}{d v} (0.61183 \dots v^{4})

Take the constant out:

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

= 0.61183 \dots \frac{d}{d v} (v^{4})

Apply the Power Rule:

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

= 0.61183 \dots \cdot 4 v^{4 - 1}

Simplify

= 2.44733 \dots v^{3}

\frac{d}{d v} (0.55457 \dots v^{3}) = 1.66372 \dots v^{2}

\frac{d}{d v} (0.55457 \dots v^{3})

Take the constant out:

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

= 0.55457 \dots \frac{d}{d v} (v^{3})

Apply the Power Rule:

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

= 0.55457 \dots \cdot 3 v^{3 - 1}

Simplify

= 1.66372 \dots v^{2}

\frac{d}{d v} (0.50267 \dots v^{2}) = 1.00535 \dots v

\frac{d}{d v} (0.50267 \dots v^{2})

Take the constant out:

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

= 0.50267 \dots \frac{d}{d v} (v^{2})

Apply the Power Rule:

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

= 0.50267 \dots \cdot 2 v^{2 - 1}

Simplify

= 1.00535 \dots v

\frac{d}{d v} (0.45563 \dots v) = 0.45563 \dots

\frac{d}{d v} (0.45563 \dots v)

Take the constant out:

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

= 0.45563 \dots \frac{d v}{d v}

Apply the common derivative:

\frac{d v}{d v} = 1

= 0.45563 \dots \cdot 1

Simplify

= 0.45563 \dots

\frac{d}{d v} (4.41299 \dots) = 0

\frac{d}{d v} (4.41299 \dots)

Derivative of a constant:

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

= 0

= 9 v^{8} + 7.25131 \dots v^{7} + 5.75111 \dots v^{6} + 4.46819 \dots v^{5} + 3.37502 \dots v^{4} + 2.44733 \dots v^{3} + 1.66372 \dots v^{2} + 1.00535 \dots v + 0.45563 \dots + 0

Simplify

= 9 v^{8} + 7.25131 \dots v^{7} + 5.75111 \dots v^{6} + 4.46819 \dots v^{5} + 3.37502 \dots v^{4} + 2.44733 \dots v^{3} + 1.66372 \dots v^{2} + 1.00535 \dots v + 0.45563 \dots

Let

v_{0} = - 5

Compute

v_{n + 1}

until

Δ v_{n + 1} < 0.000001

v_{1} = - 4.45375 \dots : Δ v_{1} = 0.54624 \dots

f (v_{0}) = (- 5)^{9} + 0.90641 \dots (- 5)^{8} + 0.82158 \dots (- 5)^{7} + 0.74469 \dots (- 5)^{6} + 0.67500 \dots (- 5)^{5} + 0.61183 \dots (- 5)^{4} + 0.55457 \dots (- 5)^{3} + 0.50267 \dots (- 5)^{2} + 0.45563 \dots (- 5) + 4.41299 \dots = - 1653389.03665 \dots

f^{^{'}} (v_{0}) = 9 (- 5)^{8} + 7.25131 \dots (- 5)^{7} + 5.75111 \dots (- 5)^{6} + 4.46819 \dots (- 5)^{5} + 3.37502 \dots (- 5)^{4} + 2.44733 \dots (- 5)^{3} + 1.66372 \dots (- 5)^{2} + 1.00535 \dots (- 5) + 0.45563 \dots = 3026854.43549 \dots

v_{1} = - 4.45375 \dots

Δ v_{1} = ∣ - 4.45375 \dots - (- 5) ∣ = 0.54624 \dots

Δ v_{1} = 0.54624 \dots

v_{2} = - 3.96802 \dots : Δ v_{2} = 0.48573 \dots

f (v_{1}) = (- 4.45375 \dots)^{9} + 0.90641 \dots (- 4.45375 \dots)^{8} + 0.82158 \dots (- 4.45375 \dots)^{7} + 0.74469 \dots (- 4.45375 \dots)^{6} + 0.67500 \dots (- 4.45375 \dots)^{5} + 0.61183 \dots (- 4.45375 \dots)^{4} + 0.55457 \dots (- 4.45375 \dots)^{3} + 0.50267 \dots (- 4.45375 \dots)^{2} + 0.45563 \dots (- 4.45375 \dots) + 4.41299 \dots = - 572909.56059 \dots

f^{^{'}} (v_{1}) = 9 (- 4.45375 \dots)^{8} + 7.25131 \dots (- 4.45375 \dots)^{7} + 5.75111 \dots (- 4.45375 \dots)^{6} + 4.46819 \dots (- 4.45375 \dots)^{5} + 3.37502 \dots (- 4.45375 \dots)^{4} + 2.44733 \dots (- 4.45375 \dots)^{3} + 1.66372 \dots (- 4.45375 \dots)^{2} + 1.00535 \dots (- 4.45375 \dots) + 0.45563 \dots = 1179476.08686 \dots

v_{2} = - 3.96802 \dots

Δ v_{2} = ∣ - 3.96802 \dots - (- 4.45375 \dots) ∣ = 0.48573 \dots

Δ v_{2} = 0.48573 \dots

v_{3} = - 3.53606 \dots : Δ v_{3} = 0.43195 \dots

f (v_{2}) = (- 3.96802 \dots)^{9} + 0.90641 \dots (- 3.96802 \dots)^{8} + 0.82158 \dots (- 3.96802 \dots)^{7} + 0.74469 \dots (- 3.96802 \dots)^{6} + 0.67500 \dots (- 3.96802 \dots)^{5} + 0.61183 \dots (- 3.96802 \dots)^{4} + 0.55457 \dots (- 3.96802 \dots)^{3} + 0.50267 \dots (- 3.96802 \dots)^{2} + 0.45563 \dots (- 3.96802 \dots) + 4.41299 \dots = - 198524.05883 \dots

f^{^{'}} (v_{2}) = 9 (- 3.96802 \dots)^{8} + 7.25131 \dots (- 3.96802 \dots)^{7} + 5.75111 \dots (- 3.96802 \dots)^{6} + 4.46819 \dots (- 3.96802 \dots)^{5} + 3.37502 \dots (- 3.96802 \dots)^{4} + 2.44733 \dots (- 3.96802 \dots)^{3} + 1.66372 \dots (- 3.96802 \dots)^{2} + 1.00535 \dots (- 3.96802 \dots) + 0.45563 \dots = 459591.06090 \dots

v_{3} = - 3.53606 \dots

Δ v_{3} = ∣ - 3.53606 \dots - (- 3.96802 \dots) ∣ = 0.43195 \dots

Δ v_{3} = 0.43195 \dots

v_{4} = - 3.15190 \dots : Δ v_{4} = 0.38416 \dots

f (v_{3}) = (- 3.53606 \dots)^{9} + 0.90641 \dots (- 3.53606 \dots)^{8} + 0.82158 \dots (- 3.53606 \dots)^{7} + 0.74469 \dots (- 3.53606 \dots)^{6} + 0.67500 \dots (- 3.53606 \dots)^{5} + 0.61183 \dots (- 3.53606 \dots)^{4} + 0.55457 \dots (- 3.53606 \dots)^{3} + 0.50267 \dots (- 3.53606 \dots)^{2} + 0.45563 \dots (- 3.53606 \dots) + 4.41299 \dots = - 68794.93716 \dots

f^{^{'}} (v_{3}) = 9 (- 3.53606 \dots)^{8} + 7.25131 \dots (- 3.53606 \dots)^{7} + 5.75111 \dots (- 3.53606 \dots)^{6} + 4.46819 \dots (- 3.53606 \dots)^{5} + 3.37502 \dots (- 3.53606 \dots)^{4} + 2.44733 \dots (- 3.53606 \dots)^{3} + 1.66372 \dots (- 3.53606 \dots)^{2} + 1.00535 \dots (- 3.53606 \dots) + 0.45563 \dots = 179076.94254 \dots

v_{4} = - 3.15190 \dots

Δ v_{4} = ∣ - 3.15190 \dots - (- 3.53606 \dots) ∣ = 0.38416 \dots

Δ v_{4} = 0.38416 \dots

v_{5} = - 2.81023 \dots : Δ v_{5} = 0.34167 \dots

f (v_{4}) = (- 3.15190 \dots)^{9} + 0.90641 \dots (- 3.15190 \dots)^{8} + 0.82158 \dots (- 3.15190 \dots)^{7} + 0.74469 \dots (- 3.15190 \dots)^{6} + 0.67500 \dots (- 3.15190 \dots)^{5} + 0.61183 \dots (- 3.15190 \dots)^{4} + 0.55457 \dots (- 3.15190 \dots)^{3} + 0.50267 \dots (- 3.15190 \dots)^{2} + 0.45563 \dots (- 3.15190 \dots) + 4.41299 \dots = - 23840.26765 \dots

f^{^{'}} (v_{4}) = 9 (- 3.15190 \dots)^{8} + 7.25131 \dots (- 3.15190 \dots)^{7} + 5.75111 \dots (- 3.15190 \dots)^{6} + 4.46819 \dots (- 3.15190 \dots)^{5} + 3.37502 \dots (- 3.15190 \dots)^{4} + 2.44733 \dots (- 3.15190 \dots)^{3} + 1.66372 \dots (- 3.15190 \dots)^{2} + 1.00535 \dots (- 3.15190 \dots) + 0.45563 \dots = 69775.21311 \dots

v_{5} = - 2.81023 \dots

Δ v_{5} = ∣ - 2.81023 \dots - (- 3.15190 \dots) ∣ = 0.34167 \dots

Δ v_{5} = 0.34167 \dots

v_{6} = - 2.50637 \dots : Δ v_{6} = 0.30385 \dots

f (v_{5}) = (- 2.81023 \dots)^{9} + 0.90641 \dots (- 2.81023 \dots)^{8} + 0.82158 \dots (- 2.81023 \dots)^{7} + 0.74469 \dots (- 2.81023 \dots)^{6} + 0.67500 \dots (- 2.81023 \dots)^{5} + 0.61183 \dots (- 2.81023 \dots)^{4} + 0.55457 \dots (- 2.81023 \dots)^{3} + 0.50267 \dots (- 2.81023 \dots)^{2} + 0.45563 \dots (- 2.81023 \dots) + 4.41299 \dots = - 8261.45550 \dots

f^{^{'}} (v_{5}) = 9 (- 2.81023 \dots)^{8} + 7.25131 \dots (- 2.81023 \dots)^{7} + 5.75111 \dots (- 2.81023 \dots)^{6} + 4.46819 \dots (- 2.81023 \dots)^{5} + 3.37502 \dots (- 2.81023 \dots)^{4} + 2.44733 \dots (- 2.81023 \dots)^{3} + 1.66372 \dots (- 2.81023 \dots)^{2} + 1.00535 \dots (- 2.81023 \dots) + 0.45563 \dots = 27188.45003 \dots

v_{6} = - 2.50637 \dots

Δ v_{6} = ∣ - 2.50637 \dots - (- 2.81023 \dots) ∣ = 0.30385 \dots

Δ v_{6} = 0.30385 \dots

v_{7} = - 2.23625 \dots : Δ v_{7} = 0.27011 \dots

f (v_{6}) = (- 2.50637 \dots)^{9} + 0.90641 \dots (- 2.50637 \dots)^{8} + 0.82158 \dots (- 2.50637 \dots)^{7} + 0.74469 \dots (- 2.50637 \dots)^{6} + 0.67500 \dots (- 2.50637 \dots)^{5} + 0.61183 \dots (- 2.50637 \dots)^{4} + 0.55457 \dots (- 2.50637 \dots)^{3} + 0.50267 \dots (- 2.50637 \dots)^{2} + 0.45563 \dots (- 2.50637 \dots) + 4.41299 \dots = - 2862.37457 \dots

f^{^{'}} (v_{6}) = 9 (- 2.50637 \dots)^{8} + 7.25131 \dots (- 2.50637 \dots)^{7} + 5.75111 \dots (- 2.50637 \dots)^{6} + 4.46819 \dots (- 2.50637 \dots)^{5} + 3.37502 \dots (- 2.50637 \dots)^{4} + 2.44733 \dots (- 2.50637 \dots)^{3} + 1.66372 \dots (- 2.50637 \dots)^{2} + 1.00535 \dots (- 2.50637 \dots) + 0.45563 \dots = 10596.88514 \dots

v_{7} = - 2.23625 \dots

Δ v_{7} = ∣ - 2.23625 \dots - (- 2.50637 \dots) ∣ = 0.27011 \dots

Δ v_{7} = 0.27011 \dots

v_{8} = - 1.99650 \dots : Δ v_{8} = 0.23975 \dots

f (v_{7}) = (- 2.23625 \dots)^{9} + 0.90641 \dots (- 2.23625 \dots)^{8} + 0.82158 \dots (- 2.23625 \dots)^{7} + 0.74469 \dots (- 2.23625 \dots)^{6} + 0.67500 \dots (- 2.23625 \dots)^{5} + 0.61183 \dots (- 2.23625 \dots)^{4} + 0.55457 \dots (- 2.23625 \dots)^{3} + 0.50267 \dots (- 2.23625 \dots)^{2} + 0.45563 \dots (- 2.23625 \dots) + 4.41299 \dots = - 991.10859 \dots

f^{^{'}} (v_{7}) = 9 (- 2.23625 \dots)^{8} + 7.25131 \dots (- 2.23625 \dots)^{7} + 5.75111 \dots (- 2.23625 \dots)^{6} + 4.46819 \dots (- 2.23625 \dots)^{5} + 3.37502 \dots (- 2.23625 \dots)^{4} + 2.44733 \dots (- 2.23625 \dots)^{3} + 1.66372 \dots (- 2.23625 \dots)^{2} + 1.00535 \dots (- 2.23625 \dots) + 0.45563 \dots = 4133.76874 \dots

v_{8} = - 1.99650 \dots

Δ v_{8} = ∣ - 1.99650 \dots - (- 2.23625 \dots) ∣ = 0.23975 \dots

Δ v_{8} = 0.23975 \dots

v_{9} = - 1.78466 \dots : Δ v_{9} = 0.21183 \dots

f (v_{8}) = (- 1.99650 \dots)^{9} + 0.90641 \dots (- 1.99650 \dots)^{8} + 0.82158 \dots (- 1.99650 \dots)^{7} + 0.74469 \dots (- 1.99650 \dots)^{6} + 0.67500 \dots (- 1.99650 \dots)^{5} + 0.61183 \dots (- 1.99650 \dots)^{4} + 0.55457 \dots (- 1.99650 \dots)^{3} + 0.50267 \dots (- 1.99650 \dots)^{2} + 0.45563 \dots (- 1.99650 \dots) + 4.41299 \dots = - 342.49576 \dots

f^{^{'}} (v_{8}) = 9 (- 1.99650 \dots)^{8} + 7.25131 \dots (- 1.99650 \dots)^{7} + 5.75111 \dots (- 1.99650 \dots)^{6} + 4.46819 \dots (- 1.99650 \dots)^{5} + 3.37502 \dots (- 1.99650 \dots)^{4} + 2.44733 \dots (- 1.99650 \dots)^{3} + 1.66372 \dots (- 1.99650 \dots)^{2} + 1.00535 \dots (- 1.99650 \dots) + 0.45563 \dots = 1616.80028 \dots

v_{9} = - 1.78466 \dots

Δ v_{9} = ∣ - 1.78466 \dots - (- 1.99650 \dots) ∣ = 0.21183 \dots

Δ v_{9} = 0.21183 \dots

v_{10} = - 1.60003 \dots : Δ v_{10} = 0.18463 \dots

f (v_{9}) = (- 1.78466 \dots)^{9} + 0.90641 \dots (- 1.78466 \dots)^{8} + 0.82158 \dots (- 1.78466 \dots)^{7} + 0.74469 \dots (- 1.78466 \dots)^{6} + 0.67500 \dots (- 1.78466 \dots)^{5} + 0.61183 \dots (- 1.78466 \dots)^{4} + 0.55457 \dots (- 1.78466 \dots)^{3} + 0.50267 \dots (- 1.78466 \dots)^{2} + 0.45563 \dots (- 1.78466 \dots) + 4.41299 \dots = - 117.65885 \dots

f^{^{'}} (v_{9}) = 9 (- 1.78466 \dots)^{8} + 7.25131 \dots (- 1.78466 \dots)^{7} + 5.75111 \dots (- 1.78466 \dots)^{6} + 4.46819 \dots (- 1.78466 \dots)^{5} + 3.37502 \dots (- 1.78466 \dots)^{4} + 2.44733 \dots (- 1.78466 \dots)^{3} + 1.66372 \dots (- 1.78466 \dots)^{2} + 1.00535 \dots (- 1.78466 \dots) + 0.45563 \dots = 637.26147 \dots

v_{10} = - 1.60003 \dots

Δ v_{10} = ∣ - 1.60003 \dots - (- 1.78466 \dots) ∣ = 0.18463 \dots

Δ v_{10} = 0.18463 \dots

v_{11} = - 1.44531 \dots : Δ v_{11} = 0.15471 \dots

f (v_{10}) = (- 1.60003 \dots)^{9} + 0.90641 \dots (- 1.60003 \dots)^{8} + 0.82158 \dots (- 1.60003 \dots)^{7} + 0.74469 \dots (- 1.60003 \dots)^{6} + 0.67500 \dots (- 1.60003 \dots)^{5} + 0.61183 \dots (- 1.60003 \dots)^{4} + 0.55457 \dots (- 1.60003 \dots)^{3} + 0.50267 \dots (- 1.60003 \dots)^{2} + 0.45563 \dots (- 1.60003 \dots) + 4.41299 \dots = - 39.72697 \dots

f^{^{'}} (v_{10}) = 9 (- 1.60003 \dots)^{8} + 7.25131 \dots (- 1.60003 \dots)^{7} + 5.75111 \dots (- 1.60003 \dots)^{6} + 4.46819 \dots (- 1.60003 \dots)^{5} + 3.37502 \dots (- 1.60003 \dots)^{4} + 2.44733 \dots (- 1.60003 \dots)^{3} + 1.66372 \dots (- 1.60003 \dots)^{2} + 1.00535 \dots (- 1.60003 \dots) + 0.45563 \dots = 256.77560 \dots

v_{11} = - 1.44531 \dots

Δ v_{11} = ∣ - 1.44531 \dots - (- 1.60003 \dots) ∣ = 0.15471 \dots

Δ v_{11} = 0.15471 \dots

v_{12} = - 1.32926 \dots : Δ v_{12} = 0.11605 \dots

f (v_{11}) = (- 1.44531 \dots)^{9} + 0.90641 \dots (- 1.44531 \dots)^{8} + 0.82158 \dots (- 1.44531 \dots)^{7} + 0.74469 \dots (- 1.44531 \dots)^{6} + 0.67500 \dots (- 1.44531 \dots)^{5} + 0.61183 \dots (- 1.44531 \dots)^{4} + 0.55457 \dots (- 1.44531 \dots)^{3} + 0.50267 \dots (- 1.44531 \dots)^{2} + 0.45563 \dots (- 1.44531 \dots) + 4.41299 \dots = - 12.75482 \dots

f^{^{'}} (v_{11}) = 9 (- 1.44531 \dots)^{8} + 7.25131 \dots (- 1.44531 \dots)^{7} + 5.75111 \dots (- 1.44531 \dots)^{6} + 4.46819 \dots (- 1.44531 \dots)^{5} + 3.37502 \dots (- 1.44531 \dots)^{4} + 2.44733 \dots (- 1.44531 \dots)^{3} + 1.66372 \dots (- 1.44531 \dots)^{2} + 1.00535 \dots (- 1.44531 \dots) + 0.45563 \dots = 109.90167 \dots

v_{12} = - 1.32926 \dots

Δ v_{12} = ∣ - 1.32926 \dots - (- 1.44531 \dots) ∣ = 0.11605 \dots

Δ v_{12} = 0.11605 \dots

v_{13} = - 1.26447 \dots : Δ v_{13} = 0.06478 \dots

f (v_{12}) = (- 1.32926 \dots)^{9} + 0.90641 \dots (- 1.32926 \dots)^{8} + 0.82158 \dots (- 1.32926 \dots)^{7} + 0.74469 \dots (- 1.32926 \dots)^{6} + 0.67500 \dots (- 1.32926 \dots)^{5} + 0.61183 \dots (- 1.32926 \dots)^{4} + 0.55457 \dots (- 1.32926 \dots)^{3} + 0.50267 \dots (- 1.32926 \dots)^{2} + 0.45563 \dots (- 1.32926 \dots) + 4.41299 \dots = - 3.53618 \dots

f^{^{'}} (v_{12}) = 9 (- 1.32926 \dots)^{8} + 7.25131 \dots (- 1.32926 \dots)^{7} + 5.75111 \dots (- 1.32926 \dots)^{6} + 4.46819 \dots (- 1.32926 \dots)^{5} + 3.37502 \dots (- 1.32926 \dots)^{4} + 2.44733 \dots (- 1.32926 \dots)^{3} + 1.66372 \dots (- 1.32926 \dots)^{2} + 1.00535 \dots (- 1.32926 \dots) + 0.45563 \dots = 54.58328 \dots

v_{13} = - 1.26447 \dots

Δ v_{13} = ∣ - 1.26447 \dots - (- 1.32926 \dots) ∣ = 0.06478 \dots

Δ v_{13} = 0.06478 \dots

v_{14} = - 1.24663 \dots : Δ v_{14} = 0.01784 \dots

f (v_{13}) = (- 1.26447 \dots)^{9} + 0.90641 \dots (- 1.26447 \dots)^{8} + 0.82158 \dots (- 1.26447 \dots)^{7} + 0.74469 \dots (- 1.26447 \dots)^{6} + 0.67500 \dots (- 1.26447 \dots)^{5} + 0.61183 \dots (- 1.26447 \dots)^{4} + 0.55457 \dots (- 1.26447 \dots)^{3} + 0.50267 \dots (- 1.26447 \dots)^{2} + 0.45563 \dots (- 1.26447 \dots) + 4.41299 \dots = - 0.64115 \dots

f^{^{'}} (v_{13}) = 9 (- 1.26447 \dots)^{8} + 7.25131 \dots (- 1.26447 \dots)^{7} + 5.75111 \dots (- 1.26447 \dots)^{6} + 4.46819 \dots (- 1.26447 \dots)^{5} + 3.37502 \dots (- 1.26447 \dots)^{4} + 2.44733 \dots (- 1.26447 \dots)^{3} + 1.66372 \dots (- 1.26447 \dots)^{2} + 1.00535 \dots (- 1.26447 \dots) + 0.45563 \dots = 35.92993 \dots

v_{14} = - 1.24663 \dots

Δ v_{14} = ∣ - 1.24663 \dots - (- 1.26447 \dots) ∣ = 0.01784 \dots

Δ v_{14} = 0.01784 \dots

v_{15} = - 1.24548 \dots : Δ v_{15} = 0.00114 \dots

f (v_{14}) = (- 1.24663 \dots)^{9} + 0.90641 \dots (- 1.24663 \dots)^{8} + 0.82158 \dots (- 1.24663 \dots)^{7} + 0.74469 \dots (- 1.24663 \dots)^{6} + 0.67500 \dots (- 1.24663 \dots)^{5} + 0.61183 \dots (- 1.24663 \dots)^{4} + 0.55457 \dots (- 1.24663 \dots)^{3} + 0.50267 \dots (- 1.24663 \dots)^{2} + 0.45563 \dots (- 1.24663 \dots) + 4.41299 \dots = - 0.03658 \dots

f^{^{'}} (v_{14}) = 9 (- 1.24663 \dots)^{8} + 7.25131 \dots (- 1.24663 \dots)^{7} + 5.75111 \dots (- 1.24663 \dots)^{6} + 4.46819 \dots (- 1.24663 \dots)^{5} + 3.37502 \dots (- 1.24663 \dots)^{4} + 2.44733 \dots (- 1.24663 \dots)^{3} + 1.66372 \dots (- 1.24663 \dots)^{2} + 1.00535 \dots (- 1.24663 \dots) + 0.45563 \dots = 31.89979 \dots

v_{15} = - 1.24548 \dots

Δ v_{15} = ∣ - 1.24548 \dots - (- 1.24663 \dots) ∣ = 0.00114 \dots

Δ v_{15} = 0.00114 \dots

v_{16} = - 1.24548 \dots : Δ v_{16} = 4.44027 E - 6

f (v_{15}) = (- 1.24548 \dots)^{9} + 0.90641 \dots (- 1.24548 \dots)^{8} + 0.82158 \dots (- 1.24548 \dots)^{7} + 0.74469 \dots (- 1.24548 \dots)^{6} + 0.67500 \dots (- 1.24548 \dots)^{5} + 0.61183 \dots (- 1.24548 \dots)^{4} + 0.55457 \dots (- 1.24548 \dots)^{3} + 0.50267 \dots (- 1.24548 \dots)^{2} + 0.45563 \dots (- 1.24548 \dots) + 4.41299 \dots = - 0.00014 \dots

f^{^{'}} (v_{15}) = 9 (- 1.24548 \dots)^{8} + 7.25131 \dots (- 1.24548 \dots)^{7} + 5.75111 \dots (- 1.24548 \dots)^{6} + 4.46819 \dots (- 1.24548 \dots)^{5} + 3.37502 \dots (- 1.24548 \dots)^{4} + 2.44733 \dots (- 1.24548 \dots)^{3} + 1.66372 \dots (- 1.24548 \dots)^{2} + 1.00535 \dots (- 1.24548 \dots) + 0.45563 \dots = 31.65496 \dots

v_{16} = - 1.24548 \dots

Δ v_{16} = ∣ - 1.24548 \dots - (- 1.24548 \dots) ∣ = 4.44027 E - 6

Δ v_{16} = 4.44027 E - 6

v_{17} = - 1.24548 \dots : Δ v_{17} = 6.62571 E - 11

f (v_{16}) = (- 1.24548 \dots)^{9} + 0.90641 \dots (- 1.24548 \dots)^{8} + 0.82158 \dots (- 1.24548 \dots)^{7} + 0.74469 \dots (- 1.24548 \dots)^{6} + 0.67500 \dots (- 1.24548 \dots)^{5} + 0.61183 \dots (- 1.24548 \dots)^{4} + 0.55457 \dots (- 1.24548 \dots)^{3} + 0.50267 \dots (- 1.24548 \dots)^{2} + 0.45563 \dots (- 1.24548 \dots) + 4.41299 \dots = - 2.0973 E - 9

f^{^{'}} (v_{16}) = 9 (- 1.24548 \dots)^{8} + 7.25131 \dots (- 1.24548 \dots)^{7} + 5.75111 \dots (- 1.24548 \dots)^{6} + 4.46819 \dots (- 1.24548 \dots)^{5} + 3.37502 \dots (- 1.24548 \dots)^{4} + 2.44733 \dots (- 1.24548 \dots)^{3} + 1.66372 \dots (- 1.24548 \dots)^{2} + 1.00535 \dots (- 1.24548 \dots) + 0.45563 \dots = 31.65401 \dots

v_{17} = - 1.24548 \dots

Δ v_{17} = ∣ - 1.24548 \dots - (- 1.24548 \dots) ∣ = 6.62571 E - 11

Δ v_{17} = 6.62571 E - 11

v \approx - 1.24548 \dots

Apply long division:

\frac{v ^{9} + 0.90641 \dots v ^{8} + 0.82158 \dots v ^{7} + 0.74469 \dots v ^{6} + 0.67500 \dots v ^{5} + 0.61183 \dots v ^{4} + 0.55457 \dots v ^{3} + 0.50267 \dots v ^{2} + 0.45563 \dots v + 4.41299 \dots}{v + 1.24548 \dots} = v^{8} - 0.33906 \dots v^{7} + 1.24388 \dots v^{6} - 0.80453 \dots v^{5} + 1.67704 \dots v^{4} - 1.47688 \dots v^{3} + 2.39401 \dots v^{2} - 2.47902 \dots v + 3.54320 \dots

v^{8} - 0.33906 \dots v^{7} + 1.24388 \dots v^{6} - 0.80453 \dots v^{5} + 1.67704 \dots v^{4} - 1.47688 \dots v^{3} + 2.39401 \dots v^{2} - 2.47902 \dots v + 3.54320 \dots \approx 0

Find one solution for

v^{8} - 0.33906 \dots v^{7} + 1.24388 \dots v^{6} - 0.80453 \dots v^{5} + 1.67704 \dots v^{4} - 1.47688 \dots v^{3} + 2.39401 \dots v^{2} - 2.47902 \dots v + 3.54320 \dots = 0

using Newton-Raphson:

No Solution for

v \in R

v^{8} - 0.33906 \dots v^{7} + 1.24388 \dots v^{6} - 0.80453 \dots v^{5} + 1.67704 \dots v^{4} - 1.47688 \dots v^{3} + 2.39401 \dots v^{2} - 2.47902 \dots v + 3.54320 \dots = 0

Newton-Raphson Approximation Definition

f (v) = v^{8} - 0.33906 \dots v^{7} + 1.24388 \dots v^{6} - 0.80453 \dots v^{5} + 1.67704 \dots v^{4} - 1.47688 \dots v^{3} + 2.39401 \dots v^{2} - 2.47902 \dots v + 3.54320 \dots

Find

f^{^{'}} (v) : 8 v^{7} - 2.37346 \dots v^{6} + 7.46332 \dots v^{5} - 4.02269 \dots v^{4} + 6.70817 \dots v^{3} - 4.43066 \dots v^{2} + 4.78802 \dots v - 2.47902 \dots

\frac{d}{d v} (v^{8} - 0.33906 \dots v^{7} + 1.24388 \dots v^{6} - 0.80453 \dots v^{5} + 1.67704 \dots v^{4} - 1.47688 \dots v^{3} + 2.39401 \dots v^{2} - 2.47902 \dots v + 3.54320 \dots)

Apply the Sum/Difference Rule:

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

= \frac{d}{d v} (v^{8}) - \frac{d}{d v} (0.33906 \dots v^{7}) + \frac{d}{d v} (1.24388 \dots v^{6}) - \frac{d}{d v} (0.80453 \dots v^{5}) + \frac{d}{d v} (1.67704 \dots v^{4}) - \frac{d}{d v} (1.47688 \dots v^{3}) + \frac{d}{d v} (2.39401 \dots v^{2}) - \frac{d}{d v} (2.47902 \dots v) + \frac{d}{d v} (3.54320 \dots)

\frac{d}{d v} (v^{8}) = 8 v^{7}

\frac{d}{d v} (v^{8})

Apply the Power Rule:

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

= 8 v^{8 - 1}

Simplify

= 8 v^{7}

\frac{d}{d v} (0.33906 \dots v^{7}) = 2.37346 \dots v^{6}

\frac{d}{d v} (0.33906 \dots v^{7})

Take the constant out:

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

= 0.33906 \dots \frac{d}{d v} (v^{7})

Apply the Power Rule:

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

= 0.33906 \dots \cdot 7 v^{7 - 1}

Simplify

= 2.37346 \dots v^{6}

\frac{d}{d v} (1.24388 \dots v^{6}) = 7.46332 \dots v^{5}

\frac{d}{d v} (1.24388 \dots v^{6})

Take the constant out:

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

= 1.24388 \dots \frac{d}{d v} (v^{6})

Apply the Power Rule:

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

= 1.24388 \dots \cdot 6 v^{6 - 1}

Simplify

= 7.46332 \dots v^{5}

\frac{d}{d v} (0.80453 \dots v^{5}) = 4.02269 \dots v^{4}

\frac{d}{d v} (0.80453 \dots v^{5})

Take the constant out:

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

= 0.80453 \dots \frac{d}{d v} (v^{5})

Apply the Power Rule:

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

= 0.80453 \dots \cdot 5 v^{5 - 1}

Simplify

= 4.02269 \dots v^{4}

\frac{d}{d v} (1.67704 \dots v^{4}) = 6.70817 \dots v^{3}

\frac{d}{d v} (1.67704 \dots v^{4})

Take the constant out:

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

= 1.67704 \dots \frac{d}{d v} (v^{4})

Apply the Power Rule:

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

= 1.67704 \dots \cdot 4 v^{4 - 1}

Simplify

= 6.70817 \dots v^{3}

\frac{d}{d v} (1.47688 \dots v^{3}) = 4.43066 \dots v^{2}

\frac{d}{d v} (1.47688 \dots v^{3})

Take the constant out:

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

= 1.47688 \dots \frac{d}{d v} (v^{3})

Apply the Power Rule:

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

= 1.47688 \dots \cdot 3 v^{3 - 1}

Simplify

= 4.43066 \dots v^{2}

\frac{d}{d v} (2.39401 \dots v^{2}) = 4.78802 \dots v

\frac{d}{d v} (2.39401 \dots v^{2})

Take the constant out:

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

= 2.39401 \dots \frac{d}{d v} (v^{2})

Apply the Power Rule:

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

= 2.39401 \dots \cdot 2 v^{2 - 1}

Simplify

= 4.78802 \dots v

\frac{d}{d v} (2.47902 \dots v) = 2.47902 \dots

\frac{d}{d v} (2.47902 \dots v)

Take the constant out:

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

= 2.47902 \dots \frac{d v}{d v}

Apply the common derivative:

\frac{d v}{d v} = 1

= 2.47902 \dots \cdot 1

Simplify

= 2.47902 \dots

\frac{d}{d v} (3.54320 \dots) = 0

\frac{d}{d v} (3.54320 \dots)

Derivative of a constant:

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

= 0

= 8 v^{7} - 2.37346 \dots v^{6} + 7.46332 \dots v^{5} - 4.02269 \dots v^{4} + 6.70817 \dots v^{3} - 4.43066 \dots v^{2} + 4.78802 \dots v - 2.47902 \dots + 0

Simplify

= 8 v^{7} - 2.37346 \dots v^{6} + 7.46332 \dots v^{5} - 4.02269 \dots v^{4} + 6.70817 \dots v^{3} - 4.43066 \dots v^{2} + 4.78802 \dots v - 2.47902 \dots

Let

v_{0} = 1

Compute

v_{n + 1}

until

Δ v_{n + 1} < 0.000001

v_{1} = 0.65147 \dots : Δ v_{1} = 0.34852 \dots

f (v_{0}) = 1^{8} - 0.33906 \dots \cdot 1^{7} + 1.24388 \dots \cdot 1^{6} - 0.80453 \dots \cdot 1^{5} + 1.67704 \dots \cdot 1^{4} - 1.47688 \dots \cdot 1^{3} + 2.39401 \dots \cdot 1^{2} - 2.47902 \dots \cdot 1 + 3.54320 \dots = 4.75863 \dots

f^{^{'}} (v_{0}) = 8 \cdot 1^{7} - 2.37346 \dots \cdot 1^{6} + 7.46332 \dots \cdot 1^{5} - 4.02269 \dots \cdot 1^{4} + 6.70817 \dots \cdot 1^{3} - 4.43066 \dots \cdot 1^{2} + 4.78802 \dots \cdot 1 - 2.47902 \dots = 13.65367 \dots

v_{1} = 0.65147 \dots

Δ v_{1} = ∣ 0.65147 \dots - 1 ∣ = 0.34852 \dots

Δ v_{1} = 0.34852 \dots

v_{2} = - 2.25263 \dots : Δ v_{2} = 2.90411 \dots

f (v_{1}) = 0.65147 \dots^{8} - 0.33906 \dots \cdot 0.65147 \dots^{7} + 1.24388 \dots \cdot 0.65147 \dots^{6} - 0.80453 \dots \cdot 0.65147 \dots^{5} + 1.67704 \dots \cdot 0.65147 \dots^{4} - 1.47688 \dots \cdot 0.65147 \dots^{3} + 2.39401 \dots \cdot 0.65147 \dots^{2} - 2.47902 \dots \cdot 0.65147 \dots + 3.54320 \dots = 2.85422 \dots

f^{^{'}} (v_{1}) = 8 \cdot 0.65147 \dots^{7} - 2.37346 \dots \cdot 0.65147 \dots^{6} + 7.46332 \dots \cdot 0.65147 \dots^{5} - 4.02269 \dots \cdot 0.65147 \dots^{4} + 6.70817 \dots \cdot 0.65147 \dots^{3} - 4.43066 \dots \cdot 0.65147 \dots^{2} + 4.78802 \dots \cdot 0.65147 \dots - 2.47902 \dots = 0.98282 \dots

v_{2} = - 2.25263 \dots

Δ v_{2} = ∣ - 2.25263 \dots - 0.65147 \dots ∣ = 2.90411 \dots

Δ v_{2} = 2.90411 \dots

v_{3} = - 1.93475 \dots : Δ v_{3} = 0.31788 \dots

f (v_{2}) = (- 2.25263 \dots)^{8} - 0.33906 \dots (- 2.25263 \dots)^{7} + 1.24388 \dots (- 2.25263 \dots)^{6} - 0.80453 \dots (- 2.25263 \dots)^{5} + 1.67704 \dots (- 2.25263 \dots)^{4} - 1.47688 \dots (- 2.25263 \dots)^{3} + 2.39401 \dots (- 2.25263 \dots)^{2} - 2.47902 \dots (- 2.25263 \dots) + 3.54320 \dots = 1053.34912 \dots

f^{^{'}} (v_{2}) = 8 (- 2.25263 \dots)^{7} - 2.37346 \dots (- 2.25263 \dots)^{6} + 7.46332 \dots (- 2.25263 \dots)^{5} - 4.02269 \dots (- 2.25263 \dots)^{4} + 6.70817 \dots (- 2.25263 \dots)^{3} - 4.43066 \dots (- 2.25263 \dots)^{2} + 4.78802 \dots (- 2.25263 \dots) - 2.47902 \dots = - 3313.66679 \dots

v_{3} = - 1.93475 \dots

Δ v_{3} = ∣ - 1.93475 \dots - (- 2.25263 \dots) ∣ = 0.31788 \dots

Δ v_{3} = 0.31788 \dots

v_{4} = - 1.64441 \dots : Δ v_{4} = 0.29034 \dots

f (v_{3}) = (- 1.93475 \dots)^{8} - 0.33906 \dots (- 1.93475 \dots)^{7} + 1.24388 \dots (- 1.93475 \dots)^{6} - 0.80453 \dots (- 1.93475 \dots)^{5} + 1.67704 \dots (- 1.93475 \dots)^{4} - 1.47688 \dots (- 1.93475 \dots)^{3} + 2.39401 \dots (- 1.93475 \dots)^{2} - 2.47902 \dots (- 1.93475 \dots) + 3.54320 \dots = 369.29768 \dots

f^{^{'}} (v_{3}) = 8 (- 1.93475 \dots)^{7} - 2.37346 \dots (- 1.93475 \dots)^{6} + 7.46332 \dots (- 1.93475 \dots)^{5} - 4.02269 \dots (- 1.93475 \dots)^{4} + 6.70817 \dots (- 1.93475 \dots)^{3} - 4.43066 \dots (- 1.93475 \dots)^{2} + 4.78802 \dots (- 1.93475 \dots) - 2.47902 \dots = - 1271.93873 \dots

v_{4} = - 1.64441 \dots

Δ v_{4} = ∣ - 1.64441 \dots - (- 1.93475 \dots) ∣ = 0.29034 \dots

Δ v_{4} = 0.29034 \dots

v_{5} = - 1.36913 \dots : Δ v_{5} = 0.27528 \dots

f (v_{4}) = (- 1.64441 \dots)^{8} - 0.33906 \dots (- 1.64441 \dots)^{7} + 1.24388 \dots (- 1.64441 \dots)^{6} - 0.80453 \dots (- 1.64441 \dots)^{5} + 1.67704 \dots (- 1.64441 \dots)^{4} - 1.47688 \dots (- 1.64441 \dots)^{3} + 2.39401 \dots (- 1.64441 \dots)^{2} - 2.47902 \dots (- 1.64441 \dots) + 3.54320 \dots = 131.68340 \dots

f^{^{'}} (v_{4}) = 8 (- 1.64441 \dots)^{7} - 2.37346 \dots (- 1.64441 \dots)^{6} + 7.46332 \dots (- 1.64441 \dots)^{5} - 4.02269 \dots (- 1.64441 \dots)^{4} + 6.70817 \dots (- 1.64441 \dots)^{3} - 4.43066 \dots (- 1.64441 \dots)^{2} + 4.78802 \dots (- 1.64441 \dots) - 2.47902 \dots = - 478.36033 \dots

v_{5} = - 1.36913 \dots

Δ v_{5} = ∣ - 1.36913 \dots - (- 1.64441 \dots) ∣ = 0.27528 \dots

Δ v_{5} = 0.27528 \dots

v_{6} = - 1.08732 \dots : Δ v_{6} = 0.28180 \dots

f (v_{5}) = (- 1.36913 \dots)^{8} - 0.33906 \dots (- 1.36913 \dots)^{7} + 1.24388 \dots (- 1.36913 \dots)^{6} - 0.80453 \dots (- 1.36913 \dots)^{5} + 1.67704 \dots (- 1.36913 \dots)^{4} - 1.47688 \dots (- 1.36913 \dots)^{3} + 2.39401 \dots (- 1.36913 \dots)^{2} - 2.47902 \dots (- 1.36913 \dots) + 3.54320 \dots = 48.57656 \dots

f^{^{'}} (v_{5}) = 8 (- 1.36913 \dots)^{7} - 2.37346 \dots (- 1.36913 \dots)^{6} + 7.46332 \dots (- 1.36913 \dots)^{5} - 4.02269 \dots (- 1.36913 \dots)^{4} + 6.70817 \dots (- 1.36913 \dots)^{3} - 4.43066 \dots (- 1.36913 \dots)^{2} + 4.78802 \dots (- 1.36913 \dots) - 2.47902 \dots = - 172.37459 \dots

v_{6} = - 1.08732 \dots

Δ v_{6} = ∣ - 1.08732 \dots - (- 1.36913 \dots) ∣ = 0.28180 \dots

Δ v_{6} = 0.28180 \dots

v_{7} = - 0.75017 \dots : Δ v_{7} = 0.33714 \dots

f (v_{6}) = (- 1.08732 \dots)^{8} - 0.33906 \dots (- 1.08732 \dots)^{7} + 1.24388 \dots (- 1.08732 \dots)^{6} - 0.80453 \dots (- 1.08732 \dots)^{5} + 1.67704 \dots (- 1.08732 \dots)^{4} - 1.47688 \dots (- 1.08732 \dots)^{3} + 2.39401 \dots (- 1.08732 \dots)^{2} - 2.47902 \dots (- 1.08732 \dots) + 3.54320 \dots = 19.15306 \dots

f^{^{'}} (v_{6}) = 8 (- 1.08732 \dots)^{7} - 2.37346 \dots (- 1.08732 \dots)^{6} + 7.46332 \dots (- 1.08732 \dots)^{5} - 4.02269 \dots (- 1.08732 \dots)^{4} + 6.70817 \dots (- 1.08732 \dots)^{3} - 4.43066 \dots (- 1.08732 \dots)^{2} + 4.78802 \dots (- 1.08732 \dots) - 2.47902 \dots = - 56.80952 \dots

v_{7} = - 0.75017 \dots

Δ v_{7} = ∣ - 0.75017 \dots - (- 1.08732 \dots) ∣ = 0.33714 \dots

Δ v_{7} = 0.33714 \dots

v_{8} = - 0.21910 \dots : Δ v_{8} = 0.53107 \dots

f (v_{7}) = (- 0.75017 \dots)^{8} - 0.33906 \dots (- 0.75017 \dots)^{7} + 1.24388 \dots (- 0.75017 \dots)^{6} - 0.80453 \dots (- 0.75017 \dots)^{5} + 1.67704 \dots (- 0.75017 \dots)^{4} - 1.47688 \dots (- 0.75017 \dots)^{3} + 2.39401 \dots (- 0.75017 \dots)^{2} - 2.47902 \dots (- 0.75017 \dots) + 3.54320 \dots = 8.46330 \dots

f^{^{'}} (v_{7}) = 8 (- 0.75017 \dots)^{7} - 2.37346 \dots (- 0.75017 \dots)^{6} + 7.46332 \dots (- 0.75017 \dots)^{5} - 4.02269 \dots (- 0.75017 \dots)^{4} + 6.70817 \dots (- 0.75017 \dots)^{3} - 4.43066 \dots (- 0.75017 \dots)^{2} + 4.78802 \dots (- 0.75017 \dots) - 2.47902 \dots = - 15.93620 \dots

v_{8} = - 0.21910 \dots

Δ v_{8} = ∣ - 0.21910 \dots - (- 0.75017 \dots) ∣ = 0.53107 \dots

Δ v_{8} = 0.53107 \dots

Cannot find solution

The solutions are

v \approx 0.90641 \dots, v \approx - 1.24548 \dots

The solutions are

v = 0, v \approx 0.90641 \dots, v \approx - 1.24548 \dots

v = 0, v \approx 0.90641 \dots, v \approx - 1.24548 \dots

Substitute back

v = u^{2},

solve for

u

Solve

u^{2} = 0 : u = 0

u^{2} = 0

Apply rule

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

u = 0

Solve

u^{2} = 0.90641 \dots : u = 0.90641 \dots, u = - 0.90641 \dots

u^{2} = 0.90641 \dots

For

x^{2} = f (a)

the solutions are

x = f (a), - f (a)

u = 0.90641 \dots, u = - 0.90641 \dots

Solve

u^{2} = - 1.24548 \dots :

No Solution for

u \in R

u^{2} = - 1.24548 \dots

x^{2}

cannot be negative for

x \in R

No Solution for u \in R

The solutions are

u = 0, u = 0.90641 \dots, u = - 0.90641 \dots

Substitute back

u = sin (x)

sin (x) = 0, sin (x) = 0.90641 \dots, sin (x) = - 0.90641 \dots

sin (x) = 0, sin (x) = 0.90641 \dots, sin (x) = - 0.90641 \dots

sin (x) = 0 : x = 2 πn, x = π + 2 πn

sin (x) = 0

General solutions for

sin (x) = 0

sin (x)

periodicity table with

2 πn

cycle:

x 0 \frac{π}{6} \frac{π}{4} \frac{π}{3} \frac{π}{2} \frac{2 π}{3} \frac{3 π}{4} \frac{5 π}{6} sin (x) 0 \frac{1}{2} \frac{2}{2} \frac{3}{2} 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} x π \frac{7 π}{6} \frac{5 π}{4} \frac{4 π}{3} \frac{3 π}{2} \frac{5 π}{3} \frac{7 π}{4} \frac{11 π}{6} sin (x) 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2}

x = 0 + 2 πn, x = π + 2 πn

x = 0 + 2 πn, x = π + 2 πn

Solve

x = 0 + 2 πn : x = 2 πn

x = 0 + 2 πn

0 + 2 πn = 2 πn

x = 2 πn

x = 2 πn, x = π + 2 πn

sin (x) = 0.90641 \dots : x = arcsin (0.90641 \dots) + 2 πn, x = π - arcsin (0.90641 \dots) + 2 πn

sin (x) = 0.90641 \dots

Apply trig inverse properties

sin (x) = 0.90641 \dots

General solutions for

sin (x) = 0.90641 \dots

sin (x) = a \Rightarrow x = arcsin (a) + 2 πn, x = π - arcsin (a) + 2 πn

x = arcsin (0.90641 \dots) + 2 πn, x = π - arcsin (0.90641 \dots) + 2 πn

x = arcsin (0.90641 \dots) + 2 πn, x = π - arcsin (0.90641 \dots) + 2 πn

sin (x) = - 0.90641 \dots : x = arcsin (- 0.90641 \dots) + 2 πn, x = π + arcsin (0.90641 \dots) + 2 πn

sin (x) = - 0.90641 \dots

Apply trig inverse properties

sin (x) = - 0.90641 \dots

General solutions for

sin (x) = - 0.90641 \dots

sin (x) = - a \Rightarrow x = arcsin (- a) + 2 πn, x = π + arcsin (a) + 2 πn

x = arcsin (- 0.90641 \dots) + 2 πn, x = π + arcsin (0.90641 \dots) + 2 πn

x = arcsin (- 0.90641 \dots) + 2 πn, x = π + arcsin (0.90641 \dots) + 2 πn

Combine all the solutions

x = 2 πn, x = π + 2 πn, x = arcsin (0.90641 \dots) + 2 πn, x = π - arcsin (0.90641 \dots) + 2 πn, x = arcsin (- 0.90641 \dots) + 2 πn, x = π + arcsin (0.90641 \dots) + 2 πn

Show solutions in decimal form

x = 2 πn, x = π + 2 πn, x = 1.25989 \dots + 2 πn, x = π - 1.25989 \dots + 2 πn, x = - 1.25989 \dots + 2 πn, x = π + 1.25989 \dots + 2 πn

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