/sci/ - Science & Math

Question on numerical methods for differential equations:
Could someone explain what's going on here?

I assume we have [math] g(x) \approx _a g(a) + g ' (a) (x-a) + \frac{ g''(a) } { 2! } (x-a) + ... [/math]
With [math] g(x) = y'(x)[/math] and [math] a = t_{ n+2 } [/math] , but what's x here, is it h? And how do they get the right hand taylor series?

Also I read that just replacing y by a polynomial of degree p and check for which p (1) is exact would also work to find the order of consistency, but it's only exact for p=0 and p=1. Why?

Question on numerical methods for differential equations:
Could someone explain what's going on here?

I assume we have [math] g(x) \approx _a g(a) + g ' (a) (x-a) + \frac{ g''(a) } { 2! } (x-a) + ... [/math]
With [math] g(x) = y'(x) [/math] and [math] a = t_{ n+2 } [/math], but what's x here, is it h? And how do they get the right hand taylor series?
Also I read that just replacing y by a polynomial of degree p and check for which p (1) is exact would also work to find the order of consistency, but it's only exact for p=0 and p=1. Why?