Derivatives and the Shapes of Graphs, Part 1

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First Derivatives and the Shape of Graphs

From “The First Derivative Test” in section 4.5 of the textbook, and this discussion of derivatives and shapes of graphs.

The First Derivative Test for Extreme Values

What’s the test and the process for carrying it out?

The key idea is that a function’s graph must rise (i.e., have a positive derivative) towards a maximum from the left and fall (have a negative derivative) away from it to the right, while the graph falls from the left towards a minimum and rises away from the right:

Thus, to determine whether ( c, f(x) ) is a minimum or a maximum of f...

1. Calculate f′(x) for some x < c (but not less than the next smaller critical point, if there is one)
2. Calculate f′(x) for some x > c (but not bigger than the next larger critical point, if there is one)
3. If the derivatives from (1) and (2) switch from negative to positive then (c,f(c)) is a minimum; if the derivatives switch from positive to negative, then the point is a maximum.

For example, the function g(x) = 1 - xe^-x has a local extreme value at x = 1. Is this extreme a minimum or maximum?

Start by finding the derivative:

This derivative only has one critical point, at x = 1, so we can test its sign on either side of 1 by picking any convenient x values. For example...

Since the derivatives change sign from negative to positive as we go from left to right through x = 1, the extreme value must be a minimum:

In doing this example, there was a typo in the first version of it: I gave g(x) as g(x) = 1 - xe^x. Applying the first derivative test to it around x = 1 produces a negative value for the derivative at x = 0, and an even more negative value at x = 2. This simply means that the derivative doesn’t change sign, and so that there is no extreme value. This can happen with real critical points (i.e, not just when there’s a mistake), since critical points are places where extremes may happen, but not places where they must happen.

Sketching Graphs from First Derivatives

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?

Lots of graphs meet these requirements, including some that aren’t continuous at 1 or 10. But they all have to have the same basic up-down-up shape:

A Game

See if you can come up with a simple line drawing, and then communicate it to someone else using only information about the derivatives of the lines over various intervals. In particular...

• 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.

For example:

• I’m thinking of a drawing described by 2 functions, f(x) and g(x).
• f(x) is relevant for -2 ≤ x ≤ 2
• f(-2) = f(2) = 0
• f′(x) = 0 for -2 ≤ x ≤ 2
• g(x) is relevant for -1 ≤ x ≤ 1
• g(-1) = g(1) = 0
• g′(x) > 0 for -1 ≤ x < 0
• g′(0) = 0
• g′(x) < 0 for 0 < x ≤ 1

What is my drawing?

Someone suggested this, which actually was what I was thinking of — a not-fine-art rising sun:

But notice that as far as the derivatives go, it could just as well have been something completely different, say a pagoda:

Next

The rising-sun-vs-pagoda example above suggests that first derivatives don’t provide a lot of information for distinguishing functions’ graphs from each other. So next I want to consider including second derivatives and concavity information for more nuanced understanding of graphs.

Please read “Concavity and Points of Inflection” and “The Second Derivative Test” in section 4.5 of the book by class time Friday.

Please also contribute to an extended version of the above game in the ongoing discussion of derivatives and the shapes of graphs.

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