Question 1: Numerical solutions often involve iterative methods such as the method of successive approximations. Accuracy depends on the number of iterations and the step size. Small errors can accumulate, leading to misconceptions about the convergence of the method.

Answer:

Question 1: Calculate the first three iterations of x = 1 + 0.5x starting from x=1. What is the approximate value after the third iteration?

Answer: 1.75

Question 2: Using the iterative formula x_{n+1} = (x_n + 3)/2, starting from x=0, find the value after 4 iterations.

Answer: 2.8125

Question 3: Plot the graph of y=2x on the grid, then approximate the solution of y=10 using iteration.

Answer: x=5

Question 4: Construct a sequence starting at x=2 with the iterative formula x_{n+1} = x_n + 0.2, and determine after 5 iterations the approximate value.

Answer: 3.0

Question 1: A student claims that increasing the number of iterations will always improve the accuracy of the solution. Is this true? Explain your reasoning.

Answer: No, because if the step size is too large, errors can accumulate, and the solution may diverge.

Question 2: A common mistake is to assume convergence after just one iteration. Why is this incorrect? Provide the correct understanding.

Answer: Convergence requires multiple iterations, and the process depends on the initial guess and the function's properties.

Question 1: Given the iterative formula x_{n+1} = 0.5x_n + 1, starting from x=0, determine after how many iterations the value will be within 0.01 of the fixed point. Show your working.

Answer: Approximately 10 iterations

Question 2: A calculator gives a sequence of approximations to solve 2x - 5 = 0 using x_{n+1} = (x_n + 5)/2. If the initial guess is 0, explain why the sequence does not converge quickly and how to improve this.

Answer: The method converges slowly because the initial guess is far from the actual root. Using a better initial guess or a different iterative formula can improve convergence.

Question 1: A bank account balance grows by 5% annually. Starting with £100, use iterative approximation to estimate the balance after 3 years, assuming compound interest. Show your calculations.

Answer: £115.76

Question 1: Derive the iterative formula for solving the equation x^3 = 2, and estimate its root using 4 iterations starting from x=1.

Answer: Approximately 1.26

Question 2: Discuss the potential errors that could arise when using iteration to find roots of nonlinear equations, and how to mitigate them.

Answer: Errors include divergence, slow convergence, or false convergence. Improving step size, initial guess, or using convergence tests can mitigate these issues.

Question 1: If x_{n+1} = 0.8x_n + 2 and x_0=0, what is the value after 6 iterations? Show your working.

Answer: 3.84

Question 2: Explain why the iterative method x_{n+1} = x_n - (f(x_n)/f'(x_n)) (Newton-Raphson) can fail or give incorrect results if not used carefully.

Answer: It can fail if the derivative is zero or near zero, or if the initial guess is far from the root, leading to divergence or incorrect roots.