Bounding the Lagrange error further? Let's say we are approximating a car's position at a time $t$ using initial position and velocity from time $t=0$. Now let's say this car cannot accelerate forward more than $2 m/s^2$ but can brake with acceleration $-12 m/s^2$. When calculating the Lagrange error, we would use the absolute value of the max acceleration, which is the braking acceleration, and say that the absolute value of the error is less than $12 * t^2 / 2$ however I was thinking you could bound this further by saying that the error must be between $-12 * t^2 / 2$ but no more than $2 * t^2 / 2$. Would this work? And if so, why wouldn't we generally do this despite it being able to further bound the error? 
 A: I know this question is a bit old, but here goes anyway:
Let's say that the displacement of the object is given by $s(t)$, which is at least linear (since the acceleration and higher derivatives can be zero) and at most a power series. Then:
$$s(t)=s(0)+s^\prime(0)t+R_1(t:0)$$
By the Lagrange error theorem, there exists some $c\in]0,t[$ such that $\displaystyle R_1(t:0)=\frac{s^{\prime\prime}(c)t^2}{2}$. Since the acceleration is at most $12\, \text{ms}^{-2}$, the absolute error is, as you say, bound by $\displaystyle\frac{12t^2}{2}=6t^2$.
You are correct when you more specifically say that $-6t^2\le R_1(t:0) \le t^2$, but this is still saying that the absolute error is bound by $6t^2$. This type of problem usually comes up in a situation in which we're trying to predict some value based on others. In this case, since we don't have complete information on the acceleration, we would most likely be more interested in the greatest error that our estimate could have than the exact range of errors possible.
We'll also often find problems involving trigonometric functions where the fact that $|\sin x|\le 1$ (for example) is used, even though we could narrow down the error range more completely by exploring the range of values of the appropriate derivative given the boundaries. I think that the point is often that a rough idea of how much error is in our approximation suffices, and a maximum provides a "worst-case" scenario that we can take into account. 
