Unit+1

=Unit 1 (Recapitulation, Review, and Differential Equations) -- parts of Chapters 5 and 6=


 * ** Blank Unit 1 Notes (Chapter 5 content) **
 * ** Blank Unit 1 Notes (Chapter 6 content) **

**Collected Homework by Book Section (//154 problems//) -- UPDATED 1/23/13 **
 * ** 5.5 (//22 problems//): ** 1, 7, 13, 19, 25, 31, 37, 43, 55, 61, 67, 73, 79, 85, 91, 97, 103, 107, 113, 125, 147, 149
 * ** 5.7 (//16 problems//) ** ** : ** 1, 7, 13, 19, 25, 31, 37, 43, 49, 55, 67, 73, 85, 91, 93, 97
 * **5.8 (//19 problems//): **1, 7, 11, 13, 19, 21, 25, 29, 31, 33, 37, 43, 49, 53, 55, 63, 65, 67, 69
 * ** 5.9 (//25 problems//): ** 1, 7, 9, 15, 19, 21, 25, 29, 31, 37, 39, 43, 47, 51, 55, 57, 59, 61, 63, 67, 69, 73, 77, 81, 83
 * ** 6.1 (//22 problems//): ** 1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 31, 37, 41, 43, 45, 47, 49, 55
 * ** 6.2 (//21 problems//): ** 1, 3, 5, 9, 13, 15, 19, 21, 23, 25, 27, 33, 37, 41, 43, 47, 49, 55, 59, 61, 65
 * ** 6.3 (//21 problems//): ** 1, 7, 9, 13, 17, 21, 23, 25, 29, 31, 33, 35, 41, 45, 47,  49, 53, 55, 59, 63, 67
 * ** 6.4 (//14 problems//): ** 1, 3, 5, 7, 9, 11, 17, 19, 21, 23, 25, 27, 31, 33

I will be collecting this book homework at the time of the Exam for Unit 1, and grading at least one problem from each section (unannounced).

=== **UNIT 1 Homework Forum -- a wide-open Google Doc where students can collaborate, share, and create a useful resource** ===


 * Answers to all odd-numbered questions from our book. **


 * Review of Calculus I Packet --** Collected


 * Solving particular differential equations --** in-class warm up for Chapter 6 concepts

**Overall Unit Thoughts**

This unit is all about reviewing where we've been (in terms of the Calculus) and heading in some interesting, new directions (applications of integration beyond two dimensions). It's the only time during this very robust course where we'll have a chance to really catch our breath and revisit some previously explored topics.

**Section 5.5 -- Integration by Substitution:**

We have a theoretical basis for all sorts of useful applications of the integral, but we need to back that up with some practical muscle (not all integrals are easy). Specifically, we need a way to integrate (i.e., find the antiderivative of) a whole bunch of complicated function types, so developing some tools for integration is really important. This section is all about the most important (for our purposes) tool, known as "u-substitution" or just "integration by substitution." This is the first of many useful integration rules we'll build upon.
 * Thoughts: **

**Section 5.7 -- The Natural Logarithmic Function: Integration**

**Thoughts:** Another in the list of rules we need in order to integrate all the functions we can -- the rule for dealing with integrals involving logarithms. Also, this section introduces (quite connected) rules for integrating trigonometric functions, all of which we will be practicing for the purpose of mastery.

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

**Thoughts:** 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.

**Section 5.8 -- Inverse Trigonometric Functions: Integration**

**Thoughts:** Just as inverse trig functions (a.k.a. arc-trig functions) were some of the last functions we learned how to differentiate in Calculus I, they are also the last major kind of function we'll be learning to integrate. By the time this section is over, you'll have a great array of rules and tools to apply to integration problems, even though we won't be able to tackle integration of //all// functions just yet (but soon!).
 * **Chapter 5 (Calc I) Quiz (up to 5.8)**

**Section 5.9 -- Hyperbolic Functions**

**Thoughts:** There are many kinds of wild functions in the world, most of which never see the light of day outside very advanced physics or engineering contexts. Hyperbolic trigonometric functions are a great example of these, but still easy enough to see the applications for which they are great tools. Just as we can define Sine and Cosine as the //x// and //y// coordinates of a point on the unit circle, we could just as easily talk about some //other// kind of conic section -- and the respective coordinates on a hyperbola are defined as the "**hyperbolic**" Sine and Cosine! Cool, huh?
 * **COMPLETED NOTES FOR SECTION 5.9**
 * **5.9 Quiz**

**Section 6.1 -- Slope Fields and Euler's Method**

**Thoughts:**

Figuring out what to do with a differential equation (i.e., an equation that involves derivatives and the variables involved) requires us to think of them as "maps" for describing how a function behaves. After all, a differential equation is just telling you what the derivative of something (a function) is at any given location; so we can develop "slope fields" (maps of a function's slope by location). Euler developed a particularly ingenious way of determining what the original function must be by using this mapping technique -- we call it Euler's Method for solving differential equations to this day.


 * **Slope Field Calculator** -- an online app for creating the slope field associated with any ordinary differential equation (ODE). It's a Java applet, so you may need to allow it to run.
 * **Euler's Method Spreadsheet** -- Euler was brilliant. His method is terrific. It is also insanely tedious. This is why we invent simple spreadsheet programming. :-) Feel free to play around with this spreadsheet, but try to leave it functional as you found it.
 * ** 6.1 In-Class Quiz 1 (Differential Equations) **
 * **6.1 Slope Field Practice** -- practice problem worksheet assigned in class
 * **6.1 In-Class Quiz 2 (Euler's Method)**

**Section 6.2 -- Growth and Decay**

**Thoughts:**

Sounds like the name of some pretentious novel, but it's actually a major topic in the study of differential equations. Considering the fact that "change" is the underlying principle of all things related to the derivative, two logical topics for application are growth and decay -- change that we see (and try to control) in our world. Building on our knowledge of implicit differentiation, we can do some lovely things with differential equations -- we just need to figure out a more general way of solving them than the straightforward integration method. Enter the notion of "separation of variables," and we discover a world of solutions at our fingertips.


 * **6.2 In-Class Quiz**

**Section 6.3 -- Differential Equations: Separation of Variables**

**Thoughts:** Though we will only be dipping our toes into the waters of solving differential equations, this section provides some hefty tools for being able to solve them. Recall, a differential equation can be as simple as a "derivative" from our previous Calc I experience, or as complex as an equation that involves first and second derivatives and multiple variables. For this section, it's all about finding ways to separate variables so that one side of the equation is integrable over one variable, while the other side is integrable over the other. Sometimes this means we have to use "tricks," such as when the differential equation is "homogeneous."
 * **6.3 At-Home Quiz (to be completed before class)**

**Section 6.4 -- The Logistic Equation**

**Thoughts:** One of the more useful and applicable (but heretofore unwieldy ) curves you will run across is the Logistic Curve. Its output always falls between zero and some "upper limit," which makes it terrific for applications where there are constraints on a system (e.g., carrying capacity for an ecosystem). Recognizing the parts of a logistic curve and how the solution is arrived at via separation of variables is an important, albeit theoretical, point in this section.
 * **6.4 In-Class Quiz**