Chapter+5

=Chapter 5=



**Homework by Book Section** (// 150 total problems **due on the day of Exam 4** //): //use **Wolfram Alpha**// //to check work//
 * **5.1** (#1 - 47 every odd, #65 - 85 e/o odd) //**-**// //**30**// **//problems//**
 * 1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29, 31, 33, 35, 37, 39, 41, 43, 45, 47, 65, 69, 73, 77, 81, 85
 * **5.2** (#23 - 29 every odd) //**-** **4 problems**//
 * 23, 25, 27, 29
 * **5.3** (#15 - 43 every odd) //**- 15 problems**//
 * 15, 17, 19, 21, 23, 25, 27, 29, 31, 33, 35, 37, 39, 41, 43
 * **5.4** (#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
 * **5.5** (#1 - 121 e/o odd, #145, 149) //**-** **33 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, 145, 149
 * **5.6** (--) //**-** **0 problems**//
 * N/A
 * **5.7** (#1 - 93 e/o odd) //**-** **24 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
 * **5.8** (#1 - 65 e/o odd) //**-** **17 problems**//
 * 1, 5, 9, 13, 17, 21, 25, 29, 33, 37, 41, 45, 49, 53, 57, 61, 65
 * **7.1** (#1 - 31 every odd, #33 - 45 e/o odd) //**-** **20 problems**// **OMITTED DUE TO TIME CONSTRAINTS**
 * 1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29, 31, 33, 37, 41, 45


 * Overall Chapter Thoughts**

This chapter introduces us to the "reverse direction" of the derivative machine -- the antiderivative -- and some of the consequences of thinking of differentiation in reverse (i.e., going from a function's derivative to somehow evaluate the function itself). This process is intimately connected with concepts of sums of areas and limit (you're not surprised there, I'm sure), and those connections made in the chapter make for some of the most inclusive narratives in all of computational mathematics. After all, whenever you have a theorem that goes by the name "The Fundamental Theorem" of something-or-other, you know you've got some gigantic concepts at play. [//As an aside, most folks forget that they've already learned one of those "fundamental theorems" before, but it's hard to get out of pre-calculus without having messed around with the **Fundamental Theorem of Algebra**. That's where, basically, you know that any polynomial has a number of roots equal to its degree -- even though those might be complex roots.//] This chapter wraps up our course's initial survey of the Calculus, and provides a jumping-off point for further courses and applications. At this point, we've covered the basics of what's known as "differential calculus" (i.e., the calculus involving thinking about how things change) and have just grazed the surface of "integral calculus" (i.e., the calculus involving finding area and sums). You'll be amazed at how much content and how many processes you've learned over such a short amount of time (and how much more there is to this subject)!


 * Section 5.1 -- Antiderivatives and Indefinite Integration:**


 * Materials:**
 * [[file:Math 151 5.1 Worksheet.pdf]]
 * [[file:Math 151 5.1 Worksheet (KEY).pdf]]
 * **Helpful video introducing antiderivatives (i.e., "indefinite integrals")**
 * **Another helpful video of examples of indefinite integrals**
 * **Examples from notes**
 * **[|5.1_notes_Ex1];**
 * **5.1_notes_Ex3_1;**
 * **5.1_notes_Ex3_2;**
 * **5.1_notes_Ex4_1;**
 * **5.1_notes_Ex4_2;**
 * **5.1_notes_Topic4;**
 * **5.1_notes_Ex5;**
 * **5.1_notes_Ex6**
 * **5.1_notes_Ex8a;**
 * **5.1_notes_Ex8b;**
 * **5.1_notes_Ex9a;**
 * **5.1_notes_Ex9b;**
 * **5.1_notes_Ex9c**

This section is the first big hurdle that folks struggle with in this section -- making sense of the "backward" direction of integration. You'd think it would be easy, what with how much time we've spent already on differentiation, to just go backwards and work all the rules in reverse; but it proves to be tricky mental acrobatics that require plenty of time and practice. We start by thinking about the "differentials" we ended last chapter on, and then extending them to invent a new process -- anti-differentiation. Specifically, we create a new nomenclature for this sort of "backward derivative" process, and call it the "indefinite integral."
 * Thoughts on Implementation:**


 * Section 5.2 -- Area:**


 * Materials:**
 * [[file:Math 151 5.2-3 Worksheet.pdf]]
 * [[file:Math 151 5.2-3 Worksheet (KEY).pdf]]
 * **5.2_notes_1;**
 * **Connecting the definite integral to area under the curve**
 * **Examples of how definite integrals represent area**
 * **Making use of the Fundamental Theorem of Calculus**

This section and the next quickly become more complicated and tedious than they need to for an introductory calculus course. Therefore, we take them both sort of at the same time. This section introduces us to the idea that, if you use familiar shapes for which we know area formulas (e.g., triangles, circles, rectangles), we can approximate the area under any curve by estimation with our known shapes. Specifically, if you use successively narrower groups of rectangles, you can get some pretty accurate estimates (upper and lower estimates) for the area between any function's curve and the //x//-axis.
 * Thoughts on Implementation:**


 * Section 5.3 -- Riemann Sums and Definite Integrals:**


 * Materials:**
 * [[file:Math 151 5.2-3 Worksheet.pdf]]
 * [[file:Math 151 5.2-3 Worksheet (KEY).pdf]]
 * **5.2_notes_1;**
 * **Connecting the definite integral to area under the curve**
 * **Examples of how definite integrals represent area**
 * **Making use of the Fundamental Theorem of Calculus**

Building off the ideas of the last section, we can improve and formalize our area estimation process more and more. Eventually, this process of successive improvements by using successively narrower rectangles leads to the use of the limit concept (remember that one?) to create one of the most interesting connections in all of mathematics -- the area under a curve is intimately linked with the antiderivative of that curve! We won't spend much time on old Riemann or some of the more subtly technical constructs, but the main idea here is very important.
 * Thoughts on Implementation:**


 * Section 5.4 -- The Fundamental Theorem of Calculus:**


 * Materials:**
 * [[file:Math 151 5.4 Worksheet.pdf]]
 * [[file:Math 151 5.4 Worksheet (KEY).pdf]]
 * **Evidence that students actually work on calculus: here, here, and here.**
 * **5.4 Notes, Ex2-1**
 * **5.4 Notes, Ex2-2**
 * **5.4 Notes, Ex2-3**
 * **5.4 Notes, Ex5**
 * **5.4 Notes, Ex6**
 * **5.4 Notes, Ex7**
 * **5.4 Notes, Ex8**

Well, here it is. The Big Idea for the chapter comes down to this: evaluate a function's antiderivative over a particular interval, and you have the area between that function's curve and the //x//-axis over that interval. Granted, it's going to take us a bit more time to spell that out and see all the things we can do with it (e.g., Mean Value Theorem for Integrals, finding the average value of a function), but that big idea actually ties together all the main concepts we've studied this term!
 * Thoughts on Implementation:**


 * Section 5.5 -- Integration by Substitution:**


 * Materials:**
 * [[file:Math 151 5.5 Worksheet.pdf]]
 * [[file:Math 151 5.5 Worksheet (KEY).pdf]]
 * **[|5.5 notes, Ex1]**
 * **5.5 notes, Ex2b**
 * **[|5.5 notes, Ex2b]**
 * **5.5 notes, Ex3b**
 * **5.5 notes, Ex5a**
 * **5.5 notes, Ex5c**
 * **5.5 notes, Ex5d**
 * **5.5 notes, Ex5e**
 * **5.5 notes, Ex5f**
 * **5.5 notes, Ex6a**
 * **5.5 notes, Ex6b**
 * **5.5 notes, Ex8**
 * **5.5 notes, Ex9a**
 * **5.5 notes, Ex9b**

Now that we have a theoretical basis for all sorts of useful applications, it's time to back that up with some practical muscle. 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." Just as we developed a series of useful differentiation rules in Chapter 3, this is the first of many useful integration rules we'll build upon.
 * Thoughts on Implementation:**


 * Section 5.6 -- Numerical Integration:**


 * Materials:**
 * A great site that let's you investigate **Simpson's Rule** (one of the main ideas in this section) to see how it works.

Mathematicians are quite fond of developing theoretical constructs and then playing with them to find as many interesting applications and implications as possible. This section develops a few of those implications, especially useful for integrating some stubbornly complicated functions. Even so, these ideas are less useful for our needs since we'll be developing other rules that can take their place very soon. Therefore, we will be skimming over this section very quickly (effectively skipping it in terms of homework and testing).
 * Thoughts on Implementation:**


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


 * Materials:**
 * [[file:Math 151 5.7 Worksheet.pdf]]
 * **[|5.7 notes, Ex1]**
 * **[|5.7 notes, Ex2]**
 * **[|5.7 notes, Ex3]**
 * **[|5.7 notes, Ex4]**
 * **[|5.7 notes, Ex5]**

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 with for the purpose of mastery.
 * Thoughts on Implementation:**


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


 * Materials:**

Just as inverse trig functions (a.k.a. arc-trig functions) were some of the last functions we learned how to differentiate in Chapter 3, 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 (that requires later Calculus courses).
 * Thoughts on Implementation:**


 * Section 7.1 -- Area of A Region Between Two Curves:**


 * Materials:**

It's really a tacked-on application section for what we've developed in Chapter 5, but this section allows us to extend our abilities of calculating area into more useful realms. If we can find the area between one function and an axis (itself a kind of "curve," though admittedly a very simple one), what's to stop us from finding the area between any two curves? The answer: only a bit of reasoning and some practice with abstract thinking. This is usually the starting point for a second-semester Calculus course, but we'll end our term on this application-based note.
 * Thoughts on Implementation:**