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Another option (I have never attempted this) would be to claim the existence of a thing called "The Lebesgue integral", list some carefully chosen theorems about it as axioms (maybe just linearity, so …

Not every property is preserved by limits. Here is a more basic situation in which the same reasoning is used: For each natural number $n$, there are only finitely many natural numbers in the interv …

Here is another attempt. Consider the function $f:\mathbb{N} \to \mathbb{R}$ defined by $f(n) = \textrm{ The number of elements in the set $\{n,n+1,n+2,...2n\}$}$. In other words $f(n) = n+1$. Cl …

In my experience, being too tightly wed to geometry was actually a hinderance to me when I was learning multivariable differential calculus. In my mind, the essential feature of the multivariable d …

3 blue 1 brown has an outstanding video on Fourier series: https://www.youtube.com/watch?v=r6sGWTCMz2k Thinking of a periodic complex valued map as a parametric curve in the complex plane, and thinkin …

One reason for this is that many beginning analysis books start with a rigorous development of the real numbers. If the chosen foundation is based on Cauchy sequences of rational numbers, then limits …

We can think of these transformations from two perspectives: as mappings of the plane or as graphs of pre/post composition of the function. For instance, the map $T: \mathbb{R}^2 \to \mathbb{R}^2$ gi …

It is impossible to say whether you will be ready to teach multivariable calculus after taking this course. Certainly many of the topics you list will employ some multivariable calculus, but using bi …

A proof is meant to convince a reader of the truth of some statement. When a mathematician is communicating an argument to another mathematician, you only include the level of rigor that you need so …

Another option which isn't geometric, but which reinforces the concept of derivative as linear approximation, is as follows. First derive (by any means) that $\frac{\textrm{d}}{\textrm{d}u} \frac{1}{u …

Depending on how much algebra you allow, you could make the exact same rectangle picture but label the sides $g(x)$ and $q(x)$ with area $f(x)$. This geometrically enforces $g(x)q(x) = f(x)$, aka $q( …