Question

Consider the polynomial

f(x) = - 0.1x⁴ - 0.15x³ - 0.5x² - 0.25x + 1.2
The true value of its derivative at x=0.5 is f'(0.5) = -0.9125. Use the forward finite difference technique to estimate the derivative numerically. Perform the numerical differencing for a range of values of step size Ax, starting from Ax = 5, reducing it by a factor of two, till Ax = 0.0098. Determine the percent error between the true value and each of the estimated values (percent error is given by
∊ = |Jestimated value - true value/true value|
converted to a percentage.) and plot it as a function of Ax. Use python to solve this problem.

Answer 1

To approximate the derivative of the given polynomial, the forward finite difference method is used with decreasing step sizes, and the percent error is calculated and plotted against each step size.

Step-by-step explanation:

To estimate the derivative of the polynomial f(x) using the forward finite difference technique, we subtract the function value at x from that at x + Δx, and then divide by Δx.

This gives us an approximation of the derivative at x. To find the percent error, we compare the estimated derivative to the true derivative, using the formula ε = |(estimated value - true value) / true value|, converted to a percentage.

For the given polynomial f(x) = - 0.1x⁴ - 0.15x³ - 0.5x² - 0.25x + 1.2, we start with a Δx of 5 and keep halving it until we get to 0.0098. At each step, we calculate the estimated derivative at x = 0.5, determine the percent error from the true value f'(0.5) = -0.9125, and plot this error against Δx.