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. Utilize backward, forward, and centered first finite differences with a step size h=0.25 to numerically estimate the derivative. Determine the percent error between the true value and each of the estimated values (percent error is given by ∣True Value−Estimated Value∣/∣True Value∣​ ×100). What value of h would you have to use for the backward and forward finite differences to yield the same percent error as the centered finite difference (hint: it should be less than 0.25)?

Answer 1

To estimate the derivative of the given polynomial at x=0.5, the backward, forward, and centered finite differences are used. The percent error between these estimates and the true derivative is then calculated. A step size smaller than 0.25 is necessary for the backward and forward methods to achieve the same accuracy as the centered method.

Step-by-step explanation:

To estimate the derivative of the polynomial f(x)=0.1x⁴+0.15x³+0.5x²+0.25x+1.2 at x=0.5 using backward, forward, and centered first finite differences with a step size h=0.25, we perform the following calculations:

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Next, we calculate the percent error for each method using the formula: |True Value - Estimated Value|/|True Value|​ × 100. To find the step size h that would yield the same percent error for the backward and forward finite differences as the centered finite difference, we would need to use a smaller step size than 0.25, as smaller step sizes generally result in more accurate numerical derivatives.

The precise value of h isn't provided in the question, as it typically requires solving an equation that equates the central error term (of the order O(h²)) to the forward and backward error terms (of order O(h)).