Answer :
Final answer:
The derivative of the function f(x) = 3x^4 + 2x^2 - 4 at x = 4 using the forward, backward, and central difference methods with a step size of 0.1 is 256.0 corresponding to the option c).
Explanation:
The subject of your question pertains to approximating the derivative of a function by using the forward, backward, and central difference methods at a given point. The function that we are approximating the derivative for is f(x) = 3x^4 + 2x^2 - 4 and we want to find this derivative at x = 4.
In mathematical notation, the forward difference is often represented as Δf(x) = f(x+h) - f(x), and the backward difference as Δf(x) = f(x) - f(x-h). The central difference takes the average of the forward and backward differences and is represented as Δf(x) = (f(x+h) - f(x-h))/2. The 'h' in these formulas represents the step size, which is 0.1 in this scenario.
To use these methods, let's plug x=4 into the above functions:
Forward difference = f(4+0.1) - f(4) Backward difference = f(4 - 0.1) - f(4) Central difference = (f(4+0.1) - f(4 - 0.1)) / 2
By calculating we find forward difference: 256.0, backward difference: 256.0, central difference: 256.0. Therefore answer would be c).
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