Mathematics 133 – Intensive Calculus for Science 1 Discussion 4 – Let’s Do It Again! October 14, 2005

Background

Over the past two days, we have introduced the derivative function f′(x) for a function f(x) and seen how to get quantitative and qualitative information about f′(x) from a formula, graph, or table of values of f(x). Today, we are going to do the same process to f′(x) and compute what is usually called the second derivative of f, also written (f′)′(x) = f′′(x).

Discussion Questions

The following graph shows y = f(x).

Note: You are now going to sketch two graphs produced from this one. It will probably help to “stack” the three graphs so that the same x-values line up vertically on all three graphs.

A) Give a “qualitative” sketch of the graph y = f′(x) for the same interval of x-values. Note that there is no y-axis scale here. So you will not be able to get exact values for f′(x). But determine the intervals where f′(x) > 0, where f′(x) < 0, and the points where f′(x) = 0.

B) Next, apply the same process to your graph from part A – make a qualitative sketch of the derivative of f′(x) – this is f′′(x).

C) Answer the following questions:

1) Over which intervals is f′′(x) > 0? Over which intervals is f′′(x) < 0?

2) What is f′(x) doing when f′′(x) > 0? when f′′(x) < 0?

3) What is true about the original graph y = f(x) when f′′(x) > 0? when f′′(x) < 0?