chapter2

=Chapter 2=

**Homework by Book Section**
 * 2.1 HW: //None//
 * 2.2 HW: #4, 8, 14, 18, 22, 24, 26, 28
 * 2.3 HW: #8, 18, 28, 34, 38, 44, 48, 56, 72, 90, 108, 110
 * 2.4 HW: #4, 10, 20, 30, 36, 40, 46, 50, 56, 60, 66 , 70, 76, 84, 92
 * 2.5 HW: #4, 8, 14, 18, 24, 28, 34, 38, 44, 48, 54, 66


 * Chapter 2 Reveiw Worksheet**
 * [[file:Math 151 Ch 2 Review.pdf]]
 * [[file:Math 151 Chapter 2 Recap (KEY).pdf]]


 * Overall Chapter Thoughts**

This chapter is about laying down the foundation for a new way of thinking about mathematics -- calculus is to algebra like Shakespeare is to the English language. Here we'll learn the first really important definition that will stick with us through the whole term: what is the relationship between the really, really small (i.e., close to zero) and the really, really big (i.e., close to infinite)?


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


 * Materials:**

Activities for Introducing Calculus Concepts


 * Thoughts on Implementation:**

Since very few students have taken calculus before this class (and to what degree of success or memory, I do not know), I find 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 at cutting lawns and calculus is more effective than other mathematical approaches to some problems). We will look at a collection of particular tasks and identify 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). In the end, you just might be genuinely surprised to see such a robust connection between empirical sciences and mathematics.


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


 * Materials:**


 * Thoughts on Implementation:**

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 of "limit" 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. It is a pedagogical pivot point for future sections.


 * Section 2.3 -- Evaluating Limits Analytically:**


 * Materials:**


 * Thoughts on Implementation:**

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:**


 * Materials:**


 * Thoughts on Implementation:**

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:**


 * Materials:**


 * Thoughts on Implementation:**

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 find 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).