Derivatives as Functions

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Anything You Want to Talk About?

Are the Canvas discussions graded? No, because that would be too much of a participation rather than mastery grade. But it’s still important to participate in the discussions, because they are the first, and for the 2/3rds of the time you can’t be in a face-to-face class, only, activity that transitions you from reading about things in the book to having to work with them and think hard about them, which is a crucial step in learning. And the ideas and questions expressed in the discussions give me a sense of what I do and don’t need to go over in the face-to-face classes.

Misc

Rejuvenation Day

Next Wednesday is the semester’s first “rejuvenation day,” i.e., a day without classes, intended as a break from academics. We won’t have a class meeting that day, and the cohort schedule will shift one day to accommodate — I’ll be careful to reflect that in announcements next week. I also won’t have a problem set to do during next week — though if you grade early in the week you’ll probably still want to meet next Monday or Tuesday to show me problem set 3, and I may hand out problem set 4 Thursday or Friday, to work on and grade during the following week. You can also use next week to catch up on meetings if you need to, e.g., problem sets 1 or 2, redos, etc. But note that I’m planning to not hold meetings on Wednesday.

SI

The next session is Sunday, 6:00 - 7:30. Please go, these sessions can be a big help, particularly if you want more in-person opportunity to learn material from this course.

Problem Set 3

Don’t forget that problem set 3 is now available. You should be working on it this week and grading it late this week or early next.

Derivative Functions

From “Derivative Functions” and “Derivatives and Continuity” in section 3.2 of the textbook, and this discussion.

Derivatives as Functions

Use the limit definition of derivative to find derivative functions for each of the following.

The limit definition of derivative is based on the slope of a secant line, and the limit of that slope as the endpoints of the line approach each other:

f(t) = 1/2 t² + t

Start with the limit definition of derivative, expand the f(x+h) and f(x) parts, and then use algebra and limit laws to find the limit:

f(s) = √(s²-1)

This goes similarly to the first example, except that different algebraic techniques are helpful:

In general, using the limit definition to differentiate a function proceeds somewhat like this:

1. (Write down the limit definition of derivative for reference.)
2. Use the function’s definition to expand “f(x+h)” and “f(x)” in the limit definition.
3. Use algebra to simplify the expression inside the limit until you can eliminate the division by 0 that comes from the limit definition.
4. Use limit laws (i.e., plug in h = 0) to evaluate the remaining limit.

Continuity

Based on their derivatives, where are the functions above continuous?

The key idea is that differentiability requires continuity: In each of the various ways a function might be discontinuous, the notion of “slope” at the point of discontinuity is undefined.

So a function must be continuous everywhere its derivative is defined, which can sometimes be easier to recognize than the original definition of continuity (limit equal to function).

For instance, noticing that the derivative of the second function above is undefined when s = -1 or 1 immediately says that the function is discontinuous at those x values, something that needed careful thinking about limits from the left and right when we looked at that function for continuity a week or so ago.

Next

Limits are a powerful way to find derivatives, but using them can get tedious. Fortunately, there are a lot of differentiation rules that are consequences of the limit definition but give you ways to find derivatives without directly finding limits.

Read “The Basic Rules,” “The Power Rule,” and “The Sum, Difference, and Constant Multiple Rules” in section 3.3 of the textbook by class time Thursday.

Also contribute to this discussion by Thursday’s class.

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