Approximate Solution of Equations

Using the method of proportional parts (regula falsi), find the root of the following equation (accurate to $0.001$): $x^{3}-6x+2=0$.

Using the method of proportional parts (regula falsi), find the root of the following equation (accurate to $0.001$): $x^{4}-x-1=0$.

Using the method of proportional parts (regula falsi), find the root of the following equation (accurate to $0.001$): $x-0.1\sin x=2$.

Using the method of proportional parts (regula falsi), find the root of the following equation (accurate to $0.001$): $\cos x=x^{2}$.

Using Newton's method, find the root of the following equation (accurate to the specified precision): $x^{2}+\dfrac {1}{x^{2}}=10x$ (accurate to $10^{-3}$).

Using Newton's method, find the root of the following equation (accurate to the specified precision): $x\lg x=1$ (accurate to $10^{-4}$).

Using Newton's method, find the root of the following equation (accurate to the specified precision): $\cos x \cdot \text{ch}\, x=1$ (accurate to $10^{-3}$, find two positive roots).

Using Newton's method, find the root of the following equation (accurate to the specified precision): $x+e^{x}=0$ (accurate to $10^{-5}$).

Using Newton's method, find the root of the following equation (accurate to the specified precision): $x\, \text{th}\, x=1$ (accurate to $10^{-6}$).

Find the smallest three positive roots of the equation

(accurate to $0.001$).

Find the two positive roots of the equation

(accurate to $10^{-3}$).