Regula falsi (method of chords / proportional parts)

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Problem 1617 from Demidovich B.P. "Сборник задач и упражнений по математическому анализу" [Dem97]More problems in Section 2.15 "Approximate Solution of Equations"

Using the method of proportional parts (regula falsi), find the root of the following equation (accurate to 0.0010.001): x3−6x+2=0x^{3}-6x+2=0.

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Problem 1618 from Demidovich B.P. "Сборник задач и упражнений по математическому анализу" [Dem97]More problems in Section 2.15 "Approximate Solution of Equations"

Using the method of proportional parts (regula falsi), find the root of the following equation (accurate to 0.0010.001): x4−x−1=0x^{4}-x-1=0.

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Problem 1619 from Demidovich B.P. "Сборник задач и упражнений по математическому анализу" [Dem97]More problems in Section 2.15 "Approximate Solution of Equations"

Using the method of proportional parts (regula falsi), find the root of the following equation (accurate to 0.0010.001): x−0.1sin⁡x=2x-0.1\sin x=2.

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Problem 1620 from Demidovich B.P. "Сборник задач и упражнений по математическому анализу" [Dem97]More problems in Section 2.15 "Approximate Solution of Equations"

Using the method of proportional parts (regula falsi), find the root of the following equation (accurate to 0.0010.001): cos⁡x=x2\cos x=x^{2}.

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Problem 1627 from Demidovich B.P. "Сборник задач и упражнений по математическому анализу" [Dem97]More problems in Section 2.15 "Approximate Solution of Equations"

Find the two positive roots of the equation

cot⁡x=1x−x2 \cot x=\dfrac {1}{x}-\dfrac {x}{2}

(accurate to 10−310^{-3}).

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Problem 1.23.45 from Кудрявцев Л.Д., Кутасов А.Д., Чехлов В.И., Шабунин М.И. "Предел. Непрерывность. Дифференцируемость." [ККЧ+03]More problems in Subsection 1.23.2 "Problems"

Using the method of chords, solve the equation with the indicated error Δ\Delta:

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(1)

x3−4x+2=0,Δ=10−3x^{3}-4 x+2 = 0, \Delta = 10^{-3};

(2)

x4+x−1=0,Δ=10−3x^{4}+x-1 = 0, \Delta = 10^{-3};

(3)

x3−3,2x2+3,1x−2,2=0,Δ=10−3x^{3}-3,2 x^{2}+3,1 x-2,2 = 0, \Delta = 10^{-3};

(4)

0,1sin⁡x=x+2,Δ=10−3;0,1 \sin x = x+2, \Delta = 10^{-3} ;

(5)

cos⁡x=x2,Δ=10−3\cos x = x^{2}, \Delta = 10^{-3};

(6)

x3−5x+1=0,Δ=10−5x^{3}-5 x+1 = 0, \Delta = 10^{-5}.

Problem 1.23.46 from Кудрявцев Л.Д., Кутасов А.Д., Чехлов В.И., Шабунин М.И. "Предел. Непрерывность. Дифференцируемость." [ККЧ+03]More problems in Subsection 1.23.2 "Problems"

Using the method of chords and the method of tangents, solve the equation with the indicated error Δ\Delta:

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(1)

x2=13,Δ=10−8x^{2} = 13, \Delta = 10^{-8};

(2)

2x3+2x2−11x+3=0,Δ=10−62 x^{3}+2 x^{2}-11 x+3 = 0, \Delta = 10^{-6};

(3)

x3+x2+2x−3=0,Δ=5⋅10−5x^{3}+x^{2}+2 x-3 = 0, \Delta = 5 \cdot 10^{-5}.

Problem 1.23.50 from Кудрявцев Л.Д., Кутасов А.Д., Чехлов В.И., Шабунин М.И. "Предел. Непрерывность. Дифференцируемость." [ККЧ+03]More problems in Subsection 1.23.2 "Problems"

Solve the equation x3+1,1x2+0,9x−1,4=0x^{3}+1,1 x^{2}+0,9 x-1,4 = 0:

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(1)

using the bisection method to an accuracy of 5⋅10−35 \cdot 10^{-3};

(2)

using the method of chords to an accuracy of 5⋅10−45 \cdot 10^{-4};

(3)

using the method of tangents to an accuracy of 5⋅10−45 \cdot 10^{-4}.

Problem 1.23.57 from Кудрявцев Л.Д., Кутасов А.Д., Чехлов В.И., Шабунин М.И. "Предел. Непрерывность. Дифференцируемость." [ККЧ+03]More problems in Subsection 1.23.2 "Problems"

What accuracy does a twofold application of the combined method give when computing the root of the equation x4−x−1=0x^{4}-x-1 = 0 starting from the interval (1,22;1,23)(1,22 ; 1,23)?

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Problem 1.23.58 from Кудрявцев Л.Д., Кутасов А.Д., Чехлов В.И., Шабунин М.И. "Предел. Непрерывность. Дифференцируемость." [ККЧ+03]More problems in Subsection 1.23.2 "Problems"

Solve the equation x3−4,1x2+6,1x−1,6=0x^{3}-4,1 x^{2}+6,1 x-1,6 = 0:

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(1)

using the bisection method to an accuracy of 5⋅10−35 \cdot 10^{-3};

(2)

using the method of chords to an accuracy of 5⋅10−45 \cdot 10^{-4};

(3)

using the method of tangents to an accuracy of 5⋅10−45 \cdot 10^{-4};

(4)

using the combined method to an accuracy of 5⋅10−45 \cdot 10^{-4}.

Problem 1.23.59 from Кудрявцев Л.Д., Кутасов А.Д., Чехлов В.И., Шабунин М.И. "Предел. Непрерывность. Дифференцируемость." [ККЧ+03]More problems in Subsection 1.23.2 "Problems"

Using the combined method, compute the greatest root of the equation x5−x−0,2=0x^{5}-x-0,2 = 0 to an accuracy of 10−410^{-4}.

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Problem 1.23.60 from Кудрявцев Л.Д., Кутасов А.Д., Чехлов В.И., Шабунин М.И. "Предел. Непрерывность. Дифференцируемость." [ККЧ+03]More problems in Subsection 1.23.2 "Problems"

Using the combined method, compute the roots of the equation

x4−3x2+75x−10000=0 x^{4}-3 x^{2}+75 x-10000 = 0

from the intervals (−11;−10)(-11 ;-10) and (9;10)(9 ; 10) to an accuracy of 10−410^{-4}.

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Problem 1.23.62 from Кудрявцев Л.Д., Кутасов А.Д., Чехлов В.И., Шабунин М.И. "Предел. Непрерывность. Дифференцируемость." [ККЧ+03]More problems in Subsection 1.23.2 "Problems"

Using the combined method, solve, with the indicated error Δ\Delta, the equation:

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(1)

2x2−e1−x=0,Δ=10−52 x^{2}-e^{1-x} = 0, \Delta = 10^{-5};

(2)

2−x−lg⁡x=0,Δ=10−62-x-\lg x = 0, \Delta = 10^{-6}.

References

[Dem97] Demidovich B.P. Сборник задач и упражнений по математическому анализу. 13-е изд., Издательство Московского университета / Издательство "ЧеРо", 1997.
[ККЧ+03] Кудрявцев Л.Д., Кутасов А.Д., Чехлов В.И., Шабунин М.И. Предел. Непрерывность. Дифференцируемость. 2-е изд., ФИЗМАТЛИТ, 2003.