But the difficulty discussed above came after this, revealing itself in the context of work on specific proofs. I’m speculating here, but perhaps one way to see it is that she was struggling with the idea of a uniform strategy; or else with the notion that a uniform strategy can be described in terms of a single (but generic) epsilon.

For what it’s worth, the analogy mentioned in note [14] did shift something for her, because she felt she understood how the algebraic proof worked.

]]>For readers interested in the Launchings columns mentioned by David, which are extremely relevant to the present piece, here are the URLS (unfortunately this blog doesn’t support links in the comments):

https://launchings.blogspot.com/2014/06/beyond-limit-i.html

http://launchings.blogspot.com/2014/08/beyond-limit-ii.html

https://launchings.blogspot.com/2014/09/beyond-limit-iii.html

On the other hand, the idea of tangent line is very useful in getting across the point that differentiability is really nothing but “local linearity”. ]]>

You might enjoy https://www.geogebra.org/m/RyPC5G7M and some of the related applets on that site

]]>The derivative of a real valued function of one (or more) real variable(s) at a point in its domain of definition is best (and most correctly) understood as providing the best approximation to the function “near” that point by a polynomial function of degree 1:

So if the function is f: R –> R, x |–> y, then the best such approximation by a polynomial of degree 1 (if there is such a beast) is of the form

y = f(a) + d(x-a),

It is easy to show that there is such a “best” approximation if and only if f is differentiable at a, and then d = f\'(a), the derivative of f at a. (I have omitted details here.)

When we represent f by its graph in the Cartesian plane, the graph has a tangent at (a, f(a)) if and only if f is differentiable at a, in which case the above equation is the equation of the tangent.

This generalises easily to multivariate functions – we just have polynomials in more variables, and the (partial) derivatives are the coefficients.

The notion of differentiability is logically prior to the notion of a tangent to the graph of a function.

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