TITLE: What are some interesting calculus facts your calculus teachers didn't teach you?
QUESTION [24 upvotes]: I recently learned an interesting fact about what the value of a Lagrange multiplier represents: suppose the maximum of some real-valued function $f(\vec{x})$ subject to a constraint $g(\vec{x})=c$ is $M$ (of course, $M$ depends on $c$), which you obtained via Lagrange multipliers (solving $\nabla f = \lambda \nabla g$).  Then, it's easy to show (using the chain rule) that the multiplier $\lambda$ can be interpreted as the change of the maximum with respect to perturbations of the level set $g=c$. That is,
$$
\lambda = \frac{d M}{dc}
$$
I think this is a pretty cool result and I never it heard about during my time as an undergraduate. 
What are some interesting calculus (or undergraduate mathematics) results nobody told you about during your calculus (or undergraduate) education?

REPLY [3 votes]: Integration by parts - visualization
Not that shocking, but my teacher never showed me this. For many people this will be familiar, but for those who never saw it: it can be an eye opener. 

Image source: Wikipedia

The area of the blue / red region is:
$$A_1=\int_{y_1}^{y_2}x(y)dy$$
$$A_2=\int_{x_1}^{x_2}y(x)dx$$
So we have: 
$$\overbrace{\int_{y_1}^{y_2}x(y)dy}^{A_1}+\overbrace{\int_{x_1}^{x_2}y(x)dx}^{A_2}=\biggl.x
> . y(x)\biggl|_{x1}^{x2} = \biggl.y . x(y)\biggl|_{y1}^{y2}$$
Assuming the curve is smooth within a neighborhood, this generalizes
  to indefinite integrals:
$$\int xdy + \int y dx = xy$$
Rearranging yields the well-known formula:
$\int xdy = xy - \int y dx$