Second derivative, unconstrained optimization in 1-d

1.3, 2.4, 4.1, 4.2


1) Main Points

Chapter 1.3 is a chapter we have already read but is good to review. The average rate of change f(b) - f(a) / b-a, also a function is increasing if the values of f(x) increase as x increases and vice versa. This chapter also explains concave up vs. down. Another way of thinking about the derivative and average rates of change is the change in distance/ the change in time. 2.4 introduces the concept of the Second Derivative. The second derivative is the derivative of the derivative marked f''. In libnitz d/dx (dy/dx) or d^2y/dx^2. The derivative tells us where the function is increasing or decreasing, the second derivative shows us where the derivative is increase or decreasing (over a certain interval). This also tells us whether a graph is concave up or down over a certain interval: if f''>0 then f is concave up there. 4.1 and 4.2 introduce ways of using derivatives to interpret graphs. We can use the signs of the derivatives to find whether the graph is increase at certain intervals or not. It also introduces local maxima and minima which are the points where f is less than/greater than the nearest areas. A critical point is any point that where f'(p)=0 or f'(p)=undefined, the critical value is the value f(p) at these points. 4.2 introduces inflection points, points where the graph changes concavity. Because the second derivative is related to the concavity, where f''=0 there is an inflection point.

2) Challenges

Reading graphs of second derivatives are difficult. It is important to remember that it is a graph of a graph of a graph, and just because the second derivative is changing one way it doesn't always mean that the function is changing in the same way. Critical points may be confusing to me, it is hard to know when a function has one and/or how many it has. There is a difference between inflection points and critical points.

3)

Being able to interpret a graph using derivatives is really important. The book gave an example of graph that was barely readable but by using derivatives we could learn a lot about how the graph's function. Interpreting graphs with this level of detail will be very important because it allows us to gain more from graphs of real life situations,