Remark: This definition will need to be generalized later to apply to curves that are not graphs of functions. Also this definition does not allow vertical lines to be tangents, whereas on geometrical grounds, vertical tangents are quite reasonable.
Let . Then for all ,
We saw in example 10.10 that does not exist. Hence, the graph of at has no tangent.
If , then in the previous example we saw that the equation of the tangent to graph at is ; i.e., the -axis is tangent to the curve. Note that in this case the tangent line crosses the curve at the point of tangency.
If then for all
Thus at each point on the curve the tangent line coincides with the curve.
Let . This is not the same as the function since the domain of is while the domain of is . (For all we have where .)
I want to investigate
. From the picture, I expect this graph to have an infinite
slope at , which means according to our definition that there is no
line at . Let
Calculate for arbitrary . Does your answer agree with your prediction?
On what intervals do you expect to be positive? On what intervals do you expect to be negative? Calculate .
On the basis of symmetry, what do you expect to be the values of , and ? For what do you expect to be zero? On the basis of your guesses and your calculated value of , draw a graph of , where is the function that assigns to a generic number in . On the basis of your graph, guess a formula for .
(Optional) Prove that your guess is correct. (Some trigonometric identities will be needed.)
For what in does exist? Sketch the graphs of and on the same set of axes.
The following definition which involves time and motion and particles is not a part of our official development and will not be used for proving any theorems.
In applications we will usually express velocity in units like . We will wait until we have developed some techniques for differentiation before we do any velocity problems.
The definition of velocity just given would have made no sense to Euclid or Aristotle. The Greek theory of proportion does not allow one to divide a length by a time, and Aristotle would no more divide a length by a time than he would add them. Question: Why is it that today in physics you are allowed to divide a length by a time, but you are not allowed to add a length to a time?
In Newton's calculus, the notion of instantaneous velocity or fluxion was taken as an undefined, intuitively understood concept, and the fluxions were calculated using methods similar to that used in the section 10.1.
The first ``rigorous'' definitions of limit of a function were given around 1820 by Bernard Bolzano (1781-1848) and Augustin Cauchy (1789-1857)[23, chapter 1]. The definition of limit of a function in terms of limits of sequences was given by Eduard Heine in 1872.