Chapter+2

= Chapter 2 =


 * Section 2.1 -- A Preview of Calculus:**

-- 2.1 Notes, version 1 Activities for Introducing Calculus Concepts
 * Materials:**


 * Thoughts on Implementation:**

I just talked about ideas and concepts in class - AJ

//SFT SAYS:// As did I. However, I think I stumbled upon a nice approach that helped bridge some gaps for some folks. Since about one quarter of my students have taken calculus before (and to what degree of success or memory, I do not know), I found it important to explain both the need for and context surrounding (and resulting from) the invention of calculus. The punchline of this brief history amounts to this: when we need to perform certain tasks (whether mowing a lawn or finding the instantaneous velocity of an accelerating object), some tools are better than others (to wit, lawn mowers are more effective than scissors and calculus is more effective than other algebraic methods). We looked at a collection of particular tasks and identified the need for a more effective method to solve/describe them (e.g., finding the tangent to a complicated curve, explaining the motion of a rocket that is constantly losing mass while its engine fires, locating the center of mass of a multi-body system). By the end, a few more heads were nodding than before and a few folks seemed genuinely surprised to see such a robust connection between empirical sciences and mathematics.


 * Section 2.2 -- Finding Limits Graphically and Numerically:**

-- 2.2 Notes, version 1
 * Materials:**


 * Thoughts on Implementation:**

I think it's important not to get too bogged down in this section, it's not as difficult a concept for them to understand. - AJT

//SFT SAYS:// I actually disagree. I think the definition portion of this section is absolutely critical. It is what the entire chapter builds upon. We start here, defining a limit in a particular way -- therefore, we're able to examine particular situations and decide whether or not a limit exists based on that definition. Later sections see added nuance to the definition of limit (i.e., limits of "twin" functions, one-sided limits, limits that approach infinite magnitudes), but we always fall back on the definition for helping determine what counts and what does not. So, foundationally, it's worth spending some time on -- modeling the use of the definition of limit along the way. So, maybe we don't technically "disagree" about this section so much as we differ on the degree of emphasis we place on it as a pedagogical pivot point for future sections.


 * Section 2.3 -- Evaluating Limits Analytically:**

-- 2.3 Notes, version 1
 * Materials:**


 * Thoughts on Implementation:**

Make sure to do an example like #108 in the homework as it's very tricky for them. - AJT

//SFT SAYS:// Agreed - some more story problems would probably be in order here. More practice with application of limits would be good, but it's mostly gruntwork at this point. It might make sense to do some real cross-over with kinematics problems from physics (i.e., like 108 and 110 from the homework that basically ask students to calculate instantaneous velocity of an accelerating body.


 * Section 2.4 -- Continuity and One-Sided Limits:**

-- 2.4 Notes, version 1
 * Materials:**


 * Thoughts on Implementation:**

The notes have too many examples... this section shouldn't take nearly as long as it took me in class. - AJT

//SFT SAYS:// Agreed, yet, students found the extensive examples to be quite helpful. Perhaps just merging 2.4 and 2.5 together so that sided-limits are introduced at the same time as limits approaching +/- infinity would make all of these examples worth doing collectively.


 * Section 2.5 -- Infinite Limits:**

-- 2.5 Notes, version 1
 * Materials:**


 * Thoughts on Implementation:**

//SFT SAYS:// If the concept of a sided-limit that approaches infinity (i.e., is unbounded) is introduced earlier (i.e., when we run into asymptotes earlier in the chapter), then this section becomes more an "additional practice" sort of thing. I found it very easy to "expand" on the definition of the limit to include a "value" of infinity. For the purposes of this introductory course, the key factor here (it seems to me) is to have students practice the skill set of evaluating limits and making use of the expanded definition of limit (i.e., both sides approach the same "value," even if that value is positive or negative infinity) to recognize when one exists, when one does not, and when there is no limit (when the function is unbounded in the same direction).