Microscopic tangency in one variable

In begining calculus you learned that the derivative measures the slope of the line tangent to a curve at a particular point, .  We begin by setting up convenient "local variables" to use to discuss this problem.  If we fix a particular in the --coordinates, we can define new parallel coordinates through this point.  The -origin is the point of tangency to the curve.

A line in the local coordinates through the local origin has equation for some slope .  Of course we seek the proper value of to make tangent to .

The Tangent as a Limit

You probably learned the derivative from the approximation

If we write the error in this limit explicitly, the approximation can be expressed as

or

where as  Δx → 0.  Intuitively we may say the error is small, ,  in the formula  

(1)

when the change in input is small,  δx ≈ 0.  The nonlinear change on the left side equals a linear change plus a term that is small compared with the input change.

The error has a direct graphical interpretation as the error measured above after magnification by .  This magnification makes the small change appear unit size and the term measures after magnification.

When we focus a powerful microscope at the point we only see the linear curve  , because is smaller than the thickness of the line.  The figure below shows a small box magnified on the right.

Figure 1.1.1: A Magnified Tangent

Figure 1

Created by Mathematica  (September 22, 2004)