Second derivative, unconstrained optimization in 1-d (10/14/2008)

1. Main Points

The major new concept in today’s reading was the second derivative (f’’), which is the derivative of dy/dx, and the different ways it can be applied for solving mathematical problems. Some things we can tell from f’’ is that if f’’>0 on an interval it means that f’ is increasing, so that the graph of f is concave up there. The reverse for f’’<0. The second derivative can also be used to test for local maxima and minima for a continuous function f, called critical points. The sign of the f’’ will also change at an inflection point, which is defined as a point at which the graph of a function f changes concavity.

2. What was challenging about this material?

The hardest part of todays rading I think was locating inflection points, maybe you could go through this at greater length during class?

3. To think of the second derivative as the rate of change of a rate of change is a really cool concept I think. The example in the reading about the senate explaining that they had not cut the defense budget, only cut the rate at which it was increasing. Essentially, the derivative of the defense budget was still positive (the budget was increasing), but the second derivative was negative (the budget’s rate of increase had slowed)