Tangents, defined poorly
You first encountered the notion of a tangent in geometry. I found the following definition of a tangent in a geometry textbook, which is probably the definition you've been carrying around in your head ever since.
A tangent to the circle is a line that contains exactly one point of .It is natural to try to generalize this definition to suit our purposes in calculus.
Bad Definition #1
A tangent to the function is a line that contains exactly one point of .
This is no good. Below, plot:
- a function,
- a point on the function, and
- the line tangent to the function at that point
Bad Definition #2 (which isn't all that bad; it just fails to describe what it means for two points to be "infinitely close")
Calculus was famously independently invented by two mathematicians, Isaac Newton and Gottfried Leibniz, in the 1600s. Leibniz described a tangent line as a line passing between two "infinitely close" points on a curve. But what does that mean?
What mathematical tool do we use to describe two things as "infinitely close"? How does Leibniz's description relate to the definition of the derivative?
Bad Definition #3
The 1828 edition of Webster's Dictionary defines a tangent as a "line which touches a curve, but which when produced, does not cut it".
This too is no good. If we were to accept this definition, functions would not have tangent lines at their inflection points. Below, plot:
- a function,
- an inflection point of the function, and
- the line tangent to the function at that inflection point,