Approximation of the tangent line

One overly simplistic way to characterize calculus might be as that part of mathematics that’s amenable to the following approach:

Approximate and then take a limit.

A common early example is the computation of the slope of a tangent line. Can you use a slider in the image below to take a guess at the slope of the tangent line?

Details

The figure shows the graph of a cubic polynomial, which we’ll call \(f\). The blue line is the line that’s tangent to the graph of \(f\) at the point \(x=4\); it’s defined by the properties that it passes through the point \((4,f(4))\) and has the same slope as the graph of the function does at that point. The question is - what is that slope?

The would be an easy question if we knew two points on the tangent line, since we could use \(\frac{\Delta y}{\Delta x}\) between the two points. We have just the one point, though - that’s where the secant line comes in.

The secant line is defined in terms of the parameter \(h\). Given \(h\), the secant line is defined to be the line that goes through the points \[(4,f(4)) \text{ and } (4+h,f(4+h)).\] Thus, its slope is \[ \frac{f(4+h) - f(4)}{h}. \] While the slope of the secant line isn’t really what we’re looking for, it is a good estimate of the slope we want. That’s where the approximation part of calculus enters in this example.

Furthermore, there’s a clear way to improve the approximation - just take the parameter \(h\) to be closer to zero. Ideally, we’d like to set \(h=0\) but that doesn’t quite work. That’s where the take a limit aspect of calculus enters in this example. Ultimately, we compute the limit as \(h\to0\) to get the exact slope.

Of course, there’s nothing special about the point \(x=4\); you can do this at any point in the domain of the function. Thus, we define a new function \(f'\) called the derivative of \(f\) defined by \[ f'(x) = \lim_{h\to0} \frac{f(x+h) - f(x)}{h}. \]