Let's break down the questions in your image step by step:

2. a) Need for calculating error in numerical computing:

In numerical computing, most solutions to mathematical problems are approximate rather than exact. The need for calculating errors arises because:

1. Approximation of real-world problems: Many real-world problems are modeled by complex mathematical equations that can't always be solved exactly. Approximations are made to make the problems computationally feasible.

2. Finite precision: Computers work with a finite precision, meaning they can only store and process numbers to a limited number of decimal places, which leads to rounding errors.

3. Iterative methods: Numerical methods often rely on iterative approaches to refine estimates. Understanding the error in each iteration helps determine when the approximation is sufficiently accurate.

4. Error propagation: Small errors in one computation step may propagate and amplify throughout the solution process. Calculating errors allows one to assess the reliability of results.

5. Stability and convergence: Calculating errors helps in assessing whether the method being used is stable and whether it converges to the correct solution over iterations.

2. b) Need for search bracket and finding the initial estimation for the equation $4x^3 - 3x^2 - 2x + 1 = 0$:

1. Why do we need a search bracket in root-finding?

   • In root-finding methods such as the bisection method, a search bracket is required to enclose the interval in which the root lies. The method iteratively halves the interval until it homes in on the root with the desired accuracy.
   • A search bracket helps ensure convergence towards the solution and provides a starting point for the algorithm.
   • Without a proper bracket, the root-finding algorithm may either fail to converge or converge to a wrong root.

2. Finding the search bracket for $4x^3 - 3x^2 - 2x + 1 = 0$: To find the search bracket, we will evaluate the function at different values of $x$ and check when the sign of the function changes (since the function must cross the x-axis, implying a root lies between those values).

   Let's try evaluating the function at some values:

   • $f(-2) = 4(-2)^3 - 3(-2)^2 - 2(-2) + 1 = -32 - 12 + 4 + 1 = -39$
   • $f(-1) = 4(-1)^3 - 3(-1)^2 - 2(-1) + 1 = -4 - 3 + 2 + 1 = -4$
   • $f(0) = 4(0)^3 - 3(0)^2 - 2(0) + 1 = 1$
   • $f(1) = 4(1)^3 - 3(1)^2 - 2(1) + 1 = 4 - 3 - 2 + 1 = 0$

   Since $f(-1) = -4$ and $f(0) = 1$, the root lies between $x = -1$ and $x = 0$, as the function changes sign in this interval. Hence, the search bracket is $[-1, 0]$.

Let me know if you need more details or clarifications!

Here are 5 related questions to expand on this topic:

1. What is the difference between absolute error and relative error in numerical computing?
2. How can you use the bisection method to solve the given equation?
3. What are other root-finding algorithms besides the bisection method?
4. How does the choice of initial guess affect the convergence of Newton's method for root finding?
5. How can error bounds help in assessing the accuracy of iterative numerical methods?

Tip: When choosing a search bracket for root-finding methods, always ensure that the function changes sign within the interval, as this guarantees the existence of a root between the bounds.