Figure 5 - A tangent line moves along a function. The slope of the tangent line is given by the derivative of the function (Wikipedia, 2022).

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

• Have a clear understanding of what a function is in mathematics
• Understand the various types of Elementary Functions and their Transformations

Limits

• Understand some of the paradoxes and challenges when dealing with the concept of infinity in mathematics
• Use algebraic techniques, tables, and graphs to evaluate the limits of various functions
• Be able to describe the behavior of functions and mathematical relationships in oral and written communication

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

• Be able to evaluate functions using the limit definition of the derivative to find the derivative value at a point and the general derivative of the function
• Understand conceptually the idea of the derivative in mathematics and in real-world applications
• Evaluate limits algebraically, graphically, and via tables
• Describe the derivative in context through oral and written communication

Derivatives - Formal Rules & Applications

• Be able to find derivatives using the Basic Derivative Rules
• Be able to find derivatives using the Product, Quotient, and Chain Rule
• Graphically connect the graph of a function to the graph of its derivative
• Use derivatives to find the local minima and maxima of functions

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.

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)

Figure 6 - Graphs of Some Common Functions (Yager, n.d.).

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:

Figure 7 - SymboLab Function Transformation Calculator Example, symbolab.com

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:

• Have a clear understanding of what a function is in mathematics
• Understand the various types of Elementary Functions and their Transformations
• Understand functions in real-world contexts

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:

Transformations_20Worksheet_201.pdf

Transformations_20Worksheet_201_20Solutions.pdf

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, {n² : 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 {n² : 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:

• Understand what a limit is in mathematics and how it applies to functions
• Understand the real-world study of infinity and its history, including paradoxes
• Understand how limit laws can be used to analyze functions at discontinuities
• Use algebraic techniques, tables, and graphs to evaluate the limits of various functions

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.

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: \((a+h, f(a+h))\)

Figure 10 - A secant line becomes a tangent line as the distance between the points goes to zero (Calculus College, 2017).

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:

• 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
• Be able to evaluate functions using the limit definition of the derivative to find the derivative value at a point and the general derivative of the function
• Understand conceptually the idea of the derivative in mathematics and in real-world applications

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.

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:

• Be able to find derivatives using the Basic Derivative Rules
• Be able to find derivatives using the Product, Quotient, and Chain Rule
• Graphically connect the graph of a function to the graph of its derivative
• Use derivatives to find the local minima and maxima of functions

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. 

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: 

Figure 16 - An inverted circular cone provides a classic use of related rates (chegg.com, n.d.).

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:

• Understand the difference between taking a derivative with respect to the variable a function is already in terms of and what happens when you take the "implicit" derivative of a different variable
• Be able to solve implicit differentiation problems to find an explicit derivative
• Be able to use the technique of implicit differentiation to applied Related Rates problems

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.

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:

Figure 19 - The first derivative test relies on what information the first derivative tells you and how that relates to the shape of the function (Cuemath.com, n.d.).

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:

Figure 20 - Critical points are an important element of understanding the derivative and its applications (byjus.com, n.d.).

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:

Figure 21 - Using the Second Derivative Test to find local extrema (CalcWorkshop.com, n.d.)

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

Figure 22 - Folding an open top box (Dawkins, 2022).

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:

• Be able to use the First and Second Derivative Tests for Local Extrema
• Understand the importance of calculus for its ability to find general solutions to local extrema problems
• Understand the role of critical points in finding local extrema
• Be able to identify intervals of increasing/decreasing and concave up/concave down
• Be able to apply local extrema to issues of optimization
• Be able to create your own optimization problems related to real-world scenarios

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.

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.

Review_20Rubric_20for_20Work.pdf

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

Knowledge Survey

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:

Figure 23 - Jonynas, S. (2022). Assessment Rubric. Created in Canva.

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.