I was able to understand all of the questions (after your guys' help, once again, thank you very much) up until this one which kind of reverses the question:

Suppose that f is a function such that $\displaystyle f(1) = 1, f '(1) = 2$, and $\displaystyle f ''(x) = \frac {1}{(1 + x^3)}$ for $\displaystyle x > -1$

a) Estimate $\displaystyle f(1.5)$ using a quadratic Taylor polynomial.

b) Find an upper bound on the approximation error made in part (a).