pellman said:
I think my trouble stems from the definition of one-forms as linear maps from a tangent space to R, which seems to have nothing to do with infinitesimal changes of a real variable. Is there another equivalent definition of one-forms?
This is not true. E.g. write the derivative in my favorite form, Weierstarß's notation for ##f\, : \,\mathbb{R}^n\longrightarrow \mathbb{R}##
$$
f(x_{0} +v )=f(x_{0})+J(v)+r(v)\, \text{ with } \,\lim_{v \to 0}\dfrac{r(v)}{\|v\|}=0
$$
What is the derivative here? It's the Jacobian matrix ##J,## a vector in this case. And what is it? It is a linear function that eats a direction ##v## and spits out a number. The direction is the tangent direction, the number is the slope, and all are evaluated at ##x_0,## and for ##f##, which means that we get different matrices at different points or for different functions. But once fixed a function and a point we get a one-form ##J##.
The same holds true in the school case ##n=1.## The derivative is the linear function, that multiplies a direction (we have only one in that case, i.e. we have a number), by the slope in the direction. E.g. let's take ##f(x):=x^3.## Then
\begin{align*}
f(x_0+\delta )&=x_0^3+3x_0^2\delta +3x_0\delta^2+\delta^3 =\underbrace{x_0^3}_{=:f(x_0)}+\underbrace{(3x_0^2)\cdot \delta}_{=:J(\delta)} +\underbrace{\delta^2(3x_0+\delta )}_{=:r(\delta )}
\end{align*}
where ##\lim_{\delta \to 0} \dfrac{r(\delta )}{|\delta| }= \lim_{\delta \to 0}\left(\delta \cdot (3x_0+\delta)\right)=0.##
You see, the derivative ##f' = J\, : \,\delta \longmapsto (3x_0^2)\cdot \delta ## is a one-form, multiplication by the slope at ##x_0.##
