Exercise Solution 4.16

The first order Taylor polynomial of a function f of a variable ^tx about a fixed point x^[0] is:

[s1]

Treating ^tz_log as the function f, we construct a second order Taylor polynomial about an arbitrary point x^[0]:

[s2]

Substituting the specific point ^t–1x for x^[0], this becomes

[s3]

which is the formula for simple return. We conclude that ^tz_simple is the first-order Taylor polynomial for ^tz_log.