Tuesday office hours

On Tuesday I will be in my office from 10-12 (which lines up with normal class and office hours). If you'd like to arrange another time, send me an email to make sure I'll be there.

Also, there will be a review session during normal class time on Thursday.

-Jesse

Some exam review problems

I. Below are some problems from the Chapter 3 review that will hopefully cover all the basics plus more:

• 89, 93 (eqn of motion and exponential growth)
• 21, 24-26, 29-34, 37-42, 46, 49, 50 (these cover everything from plain ol' chain rule through implicit differentiation, logarithmic differentiation, as well as derivatives of inverse trig functions.)
• 99, 100, 103, 105 (related rates and linear approximation)

Now, some problems from the Chapter 4 review:

• 7-14 (limits from 4.4)
• 45, 48.

And a few from 4.2 & 4.3 that should sum up the beginning of Chapter 4:

• 4.2: 19, 13, 14, 15
• 4.3: 45-52, 67

Exam III

Information on the exam is here. As always, the best way to get startedstudying is the HW. In each section tested over do the "middle" HW problems. I'll put up some supplementary problems (of a more difficult nature) later this weekend. The take home quiz should provide a good source of "fun" as well.

Hint for the quiz problem with "tan(2x)": Double angle. (edit: This doesn't make it as simple as I'd hoped... it's one of many ways to tackle this one. Enjoy!)

-Jesse

Chapter 4, section 4

Someone left a textbook in class. You can pick it up in my office, or I'll bring it to class on Friday.
-Jesse

Chapter 4, section 3

Chapter 4, section 2

Continuous functions that are not differentiable

As mentioned on Friday, the hypothesis in the Mean Value Theorem (and Rolle's Theorem) that f(x) is continuous on [a,b] and differentiable on (a,b) covers most nice functions that we deal with in class. There is one family of examples that we've dealt with a lot that have at least one point of discontinuity:

f(x) = |x-c|, where c shifts the function so that the cusp/point lands in [a,b].

What are some variants of this function? (Think of a piecewise function that looks looks like a sawtooth--lots of little absolute value functions put together. How would you define this function?)

Now, some pathological examples. The Weierstrass function (here at wikipedia) is one very interesting example of a function that is *continuous everywhere* (why?) but differentiable nowhere (a bit harder to prove). The discussion on Wikipedia is a little beyond what we've covered in class, but from the graph you get the idea that it essentially builds on the "jagged" idea of the absolute function. Weierstrass found a way to make a function that changes direction infinitely often.

Another example is due to Bolzano. Here is a link at Math World (here and a definition of the function here). The idea is similar, but Bolzano did not use continuous cosine functions in the construction. He simply extended the sawtooth idea (ad infinitum).

-Jesse