- Why do we use the Taylor series so often?
- (it lets us approximate arbitrary functions in terms of polyomials.)
- (the polynomial coeffients involve derivatives, so we can use TS when working with differential quantities.)
- Write the first four terms of a Taylor series approximation to $f(x)$ centered on point $f(x_i)$
- What do we mean when we say a finite difference approximation is second order?
- What is the notation used for saying something is second order?
- Derive the second order approximation to the second derivative evaluated at point $i$ in terms of itself and its neighbor points $i-1$ and $i+1$.
- For a central difference approximation to the first derivative, a three point stencil gave a second order approximation. How many points would be in the stencil for a fourth order approximation?
- for central derivatives, why are we considering even order approximations?
- Why would one-sided finite difference approximations be useful?
- where have we used them already?
- Describe in words how we can find arbitrary approximations to derivatives using Taylor series.
- (Choose a base point. This is where all derivatives will be evaluated. Write the function at some other grid point using the TS about the base point. Combine multiple TS evaluated at multiple grid points, all about the same base point, so that all base point terms except that involving the desired derivative cancel. The largest term that doesn’t cancel will define the error. A Taylor table is a way of formalizing this procedure and involves solving an Ax=b system.)