Version 1, August 2016 1
This format is the result of tinkering with a mixed lecture format for 3 terms. As such, it is still a work in progress and we will discuss adaptations both to the general format as well as for individual topics throughout the term. 2
As we continue to refine MATH 3, we want to standardize the course a bit more than in the past. These templates are meant to help keep all instructors roughly on the same page so that when we get to exams, there are few surprises for both instructors and students. As you ll see, this is a skeleton, there is still lots of room for your own take on the material. 3
We ve highlighted the topics that students had the most trouble with. Consequently, we now spend more time on these topics. For integration topics, most students master these just fine, the lower averages simply reflect the shorter time available to practice. 4
Class 7 begins our discussion of differentiation. At the end of class 6, we should pull together aspects of our study of limits that are most relevant to the definition of a derivative. This is a nice time to point out that while continuity is pleasing, the more interesting (and more difficult) questions in calculus deal with functions that are not obviously continuous. Recap these ideas at the beginning of Class 7 and prepare the class for a new fundamental application, the rate of change. Reading: Stewart 2.7-2.8 5
This is one possible way to introduce rates of change and link it to the slope of the tangent line and, hence, to the definition of the derivative towards the beginning of class 7. 6
Not sure I like this one so much. 7
This is a solution and discussion of the previous group problem. Deploy it if you think it is necessary. 8
At the end of class 7, we remind the students of the basic differentiation rules. They ve seen all of them but may not remember the details. The quotient and chain rules are often the ones that require some extra time. Start class 8 with some examples of using the rules in conjunction with one another. The group work on the next slide solidifies this. Reading: Stewart 3.1-3.4 9
Our real goal here is twofold. First, students need to be facile in using these rules in sometimes complex situations. Second, we want our students to connect these rules to our discussion of limits. For that, we ll focus on the power rule as an example. 10
The bulk of the rest of class 8 explores the link between limits and differentiation rules. This is a good time to emphasize the use of the theoretical pieces that underlie calculus. For this example, we should emphasize that there is only one trick and that it follows from a fundamental fact from algebra if a polynomial is zero at a point a then (x a) is a factor of that polynomial. Another technical piece that will come up here is polynomial long division students generally have a dim recollection of the technique, mostly recalling that they don t like it. But, knowing one of the factors ensures that the technique will be successful. 11
This is a hard problem for students at this level, but will stretch some of them. So far, we ve focused on using algebraic techniques to prove things, but this proof has a decidedly geometric flavor. The entire proof is in the text, so encourage students to understand it and present clearly it in their own way. Here s the proof from the book, condensed a bit. With the substitutions in the hint, and letting Δx = h, then Δu = f x + Δx f x, Δv = g x + Δx g x. We can then calculate the three shaded areas: Pink: f x g x + Δx g x Blue: g x f x + Δx f x Purple: f x + Δx f x g x + Δx g x Simplifying and summing these yields the numerator of the limit in question, f x g x + Δx f x g x + g x f x + Δx g x f x + f x + Δx g x + Δx f x g x + Δx g x f x + Δx + f x g x = f x + Δx g x + Δx f x g x. But, it is helpful to keep these separate when dividing by Δx: 12
f x + h g x + h f x g(x) Δ uv lim = lim h 0 h h 0 Δx uδv = lim h 0 Δx + vδu Δx + ΔuΔv Δx = f x g x + g x f x + f x lim h 0 Δv. As the last limit is zero, we have the results. Of course, this glosses over some subtle analysis concerning limit laws, etc. but is sufficient for our purposes. 12
At the end of Class 8, review the chain rule and introduce implicit differentiation as an application. Picking this up at the beginning of Class 9, we ll begin with some standard review problems but should end with harder problems involving using the chain rule in conjunction with other rules. 13
For the review and continuation in the beginning of class 9, it is a good idea to do several different examples as this technique is one students generally have a difficult time with. It is also a great place to introduce this technique as a way to compute derivatives of inverse functions, with the application of the arc-trigonometric functions as well as the logarithm (given the derivative of the exponential function). Possibilities: 1. Find dy dx if x3 + y 4 = 4xy. 2. Find the derivative of sin 1 (x) using the fact that if y = sin 1 (x) then x = sin(y). Reading: Stewart 3.5 14
This is a good problem to discuss at length on the board. Have groups present their answers with an eye towards demonstrating the full process how does the function in #2 break up into components based on derivative rules? How do we approach the sub-problem of finding the derivative of an inverse function? Etc. This could easily take up the rest of class time. 15
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