Calculus I
Calculus I is the gateway to collegiate mathematics. As such, Calculus I is often a prerequisite course for many majors, both science and non-science.Calculus I introduces the fundamental concept of the derivative, geometrically demonstrated in this animation showing a limit of secant lines approaching a tangent line at a point on a curve y=f(x):
Calculus I also introduces the fundamental concept of the integral, geometrically demonstrated in this animation showing the accumulation of area under a curve y=f(x) of increasing accuracy:
Curriculum: Calculus&LiveMath™
The curriculum for Calculus I is the e-textbook Calculus&LiveMath™, which is converted from the original Calculus&Mathematica™ courseware series by Davis/Porta/Uhl (published by MathEverywhere, Inc.To accompany these LiveMath™ notebooks from this e-text, we also present Video Commentary Lectures on the Basics and Tutorials notebooks in the Calculus&LiveMath™ e-text. These videos demonstrate how to use LiveMath™ as well as discuss the Calculus topics, as if reading and exploring along with the professor (which is exactly what you are doing!).
Examples of the Curriculum
Below are some PDF "print outs" of a few of the LiveMath™ notebooks from Calculus&LiveMath™ by Davis/Porta/Uhl. Included as well is an example homework notebook completed by a student in the course, demonstrating how the homework notebooks become the "common blackboards" that the students and instructor both write on in their "conversation" about the notebook.- Basics Notebook Example: 2.03.B1 - Measurements based on slicing and accumulating: Area and Volume
- Basics Notebook Commentary Movie:
iTunes or
VLC
- Homework Notebook Example: 2.03.G1 - Using the tools: Measurements of accumulation
New E-Text In Preparation
Calculus I via Distance Calculus will soon be offering a brand new, fresh, exciting Videotext as its curriculum guide through the course: That DANG Calculus! (Descriptive, Algebraic, Numerical, Graphical) by Robert Curtis, lead instructor for Distance Calculus.What is a videotext? It is like a textbook, except instead of being based upon printed information, this "text" is based upon video presentations as the core method of explaining the course topics. Instead of a huge, thick 1000-page Calculus textbook to lug around in your backpack, all of this new "videotext" can be loaded into your iPods or iPhones (and soon, the iPad!).
This new videotext features two main types of videos:
- Screencast Videos using LiveMath™
Although we are anywhere from a few miles to a few thousand miles apart, watching these screencast videos is like sitting next to the course instructor, watching his computer, learning the topics of Calculus at the same time as learning how to drive the computer algebra and graphing software LiveMath™. These LiveMath™ screencast videos make up the majority of the video presentations in the videotext.Example Videos requires either iTunes/QuickTime or the alternative VLC Media Player.
- Example Video #1: Play in iTunes or VLC
- ChalkTalk Videos: Manual Calculations
While using a computer algebra software package is a very cool way to do Calculus computations and investigations, we must also pay attention to the classical side of Calculus, and the computations that can be completed by hand with paper/pen/pencil. To be a well-rounded Calculus student, you need to be able to do calculations in both technical and manual methods.Example Videos requires either iTunes/QuickTime or the alternative VLC Media Player.
- Example Video #2: Play in iTunes or VLC
Students Make Videos, Too!
A common event in a grammar or high school is being "called to the board" to present a solution to a math homework problem. There are positive and negative aspects of this classroom activity: the negative include being nervous, embarrassed in front of your classmates, making a mistake, and just being uncomfortable with this academic rite of passage; the positive educational benefits of this activity are quite numerous, including practicing your presentation skills, demonstrating to your teacher that you thoroughly understand the math problem at hand, and providing a sense of accomplishment and confidence when you finish your board presentation.In the Distance Calculus course format, we are able to remove all of the negative aspects of this activity, and leave all of the positive qualities (and more) through the adaptation of this exercise to the Student Video Creation Solutions. Using a webcam and microphone (which many of you may already use for Chat/Skype), the student will audio/video record the presentation of a solution to a homework problem on paper, as if the student had been "called to the board". As you can shoot this solution movie multiple times until you "get it perfect", there is nothing to get nervous about. If you make a mistake during the video shoot, just start over and shoot the video again, or edit your movie to remove the incorrect piece. There are approximately 25 such video creation assignments in the Calculus I course - just enough to demonstrate a mastery of paper/pen/pencil techniques (and maybe some proofs!), but not too many that the activity becomes tiresome and a time burden.
Course Catalog Listing
Course Description: A brief review of algebra and trigonometry; coordinate systems, analytical geometry, the derivative using the definition, limits, continuity, techniques of differentiation; Mean Value theorem and its application, Applications of differentiation to extreme value problems, curve sketching and related rates problems, the integral and its properties, applications of the integral for finding area under a curve, antiderivatives, the Fundamental Theorem of Calculus.
Prerequisite: Precalculus with Trigonometry
Detailed Course Syllabus in PDF
Computer Algebra Software: LiveMath™ Maker
For the investigations of the modern portions of the course curriculum, as well as the classical topics, we utilize the computer algebra and graphing software LiveMath™ Maker.One of the great benefits of LiveMath™ is the ability to see the intermediate steps in an algebraic computation, like this one demonstrating the algebraic simplification of a difference quotient:
Through the Screencast Videos, you'll learn how to drive the LiveMath™ software as you learn the mathematics demonstrated and discussed in the videos, so there is no need to feel apprehensive about learning a new software program. Most students find the software easy to learn, and actually fun to use to explore and experiment with the calculus concepts. As the course progresses, many students find the previously cumbersome manual calculations refreshingly easy to compute in LiveMath™ (although we still practice manual computations because being well-rounded is very important!)
New Curriculum Features
Some of the exciting features of the new curriculum videotext That DANG Calculus! include:- Data, Data, Data!
The core topics of the derivative and the integral are first investigated from the point of view of raw numerical data extracted from real-life examples: Stock market prices and E-Bay auctions. We develop the ideas for the derivative and the integral based upon these numerical examples, and then move to the algebraic development and investigation - much like a good scientist, looking for patterns and concepts in the data, and extracting algebraic conclusions and formulas to describe the data in general. - Precalculus Review
Rather than just "a quick week" of review of precalculus topics, we build strong foundations (and remember forgotten concepts or weaker understandings) through an intensive 20-Assignment review of precalculus including trigonometry. If your precalculus skills are strong, you may go through this review module more quickly, while still learning the workings of the LiveMath™ software. If your precalculus skills need improvement, you will be able to spend the time and effort required to strengthen those skills. - Classical & Modern Approaches
The key to understanding Calculus in this modern technological age is to investigate the topics from both the classical side - without the use of the computer - and from a more modern side, where the use of the computer is not only helpful, but essential. Through this pairing we are able to benefit from both approaches, and achieve "Calculus literacy" in the truest sense of the phrase. - DANG = Descriptive, Algebraic, Numerical, Graphical
Good scientists look at problems from all angles; likewise, we take this reform-based 4-prong approach to investigate the calculus topics, asking at each stage: Examine this Calculus topic....- ... From the D = Descriptive viewpoint.
How can we verbally describe this topic, in narrative, written format. - ... From the A = Algebraic viewpoint.
What is the underlying algebra at work in this topic? How do we approach this topic algebraically? - ... From the N = Numerical viewpoint.
What does this Calculus topic mean in the application to a numerical data example? - ... From the G = Graphical viewpoint.
Given the algebraic and numerical players in this Calculus topic, what do the graphs tell us?
- ... From the D = Descriptive viewpoint.
- Video, Video, Video!
If you have ever been frustrated by trying to read a traditional Calculus textbook, and wondered to yourself, "I wish someone would just explain this!", then we have the answer for you: audio and video discussions of the Calculus topics that you may play once, skim through if a light review is needed, or watch repeatedly - over and over - in case repitition is required to help the concept "sink in".