Applied Calc. Cohort -
Let’s draft the outline of our OER below to include all the parts or chapters that we envision.
Applied Calculus outline first draft:
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Limits
Evaluate limits
Apply the concepts of limits and continuity to functions. -
Derivatives
Interpret Derivative as a slope of tangent line and an instantaneous rate of change / Determine the tangent line to the function at a specific given point.
Find the derivative of functions using the Power Rule, Product Rule, Quotient Rule and others…
Relate the concept of the derivative to that of a rate of change
Apply the first and second derivative tests to locate relative extrema. -
Integrals
Use techniques of integration to find particular or general antiderivatives.
Apply integration and the Fundamental Theorem of Calculus to evaluate definite integrals.
Use the concept of the integral to solve problems involving area between curves.
Describe the relation between Definite Integral and areas between graph and horizontal axis. -
Applications (could be included in previous sections instead of being a stand-alone chapter.)
Solve applied problems involving business, economics, and the social sciences.
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Thank you Karen for sharing your draft. Here are my thoughts:
Chapter 1: Limits and Continuity
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Estimating limits
- from graphs and tables
- limits by direct substitution
- properties of limit
- limits using algebraic manipulation.
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Continuity
- Continuity from graph
- Continuity using limits
- Types of discontinuities
- Removing discontinuities
Chapter 2: Derivative
- Average vs. instantaneous rate of change + Secant lines and tangent lines
- Formulas for Derivatives: Power rule: constant, sum, difference, and constant multiple
- Product rule and Quotient rule
- Combining formulas for derivatives
- Chain rule
- Second derivative, Concavity, and Inflection points
- Solving Optimization Problems
- Implicit differentiation
Chapter 3: Integrals
- Approximation of Area under the curve
- Fundamental Theorem of Calculus
- Formulas for Antiderivatives
- Substitution
- Area, volume, and the average value
I hope we can do a short zoom meeting soon and settle on a few things and commence our initial step.
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Hi all! I am getting ready to dig in and compare our outline to our working textbook. I think that it would be prudent to not delete any sections until our outline is fully agreed on. Maybe we can move them to a chapter named “Delete later” until we are sure. I teach pure calculus, so applied calculus is a little foreign to me, but I have been reviewing the Business Calculus OER that we were talking about using and I saw this in the introduction, which I think is a great reminder of what this course is trying to accomplish. I will cut and paste it from that source below:
How is Business Calculus Different?
Students who plan to go into science, engineering, or mathematics take a year-long sequence of classes that cover many of the same topics as we do in our one-quarter or one-semester course.
Here are some of the differences:
No trigonometry
We will not be using trigonometry at all in this course. The scientists and engineers need trigonometry frequently, and so a great deal of the engineering calculus course is devoted to trigonometric functions and the situations they can model.
The applications are different
The scientists and engineers learn how to apply calculus to physics problems, such as work. They do a lot of geometric applications, like finding minimum distances, volumes of revolution, or arc-lengths. In this class, we will do only a few of these (distance/velocity problems, areas between curves). On the other hand, we will learn to apply calculus in some economic and business settings, like maximizing profit or minimizing average cost, finding elasticity of demand, or finding the present value of a continuous income stream. These are applications that are seldom seen in a course for engineers.
Fewer theorems, no proofs
The focus of this course is applications rather than theory. In this course, we will use the results of some theorems, but we won’t prove any of them. When you finish this course, you should be able to solve many kinds of problems using calculus, but you won’t be prepared to go on to higher mathematics.
Less algebra
In this class, you will not need clever algebra. If you need to solve an equation, it will either be relatively simple, or you can use technology to solve it. In most cases, you won’t need exact answers; calculator numbers will be good enough.
I also think that we should keep Chapter 1 Functions and Graphs (since it already exists), but make it an optional chapter. Not all instructors will want to use it. Maybe we can name it Chapter 0 or Prerequisites.
As for applications of integrals, I already mentioned on the forum that Part 2 of the OER has a unit on this. Maybe we can get this uploaded inside this document by the same people who did the original upload.
I would be most comfortable working on the Derivative and Integral chapters and less comfortable working on the applications.
I will not do any work on the book until I hear from the rest of you, but I will try to correlate our outline to our existing source.
I hope to hear from you soon, and I am open to having an extra meeting to get a jump start on this.
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Here is my compilation of Chapter 2 Limits using the source textbook in Pressbooks and the Business Calculus textbook.
The business calculus textbook: Business Calculus
Has a single section which covers both Limits and Continuity, so I think we can trim our Chapter 2 information greatly. Here is what we currently have:
2.1 A Preview of Calculus: topics: introduction to the tangent line problem, velocity, area (I think this should be in the Derivatives Unit if we keep it.)
2.2 The Limit of a Function: topics include finding limits with tables, graphs, 1 sided limits, infinite limits, and vertical asymptotes
2.3 The Limit Laws (MUST KEEP) topics include methods for evaluating limits, but some of the 1 sided limits at the end and the squeeze theorem might not be needed in this course.
2.4 Continuity: topics include the 3 conditions and the intermediate value theorem (KEEP)
2.5 The precise definition of a limit (I think we should exclude) I will hide it in pressbooks.
2.6 Limits at infinity and Asymptotes (I think we should exclude) I will hide it in pressbooks.
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I suggest that we remove the definitions of limits, infinite limits, and limits at infinity, but keep the one-sided limits. I noticed some trigonometry examples in the textbook chapters, which we could replace with similar ones. Also, I propose that we add 2 to 5 self-check exercises for each section to help students practice.
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@karen.perilloux and @bkunwor - thank you for providing your insights here.
@karen.perilloux I like that first part of the Business Calculus text, to help us identify what is the difference between “pure” Calc and “applied” Calc. This should help us get an idea on how to adjust the OpenStax text to start. I like your thoughts with Chapter 2.
I agree with @bkunwor on removing the precise definition on limits, but I think we should keep infinite limits because of the vertical asymptote connection. Later in the text (Chapter 4) it has limits that approach infinity which I think we can remove.
I wonder if the videos from the Business Calc text may be able to be used for our resource? I think making H5P activities could be used for self-check exercises.
I find it hard to scroll around the digital document all the time, so I created a shared document (in the google drive folder) with all of the sections from Chapter 2 and copied the existing learning objectives to it so I can see them all in 1 place and print out a copy to scratch on. Please feel free to edit it. Here is the link:
https://docs.google.com/document/d/1ACydF3-vHFHBvwD6-vui2wfkO28nFY6cmc0ILZ6Xe-0/edit?usp=sharing
I agree with @Jeusea that we need infinite limits because of the vertical asymptote connection, but these are covered briefly in section 2.2. If we agree this is covered adequately, maybe we can still delete 2.6, or maybe we can take pertinent information from 2.6 and add it to the end of 2.2.
Also, do we want to keep 2.1? I’m can’t decide if I think it is too detailed for Applied Calculus or if it gives a good overview. (Reminder that I have more experience in Pure Calculus than Business Calculus, so I am open to all of your input.
And @jeusea thanks for your input in this cohort. I know you have a lot of responsibilities in this project, but with fewer members in this cohort and your experience in teaching this course we appreciate all of your input.
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Hey all -
So my post is another topic was that we are going to have to work with what was brought in for the main resource.
@karen.perilloux - Thanks for sharing the document. I think removing all of 2.1 (just hiding it in pressbook for now) is something we should do. I like all of 2.2, 2.3 (minus the squeeze theorem), removing anything in 2.4 that is not the basic 3 conditions and discontinuities, eliminating all of 2.5, and keeping 2.6. Basically, if you look at Section 2.1 in the Business Calculus resource, whatever is there should be kept at a minimum. I actually don’t see limits at infinity with asymptotes (section 2.6) so I change my mind and say hide it.
We will need to get rid of any example that is trigonometry related.
I also think the Chapter 1 is a good part to have in the resource (minus the trig chapter) to help remember some of the algebra.
