Important questions of Numerical Methods - Bsc csit 3rd sem -NM - utsabojha 

Here are some important questions covering Newton-Raphson, Bisection, and Secant methods from Numerical Methods that you can focus on:

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1. Conceptual Questions

Explain the working principle of the Newton-Raphson method.

Compare the convergence rate of Newton-Raphson, Bisection, and Secant methods.

Discuss the advantages and limitations of the Bisection method.

Why is the Newton-Raphson method considered faster but less reliable than the Bisection method?

Under what conditions does the Secant method fail?

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2. Derivation/Proof-Based Questions

Derive the formula used in the Newton-Raphson method.

Prove that the Bisection method always converges if the function is continuous on the given interval.

Derive the iterative formula for the Secant method and explain its geometric interpretation.

Show that the convergence rate of the Newton-Raphson method is quadratic.

Derive an error bound for the Bisection method after iterations.

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3. Numerical Problems

Newton-Raphson Method

1. Find a root of using the Newton-Raphson method with an initial guess .

2. Use Newton-Raphson to approximate the square root of 10.

3. Solve using Newton-Raphson with .

4. Apply the Newton-Raphson method to solve .

Bisection Method

1. Find the root of in the interval using the Bisection method.

2. Solve in the interval using the Bisection method up to 4 iterations.

3. Find the root of in the interval with a tolerance of .

4. Apply the Bisection method to find a root of within .

Secant Method

1. Solve using the Secant method with initial guesses and .

2. Apply the Secant method to find a root of with initial approximations and .

3. Use the Secant method to solve with and .

4. Find a root of using the Secant method.

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4. Algorithm and Programming Questions

Write a step-by-step algorithm for the Newton-Raphson method.

Implement the Bisection method in Python or C++ for solving .

Develop a program to solve using the Secant method.

Compare the number of iterations required by Bisection and Newton-Raphson methods for the same problem.

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These questions cover theoretical, derivational, numerical, and programming aspects, providing a comprehensive understanding of the methods.