Chapter+3

=Chapter 3=



**Homework by Book Section** (//180 total problems **due on the day of Exam 2, part 2** //)
 * **3.1** (#1 - 101 e/o odd) //**-**// **//26 problems//**
 * 1, 5, 9, 13, 17, 21, 25, 29, 33, 37, 41, 45, 49, 53, 57, 61, 65, 69, 73, 77, 81, 85, 89, 93, 97, 101
 * **3.2** (#1 - 109 e/o odd) //**-** **28 problems**//
 * 1, 5, 9, 13, 17, 21, 25, 29, 33, 37, 41, 45, 49, 53, 57, 61, 65, 69, 73, 77, 81, 85, 89, 93, 97, 101, 105, 109
 * **3.3** (#1 - 139 e/o odd) //**- 35 problems**//
 * 1, 5, 9, 13, 17, 21, 25, 29, 33, 37, 41, 45, 49, 53, 57, 61, 65, 69, 73, 77, 81, 85, 89, 93, 97, 101, 105, 109, 113, 117, 121, 125, 129, 133, 137
 * **3.4** (#1 - 169 e/o odd) //**- 43 problems**//
 * 1, 5, 9, 13, 17, 21, 25, 29, 33, 37, 41, 45, 49, 53, 57, 61, 65, 69, 73, 77, 81, 85, 89, 93, 97, 101, 105, 109, 113, 117, 121, 125, 129, 133, 137, 141, 145, 149, 153, 157, 161, 165, 169
 * **3.5** (#1 - 89 e/o odd) //**-** **23 problems**//
 * 1, 5, 9, 13, 17, 21, 25, 29, 33, 37, 41, 45, 49, 53, 57, 61, 65, 69, 73, 77, 81, 85, 89
 * **3.6** (#11, 13, 19 – 51 e/o odd, 65, 67, 69) //**-** **14 problems**//
 * 11, //**13**//, 19, 23, 27, 31, 35, 39, 43, 47, 51, 65, 67, 69 ** (UPDATED: 3/1/10) **
 * **3.7** (#1 – 43 e/o odd) //**-** **11** **problems**//
 * 1, 5, 9, 13, 17, 21, 25, 29, 33, 37, 41


 * In-Class "Pop" Quiz** - A screen capture of the in-class surprise quiz we used as a review of some major concepts.


 * Overall Chapter Thoughts**

This chapter is all about extending the concept of "limit" into some new, and wonderfully useful territory. By starting with the limit of the difference quotient as a means for solving the problem of "what happens to a secant line when the two points become infinitely close together," we will develop the concept of "derivative" and a grammar for using it in concert with the algebraic, graphical, numerical/tabular, and linguistic modes of functions of all types.


 * Section 3.1 -- The Derivative and the Tangent Line Problem:**


 * Materials:**
 * [[file:Math 151 3.1 worksheet.pdf]]
 * [[file:Math 151 3.1 Worksheet KEY.pdf]]

This section is critical in the development of the overall "narrative" of the calculus course. This is the pivot point where we make use of our new equipment (i.e., the concept of limit and continuity of functions) to solve the problem of finding instantaneous rates of change (i.e., the derivative). Unfortunately, using the limit definition of derivative is often seen as unnecessary once you know the "short cuts" (e.g., the power rule), which tends to make this section one of the most reviled by students -- yet, it is precisely the limit-based definition of the derivative that allows for a continuous development throughout the course (and beyond). Time and practice are critical components of the recipe for developing understanding here.
 * Thoughts on Implementation:**


 * Section 3.2 -- Basic Differentiation Rules and Rates of Change:**


 * Materials:**
 * [[file:Math 151 3.2 Worksheet.pdf]]
 * [[file:Math 151 3.2 Worksheet KEY.pdf]]

While this section is relatively brief (in terms of the theory), it covers loads of ground in terms of applications. Here we learn all about the most common and important "short cuts" for finding derivatives of some of the most common functions: polynomial, exponential, and trigonometric functions. Many rules to learn and practice.
 * Thoughts on Implementation:**


 * Section 3.3 -- Product and Quotient Rules and Higher-Order Derivatives:**


 * Materials:**
 * [[file:Math 151 3.3 Worksheet.pdf]]
 * [[file:Math 151 3.3 Worksheet KEY.pdf]]

Much like the previous section, we're busy expanding our repertoire here -- figuring out how to differentiate (the verb form of 'derivative') more complex functions (e.g., functions that are the product or quotient of two familiar types). More rules to learn and practice...
 * Thoughts on Implementation:**


 * Section 3.4 -- The Chain Rule:**


 * Materials:**
 * [[file:Math 151 3.4 Worksheet.pdf]]
 * [[file:Math 151 3.4 Worksheet KEY.pdf]]
 * Screen capture of **3.4 in-class board work**

This section introduces probably the most important rule for differentiation, the chain rule. With this rule, we can now differentiate practically (with a few notable exceptions) any function -- a powerful claim, indeed! However, the rule tends to be misunderstood until practiced extensively, and there are several variations of the rule that appear different only because of different notation (e.g., function notation vs. differential notation).
 * Thoughts on Implementation:**


 * Section 3.5 -- Implicit Differentiation:**


 * Materials:**
 * [[file:Math 151 3.5 Worksheet.pdf]]
 * [[file:Math 151 3.5 HW Quiz KEY.pdf]]
 * **A brief diagrammatical explanation for the need for implicit differentiation**
 * Screen capture of **3.5 in-class board work**

This section take a bit of a curve into some interesting territory; namely, what happens if we want to differentiate things that are not functions (i.e., more than one output for some inputs). At first this might seem contradictory, since we did spend all of our time up to this point working specifically with functions, but this section helps show how the derivative (as a mathematical operator) can be applied to all types of expressions -- not just functions. Some interesting applications result.
 * Thoughts on Implementation:**


 * Section 3.6 -- Derivatives of Inverse Functions:**


 * Materials:**
 * [[file:Math 151 3.6 Worksheet.pdf]]
 * [[file:Math 151 3.6 Worksheet KEY.pdf]]

While not a major focus for our course, this section introduces the rules for differentiating inverse trigonometric functions, as well as some useful strategies for dealing with inverse functions overall.
 * Thoughts on Implementation:**


 * Section 3.7 -- Related Rates:**


 * Materials:**
 * [[file:Math 151 3.7 Worksheet.pdf]]
 * [[file:Math 151 3.7 Worksheet KEY.pdf]]
 * **Related Rates Practice Problems** - some extra problems I put together to use in class and for practice
 * Screen capture of **3.7 in-class board work**

One of the most important applications of the derivative for beginning calculus courses is the related-rate type problem. These are all premised on the calculus-based realization that, if variables can change in value over time, we can think about each variable in terms of its rate of change as well as in terms of specific values at particular points in time. The set-up for some of these problems can appear brutal at first, but they all tend to conform to the same basic format (e.g., what static equations do we know, and what happens when we differentiate those with respect to a particular variable such as time?).
 * Thoughts on Implementation:**


 * Section 3.8 -- Newton's Method:**


 * Materials:**
 * [[file:Math 151 3.8 Worksheet.pdf]]
 * [[file:Math 151 3.8 Worksheet KEY.pdf]]

For the purposes of time, and given that this is a survey course, we will likely skip this section. While Newton's method for approximation of zeros is interesting (a proto-computer program!) from an historical perspective, it has become less useful for contemporary students of the calculus. However, I still strongly recommend that folks interested in going on to further mathematical or computer-related fields spend some time familiarizing themselves with this crafty technique, as it is a prime example of human ingenuity and the value of mathematical tools for solving highly complex problems.
 * Thoughts on Implementation:**