Q1. Answer the following parts.
- (a)) Define true error and relative true error. (160 words)
- (b)) The approximate value of the derivative of a function f(x)at x can be calculated using the formula: df(x)/dx = (f(x + h) – f(x))/h. Calculate df(x)/dx at x = 2 for f(x) = 2x² + exp(3x) for (i) h = 0.5 and (ii) h = 0.05. Determine the true error and relative true error in each case. (640 words)
Key Points:
- True error is the absolute difference between true and approximate values.
- Relative true error normalizes true error by the true value, often as a percentage.
- f'(x) for f(x)=2x²+exp(3x) is 4x+3exp(3x); true value at x=2 is 1218.286.
- Numerical derivative approximation uses (f(x+h)-f(x))/h (forward difference).
Answer: In computational physics, understanding and quantifying errors are fundamental for assessing the reliability of numerical methods. This question explores two critical error metrics: true error and relative true error. These metrics provide insights into the accuracy of approximations by comparing them against known true values. We will define these errors and then apply them to analyze the accuracy of a numerical differentiation method for a given function, demonstrating how step size (h) influe...