Final answer:
The order of accuracy for the central divided difference formula for approximating the first derivative of a continuous function is O(h^2), which implies the error decreases quadratically as step size h decreases. The formula is applicable when the function and its first derivative are continuous.
Step-by-step explanation:
The question asks for the order of accuracy of the central divided difference formula for approximating the first derivative of a continuous function. The formula given is (f(x + h) - f(x – h)) / (2h), which is used to estimate f'(x). The order of accuracy for this approximation is O(h2), which means that the error in the approximation decreases quadratically as the step size h becomes smaller. Since y(x) must be a continuous function and the first derivative with respect to space, dy(x)/dx, also needs to be continuous, it allows the use of the central divided difference formula accurately.
When considering the behavior of functions and their derivatives, properties like whether a function produces an odd or even function can be useful. For instance, the integral of an odd function over all space is zero, which can simplify the calculation of certain integrals. It's also important to note from the mathematical principles that partial derivatives can be computed and in cases of functions involving multiple variables, each term is differentiated separately to find the total derivative. The accurate order for the given central divided difference formula is O(h2) and this approximation is valid under certain conditions of function continuity and it agrees well with experimental results without being affected by temperature changes, as indicated in the provided information.