Approximated distance between two points on a surface

Question

I'm reading this paper and I don't understand this line because I haven't the book (I can't look up Theorem 7.4.2) . How come that the distance between $P_i$ and $P_{i-1}$ is calculated as follows?

Answer 1

By the Pythagorean Theorem, the distance between $P_i$ and $P_{i-1}$ is $$\sqrt{(\Delta x)^2+(\Delta y)^2}=\sqrt{1+\Bigl({{\Delta y\over \Delta x}\Bigr)^2}}\cdot\Delta x.$$ By the Mean Value Theorem, ${\Delta y\over \Delta x}=f'(x_i^*)$ for some $x_i^*$ in $[x_i,x_{i-1}]$; which gives your expression.