Linear Approximations Part 1 - Interpolating between Empirical Data

Let's have a look at the idea behind linear approximations.

We know that at the point a on the curve f(x), the derivative at a, f'(a) is the slope (gradient) of the curve at that point.

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The tangent line L(x) at the point a is given by:

Now, if we zoom in really close to the point a, we can see that above and below the point of tangency, where L(x) and f(x) touch, the curve lies very close to the tangent line.

If we keep zooming in toward the point a, f(x) looks more and more like L(x), to the extent that we can say...

...near the point (a, f(a)). So we consider L(x) to be the linear approximation of f(x) around f(a).

Ok, big deal! Why bother with this?

Well, in some cases we might know the value f(a), but it might be difficult, if not impossible to calculate nearby values. For instance, we might have a set of empirical data points from an experiment. We can use linear approximation to estimate a value between data points.

Let's do an example then...

The table below tracks the population of Nepal (in millions) at 5 year intervals at June 30 of the given year. We did not record the population at mid-year 1984, but we can use linear approximation the estimate the population at this time.

t	1980	1985	1990	1995	2000
N(t)	15.0	17.0	19.3	22.0	24.9

First, we need to find or estimate the slope N'(a) for the year 1985, given this data point. According to the definition of the derivative, we have...

But the only measurement we have before 1985 is from 1980, so we can only estimate the slope by the difference quotient between these two data points:

Beware that we've just approximated the instantaneous population rate of change as an average rate of change between the years 1980 and 1985. So with this estimate, by the linear approximation formula...

...we estimate the population of Nepal at mid year 1984 to be 16.6 million.

A graphic of this solution is shown below:

Note: A smooth curve has been fitted to the dataset as shown in the graph.

This problem was adapted from an exercise in Calculus 5th Edition by James Stewart.

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