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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 grounding in the fundamentals presented here. To have a conceptual understanding of functions and graphs 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, check work done by hand, and use some of the calculation tools that may be useful on the university examination of choice, but make sure everything could theoretically be done by hand.
Lesson 2: Relations and Functions
While it may seem like an issue of semantics, I encourage my students to use, and try to use exact terminology 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:
There are also the standard sets that have defined bold-face letters: Real numbers, Rational numbers, Integers, 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: 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 . This is clearer because the function is placed into the function 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.
- Population and monetary (interest, investment) data sets are almost always modeled with exponential 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 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.
Something to remember is that the functions are not meant to be perfect reflections of observed phenomena, 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.
This is also a great opportunity to use some of the features of calculators and other computer math systems. There is no harm in teaching the concepts and solving problems numerically with the calculator performing 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 but the most advanced computers systems have no chance of solving. Finally, it is good to have the students 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.
Lesson 6: Evaluating limits
The most common thing for students to want to do at this point is to apply the techniques used to illustrate 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:
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 rate that there is a factor of the numerator approaching zero then no asymptote can occur, as there isn’t the chance for values to become very large by being divided by a very small number. Therefore, if there are matching factors top and bottom, there will be a point discontinuity, but no asymptote (this is why factoring and canceling for limits works).
Special functions: The most common one to look at here is the integer step function, notated , which takes the decimal truncated value of , making it complete “steps” as increases. Another common one for calculus is Dirichelt’s function, which take the value zero for irrational numbers, and the value of the rational number for each one. This function is only continuous at . Most of the special functions, however, are fairly trivial at this point and are more useful for showing concepts than being used for anything in particular.
Lesson 8: Infinite Limits
There is a lot of mathematical language that is typically used for infinite limits. It will be of use to introduce students to the terminology you, and texts, are going to use that students have not yet heard.
Some key vocabulary:
- End Behavior: The activity of a function way, way out in either direction. The temptation will be to establish a certain number that is large, or small, enough but some limits converge very slowly, so it is important to stress that clues about end behavior can be found with very large numbers, but actual end behavior is an analytic concept.
- Dominates: When we have rational expressions we tend to look at where the variables are and how the numerator and denominator act. In the simple case we can see that the only thing changing is that the denominator is getting very large, and is dominating, and therefore sending the limit to zero.
- Indeterminate Form: If one breaks down the first word it is clear, but it is worth stressing that indeterminate forms are the expressions where no clue is given to the behavior of the function. Typical indeterminate forms are anything divided by zero, infinity plus or minus infinity, and infinity divided by infinity. Due to the conceptual nature of infinity (it’s not a number!) none of these can say exactly what is going on with the answer.
- Gets Large, Gets Small: Again these are conceptual descriptions of what is happening to numbers. The tendency is to think of infinite limits as a sequence of increasing (or decreasing) variable values. The behavior of this informal sequence is often described as getting large, or getting small.
There are a number of conceptual analytic themes for students to understand at this point. They should get a sense that large values in denominators tend to zero, and large values in numerators diverge. There is no hurry to teach many of the specific techniques for evaluating limits at this time as they are covered in later sections. The only tool I might teach at this time is the polynomial rule, where the large exponent dominates, or if the degree of both top and bottom functions are the same the limit is the fraction of leading coefficients. This is covered later, and is clearest as a consequence of l’Hopitals rule, but is a handy and easy tool to start using immediately.
Lesson 9: Tangent lines and rates of change
The most important concept is to understand that the derivative is the slope function. A nice aspect of differential calculus is the relationship between all the concepts, and some ideas from algebra in years past.
Before students begin to develop the formula for the slope of the tangent line they need to have a strong understanding of what the tangent line to a curve is. This is accomplished quickly with physical items. Students can have fun taking straightedges to various curved surfaces in the classroom like sports balls, balloons or any other curved surface. If available, taking a large pole outside to a hill or other curve can be fun. Simply saying that the line touches the curve at a single point is not sufficient (try giving that definition to a classroom of new calculus students and have them sketch a tangent line.) It is also worth noting that strictly speaking, it is possible to draw a tangent line that crosses a graph at multiple points, and a non-tangent line that only intersects at that single point. The idea of a line resting on the curve, staying on a single side and not intersecting the curve at any point near by (except in the case of points of inflection) may take some time for students to understand.
It is not unreasonable for students to come up with the tangent slope function on their own. Given a linear function, students should have no trouble calculating the slope. Now give them a curve and ask them to find the slope at a point. Some may try to sketch a tangent line and find the slope of that line. Not a bad idea, but they should know that this is circular, that is, the goal is to find the slope to properly draw the line. The first thing students should realize is that two points are needed for a slope, so two points must be chosen. They can do so to find an estimation, and then some students can be selected to show how students who chose closer points appear to have better approximations. While they may not come up with exactly the standard form used for the tangent line slope, combining the new concept of the limit with the “closer points” concept just figured out, students should have a pretty good definition for the derivative. Then the next step is only in attaching the standard notation, usually using the standard diagram.
Something to stress throughout the course is that Rate Slope Derivative. This especially helps later for related rate and other applied problems.
Lesson 10: The Derivative
Teaching the definition and the conceptual rule is a little bit strange. So much of the first year involves taking derivatives, yet students seem to run through this section, and then forget about it as soon as the specialized techniques are presented. As it is, other than a couple of exercises, there are few instances where they will use it. However, those instances are important. The classic exercise is to use the definition a few times for very simple polynomials, like . Make sure students clearly show each and every step when working these problems. They will all use a very similar process of expanding out and canceling the numerator. They can then extend this process to the general form for the power rule. It will include some undetermined terms in the middle, but students should recognize how those will cancel.
Another common use of the definition of the derivative is in finding limits that look like derivatives, and using the derivative function to evaluate those limits. Here is an example:
This could be a fairly involved limit, but if you can see that this is the definition of the derivative, we can actually write the limit as follows:
If the student then takes the derivative of (knowing that learning this derivative is presented in a later chapter) and then evaluates the derivative at pi over , the limit will be found. This type of problem is nearly guaranteed to show on the AP examination.
Lesson 11: Techniques of Differentiation
The first technique presented, the derivative of a constant, may seem trivial to students, but it is a place where students make mistakes. The problem is not with simple examples using familiar number, but rather what I call “sneaky” numbers. Often times in physics, or other applied problems, there will be many constants that have either letter names, like c for the speed of light, or quantities that can change from problem to problem, but are not variables. An example is the formula for conservation of momentum of a ballistic pendulum (a projectile colliding with a stationary weight at the end of an arm).
Where is the velocity of the projectile, is the mass of the projectile, is the mass of the pendulum weight, is the gravitational constant and is the height above the center of mass at rest of the pendulum. Some of these can be treated as variables, depending of what is observed and what is being asked. The constant is always going to be the same value on earth, so it is always a number. So in any problem, there are two variables here, and numbers masquerading as variables. Slightly more common is the trouble encountered with and which are again constants unless being acted on by a variable.
Students should see the utility of the power rule immediately. Because it is easy to use, if any algebra can be done to use it more often, then it should be done. The most common algebraic changes to make are changing expressions in the denominator to negative exponents and using fractional exponents for square roots. I have observed students making mistakes applying the quotient rule at a much higher rate than when applying the product rule. If it can be done, change the rational expression to a product using negative exponents.
One of the things that can be done to help students with remembering the quotient rule is forcing students to learn, and apply, the product rule in a particular manner. While it doesn’t make a difference for the product rule which order the derivatives is take, the subtraction in the quotient rule makes it so that the terms can’t be switched. Forcing students to think of the product rule as “derivative of the first times the second, plus the first times derivative of the second” then it’s just a simple change for the quotient rule by replacing the plus with a minus and dividing by the denominator squared.
Lesson 12: Derivatives of Trig Functions
The trig functions do require a certain degree of memorization. It is up to the instructor, and the students, to decide what they wish to memorize, and what to work out. In my personal experience, I have found it easier, and more useful, to memorize trig identities, double and half angle formulae, and how all of the other standard trig functions can be expressed in terms of sine and cosine. I never did, and still don’t have the derivatives of any of the trig functions memorized beyond sine and cosine. If I need to take the derivative of tangent, I convert it to sine over cosine and apply the quotient rule. The advantage of this method is that it involves less overall memorization, the information memorized is applicable to more types of problems than only derivatives and is probably more flexible for solving new problems that don’t conform to any of the standard derivatives.
A couple of disadvantages are that I will often end up doing more work than someone who knows the standard derivatives. Another is that knowing all of the standard trig derivatives helps when it comes time to find anti-derivatives, as it will often be helpful to quickly identify functions that have easy anti-derivatives. Another advantage to memorizing all of the standard trig functions is that the current section will be easier to teach, and probably faster for students to learn. In my classroom I endeavor to teach both, and allow students to choose.
Getting solutions in exact terms for trig functions is a challenge for many students. Most students will seek to use their calculators to evaluate nearly any numerical answer, which in the case of trig functions often leads to non-exact answers, sometimes in a different form than requested. Students will likely need a refresher on the standard unit circle values for the trig functions, and it may be useful for the students to have a ready reference. Another thing for students to start to recognize is when exact answers are needed. On multiple choice tests the answer can give clues on what needs to be done. If the answers are all with decimal approximations, then there is no need to worry about exact answers and calculators should be utilized to the fullest extent.
Lesson 13: The Chain Rule
Compositions are sometimes the least familiar method of combining functions to students. The other operations are more familiar from having used them with numbers. Many functions that students have worked with in the past can be deconstructed as a compositions of two, or many, functions, even ones that seem fairly simple. Since only the most basic functions have known derivatives, the chain rule gets applied very frequently. Combine this with the lack of familiarity with compositions and students have many little struggles.
First is in identifying that the chain rule needs to be applied. There are a few clues: parenthesis, radicals, and exponents are the usual places to look. If there is anything more than a simple variable, then the chain rule will need to be applied. Another way to look at it is that the chain rule can always be applied. This is also useful to start setting up for implicit differentiation, as it sets up why the differential term gets chained out from each variable. This way, if the derivative of the variable is anything more than a then the chain rule will need to be applied.
Next is in understanding what the two functions involved are. Many textbook examples are not particularly helpful for understanding the mechanics of the chain rule as they keep referring back to the composition notation that students are not particularly comfortable with. Sometimes the idea of “inside” and “outside” functions can be used. This is probably the clearest way to think about functions involving parenthesis, or inside of radicals, trig and log functions. This can get confusing when the composed function is in the exponent. Sometimes it helps to think of “little” and “big” functions, where the little function is inside of the big one. This is maybe not as clear for parenthesis, but can be helpful for exponents.
It is always a good idea to try to get the class to use clear language when asking questions, or presenting solutions. If students are all using clear descriptions of what the composed functions are, even if it is not a single standard among the whole class, it will help all students understand how to identify where and how to apply the chain rule.
Lesson 14: Implicit Differentiation
This seems to be a stumbling point for many students. Technically speaking there is nothing new going on here. Implicit differentiation is really just an instance of the chain rule applied to each variable, where one variable is not defined explicitly. The problem is, and this is not unique to this section, that to make things easier to teach and learn in the earlier sections, not everything is exactly written out in full technical form. I’m not advocating doing so, as it would turn simple problems into massive undertakings.
It is always useful to see if the function can, in any way, be solved explicitly for one of the variables. Once getting an explicit function is ruled out, it’s useful to make sure all of the various rules are identified that will need to be invoked. The chain rule is a given, and will frequently be required multiple times for each term. Quotient and product rules are also often needed. All of these will be combined, so keeping track of each will be quite a task.
This is a useful section to spend lots of time with class wide examples. Start out by solving problems with students following along, and progress towards having the class work problems with step by step check-ins to make sure everyone is getting fast feedback on the example problems. Students should begin to see there is a sort of a rhythm to the problems. The process of applying the rules, collecting the terms with a in them on one side, and the other terms on the other side of the equals sign and then dividing to get the derivative is going to be very similar among all of these problems.
Another thing for students to understand is that these problems require a point, rather than an value, if a numerical derivative is to be computed. This is especially true as the implicit expressions are not always going to be functions, and may have multiple values for each . Sometimes the original function will need to be revisited in order to get the point if some information is not given in the problem.
Lesson 15: Linearization and Newton's Method
The description at the start of the text of “zooming-in” is a great idea for how to illustrate the idea for the class. Use a graphing calculator or a computer program to show that nearly any function “looks” like a straight line if you get in close enough. (As a counter example, it can be useful to show some non-differentiable functions to show that differentiability is a necessary requirement for a linear approximation. Some functions to look at would be the absolute value function, which will always have a sharp point at , and which will just about always look the same about the origin because it increases in frequency.) After showing that the function is nearly linear after zooming in, find the value of the derivative and use the point to also graph the linear function that approximates the original function at that point. Students can then use the trace function, with the up and down buttons, to switch between the two functions to see how close they really are.
Students may make the mistake of thinking that the approximations they find are good everywhere. It is important to stress that linear approximations get worse the further away they get from the point chosen. There will be better approximation techniques, and encourage students to try to think of how they might do so.
In the age of calculators everywhere this chapter may be a tough sell. There isn’t any trouble in finding quantities like these days. The key here, and it is useful to let the students in on this, is that using calculus to approximate functions and values is a recurring theme. This is not a chapter to be ignored, as the ideas here will be expanded on later in more complicated problems.
Lesson 16: Related Rates
Students can be guaranteed to encounter a couple of these problems on the AP examination. Often, there is a related rate question in the free response section. Related rate problems have many steps, involving many variables and can be a little bit of a logistical challenge for students. There are a couple of things you can do to help out.
First, many related rate problems require area, distance and volume formulae. Some of these students will have memorized, like the pythagorean theorem. Others will not be so familiar, like the volume of a pyramid or the surface area of a sphere. Having a poster or individual cheat sheets of common formulae will help students learn and recognize what they will need to solve the problems. Second, this is a great opportunity for “we do” instruction where the class is working on the same problem together with check-ins after every step. This will help students learn the mechanics of the problems without getting stuck.
There are not many different kinds of problems that are typically seen in a first year calculus class. Make sure the students have seen a couple of examples of each. Those are:
- Tracking a straight moving object at an angle (or the sliding ladder)
- Inflating spherical balloons
- Filling various geometric containers
- Temperature change in a steady environment
If a few examples are seen of each then identifying how to solve each problem will become easier later.
Lesson 17: Extrema and the Mean Value Theorem
It is often a dilemma for teachers of a first year calculus course to decide how much of the classic analytic proofs to present. None of the proofs are necessary to understand how to apply each of the theorae. Furthermore, there are only a select few students who will be moving on to levels of mathematics where the mechanics of the proofs are helpful. However, as an applied mathematician, not all of calculus is applied and the ideas and mechanics of the proofs are valuable. How much of each proof to present is up to the instructor; it depends on the ability level of the class and where it looks like the students may go in to the future.
If proof are presented to the class, the traditional process has been to present them in lecture format with students taking notes and following along. The idea is that the proofs are challenging and just the exposure will rub off on the students. I don’t believe this is helpful. Students tend to tune out long lecture type presentations. Some sort of interaction is needed with the material to make it worth the time. Students are capable of writing the proofs themselves at this point. They may not be perfectly rigorous, but using previous theorae, they can be done. Some ways of helping students along is with starting them off with the first couple of lines, telling them some of the theorae they will need to apply, giving the students the trickiest part, or proving a similar result and having the students complete a corollary.
Applications of theorae need to be presented whether the proofs are taught or not. Most of the proofs here are existence proofs, showing the existence of certain properties of functions. This is especially true of the mean value theorem, which will probably be the most applied in this section.
Lesson 18: The First Derivative Test
The first derivative test is a huge foundation for the analysis of functions to come. Also, there are a couple of things that can be confusing if there is not a good foundation for the concept presented here. The students will learn that the first derivative can be used to find maxima and minima. More important is to understand that the possible critical points are where the slope of the function, and therefore the derivative of the function, is zero. This helps with the confusion that sometimes happens when looking at functions that have a slope of zero, but do not possess a maximum or minimum at that point. For instance at . The derivative at zero is zero, but there is no extrema at that point.
One tricky thing for many students is to interpret the graph of the derivative. Given a plot of a function, it is not hard for students to identify increasing and decreasing intervals. What does seem to be tough is to reflect the slopes on a new graph. There seems to be a mental block in drawing a new line that is positive and decreasing at a point where the original function has negative values and is increasing. Be sure to spend some structured time helping students to fully understand how to create and understand these graphs. The concepts that they will learn will help with this section, and others down the road.
Lesson 19: Second Derivative Test
Here it is absolutely critical that students have some sort of organization technique for all of the information for each interval. After taking the first and second derivative tests students will know where there are maxima, minima, intervals where the function is decreasing, increasing, and points of inflection. The problem solving guide has a recommendation for how to keep track of all the information using tables.
Again, this is sometimes considered to be an antiquated chapter. With graphing calculators readily available, students do not need all of the support to graph a function, unless they are strictly doing it by hand. For these reasons, I somewhat devalue the sketching aspect for the derivative tests. One things that is very important to understand is that the derivatives test will sometimes show information that might not show up on calculators due to the domain used, or the scaling. It is not uncommon for test to specifically choose functions that may not show all extrema on a standard graphing screen. Also, a little bit of work can help in determining the best scaling for the window to show the graph on a calculator or computer grapher.
Lesson 20: Limits at Infinity
L’Hopital’s rule is a student favorite. Limits are often tedious to find by hand, involving either complicated comparison rules or non-trivial amounts of algebra. L’Hopital’s rule is a relatively quick and easy way to compute limits. The issue is going not going to be getting students comfortable with the rule, but rather holding them to only use it when it is permitted. Stressing the fact that the rule can only be applied for or , and that it is not allowed for finding the derivative using the limit definition is important.
A very nice application, and then shortcut for infinite limits, of l’Hopital’s rule is looking at rational expressions. Give the students some examples, like:
All of these are indeterminate and can be found using l’Hopital’s rule. More importantly is finding the pattern, which students should be able to do after not too many questions similar to these. The rational function shortcut is that only the leading term is important. Since the behavior out at very, very large or very, very small numbers means that the largest degree will make so much of an impact that none of the other terms will have any effect. Along those lines, the larger degree in either the top or bottom will dominate the whole expression. If the polynomial on top has a larger degree, the limit diverges. If the polynomial on the bottom has the greater degree, the limit converges to zero. If the degree of each is the same, then the fraction of the leading coefficient is the value of the limit. This is verified with l’Hopital’s rule as the coefficients are the only thing left after multiple iterations of the rule. A nice challenge for an advanced student is to show this rule works in general terms.
Lesson 21: Analyzing the Graphs of a Function
The table that is presented in the text should provide some help in getting students started on this section. Some students may find it tedious, and it is true that there is no standard format to present the information from text to text, class to class and test to test. There will frequently be free response questions that ask for some parts of these tables. Therefore, it is good practice to be able to fill in the table. I would not get too worried about the form students present the intervals in. There are a number of standard ways to do so, and teaching a new notation is probably not the best use of time at this point. Clarity is the key, not standardization.
The text restricts the domain in the subsection regarding trig functions. There are two considerations about restricting domains. Frequently the question students will need to answer is related to an applied problem where the a minimum, or interval of increasing values, for instance, is asked for. In applied problems there will often be limits placed on the domain from physical (negative time, etc.) or logical (the race ends at ) that will be reflected in the domain. With all models, the function will return values outside of this interval, but they are meaningless, as the model only holds for the interval specified.
The consideration in the text is slightly different. Periodic functions will have infinite local max, min and intervals increasing and decreasing. Therefore, with minor exception, if one or two periods are analyzed the whole function is described, as it will be a repeat of the same values or behavior. One trouble is in making sure that the domain restriction allows for at least one period. After examining the function on this period, it is possible to construct the critical information for the entire graph. It requires the use of describing critical points as a sequence, and may add an unnecessary level of complexity for students at this point. If you have students who are likely to continue with math, engineering or physics, it is a very useful exercise to have them complete.
Lesson 22: Optimization
Optimization problems, along with related rate questions, are the heart of the applied questions in the first year of calculus. In many cases, the two are interrelated; if the question is asking for the optimization of a rate, then there will often be a related rate question embedded. Students should be made aware of this, and should refresh their memories regarding related rate questions. I have in the past held back on the related rate chapter and presented it immediately before the optimization chapter more or less showing them as one unit. The advantage is that students are doing the same types of problems in a logical progression. The down side is that these are two sections that first year students need above average effort and time to master. It can be mentally taxing and frustrating.
Like related rate questions, there are a few classical problems that should each be covered so students have confidence when encountering them. They are:
- Optimizing volume to surface area, or perimeter to area (the most classic question, and a fun one, is minimizing surface area for a soda can it can be shown that the standard size is the optimal dimensions)
- Optimizing the pathway with different rates (the walking problem, the road problem)
- Chemistry or Pharmacy problems (rates of reactions, yields)
- Cost/profit problems (minimizing warehousing costs, maximizing profit)
There are, of course, many problems possible. These are the ones I have seen most frequently, in descending order of frequency. Students should be, at the least, very familiar with the first two on the list. It is nearly possible to guarantee one, or both, problems on a standard exam.
Clear organization of facts is very important. Because optimization problem inevitably include substantial information about the problem, and much of it is not actual content. Also in the cases of volume, area and perimeter the functions are frequently not included in the problem. Pulling the information of what shape is needed and then connecting to the correct formula is a skill that needs to be developed.
Lesson 23: Approximation Errors
The text here relies mostly on techniques with the graphing calculator. Sometimes teaching with the TI calculators can be a bit of a challenge. The menus and buttons can be a challenge to negotiate for students who are not completely familiar with the calculators. It is advisable to get students more comfortable any way possible as the calculator is a necessary tool for many of the standard exams. Guiding the class along is partially dependent on what sort of technology you have. The overhead projector adapter is very helpful in keeping the class together. Another option is using an emulator on a computer attached to a projector. There are some serious issues with this, as the ROM for the calculators is protected property. There are assorted guides available online, but this is not for the technologically faint of heart. A buddy system works very well, where a student who is very comfortable with the menus of the calculator with a student who may need some help in finding all of the functions.
The presentation of taylor polynomials here is truly optional. It is not a topic on the AB AP exam. Furthermore, some teachers (and as a consequence, their students) are not comfortable with using tools and theorae that are not fully presented. It is, however, good to show that there is a world beyond linear approximations, and that truly there is nothing needed for taylor polynomials beyond being able to take a derivative. Therefore, it is good to present here, but do be careful and honest about the “hand waving” that needs to take place here.
Lesson 24: Indefinite Integrals
Sometimes teachers can get carried away with using too precise of language. An early example is in introducing a chapter on functions in algebra . The teacher swears they are important (which they are) and the students can’t understand how is different from the equations they have been using the while time. I see a similar deal going on with antiderivatives and integrals. Is there really a difference between the two? Technically, yes. In a first year calculus class, no. Sometimes I will have students refer to individual parts of the integration process as antidiferentiation. For instance, if parts is needed to take the integral, part of the process is taking the antiderivative of one of the parts. Most classes, however, will use the two terms interchangeably, and there really isn’t a problem with that.
I would recommend using the lower case, upper case notation before introducing the integral symbol. It is important for students to have a level of comfort in taking the antiderivatives before the added notation complexities. Also, if you choose to introduce basic differential equations at this time, this is the notation that will make it more clear. It also helps with some of the notation issues for the fundamental theorem. A notation issue you have to be strict about is the term at the end of each integral. This is important for substitution and parts later.
Lesson 25: The Initial Value Problem
Something students are going to have trouble with is when to put the initial values in to compute the constant left from integration. Even more challenging is when there are multiple sets of information, like for differential equations starting from acceleration and getting both the velocity and the position functions. Another common problem is where there are multiple constants and we combine them to create a new constant, that looks exactly like the old one. The thing to remind student of here is that these constants are a sort of hybrid variable. They are a yet undetermined quantity, but we will have a chance to find their value later. A good example, building on prior experience, is to have them work is finding the equation for a line using the slope intercept form. The intercept is just like the constant of integration and we will find each of them exactly the same way.
There is a great deal of importance of initial value information. In physics it is very common to use easy to measure quantities and models to find information that would be tough to observe. A big part of this process is setting up the initial conditions for the experiment. It is also worth noting that while it is most common to use the true initial values, usually time is , any observable point will work to find the constant of integration.
Lesson 26: The Area Problem
The tough part for students is usually understanding where the height of the rectangle is coming from. There sometimes seems to be a disconnection from their easy understanding that the area will be base times height, and where those quantities are coming from in the rectangle area process. I often observe students hoping for, and then exclusively using, more of a formula than develop a true understanding of the problem. I would try to spend enough time to make sure that the students understand that the function value is the height, and how the left, right, and middle methods change where the function is evaluated.
The toughest part about introducing sigma notation for sums is the indexing. Without significant experience, students get confused about how different terms, with different subscripts, are related, why there seem to be more than one variable and how it all fits together. Scaffolding work with sequences and their subscripts will keep it so that it is only a single new notation issue at a time. It will also be helpful to work with series without the subscript notation for each of the endpoints for the rectangles. The fewer variables at a time students have to work with the easier time they will have. Working with indices, variables and new symbols can be overwhelming.
Something to consider is how much to focus on sums. There are frequently a few questions on exams that ask for the estimation of areas using different approximations. None of these, however, require an understanding or use of the summation notation. The notation, along with the rules for finding values of infinite sums, is really used to establish a definition for the integral. It can be presented as such. Make sure, however, that whether they use the summation notation or not, the students are comfortable with finding area approximations using different shapes.
Lesson 27: Definite Integrals
What is a Riemann sum and why do they get such a name? At least that I what I wondered as a first year student. Typically if a theorem or a rule is important, we will give it a title, like Mean Value Theorem. If it is really important it gets a name, like Fermat’s last, or the Pythagorean theorem. Riemann is one of the giants of math, so these sums must be really important, right? Well, not really, and it may be useful to let students know that this is the case. It’s not that Riemann sums are not important, it’s just that there really isn’t anything unique about them. It is a fancy name for a summation of rectangular areas to approximate the area under the curve. Really that’s it. Students in calculus have learned some of the decoding techniques to identify important information and discard others, which is a very good skill and shows high academic literacy. This happens to go against the rule and you can save your students some trouble by alerting them to the fact that this term is worth knowing, and it will be referred to at various times, but it is of small utility for this course. Especially because understanding of infinite series is needed to find all but the most basic integrals using Riemann sums they should not be stressed at this time.
In some ways it is easier for students to think of definite integrals as an instance of indefinite integrals, even though from a completely analytical perspective this is a bit backwards. Since the easiest way to evaluate definite integrals is to take the anti-derivative and evaluate at the endpoints, there is no reason to evaluate definite integrals in another fashion at this time. There are other sections that focus on definite approximations using different rules.
Something that is mentioned in the text, but is worthy of reiteration, is that the definite integral does not give total area between the curve and the axis. You can think of it as net area, and total area, or area under the curve, or a number of other ways of saying it, but it is important that students are aware of this distinction and take a second to make sure they are clear on what the question is asking for. Not every question wants total area, so there is no blanket statement that can be made, just warnings to be careful and detailed about answering the question.
Lesson 28: Evaluating Definite Integrals
There is a little bit of a divergence here between the strictly pure and the applied crowds. The applied group, with the focus on answer the question presented in the simplest possible way, will teach that definite integrals are just like indefinite integrals with the extra step of evaluating the endpoints and subtracting. The pure crowd, with the focus on theory, will insist that definite integrals and numerical methods are what came first, and that the fundamental theorae are what connect this process to indefinite integrals. I tend to the former for a first year class, even though my background is decidedly pure. Few students will be continuing in that path, and if they do they will have the chance to learn everything in the strict manner in their real analysis class, and get to curse their high school teacher who clearly didn’t know any better. With that said, likely the best thing for your class is to present definite integrals as an instance of integration.
Definite integrals do have a nice side effect of being able to be evaluated easily by nearly all graphing calculators and computer solvers. Which brings up an important tip for the AP examination. If there is a problem in the calculator legal section of the test calling for a definite integral, by all means use the calculator to solve it. There are no bonus points for doing extra work. The only problem could arise when all of the options are exact answers, but even still students can get a decimal approximation for each answer and compare to the solution found for the definite integral on the calculator.
Lesson 29: Integration by Substitution
This is one of the key chapters in a first year class. Very few integrals can be evaluated without substitution. Furthermore, the practice of variable substitution is a useful skill for solving all kinds of problems, even outside of calculus. It is a skill that many student are not comfortable with, so that is a good place to start from. Even outside of calculus problems, students should practice changing variables to find solutions.
Students can run into organization problems with substitution. With complicated composite functions and their derivatives space on the page can be at a premium. There ar