Derivatives and Shapes of Graphs Discussion

(The following is/are the initial prompt(s) for an online discussion; students may have posted responses, and prompts for further discussion may have been added, but these things are not shown.)

Since a function’s derivative gives the instantaneous slope of the function’s graph, it’s not surprising that there are connections between derivatives and the shapes or behavior of functions’ graphs. Section 4.5 in our textbook explores these connections in more depth, and this discussion is a place where you can start working with and talking about the ideas.

To start with, here are examples of two ways information from first derivatives can be used to understand a function. (It would be nice if different people contributed at least the first answer, comment, or question about each example.)

The function

has a local extreme value at x = 1. Is this extreme a minimum or maximum?

Suppose all you know about some function f is the following information about its derivative:

• f′(x) > 0 whenever x < 1
• f′(x) < 0 whenever 1 < x < 10
• f′(x) > 0 whenever x > 10

What, if anything, can you say about what the graph of f(x) looks like? How many different graphs can you sketch that satisfy the derivative requirements? (You should be able to attach photos or computer sketches of graphs to your posts if you want to show exactly what you have in mind.)