# Approximations

#### Prerequisites

A key component of learning to build mathematical models of physical systems is the idea of approximation. There are two components to the meaning of approximation in mathematics. A mathematical approach is said to be an approximation if the mathematics being used is

1. not precise as it could be (either mathematically or in matching the data)
2. part of a more complete system that allows (in principle) corrections
3. useful for describing a physical system.

Unfortunately, in everyday speech (as for example in dictionary.com) only the first condition is stated. Conditions 2 and 3 are what make approximation an essential tool in the tool belt of any mathematical modeler. This is why the phrase "a good approximation", that we will be using over and over again, seems at first glance a self-contradiction.

You may not have seen this usage of the term "approximation" even if you have previously taken many introductory math and science courses. A quick check of introductory biology texts and even an introductory calculus text for biology shows that the term does not appear in the index of the biology texts and only appears in the common-speech form in the calculus text. So let's take a look at a couple of specific cases to see how the idea plays out in practice.

## Approximation 1: The derivative

When we consider the derivative of a function $f(x)$, we start by looking at a small change in $x$, $\Delta x$, and seeing how much of a change in $f$ results, $\Delta f$. The derivative is defined as the slope

$$\frac{df}{dx} = \lim_{\Delta x \rightarrow 0} \frac{\Delta f}{\Delta x}$$

The algebraic limit is kind of tricky to get your head around, but graphically it makes a lot of sense. The $\Delta$ stuff creates a small triangle around the point $x$ where we are calculating the derivative (shown in blue). The derivative is approximately the ratio of the delta terms. This is shown geometrically as the "chord" (the hypotenuse of the little triangle cutting across the curve of $f$). The actual derivative is the slope of the tangent line to the curve at that point (shown in red). We see that it doesn't have exactly the same slope as the blue hypotenuse of the little triangle of the deltas, but it's pretty close: "a good approximation".

It's an approximation because we are keeping the effects in calculating $\Delta f$ that are proportional to $\Delta x$ but we ignore terms proportional to $(\Delta x)^2$ and higher powers. (See "Derivative as algebra" in the page What is a derivative, anyway?) We could if we wanted include more terms in powers of $\Delta x$, treating $\frac{df}{dx}$ as an unknown and solving for it, but if  $\Delta x$ is small enough our approximation should do fine.

## Approximation 2: The sine function

A second approximation that we'll be using in this class is The small angle approximation. This says, "if you measure an angle in radians and the angle is small enough, then

$$\sin{\theta} \approx \theta$$.

You can see this by trying it out on your calculator, or by looking a the comparison of a graph of the sine function, $y = \sin{\theta}$ in green compared to the linear graph, $y = \theta$, (in red) shown at the right.

This is an mathematical approximation because, although it's not precise, it's "good enough" for certain situations (for $\theta \lt \pi/6$, say) and we can correct our approximation if we need to.

## Approximations in biology

Many mathematical models in biology can be considered as approximations. Two places where mathematical have been extensively applied are in neuroscience and in population dynamics (ecology and evolution).

In neuroscience, the Hodgkin-Huxley model creates a mathematical model of signal propagation on a neuron's axon using a physical model using batteries, resistances, and capacitors. While this model gives a reasonable description of action potentials, improved experimental information have led to more complex models that are more accurate and to which the HH model can be seen as an approximation.

In population dynamics, the Lotka-Volterra model creates a model of the interaction of a predator and its prey. It's a fairly simple model (two coupled ordinary differential equations) that ignores many important biological factors (such as what the prey itself prey's on and the fact that individuals are discrete and breed in cycles). Nonetheless, iit gives useful insights into the kinds of phenomena that can occur as a result of a two-species interaction, such as large oscillations. It can be considering an an approximation to a more complete model.

In all the four cases discussed above, the simplifications of the full system, whether mathematical or biological, can be seen as part of a more complete description that could potentially be used to provide corrections; and, they all are highly useful when used in the right places.

The concept of approximation is somewhat complementary to our idea of estimation. We need to make approximations to create our estimations (to decide what to ignore) and we need to carry out estimations to create good approximations (in order to decide whether it is legitimate to ignore some corrections). It is an essential component of our process of mathematical modeling.

Joe Redish 8/13/18

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