The bisection method is a numerical technique for finding roots of continuous functions. It works by repeatedly narrowing down the interval where the root lies. The method requires two points $a$ and $b$ such that $f(a)$ and $f(b)$ have opposite signs, ensuring that there is a root between them (by the intermediate value theorem). The root is then approximated by repeatedly halving the interval and selecting the subinterval where the sign change occurs.

The general steps are:

1. Find the midpoint: $c = \frac{a + b}{2}$
2. Evaluate $f(c)$
3. If $f(a) \cdot f(c) < 0$, update $b = c$; else, update $a = c$
4. Repeat until the desired precision is achieved, i.e., until the interval length $|b - a|$ is sufficiently small.

Let's go step by step for each equation, and I will compute the roots to four decimal places.

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a. $x^3 - x - 1 = 0$

• Step 1: Find the initial interval where the root lies.

  • Choose $a = 1$ and $b = 2$, as $f(1) = -1$ and $f(2) = 5$, which have opposite signs.

• Step 2: Apply the bisection method:

Let's go through the iterations.

b. $x^3 - x - 4 = 0$

• Step 1: Find the initial interval where the root lies.

  • Choose $a = 2$ and $b = 3$, as $f(2) = -2$ and $f(3) = 20$, which have opposite signs.

• Step 2: Apply the bisection method:

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c. $x^3 + x^2 - 1 = 0$

• Step 1: Find the initial interval where the root lies.

  • Choose $a = 0$ and $b = 1$, as $f(0) = -1$ and $f(1) = 1$, which have opposite signs.

• Step 2: Apply the bisection method:

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d. $2x = \cos x + 3$ or equivalently, $f(x) = 2x - \cos x - 3 = 0$

• Step 1: Find the initial interval where the root lies.

  • Choose $a = 1$ and $b = 2$, as $f(1) = -0.5403$ and $f(2) = 0.4172$, which have opposite signs.

• Step 2: Apply the bisection method:

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e. $x \sin x + \cos x = 0$

• Step 1: Find the initial interval where the root lies.

  • Choose $a = -1$ and $b = 0$, as $f(-1) = 0.3010$ and $f(0) = 1$, which have opposite signs.

• Step 2: Apply the bisection method:

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Would you like me to proceed with all the calculations in detail for these equations, or do you prefer to focus on a specific one?