The Tangent Line Approximation

As with one-variable calculus, linear functions, being so simple, are the starting point for approximating a function. This involves calculating the tangent line. The generic equation for a line is y = mx + b. For this line to be tangent to the graph of the function f(x) at the point (x₀, f(x₀)) the slope of the line must be the same as the derivative of the function at this point. Thus, m = f'(x₀). If we rearrange the constants in the form of the line, we can write the equation as y = m(x - x₀) + b. For this line to touch the graph of the function at the point specified, we should choose b = f(x₀). Thus, we can approximate a function, f, of one variable simply by knowing one point on the function and the slope at that point:

$$f(x) \approx f(x_0) + f'(x_0) (x - x_0).$$

This is a ``first order approximation'' to the function. It is called first order because it is a polynomial of degree one which approximates the function. This is not a very accurate approximation, but if we stay ``close'' to the initial point (x₀,f(x₀)) then the approximation will be close to the actual value of the function. How close is good enough is up to the individual application. It usually depends on how rapidly the function changes slope. Notice that near the point of tangency in the graph below, both the function (in red) and the tangent line (white) provide similar (if not identical) values for the function. Far from this point, though, the tangent line is nowhere close to the function.

Suppose we know that the point (1,5) is on a function and the slope of the function is -2 at this point. If we need to guess a value for f(1.5) we use the first order approximation:

$$f(1.5) \approx f(1) + f'(1) (1.5 - 1) = 5 + (-2)(1.5 - 1) = 4.$$