Discussion of Linearization and Differentials
The idea behind linearization of a function and differentials is that the tangent line to a function at a point is a very good approximation to the function near the point.
Consider a function f(x). Recall that the tangent line at a point 'a' has slope f '(a) and passes through the point (a, f(a)). That is, the tangent line hits the graph of the function at the point 'a'. We now have a slope and a point, so we can write the equation of the tangent line as: y - y 1 = m(x - x 1 ) or y - f(a) = f '(a)(x - a). Equivalently, we have
y = f '(a)(x - a) + f(a)
We therefore define the linearization of f at the point x = a to be
L(x) = f '(a)(x - a) + f(a)
As an exercise to see the power of the linear approximation of a function, pick a random function and find the equation of the tangent line at a particular point. Use your graphing calculator to graph both the function and the tangent line. You will see that when x is near the point, the line and the function almost coincide.
This idea of linear approximation can help us to approximate complicated functions. For example, in the webwork, many of you were asked to approximate cos(1) by basing your approximation at pi/3. You cannot calculate cos(1) by hand. You have to have a calculator. However, in only a couple steps, you can write down the tangent line and the resulting approximation since you know cos(pi/3) and sin(pi/3). This is the power of linear approximation.
Sometimes, we also wish to know how much the function value changes if we change the x value slightly. For this, we can look at linearization. We know that f(x) is approximately f'(a)(x - a) + f(a), so f(x) - f(a) is approximately f'(a)(x-a). The quantity f(x) - f(a) is the change in y value and x - a is the change in x value, so we have that the change in y is approximately f'(a) multiplied by the change in x.
Finally, this brings us to the definition of differentials . The differential of x is simply defined to be the change in x. We then define the differential of y to be
dy = f'(x)dx
The discussion above shows that dy is approximately the change in y, so we can use this in our approximations.
In the end, linearization and differentials are the same concept. Some problems are easier when viewed in terms of differentials but the concept is the same.