Learning about approximations

Today the second my son was studying in his calculus book was about using first and second derivatives to approximate values of functions.

I had him work through 6 problems in the book – 3 of which had answers in the book so he could check his work. Then we moved on to talking about the problems he did for which he was unable to check the answer.

The first problem involved finding the approximate value of (1.97)^6 using a linear approximation:

The next problem was finding the value of \cos(31.5^o) again using a linear approximation:

Instead of working through the 3rd problem, I asked him to use a quadratic approximation to find the approximate value of \cos(31.5^o).