Chapter+4

=Chapter 4=



**Homework by Book Section** (//130 total problems **due on the day of Exam 3** //): //use **Wolfram Alpha**// //to check work//
 * **4.1** (#1 - 65 e/o odd, #67 - 73 every odd) //**-**// **//21 problems//**
 * 1, 5, 9, 13, 17, 21, 25, 29, 33, 37, 41, 45, 49, 53, 57, 61, 65, 67, 69, 71, 73
 * **4.2** (#1 - 61 e/o odd, #63 - 67 every odd) //**-** **19 problems**//
 * 1, 5, 9, 13, 17, 21, 25, 29, 33, 37, 41, 45, 49, 53, 57, 61, 63, 65, 67
 * **4.3** (#1 - 105 e/o odd) //**- 27 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
 * **4.4** (#1 - 85 e/o odd) //**- 22 problems**//
 * 1, 5, 9, 13, 17, 21, 25, 29, 33, 37, 41, 45, 49, 53, 57, 61, 65, 69, 73, 77, 81, 85
 * **4.5** (#1 - 93 e/o odd, #95 - 105 every odd) //**-** **30 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, 95, 97, 99, 101, 103, 105
 * **4.6** (--) //**-** **0 problems**//
 * N/A
 * **4.7** (#1 - 41 e/o odd) //**-** **11 problems**//
 * 1, 5, 9, 13, 17, 21, 25, 29, 33, 37, 41


 * Practice Problems on various Chapter 4 Topics**


 * Overall Chapter Thoughts**

This chapter is all about applying the derivative concept in some important and useful ways. We'll start by analyzing graphs of functions in order to do some interesting things. We can use the derivative of a function to determine the function's maximum and minimum values (i.e., highest and lowest points) as well as its general behavior over large intervals. We can use our knowledge of limits and derivatives to identify a function's asymptotes. Combined with a strong pre-calculus understanding of general function types, horizontal and vertical stretches and shifts, we can quickly and easily sketch a graph of practically any function as a result of examining its analytic expression (i.e., it's equation). Also, we can talk about those graphs and their shapes (e.g., concavity, inflection) in predictable ways by applying the derivative operator. Finally, we'll use our mastery of differentiation to solve complex optimization problems in a variety of contexts (e.g., what dimensions of a soup can maximize the volume with the smallest amount of metal used?).


 * Section 4.1 -- Extrema on an Interval:**


 * Materials:**
 * [[file:Math 151 4.1 Worksheet.pdf]]
 * [[file:Math 151 4.1 Worksheet (KEY).pdf]]
 * Working through solution for **4.1 worksheet problem #4**
 * An example of a definition that (even though it makes no sense) can be used in a mathematical way to test something: **the Blarthog **
 * A quick concept-based **quiz for 4.1 (and a little of 4.2)**

In this section, we add more "jargon" to our vocabulary and begin to talk about the "extrema" of functions (maximum and minimum points over specified intervals). In order to do this, we talk about distinctions between open and closed intervals, as well as relative and absolute extrema. Specifically, we spend time defining relative extrema (potentially any point //within// //an interval//, not including the endpoints) as well as absolute extrema (potentially any point //on an interval//, including the endpoints). This careful use of definitions is what mathematics is all about -- we can reason **deductively** only when we base our reasoning on well-defined principles, theorems, and axioms.
 * Thoughts on Implementation:**


 * Section 4.2 -- Rolle's Theorem and the Mean Value Theorem:**


 * Materials:**
 * [[file:Math 151 4.2 Worksheet.pdf]]
 * [[file:Math 151 4.2 Worksheet (KEY).pdf]]
 * **Example 1 from the 4.2 notes**
 * **Other Examples from 4.2 notes (4.2_notes_1.png; 4.2_notes_2.png; 4.2_notes_3.png)**

Based on our new-found skills -- i.e., the ability to find extrema on intervals -- we are able to generalize about the behavior of specific kinds of functions (Rolle's Theorem) and even globally about all functions (Mean Value Theorem). These theorems become important tools for being able to work many kinds of problems, especially some interesting applications in the everyday world (e.g., how can the cops PROVE you were speeding?).
 * Thoughts on Implementation:**


 * Section 4.3 -- Increasing and Decreasing Functions and the First Derivative Test:**


 * Materials:**
 * [[file:Math 151 4.3 Worksheet.pdf]]
 * [[file:Math 151 4.3 Worksheet (KEY).pdf]]
 * **Examples from 4.3 notes ([|4.3_notes_1]; [|4.3_notes_2]; [|4.3_notes_3];** **4.3_notes_4****)**
 * **Quick quiz on 4.3 and other sundries**

Though we've already hinted at the idea of increasing and decreasing as ways of categorizing the behavior of functions, this section formalizes the way we talk about it. In addition, we develop a new tool for locating and identifying relative extrema of a function. These concepts don't seem like a big deal at first, but there are //many// applications for them in all fields of quantitative science.
 * Thoughts on Implementation:**


 * Section 4.4 -- Concavity and the Second Derivative Test:**


 * Materials:**
 * [[file:Math 151 4.4 Worksheet.pdf]]
 * [[file:Math 151 4.4 Worksheet (KEY).pdf]]
 * **Images of student work on the 4.4 Worksheet**
 * **Examples from 4.4 notes (4.4_notes_2; 4.4_notes_3)**
 * **Quick quiz on 4.4**

Just as we found that the first derivative of a function told us important things about how the function behaves over any given interval, we can also learn a great deal about a function by looking at its //second// derivative. Specifically, the concept of concavity (e.g., how a parabola can be "happy" or "sad," depending on which direction the curve goes) affords a great deal of useful applications. And, just as before, we can develop a test making use of the properties of the second derivative.
 * Thoughts on Implementation:**


 * Section 4.5 -- Limits at Infinity:**


 * Materials:**
 * [[file:Math 151 4.5 Worksheet.pdf]]
 * [[file:Math 151 4.5 Worksheet (KEY).pdf]]
 * **Examples from the notes (****4.5_notes_1)**

I always thought the title for this section would make a great rock band name. Regardless, this section is all about applying what we know about function behavior and limits to better understand and predict the existence of horizontal asymptotes, as well as to explain, in general, how functions behave when we imagine letting input (//x//-) values get enormously positive or negative. There are many applications for using mathematical models and extending them to make predictions, and we also get a chance to play around with the always-entertaining concepts of **infinity** and **zero**.
 * Thoughts on Implementation:**


 * Section 4.6 -- A Summary of Curve Sketching:**


 * Materials:**
 * An **interesting site** on curve sketching that walks you through all the techniques we've learned so far.

In the "old days," before we had fancy graphing calculators or wonderful computer algebra systems (like **Wolfram Alpha**), it was REALLY hard to make reasonably accurate sketches of functions that were much more complicated than the standard general types (e.g., squared, cubed, square-root, etc.). Armed with all the additional information we can gather by using the first and second derivative, as well as being able to locate all horizontal asymptotes, we are able to create amazingly accurate graph sketches without resorting to the use of calculators or computers. Of course, the //reasonable// reader will now ask that most important (and frustrating) of all questions, "//why would we not use the technology if we have it?//" Good question, dear reader (and one that has **occupied lots of time and energy** in the mathematics education world for years)! My answer is simple -- we //will//, but knowing how to do such a thing is not a skill one should forgo based merely on the existence of alternative solution methods. The same argument applies here as it does to modes of transportation; we don't always have a helicopter handy, so sometimes it's good to make use of other methods (e.g., recall how to walk). ;-) Therefore, we'll be briefly surveying this section without homework.
 * Thoughts on Implementation:**


 * Section 4.7 -- Optimization Problems:**


 * Materials:**
 * [[file:Math 151 4.7 Worksheet.pdf]]
 * [[file:Math 151 4.7 Worksheet (KEY).pdf]]
 * **Examples from 4.7 Notes (4.7_notes_Ex3; 4.7_Notes_Ex4_1;** **4.7_Notes_Ex4_2; 4.7_notes_Ex8; 4.7_notes_Ex10)**
 * **Setting up worksheet problems (****4.7_Worksheet_#4;** **4.7_Worksheet_#6)**

Just like every good story has to have a villain, every good mathematics textbook chapter has to have that section that's all about story problems. Well, that's this section, though these are some really cool problems (admittedly, this is coming from a self-avowed nerd) about a variety of max/min situations (situations involving the need to maximize or minimize one aspect with respect to others). From figuring out how to maximize the volume of a cardboard box while using the least amount of cardboard to make it, to finding the minimum distance at which you need to place a camera to capture an entire painting in-frame; these problems may seem quite different at first, but their solutions tend to boil down to the same Calculus-based patterns and techniques.
 * Thoughts on Implementation:**


 * Section 4.8 -- Differentials:**


 * Materials:**
 * [[file:Math 151 4.8 Worksheet.pdf]]

This section is the last in our course's pass through the territory of derivative applications (though there are many more applications for that useful operator to be learned in subsequent Calculus and science courses). It also serves as a nice bridge into our next (and final) thread for the course. It stems from the application of the original tangent-line-based definition of the derivative and builds on the way we talk about and use the differential (e.g., //dy/dx//) compared to the previously familiar concept of "slope." This has some REALLY important applications in the measurement-intensive world of physics, chemistry, and the many sub-disciplines of engineering, and quite interestingly connects to the very way in which we view the //nature of knowledge// (often called "//**epistemology**//").
 * Thoughts on Implementation:**