Linear Approximations and Differentials

If we have a function f(x) with a value associated with $f(x_0)$ that is difficult to obtain exactly, we may wish to approximate this. We already have all the tools in hand to do this! Recall that one way to conceptualize the derivative is as the slope of the tangent line passing through a point. If we obtain a tangent line near a point on f that is easy to obtain such as $f(x_1)$ and such that $|x_0–x_1|<\epsilon$ is small enough then we can obtain a good approximation of $f(x_0)$ by using the line tangent to $f(x_1)$. Let us consider an example:

There are some nuances to consider here. For instance, if the first derivative is large near a point, then it is likely that the linear approximation is only valid in a very small neighborhood of that point. Secondly, if the function is oscillatory in nature, it is also likely that linear approximation may not be ideal. This example even suggests that we might somehow better approximate functional values by incorporating higher level derivatives. The second derivative tells us how fast the first derivative is changing and we may be able to further correct for this with the right sort of manipulations. We see power series and Taylor’s theorem which do exactly this later!

We often write $y^{\prime }=\frac{dy}{dx}=f^{\prime }(x)$ and consider them equivalent notations. They are not exactly equivalent and they draw attention to different aspects of the derivative. Recall that we established the derivative as the limit of the secant line approximation. The notation $\frac{dy}{dx}$ captures this idea very succinctly. Consider the following example.

What do I mean by a nonlinear response? If our response were linear then uncertainties of .1 in the independent parameter would always result in the same uncertainties in the dependent parameter. However, if we consider the above example near 10.0, then we find that $dy=30(10.0{)}^2(.1)=300$. Here the same movement in the independent parameter creates vastly different uncertainties in the dependent parameters depending upon what value the independent parameter has. In the linear response case, this is not what happens- only the uncertainty in the independent parameter is necessary to determine the uncertainty in the dependent parameter. Often, this results in scientists operating in what is sometimes called the linear regime of a detector.

Another interesting point that you may already be aware of is that we can repeat an experiment and obtain a standard deviation. How does this relate to the above calculations and why might we use one or the other? Sometimes an experiments is extremely difficult or costly to do and tracking the instrument precision is more reasonable than repeating the experiment. Sometimes an experiment is not repeatable! You may be familiar with observational experiments where repetition is impossible because the event is rare or inflicting the repeated experiment is unethical (think diseases!).

Hopefully, you are equally aware that one of the principal delimiters of science is that with every measurement you get an inherent uncertainty. A principal sign of junk science is a measurement with no reported uncertainty. An additional point is that our calculated uncertainty is usually more precise than an actual uncertainty generated by repeated measurements. Standard deviations are always better than calculated uncertainties, however, sometimes the best you can do is calculated uncertainties.