On page 55-56 of Gelfand and Fomin, the authors claim that the photo on page 55 justifies the formulas at the top of 56. I’ve been dwelling on this for a while, because I really don’t see how the picture implies the formulas on 56, or what those formulas totally represent. Could anyone clarify what they’re trying to say here?
Page 54 (https://i.stack.imgur.com/GgZIs.jpg)! Page 55 (https://i.stack.imgur.com/u473G.jpg)! Page 56 (https://i.stack.imgur.com/b8oM2.jpg)
Thank you!
Let's look at the approximation at the left point $x_0$. We would like to describe the variation of $y$ from $y(x_0)$ to $y^*(x_0+\delta x_0)$, which is $$ \delta y_0=y^*(x_0+\delta x_0)-y(x_0). $$ From the Figure 4 it is clear that this variation consists of two parts: (1) the vertical part of the parallelogram and (2) the increase due to the slope of $y(x)$ at $x=x_0$.
The part (1) is simply $h(x_0)$. The part (2) is approximated by the tangent line as $y'(x_0)\delta x_0$. It gives the total increase as $$ \delta y_0\sim h(x_0)+y'(x_0)\delta x_0\quad\Rightarrow\quad h(x_0)\sim\delta y_0-y'(x_0)\delta x_0. $$ Similarly for the right point.