Looking at Figure 2, we see the first order results from the difference approximations plotted alongside the analytical first derivative. What is interesting to note here is that the forward approximation seems to underestimate the derivative at x values below zero, and overestimate it at x values above zero. The reverse is true for the backward approximation, which overestimates the derivative at x values below zero and underestimates it at x values above zero. This makes sense, as the forward approximation takes the average slope between a point at the current x value and a point one "step" ahead of the current x value. So when the derivative is decreasing (as it is between x = -2 and x = 0), the forward approximation should return underestimations as it is factoring in a smaller f(x) value one step ahead. When the function is increasing, as it does after x = 0, the forward approximation will overestimate because it is factoring in larger increases at greater values of x.

As we can see visually, the centered finite approximation appears to be extremely close to the first derivative. As we get into error calculations, we will observe how the centered-finite approach is comparatively more accurate at predicting the analytical solutions relative to the forward and backward approaches.

Looking at Figure 5, we see how the center finite approximation most accurately predicts the behavior of the analytical second derivative. As we will observe in our error graphs, the center approximation is actually identical to the second derivative at all x values on this interval.

The backward approximation is always an underestimation of the second derivative, which makes sense given the second derivative has a constant slope upward (around 6). Because the backward approximation is always using the current and the previous value of x to estimate change, it will always underestimate the function in the case where the function is increasing at a constant rate. The same logic applies for the forward approximation. If the forward approximation uses the average rate of change between the current value of x and a future value of x, then at any point x the forward approximation will be too high as it factors in larger outputs of the function.