CK-12 F OUNDATION CK-12 Calculus Teacher’s Edition Dreyfuss Narasimhan Prolo CK-12 Foundation is a non-profit organization with a mission to reduce the cost of textbook materials for the K-12 market both in the U.S. and worldwide. Using an open-content, web-based collaborative model termed the “FlexBook,” CK-12 intends to pioneer the generation and distribution of high-quality educationalcontentthatwillservebothascoretextaswellasprovideanadaptiveenvironmentforlearning, powered through the FlexBook Platform™. Copyright © 2011 CK-12 Foundation, www.ck12.org Except as otherwise noted, all CK-12 Content (including CK-12 Curriculum Material) is made available to Users in accordance with the Creative Commons Attribution/Non-Commercial/Share Alike 3.0 Un- ported (CC-by-NC-SA) License (http://creativecommons.org/licenses/by-nc-sa/3.0/), as amended and updated by Creative Commons from time to time (the “CC License”), which is incorporated herein by this reference. 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Printed: June 28, 2011 Authors Andrew Dreyfuss, Ramesh Narasimhan, Jared Prolo i www.ck12.org Contents 1 Calculus TE - Teaching Tips 1 1.1 Calculus TE Teaching Tips . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 2 Calculus TE - Common Errors 27 2.1 Functions, Limits, and Continuity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 2.2 Differentiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 2.3 Applications of Derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 2.4 Integration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 2.5 Applications of Integration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 2.6 Transcendental Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66 2.7 Integration Techniques . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71 2.8 Infinite Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76 3 Calculus TE - Enrichment 84 3.1 Functions, Limits, and Continuity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84 3.2 Differentiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91 3.3 Applications of Derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97 3.4 Integration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102 3.5 Applications of Definite Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108 3.6 Transcendental Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113 3.7 Integration Techniques . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117 3.8 Infinite Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121 4 Calculus TE - Differentiated Instruction 127 4.1 Functions, Limits, and Continuity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127 4.2 Derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140 4.3 Applications of Derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151 4.4 Integration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160 www.ck12.org ii 4.5 Applications of Definite Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168 4.6 Transcendental Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177 4.7 Integration Techniques . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182 4.8 Infinite Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191 5 Calculus TE - Problem Solving 201 5.1 Functions, Limits, and Continuity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 204 5.2 Differentiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211 5.3 Applications of Derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217 5.4 Integration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 225 5.5 Applications of Integration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 229 5.6 Transcendental Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 235 5.7 Integration Techniques . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 237 5.8 Infinite Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 244 iii www.ck12.org www.ck12.org iv Chapter 1 Calculus TE - Teaching Tips 1.1 Calculus TE Teaching Tips This Calculus Teaching Tips FlexBook is one of seven Teacher’s Edition FlexBooks that accompany the CK-12 Foundation’s Calculus Student Edition. To receive information regarding upcoming FlexBooks or to receive the available Assessment and Solution Key FlexBooks for this program please write to us at [email protected]. Lesson 1: Equations and Graphs It is almost cliché how math courses start out with a review of material from previous years. Students are out of practice and never seem to have either been taught, or don’t remember what has happened in previous classes (and will always claim to have not been taught it if they don’t remember). There are two considerations here as the calculus course starts. First, a complete calculus course is a full years worth of university material. This means that the course is conducted at a faster pace than high school students are used to. Compounding the problem for many classes is the even shorter year with the AP examination. Therefore, it is dangerous to get bogged down in the preliminaries. However, a strong case can be made that not much can be accomplished in a calculus class without a firm groundinginthefundamentalspresentedhere. Tohaveaconceptualunderstandingoffunctionsandgraphs is essential to gaining mastery of the basis for the limit, derivative and integral. In case of limited time, the key idea that needs to be driven home is how the relationship between the two variables creates a graph, and what the line means. The way that limits, derivatives and integrals are presented in a first course of calculus is all graphical. If students do not understand what they are looking at when the text later talks about zooming in on an area, strictly increasing or looking at activity at a minimum or maximum, to name a few examples, the key concepts will be lost. Graphing calculators can be valuable tools at this point, especially as they allow for fast manipulation of accurate graphs. There is some danger in relying too much on the graphing calculator, however. I have observed students who have done all of their graphing since linear function on graphing calculators and they end up with some peculiar habits. The most noticeable of which is losing track of the activity of a function outside of the domain graphed, lack of understanding of what happens near vertical asymptotes (the calculator often shows a continuous line), and an over reliance on guess and check methods, especially when the student gets to the chapter on extrema. Use the graphing tool to illustrate some key concepts quickly,checkworkdonebyhand,andusesomeofthecalculationtoolsthatmaybeusefulontheuniversity 1 www.ck12.org examination of choice, but make sure everything could theoretically be done by hand. Lesson 2: Relations and Functions Whileitmayseemlikeanissueofsemantics, Iencouragemystudentstouse, andtrytouseexactterminol- ogy when talking about mathematical relationships. The terms “expression”, “equation”, and “function” all have specific meaning. Students will often confuse them, or believe they can be used interchangeably. Knowing the difference pays off later in sections on inverse and transcendental functions. It is also useful when it comes to writing clear solutions, especially those with prose attached, because they author can then be absolutely clear, presenting work in an easy to follow manner. There is some inconsistency in the way students are taught to express intervals; the topic is pertinent here in expressing the domain and range of functions. The text uses mostly the inequality notation to state which numbers act as endpoints for each variable. Another option is to use the strict set notation with the parenthesis for not inclusive intervals and brackets for inclusive intervals, with the union set operator to join discreet intervals. Example: D = {−3 < x ≤ 0,1 ≤ x < 2} = (−3,0]∪[1,2) There are also the standard sets that have defined bold-face letters: R = Real numbers, Q = Rational numbers, Z = Integers, N = Natural numbers. None of this is important to drive home to students except for the fact that a textbook, or instructor, often chooses one notation method and sticks to it. Different texts and classes may have different notations so students should be at least aware of the different choices. Speaking of notation, the different forms for writing the operation of composition for functions is a source of potential confusion for many students. The operator: (f ◦g)(x) tends to cause all kinds of problems. First, it looks like even more of a product than a single function. Second, we do everything left to right, but the action here is more right to left, made even worse by the fact that composition is one of the few non communicative operations that students have yet come across. Please use, and have students convert to, the nested notation, where the previously mentioned operation is equivalent to f(g(x)). This is clearer because the function g(x) is placed into the function f as if it were the variable, just like the composition is written. Lesson 3: Models and Data One of the tough things for students to do at this point is to have a sense for function behavior given a set of data points. The best tool is experience, of which the students are at a disadvantage. There are a few rules of thumb to help them out. • Populationandmonetary(interest,investment)datasetsarealmostalwaysmodeledwithexponential functions. • Repeating data sets, like measurements taken every hour for a day, every month for a year etc., are almost always modeled by periodic functions. • Look at the difference in endpoints for suspected linear functions. The change in values on each extreme end will be the same for linear functions and no others. The text recommends plotting the point in either a calculator or by hand to choose a model based on the shape of the graph. This is often a useful task, but one with a chance to be misleading. The scaling of each axis can determine the shape of the graph sometimes more than the data points themselves. There is no clear rule for determining the correct scaling, other than choose endpoints far enough to show all the www.ck12.org 2 data points, so again experience and trial and error are the best tools. It is useful to use different scaling to see if it appears to change the shape of the graph. Linear functions will always appear to be linear, regardless of scaling (unless the data points vary substantially and you are zoomed in very “close”), where other functions may appear to be linear at some scales, but their curves will appear at others. Also, filling the screen as best you can will often help. Somethingtorememberisthatthefunctionsarenotmeanttobeperfectreflectionsofobservedphenomena, but useable models for a defined range. Negative time may not make sense, and the quadratic function that models a falling object fails to model correctly after the time at which the object comes to rest after hitting the ground. Students should always keep in mind that models are just that, and restrictions are useful to note. Lesson 4: The Calculus I sometimes joke with my students that calculus is an hour and a half of content that we manage to stretch out over two or three years. There is a nugget of truth to it—the central concepts are not complicated. The chapter presented here illustrates the basic concepts and alleviates some of the chicken-egg situations that sometimes happen. Calculus is the science of “close enough”. Before presenting the words derivative, integral and limit, it can be a fun and useful activity to look at some of the everyday situations where smaller and smaller iterations are used for measurement. Things like mapping the ocean floor, finding volumes for figures, and using data points to make a smooth curve all give insight to the basic concepts presented here. Thisisalsoagreatopportunitytousesomeofthefeaturesofcalculatorsandothercomputermathsystems. Thereisnoharminteachingtheconceptsandsolvingproblemsnumericallywiththecalculatorperforming the “magic”. Some teachers and classes have the philosophy that you need to be able to do everything by hand before using a computer’s assistance. I don’t agree for the following reasons. First, there is no “hiding” technology from the students these days. Second, there are plenty of problems where everything butthemostadvancedcomputerssystemshavenochanceofsolving. Finally,itisgoodtohavethestudents used to using calculators now for every problem where it makes sense. There are calculator mandatory sections on the AP exam, and it makes no sense not to use a calculator for some of the problems. Lesson 5: Limits The chapter starts out with evaluating limits using a calculator for assistance. There is no reason not to do this; it is a very efficient way of evaluating some numerical limits. The most common trouble is when an exact irrational number is needed, the calculator will only return a decimal and the student may or may not know what that number is. Another problem that I have seen is that students over use the close number technique with the calculators. It is good to always have a backup in case of total confusion, but going to the calculator every time is time consuming, and will not be allowed on calculator illegal test sections. All of the same applies in using the zoom rather than the table or iterating evaluations. A decision needs to be made about how strict of a definition for limits will be presented. Limits as a concept are relatively easy to understand, but involve a tricky definition. A first year student typically will have a hard time understanding “small enough” and “large enough” comparisons that seem arbitrarily made up. The definition is never really used in a first year class, so a strict definition is rarely presented in texts, as is true here. An advanced class, however, may need to see the formal definition, or have a little more interaction with the definition that is presented in this text. There are a select few functions and situations that are run into where there is no limit where it seems like there should be one, and the only way to show it is with the formal definition. 3 www.ck12.org Lesson 6: Evaluating limits Themostcommonthingforstudentstowanttodoatthispointistoapplythetechniquesusedtoillustrate the derivative and limit conceptually. While there is real value in using the calculator to show the concept behind limits, for some reason students seem to latch onto the zoom over and over, or table technique when they run into any difficulty. It is not a bad thing to always have an “out” in complicated situations, as finding an answer is always better than not finding one. The problem is in accuracy, if the answer is expected to be in exact form for an irrational number, and time. Time is the big one here, as students are likely entering the first of some years of tests where every level of student is likely to be under stress to finish within the time limit. The calculator techniques frequently take extra time, and can really cause trouble for the overall score on the test. If there is a technique to focus on, it is finding the limits of rational functions. There are two reasons for this. First, they are common problems on standard examinations, like the AP exam. They also tend to be some of the “easier” problems, but like any problem, are only easy if you are confident in the method of solution. Where students may lack some confidence is in the high powered algebraic manipulation needed for some problems to find factors for each polynomial to cancel. Students should be given ample time to practice, and should have a safe environment to ask questions, as many will be afraid, remembering that many of the answers will be from an Algebra I class. Second, the techniques used for finding limits of rational functions are often the very same techniques that will be used later in finding derivatives using the limit definition. If students have the confidence to tackle these problems, it will make teaching this later chapter much easier, as the focus will be more on specific application and concepts. Lesson 7: Continuity There is sometimes a habit to brush off one sided limits. They are taught at this time, but seem to then be forgotten about for a long period of time. Later topics do revisit them, but often times in proofs and justifications for rules that students do not often directly interact with. Another problem with one sided limits is that many of the techniques used for evaluating limits already learned are not applicable for one sided limits (unless the one sided limit matches the two sided limit, of course). Sometimes this means that more brute force methods, or computers and calculators, are used which many instructors feel is less important or desirable than the analytic techniques. They are important, and they should be understood, but at the same time, without context, they may not stick and are best considered here in the context of continuity. In teaching, it is sometimes useful to have a library of functions that have different kinds of discontinuities. Here is a primer on how to write examples of each: Piecewise discontinuities: These are probably the easiest to write, and the easiest to identify. Any type of function that is defined differently for different intervals often has discontinuities. An interesting thing about piecewise functions is that a favorite question on the standard exams is to identify a coefficient that makes a piecewise function continuous. Example: { } x2 for x < 3 f(x) = −2x+c for x ≥ 3 Where the students will be asked to find the c that makes the functions “match”. An added level of complexity is to have the function given undefined at the endpoint necessitating the use of a limit. Functions with vertical asymptotes: These are going to occur most frequently in rational functions, but happen anytime the denominator of a function equal to zero. (there is an exception, see the next example) Rational expressions with removable discontinuities: If the denominator is approaching zero at the same www.ck12.org 4