## ABSTRACT

We begin by looking at ﬁnite diﬀerence approximations for ordinary diﬀerential equations, as they involve most of the pertinent ideas from PDE methods, yet are easier to implement. The ﬁrst question we ask is the somewhat obvious question of how to approximate the ﬁrst derivative. But remember the deﬁnition of f ′ using the diﬀerence quotient,

f ′(x) = lim h→0

f(x+ h)− f(x) h

.