Assignment 26 (assigned 3/31, due 4/2)

1. On Friday 3/28, we discussed the proof of the product rule for derivatives in class.  I wrote out the computational part of the proof and pointed out that to write the proof correctly, it would be necessary to put the steps in a different order (as we did earlier in the same class period for the sum rule for derivatives).  Your assignment is to write a detailed know-show table (also called a proof in statement-reason format, or a two-column proof) of the product rule for derivatives.  That is, in the left hand column you should have numbered statements, which can either be formulas or sentences using words.  (You do not have to write complete sentences with capital letters and periods, I only have to understand what you mean.)  In the right hand column you should give the reason why the statement is true.  The reason will usually be a theorem that we have previously proved.  If the reason is a well known fact from algebra, you can just say "by algebra" as the reason.  If you want to refer to an earlier statement in the left hand column as one of the reasons for a statement, you can do so by giving the line number.  (It is not really necessary to do this.  It is understood that anything written earlier in the statement column can be used as one of the reasons for a new statement.  The reasons column should be used primarily for reasons that are not among the earlier statements in the statement column.  But sometimes it may be helpful to refer to a line number, especially if you are using a statement that occurred much earlier in the proof.)

2. On Mondy 3/31 I presented half of the proof of the theorem that at an interior maximum point where a function is differentiable, the derivative is zero.  (The half that I proved was that the derivative was not positive.)  Prove the other half (that the derivative is not negative).

3. Without providing a detailed proof, describe what changes would need to be made in order to show that at an interior minimum point, the derivative is not positive.