Question 1: Calculate the value of x using the iterative formula x(n+1) = (x(n) + 10/x(n)) / 2, starting with x(0) = 5, after 3 iterations.

Answer: 3.3166

Question 2: Use the iterative process x(n+1) = (x(n) + 50/x(n)) / 2, starting from x(0) = 10, to approximate √50 after 4 iterations.

Answer: 7.071

Question 3: Apply the Newton-Raphson iteration for solving x^2 - 25 = 0, starting with x(0) = 6, after 2 iterations.

Answer: 5.0006

Question 4: Construct an iterative process to approximate the cube root of 8, starting with an initial guess of 2. Show the first two iterations.

Answer: x1=2.5, x2=2.659

Question 5: A worker estimates the number of items produced daily by repeatedly adjusting their estimate using the formula: estimate(n+1) = (estimate(n) + 200/estimate(n))/2. Starting with estimate(0)=50, what is the estimate after 3 iterations?

Answer: 63.636

Question 6: Explain why the iterative process x(n+1) = (x(n) + 100/x(n))/2 converges to √100, and identify the limit.

Answer: It converges because the process is a form of the Newton-Raphson method for √100, and the limit is 10.

Question 7: A manufacturer uses an iterative method to adjust the quantity x of a product to meet a target revenue. The formula is x(n+1) = x(n) - (p(x(n)) - R)/p'(x(n)), where p(x) is the price function, R is target revenue. Describe conceptually how iteration helps reach the desired quantity.

Answer: Iteration refines x by approximating the solution of p(x)=R, using derivatives to improve accuracy each step until the desired revenue is achieved.

Question 8: Estimate √200 using the iterative method x(n+1) = (x(n) + 200/x(n))/2, starting with x(0)=14, after 3 iterations.

Answer: 14.177

Question 9: Identify the common mistake in the following iterative calculation for √50 when starting with x(0)=10: x(n+1) = (x(n) + 50/x(n))/2, and after 2 iterations, students get 6.5 and 4.0. Explain and correct it.

Answer: Mistake: students may misapply the formula or forget to update x each iteration. Correct approach is to carefully compute each step sequentially.

Question 10: Use the calculation process to find the approximate cube root of 27 using the iterative formula x(n+1) = (2/3)x(n) + (1/3)(27/x(n)^2), starting from x(0)=3.

Answer: x1=3, x2=3, x3≈3