The goal of this Learning Module is to introduce students to the concept of the limit and understand its application to the derivative of a function of a single variable. Then, having understood that theoretical framework, students will explore key applications of differentiation, including finding local extrema, related rates, and optimization. This Learning Module seeks to adopt wherever possible the Knowledge Process and multiliteracies framework in order to better connect calculus to its applications in the real world and to questions of significance for the learners.
mathematics, calculus, limit, functions, derivative, rates of change, instantaneous rates of change, slope, optimization, related rates, extrema
The goal of this Learning Module is to try and adapt the core principles of the Knowledge Process alongside a multiliterate approach to a rigorous high school mathematics course. Both the knowledge process and the multiliteracies framework are ideas developed by Kalantzis, Cope, and others over the past several decades. Specifically, I hope to use it as a framework for building a Learning Module around the concepts of the limit and its application to the derivative as seen in single variable calculus.
Wherever possible the goal is to align the individual Updates and Learning Outcomes along the four cardinal directions of the Knowledge Process: conceptualizing, experiencing, analyzing, and applying (Kalantzis & Cope, 2014, p. 111 - 113).
As well as incorporate to the greatest extent possible a wide range of meaning forms, with the goal of creating a more multiliterate and relevant mathematics classroom. The multiforms of meaning are diagramed below:
The ultimate aim is to have these multiform meanings intersect, as Kalantzis & Cope (2023) outline, with the multisituational forms of being, in order to reinforce the applicability of the mathematical concepts to the larger world and to increase their intrinsic meaning to the learners.
I have taught high school mathematics, specifically calculus, for over a decade. During that time my pedagogical approach has been largely unchanged, as has my rationale for that intransigence: the work is hard enough, the pace is fast enough, that we just don't have time for anything else. And moreover, I'm a pretty good didactic teacher: my year-end student feedback surveys are kind, when my students take the AP test they typically do well, and I'm at the point in my career where I can anticipate questions almost before students ask them. But when my current school moved away from the official AP designation over five years ago, an opportunity opened up to explore calculus not as a race to a test in May, but as an incredibly powerful framework with which to analyze the world. By employing a multiliterate, Learning-by-Design approach to this Learning Module I hope to deepen my own pedagogy and explore new directions to take my own classroom.
A "calculus cheat sheet" taken from Pinterest shows the great collection of facts that a typical high school calculus student must know. It is worth pointing out though that this is arguably about half - this sheet does not include integral calculus. Our aim is to avoid this sort of one-dimensional, static knowledge artifact that is designed exclusively for some sort of summative assessment.
This Learning Module is designed to introduce you to the concept of the limit and the derivative as seen in single-variable calculus. This Module is designed for students with a strong foundation in previous high school mathematics classes, including Algebra, Geometry, Algebra II, and Trigonometry and/or Precalculus. The Module is designed to take 7 - 9 weeks to complete. It is designed to be a majority of "asynchronous" and self-paced work, but there are components that assume some synchronous meetings, either online or in person.
By the end of this Module, you should:
Functions
Limits
Derivatives - Conceptual
Understand the role of the limit in finding a tangent line to a curve at a point
Understand how the secant line between two points on a function relates to the average rate of change, and how the tangent line relates to the instantaneous rate of change
Derivatives - Formal Rules & Applications
These criteria follow the Standards set out by the AP College Board under the BC Course and Exam Description.
Participation Requirements
Required Media: Read the assigned textbook reading or watch the assigned videos for each Update.
Comment on Admin Updates: Comment on all six admin updates, responding to the prompt provided. Comments should be at least 50 words.
Updates: Create six of your own updates, each one a response to one of the six Admin Updates. Choose three from category (1) and three from category (2). In general, updates should be about 200 words for reports, 5 - 10 minutes for videos, and 10 questions for problem sets, unless otherwise specified.
Comments on Peer Updates: You should respond to six Peer Updates from category (1), and six Peer Updates from category (2). Comments to Peer Updates of Category (2) are in the form of your own attempt at completing the problem set they have created and verifying their own provided answers for accuracy.
Peer Reviewed Work: Create your own Peer Reviewed Work. Your first Draft should be a nearly complete work, ready for Peer Reviews. You are expected to incorporate and comment on the two Peer Reviews you receive on your own. Then, make all necessary edits and submit a Final Draft of your work. Finally, perform a Self-Review of your work and then Review the Reviews of your own work.
Peer Reviews: Each student is expected to complete two Peer Reviews of others' works.
Materials
Access to the internet through a web browser is necessary to take in the entirety of some of these online videos and links.
A calculator or some app that works as a calculator
Supplemental Materials: Ruler, household objects, pencil and paper, protractor, and compass.
Frequently Cited Resources
Calculus 1 by OpenStax - an open textbook covering the fundamentals of single variable calculus and beyond.
Calculus 1 by Paul's Online Math Notes - a great student or instructor resource with worked examples.
Calculus Problems by UC Davis Mathematics - a selection of sample problems with worked solutions.
Calculus 1 by Khan Academy - videos covering the fundamentals of calculus with sample problems and worked solutions.
Videos by The Organic Chemistry Tutor - YouTube videos on various math and science topics.
Videos by 3Blue1Brown - YouTube videos on math topics with an emphasis toward higher-end problems and advanced abstract mathematics.
Videos by NancyPi - YouTube videos on a variety of math topics.
Videos on CalcWorkshop - A selection of calculus videos.
WolframAlpha - A symbolic search and knowledge engine, capable of showing you step-by-step derivatives.
SymboLab - Another powerful symbolic knowledge and computation engine.
Calculus is a mathematical concept richly alive in the world today, ubiquitous as a framework for analyzing functions and complex mathematical spaces, as well as in engineering, the sciences, computer graphics, and much more. The canonical story of calculus is embodied learning: an apple fell on old Newton's head and he was struck by a connection between that and the movement of the celestial bodies. Understanding calculus at its most profound levels requires that students and educators employ all of the methods of knowledge acquisition and generation at their disposal.
By using a multiliterate Learning-by-Design framework, students will be able to connect their work more immediately and authentically to the real world and the issues of concern to them. In this framework, students are asked to take a lead role in their own learning, to pace themselves through the material, and then to be the creators of knowledge artifacts. Through doing so, students will be able to build frameworks of understanding that demonstrate the intrinsic power of mathematics as a driver of the development of epistemic capital (Kalantzis & Cope, 2023, p. 3).
Of paramount importance is that students come away with an understanding of how the derivative is a ubiquitous tool of mathematics and therefore of modern sciences, engineering, economics, and many, many other fields. The applications we look at here are limited in complexity only through our limit to a single variable and this is a very introductory look at these concepts. Taking these ideas further can be done without the underlying mathematics and again it is important to stress how these concepts are powerful tools for individuals to be able to understand in order to navigate high-level quantitative conceptual frameworks with more ease.
The derivative as the instantaneous rate of change should continue to be stressed throughout, and it should be reconnected back to the idea of a slope between two points - with limits bridging that gap between two points and the convergence to one point.
It is also critically important that students feel empowered in their own learning - they not only have the tools to learn and understand concepts but also to use that knowledge and apply it in new ways or to create questions of their own.
Finally, a note about formatting: as noted in the Learner portion of the Overview, this course is designed primarily to take place with much of the work done "asynchronously" as Learners explore the various Updates at their own pace, utilizing as much of the supplemental material provided as needed. However, there are activities that assume a synchronous meeting session, which for many of the activities could be done either online in small break-out rooms, or in person. How the rest of the synchronous time is structured is left to the discretion of the Instructor, though some ideas include: concept review, example problem work, student-lead explanations of concepts with instructor-provided corrections, a traditional lecture, small-group work, and collaboration, or a sharing out of Updates and Peer Reviewed Works.
The "Required Media" presents the essential expanded conceptual framework either as written text in an online textbook or through a series of video lessons of various MOOCs or YouTube channels. I would encourage you to require at least one of those two or to present the same material in a synchronous or asynchronous lesson of your own.
Functions are a deeply embedded concept in the world of mathematics. To begin, a few definitions of functions may be helpful:
Encyclopedia Britannica definition of functions in mathematics:
an expression, rule, or law that defines a relationship between one variable (the independent variable) and another variable (the dependent variable). Functions are ubiquitous in mathematics and are essential for formulating physical relationships in the sciences. The modern definition of function was first given in 1837 by the German mathematician Peter Dirichlet. (Encyclopedia Briannica, 2022)
WolframAlpha definition and examples:
In mathematics, a function is defined as a relation, numerical or symbolic, between a set of inputs (known as the function's domain) and a set of potential outputs (the function's codomain). (WolframAlpha, 2022)
Functions are used in mathematics to describe both pure mathematical relationships, but also very often to then describe the world.
Within the wide-ranging possibilities of all functions, we will primarily keep our focus on the families of elementary functions and a few others, which include
The class of functions consisting of the polynomials, the exponential functions, the logarithmic functions, the trigonometric functions, the inverse trigonometric functions, and the functions obtained from those listed by the four arithmetic operations and by superposition (formation of a composite function), applied finitely many times." (Encyclopedia of Mathematics, 2011)
The table of functions above comes from a Name That Function activity in Desmos.
To create more complex functions from these families of elementary functions, a number of transformations and compositions can be performed.
Basic Function Transformations from MathIsFun
Basic Function Transformations from PurpleMath
SymboLab's Function Transformation Calculator decomposes functions into descriptions of their transformations, as shown below:
More details on the transformations of functions can be shown with the SymboLab link above.
Khan Academy Lesson on Function Transformations
Khan Academy Identify Function Transformations Practice
GeoGebra Activity on Function Transformations
Essential Media:
Chapter 1.1 - 1.5 from OpenStax Calculus Volume 1
OR
Function Transformation on Khan Academy
Learning Objectives:
Comment:
Describe a place or time in your life when you encountered a mathematical function or mathematical relationship. The article from Education World (also linked above)on functions in the real world may inspire you here, but please use your own example.
Update:
(1) Research a person, time, or place in mathematical history and create an Update about it. This Update should be at least 500 words, include at least 3 media elements, and properly cite all sources both in text and in the reference. Try as much as possible to pick a person, time, or place of some personal relevance or curiosity.
OR
(2) Create a "worksheet" of your own with a mix of open-ended conceptual questions; for example, "What are functions?" alongside a mix of applied problems like the ones included in the worksheets below. The length should be no more than 10 problems, with between 40% and 60% being open-ended and conceptual questions. For your applied problems consider the difficulty and range of functions and transformation types. Include at the end an Answer Key with model answers to the conceptual questions and worked-out solutions to the applied problems.
OR
(2) Create your own online math learning video like some of the ones you have seen here, in which you explain one specific aspect of the content covered and then work through at least two example problems not already covered in the material provided. Topics could include a specific function and its graph, a type of transformation, or a combination of those topics. Your video should be 5- 10 minutes long and though it does not need to be rehearsed it should be well-prepped with little wasted air time.
Supplemental Practice:
This first Admin Update is principally concerned with grounding students with a review of what they should already know, but perhaps have not seen all laid out together in this way. Stress the ideas of Parent Functions or Families of Functions and how other functions can be arrived at via their transformations. Have students name these transformations and it can be helpful to use a graphing program like Desmos to show that the same transformations affect all graphs in similar ways:
Activity:
Have students identify what is happening to the graph as each of the sliders is moved in this Desmos link. Make a conclusion about what type of transformation each of them entails.
Learning Objectives
- Students understand the definition of a function and can relate several examples of functions in mathematics.
- Students are able to manipulate functions through transformations and can identify parent functions of transformed graphs and 'work backward' to find the underlying function of a graphed, transformed function as well as the variables used in each of the transformations.
- Students understand the ubiquity of functions not just in mathematics but in the real world. Students can begin to describe relationships in the real world through the lens of "functions" and make connections to the different types of functions that can be used to describe different real-world scenarios.
- Students can express a function algebraically, graphically, via a table, and explained in written or spoken words.
Estimated Time: 1 Week
The concept of infinity can cover a wide range of meanings, from the mathematical to the metaphysical. The Encyclopedia Britannica offers at least three separate colloquial types:
infinity, the concept of something that is unlimited, endless, without bound. The common symbol for infinity, ∞, was invented by the English mathematician John Wallis in 1655. Three main types of infinity may be distinguished: the mathematical, the physical, and the metaphysical. Mathematical infinities occur, for instance, as the number of points on a continuous line or as the size of the endless sequence of counting numbers: 1, 2, 3,…. Spatial and temporal concepts of infinity occur in physics when one asks if there are infinitely many stars or if the universe will last forever. In a metaphysical discussion of God or the Absolute, there are questions of whether an ultimate entity must be infinite and whether lesser things could be infinite as well. (Encyclopedia Britannica, 2022)
The concept of infinity defies much of our finite understandings of the Universe and leads to many paradoxes. One famous example is Galileo's Paradox:
Galileo's paradox of infinity involves comparing the set of natural numbers, N, and the set of squares, {n^{2} : n ∈ N}. Galileo (1638) sets up a one-to-one correspondence between these sets; on this basis, the number of the elements of N is considered to be equal to the number of the elements of {n^{2} : n ∈ N}. It also characterizes the set of squares as smaller than the set of natural numbers, since ``there are many more numbers than squares". As a result, it concludes that infinities cannot be compared in terms of greater--lesser and the law of trichotomy does not apply to them. (Błaszczyk, 2021)
A quote from Galileo's Two New Sciences:
Simplicio: Here a difficulty presents itself which appears to me insoluble. Since it is clear that we may have one line greater than another, each containing an infinite number of points, we are forced to admit that, within one and the same class, we may have something greater than infinity, because the infinity of points in the long line is greater than the infinity of points in the short line. This assigning to an infinite quantity a value greater than infinity is quite beyond my comprehension. (Galileo, 1954 (1638), p. 31)
Gregor Cantor would seek to resolve many of these dilemmas with his formal work on the cardinalities of infinity. "Mathematicians Measure Infinity and Find They are Equal" in QuantumMagazine describes though how he would leave open a question about intermediate levels of infinity, known as the "continuum hypothesis," that would be worked on and "put to rest" by the work of Paul Cohen and others.
Figure 7 - A video about sizes of infinity, also known as cardinalities (Doing Maths, 2022).
The above video provides an overview of some of these topics.
Functions & Limits
To deal explicitly with the notion of infinity in regard to mathematical functions, the notion of the limit needs to be introduced. The precise mathematics that underlies limits can be found in the subject of real analysis and what are known as "epsilon-delta proofs."
Video of Epsilon-Delta Definition of the Limit Proofs
Figure 8 - A video describing delta-epsilon proofs (The Math Sorcerer, 2015).
Finding Limits of Functions
The following resources provide a number of properties of limits as well as techniques for solving to find the limit of functions, specifically at values of the function that are not defined. For example, using algebraic techniques, we can find a limiting value for
\(f(x)= \frac{x-3}{x^{2}-9}\)at the value \(x=3\) as well as understand why this x value produces a "hole" or removable discontinuity while the x value of -3 creates a vertical asymptote.
Khan Academy on Limits
Brilliant.org on Limits of Functions
OpenStax on The Limit of a Function
OnlineMathLearning.com on Limits of Functions with Problems
Essential Media:
Chapter 2.1 - 2.5 of OpenStax Calculus Volume 1
OR
Videos 29 - 56 of The Organic Chemistry Tutor's New Calculus Video Playlist
Learning Objectives:
Comment:
Describe a situation in the real world in which you have encountered a type of limit - a value that you can continue to head toward, but which does not exist itself. Or a way in which you travel toward something and the result is asymptotic behavior toward infinity.
Update:
(1) Expand upon one of the topics included in either the Wikipedia, Stanford Encyclopedia of Philosophy, or Dartmouth Mathematics about infinity. Your Update could be about a person, concept, place of rapid or sustained mathematical development, culture, or development within the thinking about the concept of infinity, among other possibilities. The Update should be at least 500 words, include at least 2 media elements, and properly reference all sources used.
OR
(1) Create a short video acting out or describing in detail one of the paradoxes discussed in this Update and listed here in this Wikipedia article. Upload or embed your video into your Update. You are also welcome to create your own infinite paradox or adapt an existing one with new characters, settings, or situations, but with the underlying principle. If you do one of these latter options please be sure to explicitly reference your original paradox. Be sure to include any references you use in the body of your Update. The video should be 5 - 10 minutes long.
OR
(2) Create a Problem Set of formal limit questions, including at least 10 questions with a mix of conceptual questions as well as function-analysis questions that ask you to find the limit of various functions, oftentimes at points that are not within the domain of the original function (holes, asymptotes). Provide an answer key.
It is important to remember here how much of a conceptual leap this work with infinities can be for first-time students. These were big, abstract and in some ways revolutionary ideas for their time (every time they came up, which was again and again throughout history!) so take things slow, be open to lots of tough questions you may not be able to answer, and as always, patience!
This is also an opportunity for students to get to explore some interesting paradoxes and ideas of mathematics, and some students may really take to the issue of cardinalities of infinity and the rest - encourage as much outside research as the students have time for!
Activity:
Have students pair up and each will take turns in the two roles: Function Dreamer, Function Creator. The Dreamer's job is to imagine a function with characteristics that they like, related to limits. For instance, the Function Dreamer may want a function with a horizontal asymptote at \(y=2\) and vertical asymptotes at \(x=-1, 0, 1\). The Creator's job is to come up with a function that has all of those properties or states if they think such a function cannot exist.
Learning Objectives:
- Be able to correctly use limit notation to describe the limits of functions
- Be able to use graphs, tables, and algebraic methods to find the limit of various functions
- Be able to use one-sided and two-sided limits to analyze functions
- Understand that a limit at a value does not mean the function exists at that value. Understand the role limits play in mathematics, how they came about, and some of the earliest controversies when talking about infinity, something limits oftentimes deal with
- Look for real-world connections and applications of the limit and categories of infinities
Estimated Time: 1 - 2 Weeks
The framework of limits allows us to answer an important question in mathematics: what is the instantaneous rate of change of a function?
Although it can at first seem a contradiction to consider something both changing and frozen at a single instant in time, we are in fact exposed to this paradox every day.
Figure 9 - A video about the paradox inherent in the derivative (3Blue1Brown, 2017).
The speedometer on a car or bus gives the driver their instantaneous velocity. A speedometer might read "45 kilometers per hour" but if you did not travel at that speed for a full hour you would not go 45 kilometers. The implicit understanding here is that the speedometer is telling you your changing moment-to-moment instantaneous velocity.
For more information on how a speedometer works in practice, howacarworks.com is a good place to start.
To find the instantaneous rate of change of a function we can start by finding the slope between two points on a line: \(m = \frac{y_2-y_1}{x_2-x_1}\) and this would be our average rate of change and graphically would be the slope of the secant line connecting those two points on the function. Then, using the power of limits, we can take the limit as \(x_2 \rightarrow x_1\)and analyze these for answers using the tools developed in Update 2. This process can be seen in the animation below where instead of discussing two points, we discuss one point, \((a, f(a))\) and a second point some arbitrary but decreasing distance h away: \((a+h, f(a+h))\)
For another example, the slope of the function \(f(x)=x^2\) at the point \((3, 9)\) can be expressed as
\(\lim\limits_{x \to 3} \frac{x^2-9}{x-3}\), the limit of the slope between that point and an arbitrary point \((x, x^2)\) that is getting infinitely close to \((3,9)\). Through our work with limits we can expand the difference of perfect squares on the numerator, cancel with the denominator, and solve by direct evaluation, yielding a final result of \(\lim\limits_{x \to 3} \frac{x^2-9}{x-3}=\lim\limits_{x \to 3} (x+3)=(3+3)=6\), meaning that our function \(f(x)=x^2\) has an instantaneous rate of change of 6 at \(x=3\). This instantaneous rate of change is also the slope of the tangent line to the function at that point.
In calculus, this instantaneous rate of change and slope of the tangent line at a point is given an important name: the derivative, and for a given function \(f(x)\) its derivative function is denoted \(f'(x)\). The definition of the derivative is often given one of two ways:
\(f'(x)=\lim\limits_{x \to a} \frac{f(x)-f(a)}{x-a}\)
\(f'(x)=\lim\limits_{h \to 0} \frac{f(x+h)-f(x)}{h}\)
Follow the links below for more information on The Limit Definition of the Derivative and how to use it to find the slope function (the derivative) of a parent function, or to find the slope at a single point:
Paul's Online Math Notes on The Definition of the Derivative
CalcWorkshop on Limit Definition of the Derivative
KhanAcademy on Limit Definition of the Derivative
Figure 11 - Definition of the derivative (TheOrganicChemistryTutor, 2018).
You can also check out SymboLab's Definition of the Derivative Calculator
By employing limits to talk about the slope between two points as they converge on top of one another, we can understand in more abstract terms what the speedometer in a car is telling us, that is, what we mean when we say that our velocity is our instantaneous rate of change. This rate of change on a graph is the slope of the graph not between two points, but at a single point. The instantaneous rate of change of a function at a point is its tangent line slope at that point.
Essential Media:
NancyPi's video on the Definition of the Derivative
OR
KhanAcademy's video on the Definition of the Derivative
OR
OpenStax Chapter 3 Introduction - 3.2 The Derivative as a Function
Learning Objectives:
Comment:
Answer the following two questions: What is an example of an instantaneous rate of change that you have encountered in your daily life outside of the speedometer reading of a vehicle? What are the units of this rate of change? Also: If you were to talk about the way that your velocity is changing, what would be your units? What is the name given to this rate of change in your velocity?
Update:
(1) How close can you get to measuring an instantaneous velocity in real life? Record a video of you attempting to do so. You will need a stopwatch (one on a phone will do fine), something that moves, and a way to measure the distance that is appropriate for the speed and time your object will be moving. Once you have a moving object record pairs of distance/time measurements, attempting to get those measurements as close together as possible. You can work with a partner and one of you can be the 'moving object.'
OR
(1) Research the history of calculus, its early theoreticians, and its development within the context of the relevant time. Focus on what types of questions was calculus interested in answering, how it was received, and how it began to develop afterward. Include at least 3 media elements and cite any sources you use.
OR
(2) Construct a problem set of questions relating to the Limit Definition of the Derivative and the Tangent Line. Include a mix of conceptual as well as directly solvable problems. Include an answer key with model answers. The problem set should be at least 10 questions long.
Hopefully with enough grounding in work with limits will have left students feeling comfortable for the move ahead: using limits to find the instantaneous rate of change of a function. This is a critical time in that it is the first time we are really talking about the derivative in any formal sense - so again, go slow, and have patience. These were again ideas that took a long time to develop in human history and it is not surprising that many high school students feel a bit of over-awed dread when they hear the term "calculus."
Remind students that they have seen what we are talking about every day - speedometers, instantaneous velocity, and speed are all nothing new. When we see rockets launch from their pads we know that they are going faster at every moment - they have an acceleration, a velocity, a position function, and other orders of derivative functions on top of those.
Activity:
The activity mentioned for the learner Update can also be done in class - finding a way to get as close as you can to measuring the instantaneous velocity of something. Alternatively, look around the school for something like a water fountain or faucet and think about how you might be able to measure the instantaneous flow rate of the water coming out. How close can you get? How does taking closer and closer measurements to change your predicted instantaneous flow rate?
Learning Objectives
- Understand the meaning of instantaneous rate of change and its connection to a tangent line and the slope at a point
- Be able to use the Limit Definition of the Derivative to find the derivative of a function at a point and to find the derivative function itself
- Be able to estimate a derivative from a table or graph of a function
- Be able to explain in words the connections between velocity, the derivative, slope, and tangent lines
Estimated Time: 1 Week
As would have quickly become obvious, the limit definition of the derivative can be quite cumbersome, and problems can become overly difficult to solve using the algebraic manipulation of expressions resulting from using the limit definition. So instead a large number of rules have been found over time that can be learned and employed to find the derivatives of the Elementary Functions and their composites.
Figure 12 - Rules for differentiation (The Organic Chemistry Tutor, 2019).
Basic Derivative Rules
The Basic Derivative Rules, by The Organic Chemistry Tutor
Basic Derivative Rules and Proofs by SFU
Derivative Rules from MathIsFun
The Power Rule on Khan Academy
Derivative of Constant, Addition, Difference, and Constant Multiples on Khan Academy
Differentiating Polynomials on Khan Academy
Derivatives of Sine/Cosine on Khan Academy
The Product, Quotient, and Chain Rules
The Product Rule on Khan Academy
The Quotient Rule on Khan Academy
The Chain Rule on Khan Academy
(Note that all above Khan Academy videos can be found under the Calculus AB page)
Combined Rules from CalcWorkshop
The Chain Rule by The Organic Chemistry Tutor, included below:
Figure 13 - The Chain Rule (The Organic Chemistry Tutor, 2018).
A visual proof of the product rule can also be helpful:
Figure 14 - Visual proof of the product rule (Veitch, B., 2013).
Another visual proof of the product and chain rule by 3Blue1Brown:
Figure 15 - Visualizing the chain and product rule (3Blue1Brown, 2017).
With all of these formal derivative rules to work with, we can now begin to look toward more application-oriented uses of the derivative. Keep in mind that we will have to continue to limit ourselves to functions and phenomena expressable as a single variable. After single-variable calculus comes multivariable, and this then opens up ways to model increasingly more complex and realistic scenarios.
Learning Objectives:
Essential Media:
TheOrganicChemistryTutor's Basic Rules for Derivatives, Product Rule, Quotient Rule, and Chain Rule
OR
OpenStax Chapter 3.3 - 3.7
Comment:
The Chain Rule allows us to find the derivative of compound functions, and in doing so we can now take the derivative of ever-more-complicated functions. Can you come up with an example in the real world of a "function inside of a function"? Consider nested processes and nested levels of causality. Assign letters to your function and variables, for example: "Let x be the amount of fertilizer used on a farm. Then \(f(x)\) represents the amount of corn, \(f\), as a function of the amount of fertilizer used. The amount of high fructose corn syrup that can be processed from a given weight of corn is given by the function \(w\), which is a function of the amount of corn which is a function of the amount of fertilizer used. The output of corn syrup is dependent upon the amount of fertilizer: \(w(f(x))\).
Update:
(1) Create an educational video that outlines one of the limit proofs of the properties of the derivatives. You can base your video on Paul's Online Math Notes page on Derivative Proofs or any other accurate source (be sure to cite). The video should be at least 5 minutes and though it does not need to be entirely rehearsed there should be little downtime in the video and it should feel prepared beforehand.
(2) Create a problem set about the Rules for Derivatives that mix conceptual and applied problem questions together. There should be a slight focus on the applied problems for this set as there are a lot of practical differentiation skills here to practice and learn. Include at least ten questions and include an answer key.
With a conceptual framework under our belt, we need a lot more rules to be able to work with functions in an analytically meaningful way - so sometimes there is just no getting around needing a bunch of new Rules. I'd use this opportunity to have students work on as many problems as possible. If you have the whiteboard space it can be incredibly helpful to have students work in small groups at the board - allowing you to move about the room and see how the students both work together and work through problems.
Stress that rather than think about memorizing they should expect to remember the rules more and more as they practice with them - so again, get them to it! This can be a fun time for a calculus student as well - things may be starting to click, and sometimes the move away from more conceptual work and back into the "apply the rules like so and so" can be a relief to students who would never admit it but at this point miss a bit of factoring or algebraic manipulation.
Activity:
Once you have covered all of the rules up through the Chain Rule, have students form into small groups and head to the board. Have them draw a large Venn Diagram with the following categories: Power Rule, Chain Rule, Product Rule, and Quotient Rule. Start listing off functions, one at a time, and have students categorize them into the Venn diagram based on what rules would be required to take the derivative of each. For instance, \(\frac{d}{dx}\left(sin(x^2)\cdot{5x^3}\right)\) requires the Power Rule, Product Rule, and Chain Rule, so it would go at the intersection of all three. Interesting cases include ones where rational expressions simplify or where algebraic expressions can be expanded to avoid chain rules, among others.
Learning Objectives:
- Be able to identify appropriate usages of the Power, Product, Quotient, and Chain Rule
- Be able to use the Power Rule for rational exponents and negative exponents
- Be able to combine various derivative rules in order to find the derivative of more complicated functions
Estimated Time: 2 Weeks
One interesting application of the derivative is in finding the way that two rates are related. For example, consider the following inverted circular cone which has water flowing into it at a constant rate:
A question to ask is: at what rate is the water level rising at any given moment? If you ponder the shape of the cone and consider the constant flow rate, you may deduce that for a fixed flow rate the cone will fill most rapidly at the bottom where the cone is narrowest, and will fill slowest near the top where the cone is widest. This is a question of how one rate, the inflow of water, relates to another rate, the changing height. Calculus and the derivative can help us answer these types of questions!
Implicit Derivatives
To do so we must first introduce the idea of an implicit derivative, itself an interesting application of the chain rule. We take two variables, for example, x and y, and treat one as a function of the other. For something like \(y^2\), if you were to differentiate it with respect to its original variable, y, you have a straightforward derivative: \(\frac{d}{dy}y^2=2y\)
But if you were to differentiate \(y^2\) with respect to a new variable, like x, then we are forced to treat y as a function implicitly defined in terms of x and must as stated above, employ the chain rule:
\(\frac{d}{dx}y^2=\frac{d}{dy}y(x)^2=2y(x)\cdot{y'(x)}\) which is often simplified to: \(\frac{d}{dx}y^2=2yy'\)
Further examples can be found in the links below.
Khan Academy Video on Implicit Differentiation
Paul's Online Math Notes on Implicit Differentiation
The Organic Chemistry Tutor on Implicit Differentiation
OpenStax on Implicit Differentiation
SymboLab's Implicit Derivative Calculator with Steps
The video below by 3Blue1Brown provides a more in-depth look at implicit differentiation with a focus on the conceptual level:
Figure 17 - Implicit differentiation (3Blue1Brown, (2018).
Related Rates
With the concept of implicit derivatives to employ we can now return to the question of how two rates are related to one another. Let's take a simple example, the area of a square whose side length's are x. The area of the square then is \(A(x)=x^2\). So for any given side length, we can find the corresponding area. But calculus and the concept of the derivative allows us to put this situation into motion, and we now imagine that our square is growing in size.
We can call this rate of growth of the side \(\frac{dx}{dt}\)using the language of calculus and the derivative. This is the change in the length of x as a function of the change in time. But our area function doesn't include time as a variable! What we do then is assume that implicitly, the side length was always a function of time, because after all, it is changing as time goes on. So we take the derivative of \(A(x(t))=x(t)^2\) with respect to time: \(\frac{d}{dt}A(x(t))=\frac{d}{dt}x(t)^2\) which can be solved: \(\frac{d}{dt}A(x(t))=\frac{dA}{dt}=\frac{d}{dt}x(t)^2=2x(t)\frac{dx}{dt}\), and so we have an explicit expression connecting the change in area, \(\frac{dA}{dt}\), to both the rate of change of the side length, \(\frac{dx}{dt}\) and how big the side length is at that moment, \(x\).
Khan Academy on Related Rates
Paul's Math Notes on Related Rates
The Organic Chemistry Tutor on Related Rates
UC Davis Mathematics on Related Rates
OpenStax on Related Rates
Related Rates are just one of the multitude of applications of derivatives and calculus in general. Our final Update of this Learning Module will take us through another critical one: finding minima, maxima, and using this to optimize real-world relationships.
Essential Media:
TheOrganicChemistryTutor on Implicit Differentiation and Related Rates
OR
OpenStax on Implicit Differentiation and Related Rates
Learning Objectives:
Comment:
Give an example of a "related rate" that you have encountered in your own life. This can be as practical or as whimsical as you want. Explain how the rates are related and what the units of the related rates are.
Update:
(1) Create a video in which you act out or demonstrate one of the related rates. Do your best to try and capture measurements of the rates at an individual moment in time at least twice to show how those rates change. For example, if you are filling a cone at a constant rate, show that the rate of change in the height is different when a measurement is taken at the bottom versus when one is taken at the top.
(2) Create your own worksheet of at least 4 unique Related Rates problems. Try and come up with entirely new scenarios from the ones you have seen, or create more elaborate stories surrounding the problems. Include an Answer Key.
Implicit Derivatives and Related Rates can both represent challenging topics for students. It helps to understand the derivative here at a conceptual level, and therefore be more comfortable with the idea of "taking the derivative" with respect to a new variable. Continue to stress the idea that we are considering one or more of the variables in an expression as implicit functions of some other variable, usually \(x\) or \(time\) in the case of Related Rates
Implicit Derivatives
One idea is to frame it as taking the derivative of the same expression but with respect to different variables:
\(\frac{d}{dy}y^2=2y\)
But what about
\(\frac{d}{dx}y^2\)
It only makes sense to consider the rate of change of y with respect to x if y was somehow a function of x, which we would write as \(y(x)\), and then our derivative expression becomes:
\(\frac{d}{dx}\big(y(x)^2\big)\)
Pause here and make sure that students can see and understand the difference between the above two expressions. Furthermore, can they then make the connection to why we need the Chain Rule for the second expression, but only the Power Rule for the first? Finally, what is the inside and what is the outside function in this case? Further examples may also be helpful.
\(\frac{d}{dx}\left(y(x)^2\right)= \Big(2\cdot{y'(x)\Big)}\cdot{y'}\)
The brackets on the right are not necessary but help delineate the \(f'\big(g(x)\big)\) part of the Chain Rule.
Activity:
Have students partner up and create simple Implicit Derivative questions of two variables, for example: \(2y^2x=x^2+3y\) or \(sin(xy)=y^2-x\). Stress to not make them too challenging, which is a function of how many nested rules they need to use, but also not trivially simple. Aim for the use of at least one derivative rule, but no more than 2 or 3 of Product, Quotient, Power, Chain. Once they have their questions, swap, and try to find an explicit derivative expression, \(\frac{dy}{dx}\)
Related Rates
Stress the connection between Related Rates and Implicit Differentiation: in the same way we treated y as an implicitly defined function of x, we will now treat some new variable, dependent upon the situation, as a function of time. So for example, we can take the volume of a sphere, which is a function of its radius, \(V(r)=\frac{4}{3}\pi{r^3}\), and now imagine that we had a sphere whose radius was growing as a function of time. Now we would not be trying to find out how the volume changes as a function of the radius, which would be \(\frac{dV}{dtr}=\frac{d}{dr}V(r)=\frac{d}{dr}\Big(\frac{4}{3}\pi{r^3}\Big)=4\pi{r^2}\)
Instead, we are taking the derivative of that same volume expression, but now where we encounter the radius, we treat that as a function itself, one of time, and must use the Chain Rule accordingly:
\(\frac{dV}{dt}=\frac{d}{dt}V(r)=\frac{d}{dt}V\big(r(t)\big)=\frac{d}{dt}\big(\frac{4}{3}\pi{r(t)^3}\big)=4\pi{r^2}\frac{dr}{dt}\)
Activity:
This can be an opportunity to allow students some creativity and have them create their own Related Rates scenarios. Start with ones that adhere closely to examples already provided in class, but with more whimsy or crazier numbers. If some students are up to the challenge they can begin to look for more challenging Related Rates relationships - other volumes, shapes, or geometrical relationships, etc.
Learning Objectives:
- Understand how Related Rates is an application of the Chain Rule
- Be able to find the explicit derivative of an implicitly defined function of two variables
- Understand the connection between Implicit Differentiation and Related Rates
- Be able to analyze word problems to find the underlying function relationship and then use the technique of Related Rates to solve for unknown quantities
- Be able to make connections to the real world about the many instances of rates of change being dependent upon one another
Estimated Time: 1 - 2 Weeks
The final application of derivatives that we will look at is its use in finding the relative minima and maxima of a function, and the application of this to optimization. Calculus and the derivative give us the tool to find the local highs and lows of any function that we can take the derivative of. This is an incredibly powerful tool because functions describe the world. And calculus allows us to find things like the highest point in a rocket trajectory, or a steady-state equilibrium for a complex system, it can tell us how to maximize profit, minimize loss, minimize the material to pack a certain volume, and a nearly infinite number of other applications that pervade our daily lives.
Critical Points & The First Derivative Test for Local Extrema
Although there are many nuances to explore, we can begin with a very simple definition of a critical point in calculus: a critical point, c, is a place where the first derivative equals zero or does not exist. Now, a good question is: why are these points critical? To begin to understand why we can look at the following from CUEMATH:
A local maximum or minimum is a local "flat" area of the graph. For a maximum, the function had to be increasing (getting bigger) before, then was zero, and then began to decrease. A local minimum works in reverse.
A point where the first derivative does not exist can also be a local maximum or minimum, as this image from BYJU'S demonstrates:
For more detail on how to use what is known as The First Derivative Test for Local Extrema, see the links below:
Khan Academy on First Derivative Test
Wolfram MathWorld on First Derivative Test
BYJUS on First Derivative Test
Paul's Online Math Notes on Shape of Curves and the First Derivative Test
OpenStax on Shapes of Curves and the First Derivative Test and Maxima and Minima
The Second Derivative Test for Local Extrema
By employing the second derivative and concavity we can often make the determination of minima and maxima easier: if a second derivative exists at a critical point, c, then a positive second derivative (concave up) at c indicates a local minimum, and a negative second derivative (concave down) at c indicates a local maximum. Details can be found in the OpenStax link above, or via the following links:
Paul's Online Math Notes on Shape of Curves and the Second Derivative Test
Wolfram MathWorld on The Second Derivative Test
CalcWorkshop on The Second Derivative Test
KhanAcademy on The Second Derivative Test
The image below from CalcWorkshop demonstrates this principle:
Optimization
What if the function we are seeking to find the minimum or maximum of has a real-world significance? As noted at the beginning of this Update, the derivative is used in a process known as optimization to model real-world scenarios with a function and then use the derivative to find its maximum or minimum. There are a number of unique problems within the category of single-variable optimization, but they all maintain a similar structure: identify knowns and unknowns and express these as mathematical and derivative quantities or variables - oftentimes drawing a diagram or model, or sketch of the situation to aid this process. Relate all of these quantities together through a function, and lastly, take the derivative of that function and use the First or Second Derivative Test to find the local extrema. Oftentimes easier said than done!
Paul's Online Math Notes on Optimization
OpenStax on Applied Optimization Problems
CalcWorkshop on Optimization
KhanAcademy on Optimization Problems
Above is a classic optimization problem: cutting the corners off a box and folding up its resulting sides in order to maximize the volume of the box for a given size of cardboard. For more information on how to solve it, check out the link above to Paul's Online Math Notes, Example 5 of Optimization.
Optimization is just one of the many applications of differential calculus and calculus in general.
Learning Objectives:
Essential Reading:
KhanAcademy on First Derivative Test, Second Derivative Test, and Optimization
OR
OpenStax on First Derivative Test, Second Derivative Test, and Optimization
Comment:
What is a situation in your own life when you would have liked to optimize something? Could this have been something reasonably or realistically presented as a function of one or possibly multiple variables? Describe such a function and what optimization of it might look like.
Update:
(1) Research applications of calculus to a field of your choice. Provide some examples of how they are used in context. Be on the lookout for partial derivatives which look a little different but are conceptually very similar.
(2) Create at least five unique optimization problems of your own. Include all necessary information and diagrams or pictures where helpful and appropriate. Include an Answer Key.
This final Update is perhaps the most applicable to the real world and gets to one of the most powerful tools of calculus: the ability to find local maxima and minima of functions. Stress to students that Optimization is just an applied version of finding minima and maxima - the trick is to just find the underlying function that describes the real-world dynamic at play.
Throughout this lesson, it is important to have students continue to make connections between the graph of the function and the graph of its derivative. This is a good time to start introducing the activity of drawing a derivative graph from a sketch of a function on the board - this is a great warm-up activity for every class.
Take time with the First Derivative Test for Local Extrema and do not introduce the Second Derivative Test until students feel very comfortable with the First. Also, be ready to answer questions or address confusion about the Second Derivative Test and Points of Inflection and Intervals of Concavity.
For Optimization, stress the importance of drawing pictures, writing out what you know, and using the tools and language of calculus. In addition, ask students what the difference is between Related Rates problems and Optimization problems, stressing the differences in form and what we are talking about. For example, the "answer" in a Related Rates problem is a rate of change of some sort. For Optimization, it is more often a fixed amount or optimal value.
Activity:
Have students research optimization applications out in the real world. Many may be beyond the scope of the class, but they can understand the general principles if they understand that the goal is to model something with a function (probably of many variables for the real world) and then use derivatives to optimize it. Alternatively, have students come up with their own fanciful optimization problems similar to the exercise with Related Rates.
Learning Objectives:
- Understand critical points in relation to derivatives and their role in finding relative extrema
- Be able to use the First Derivative Test to find local extreme values
- Be able to use the first derivative to find intervals of increasing and decreasing
- Be able to use the Second Derivative Test to find local extreme values
- Be able to use the second derivative to find intervals of concavity and points of inflection
- Understand the role of calculus in finding optimum solutions in the real world
- Be able to use the technique of Optimization in word problems
- Make connections between calculus and applications in the real world
Estimated Time: 2 Weeks
Math Research Project
The purpose of this Peer Reviewed Project is to allow you the freedom to explore a topic within mathematics that is of interest to you, along two broad categories: historical or conceptual. You may investigate either a period, person, or place in mathematical history, or, you can research a field of advanced mathematics (beyond calculus) at the undergraduate level, graduate level, or higher. Examples for the two topics are given below.
Topic 1: A famous or non-famous mathematicians or groups of mathematicians, a particular school, country, or geographic region of importance to mathematical development, a particular time period of mathematical development and societal change, a significant controversy in mathematics or applied mathematics, a famous feud between two mathematicians or two schools of mathematical though, or anything broadly within this vain. It is important throughout to answer: what questions were these people or times trying to answer? What was their conception of the world and how did that influence the work? Who influenced them/it/when and what was the lasting influence?
Topic 2: Linear Algebra, Analysis, Real Analysis, Complex Analysis, Abstract Algebra, Group Theory, Graph Theory, Knot Theory, Chaos & Dynamics, Fractals, Topology, Differential Equations, Number Theory, Set Theory, Combinatorics, Statistics, or other fields with instructor approval. Your paper should include an explanation of what types of questions this branch of mathematics is trying to answer, the major mathematicians known within this field, and potential applications to the real world, including the sciences.
Requirements
Your finished Work should include an Introduction, Personal Alignment, Main Body, Conclusion, and References Section. The Main Body should be at least 1000 words, include at least 5 media elements, and properly cite all sources. Your Personal Alignment should explain why you chose the topic you did and its personal relevance to you.
Rubric & Assessment
The rubric below can guide you in the creation of your work. Your overall grade on the Work is related to how well it adheres to the core directions of the Knowledge Process. The rubric is adapted from the Work 1 rubric by Kalantzis & Cope. You are expected to complete two Peer Reviews of other works and will have your own reviewed twice as well.
One of the major goals of this Peer Reviewed Work is to give students an opportunity to see mathematics as part of a larger societal context: how it fits into a broader human history, or how it is alive and growing today as an existing body of work.
If your school has a library and librarians that students can access online or in person, they can be a great resource in helping students take on the task of researching a topic that may not have as many ready google results as some other areas that they have looked at before.
Encourage students to pick topics of personal or connection to them. Wherever possible allow time for peer discussion and feedback sessions so that they can help each other with positive feedback and further questions.
Please complete the following survey at the beginning of the course so that the Instructor can better understand your mathematical background. Please take some time to answer the survey and answer as honestly as you can - the feedback you give here is essential in shaping future iterations of this course
The questions for the opening Knowledge Survey are provided below. The hope is to get students to begin to think about themselves as math learners and reflect critically upon their own experiences in the math classroom: what were the positives, what were the negatives, and do they see themselves as a "math person"?
Knowledge Survey
For The Limit & The Derivative
- Name
- Preferred Pronouns
- Previous Math Course
- Describe your Previous Math Course in your own words: what were the topics, types of problems, etc.
- In your own words, define the word "function" in the real world and in mathematics (if there is a difference in your opinion)
- What words would you use to describe yourself as a math student?
- What words would you use to describe your previous math course?
- To what extent did your previous math course emphasize collaboration?
- To what extent did your previous math course emphasize independent problem-solving?
- To what extent did your previous math course emphasize real-world applications?
- Do you feel that a math classroom is a place where your voice can be heard or your identity shared safely? Please explain to whatever extent you feel comfortable.
- Describe the characteristics of your favorite classroom experience so far (it does not have to be mathematics). Please explain why this was such a memorable or enjoyable experience.
- Do you consider yourself to be a "math person"? Why or why not? Was there ever a time when you would have answered differently, and if so, what changed?
- What do you hope to learn from this math course? Or, what do you hope to "get out of it?"
Participation Requirements
(Also included under Overview & Learning Outcomes for clarity)
Comment on Admin Updates: Comment on all six admin updates, responding to the prompt provided. Comments should be at least 50 words.
Updates: Create six of your own updates, each one a response to one of the six Admin Updates. Choose three from category (1) and three from category (2). In general, updates should be about 200 words for reports, 5 - 10 minutes for videos, and 10 questions for problem sets, unless otherwise specified.
Comments on Peer Updates: You should respond to six Peer Updates from category (1), and six Peer Updates from category (2). Comments to Peer Updates of Category (2) are in the form of your own attempt at completing the problem set they have created and verifying their own provided answers for accuracy.
Peer Reviewed Work: Your first Draft should be a nearly complete work, ready for Peer Reviews. You are expected to incorporate and comment on the three Peer Reviews you receive on your own. Then, make all necessary edits and submit a Final Draft of your work. Finally, perform a Self-Review of your work and then Review the Reviews of your own work.
You can use the following checklist, created in Canva, to aid in keeping track of your progress:
Evaluation
Evaluation is based on the above Participation Requirements. Accuracy of your work in constructed problem sets is important and will arise from the process of your peers' attempts to answer your problem sets and their work to verify your answer keys, as well as the instructor overview.
The evaluation of your Peer Reviewed Work is based on the Rubric provided in the relevant Project section.
Evaluation in this course is primarily through formative assessments - comments, updates, and responses to updates are all meant to be formative in that they are not graded on a strictly right or wrong basis but instead on effort and adherence to the grading rubric.
That being said, there is nothing inherent with this Learning Module that prevents summative assessments at the end of each Update, which could take the form of individual or group assessments (test) or problem sets, which would be more open-ended and long-form answers.
As the instructor it is important to keep a close watch on the Update 2's - make sure that students are creating substantial problem sets and that when students are finding any errors when they go and attempt to correct them. If a student continues to make errors on their own problem sets you might encourage them to try more elementary functions with fewer transformations or to revisit some of the learning material.
You might also consider administering summative assessments, but then allowing for substantial corrections, even multiple corrections, until the finished product is completely correct. Because of the open-ended time frame of these learning modules students have some flexibility in revisiting topics.
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3Blue1Brown. (2017) Visualizing the chain and product rule [Video]. YouTube. https://www.youtube.com/watch?v=YG15m2VwSjA&t=243s
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