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In elementary textbooks on differential calculus, the notion of derivative number at, say, $x=a$ (i.e. $f'(a)$) is sometimes introduced via a parallelism between the analytic process and the geometric process, saying: while the difference quotient ($ \frac { f(a+h) - f(a)} { h} $) goes to a determinate limit (as $h$ goes to $0$), the straight line passing through $(a, f(a))$ and $(a+h, f(a+h))$ tends to a " limiting position", namely the tangent line to the graph of function $ f$ at point $(a, f(a))$.

Of course, this approach is intuitive, and using a graphing calculator we "see" that the line defined by $ y-f(a) = f'(a) ( x-a)$ is tangent to the graph of function $f$.

Here we seem to face an alternative: (1) either the tangent is defined as the line satisfying the above equation ( and in that case my question is pointless), or (2) we have a concept of tangent line that is independent from differential calculus , in such a way that the claim "the line passing through $(a, f(a))$ and having a slope equal to $f'(a)$ is tangent to the graph of $f$" has a substantial content.

In short: is it possible to prove, as an informative (and not simply definitional) claim that the line defined by $ y-f(a) = f'(a) (x-a)$ is the tangent to the graph of $f$ passing through $(a, f(a)$? And if so, which concept of "tangent " can be used to this effect?

Vince Vickler
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    "How do we know that the derivative number is actually the slope of the tangent line." Because that is how the tangent line is defined. – Xander Henderson Jul 30 '23 at 16:31
  • Having now read your question, how do you define a "tangent line" without calculus? If it isn't defined as the limit of secant lines, how is it defined? – Xander Henderson Jul 30 '23 at 16:33
  • I don't understand the question. You have to start with definitions. Once you have defined what a tangent line is, we can start to ask about how to prove different properties of that definition. – Xander Henderson Jul 30 '23 at 16:34
  • This is what I'm looking for, a definition of tangent line that does not involve the notion of derivative number, so that the claim becomes " substantial". – Vince Vickler Jul 30 '23 at 16:34
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    Related: https://math.stackexchange.com/q/4028997/42969 – Martin R Jul 30 '23 at 16:36
  • Well, such definitions do exist. For example, if $p$ is a polynomial and $a$ is a real number, then there exists a unique line $\ell$ of the form $\ell(x) = mx + k$ such that $(p-\ell)(x) = q(x)(x-a)^2$, where $q$ is a polynomial. It is reasonable to define $\ell$ to be the tangent line to $p$ at $a$. – Xander Henderson Jul 30 '23 at 16:38
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    And, as @MartinR points out, there are other ways of talking about tangency, but most of these are either non-standard (like the one I presented above), or more sophisticated, and generally require building a lot more theory before you can get to a definition. (And you aren't really going to get very far away from some notion of "derivative", anyway). – Xander Henderson Jul 30 '23 at 16:39
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    how about the following intuitive definition; A line L is tangent at (a,f(a)) if there is an open neighbourhood U such that either U's intersection with L is $L\cap U$ or the point (a,f(a)). – LBE Jul 30 '23 at 16:41
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    @LBE What line is "tangent" to $x \mapsto x^2 \sin(1/x)$ at zero? – Xander Henderson Jul 30 '23 at 16:42
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    whoops i meant $U\cap L\cap graph$ is either the line segment of the point – LBE Jul 30 '23 at 16:43
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    @LBE: Xander's function is still a counterexample -- the intersection of the graph with the tangent at $0$ contains infinitely many isolated points for any neighbourhood $U$ of $(0,0)$. – TonyK Jul 30 '23 at 16:45
  • sure it is, it was not a reply to Xander, it was just a correction of my comment. i wonder what the largest class of functions is that this "definition" works for – LBE Jul 30 '23 at 17:03
  • I think the most satisfactory connection between the analytic & geometric worlds in this case is for convex functions where tangency means a supporting hyperplane (line). – copper.hat Jul 30 '23 at 17:14

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