Derivatives and the Shapes of Graphs, Part 2

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Sunday 6:00 - 7:30. Review of the past week and a chance to go over problem set questions. Watch for an announcement Sunday afternoon with a Zoom link.

Derivatives and Graphs

Particularly based on “Concavity and Points of Inflection” and “The Second Derivative Test” in section 4.5 of the book and the ongoing discussion of derivatives and the shapes of graphs.

The Second Derivative Test

What is it and how does it work?

It’s a way to tell whether a critical point corresponds to a minimum or maximum (or neither), based on how the second derivative indicates the “concavity” of a function.

“Concavity” is whether a function is bending in a way that makes its graph head more upward (concave up, and notice that even a graph that’s decreasing can be concave up, if the rate of decrease is slowing down) or bending to head more downward (concave down).

If a graph is concave up, then its derivative is increasing, and so its second derivative is positive. Similarly, if a graph is concave down then its second derivative is negative:

As a special case, points where concavity changes from up to down or vice versa have second derivatives of 0 (assuming the second derivative is continuous). Such points are called “inflection points.”

Notice how an extreme value that occurs where a graph is concave down has to be a local maximum, and one that occurs where a graph is concave up has to be a local minimum. Thus the second derivative test:

• If f′′(c) < 0 then f(c) is a maximum
• If f′′(c) > 0 then f(c) is a minimum

How would this test work with yesterday’s example of an extreme point of 1 - xe^-x at 1?

Start by finding the second derivative:

Then plug x = 1 into the second derivative and see what the sign is:

Since the second derivative is positive at x = 1, the critical point corresponds to a minimum.

You could also reach the same conclusion, for essentially the same reason, if you were able realize by inspecting it that the first derivative is positive on both sides of x = 1.

What about the critical point of x⁵ at 0 that turns out not to be an extreme point?

This time the second derivative is 0 at x = 0, so the second derivative test doesn’t tell you whether 0⁵ is a minimum or maximum.

The Graphs Game

A game in which one player (or team) tries to communicate a simple line drawing to another player or team by describing only the derivatives of the lines involved. See yesterday’s class and the discussion for further description of the game and rules, but the key rules are...

• You can describe one function per continuous line in your drawing.
• For each function, you can define an interval over which it’s relevant.
• You can define the function’s value at the endpoints of that interval, but nowhere else.
• You can divide the interval over which each function is relevant into subintervals or individual points, and for each subinterval or point say whether the function’s derivative is negative, zero, or positive.
• Similarly, you can say whether the function’s second derivative is negative, zero, or positive over each subinterval or point.

We came up with 2 examples, in both cases having teams of students give me descriptions of drawings and seeing if I could reproduce them.

The first was meant to be the letter w, and the sketch I came up with looked like...

The second was a “squiggle,” and I came up with...

Next

The “end behavior” of graphs, i.e., how they behave as x approaches ∞ or -∞.

Please read “Limits at Infinity” in section 4.6 of the textbook.

Also contribute to this discussion of limits at infinity.

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