Letter to Students
Why study calculus? Is it because you want to be a doctor, an engineer, a forensic scientist, a biologist, or a mathematician? Or is it because someone told you that it would be "good for you"? Well, all of those are certainly legitimate reasons. But if you aren't quite sure "Why calculus?" here is another reason: It is one of the greatest intellectual achievements of humankind, and, even more to the point, it is easily accessible to anyone who already has studied algebra, trigonometry, and geometry.
Gottfried Leibniz and Isaac Newton independently invented Calculus. Although the notation, terminology, and rigor may have changed or been standardized over the years, the concepts that we will study have not changed fundamentally since the work of the co-inventors in the 17th century.
We will begin Calculus with the study of Rates of Change and what are called derivatives. Rates of Change are all around us. For example, velocity is the rate of change of distance with respect to time; and acceleration is the rate of change of velocity with respect to time. So, we experience derivatives every time we ride in a car, or fly in an airplane. Calculus can also be used to compute areas and volumes of odd shapes with what are known as integrals. Thus, calculus has a place in architecture too.
That calculus is important in real-world applications is not in dispute. In fact, there are so many applications of calculus that studying it really is "good for you." Besides engaging in a worthwhile intellectual endeavor, you will be helping to keep your future career options open. That, from a practical point of view, is not a bad reason at all.
Moreover, studying calculus can be a lot of fun, but we will let you be the judge of that.
How to study calculus. Calculus has a lot of rules. We admit it! Find the derivative of this function, or the integral of that. However, you should train yourself to keep the rules in perspective: Always put the concepts first. When you study a given calculus concept, ask yourself three questions:
- What is the picture (a graph or sketch) that I should have in mind?
- What is the theorem or formula that gives a statement of the concept?
- Do I know how to use the concept in different situations and with different numbers?
The above is what we might call "high-level thinking." It provides a way for you to stay oriented, to know what you are learning, and for what purpose. The rest is in the details, important though they are. The combined process is a lot like using a map to hike through the woods. Yes, you have to look at individual trees and their proximity to local landmarks to find your way, but you always want to keep the map and a compass handy to give an overview of the whole trip. So it goes with the study of calculus concepts. Always know where you are going, the direction to take to get there, and how to accomplish the task. The proof, though, is in getting there!
Practice, practice, practice. In the end, there is no other way. You cannot learn calculus by reading about it. You have to take a pencil and paper and work problems. Learning calculus, or mathematics in general, is a participatory activity. You have to do it to learn it, to make it your own.
Using Calculus on Demand to study calculus.
- The Index page of COD (i.e., the page you see when you go to the web site with your Browser) has a picture of a Leibniz Wheel in the upper left-hand corner. Click on the Leibniz Wheel to find out what it is, and how it represents our philosophy of combining theory with practice in the study of calculus in a computing environment.
- COD is a Calculus I course that gives an introduction to differential and integral Calculus that comprises roughly to the AB part of the AP exam. You will find the topics listed in the green left-hand sidebar of the COD index page. Clicking on a topic will take you to the Lecture-page on that topic.
- Lecture pages: All the Lecture pages have the same layout. There is a short statement of the main thrust of the lecture, followed by these items:
- Quick Question: This section is intended to get you thinking about the topic. Sometimes you can answer this question before studying the material of the Lecture, other times not. If you have answered the question, you can check its correctness by clicking Answer. If you don't think you know the answer to the Quick Question, it is better to wait and try the question again after you have studied the Textbook section, which should allow you to answer the question. As before you then can check the correctness of your answer by clicking Answer.
- Textbook: Here you will find a link to on-line material from a textbook. At present, we are using the book Principles of Calculus Modeling: An Interactive Approach by Donald Kreider and Dwight Lahr. When you click on the topic in the Textbook section of the Lecture page, you will get a list of the material from the sections of this book corresponding to the COD topics. If you click a topic, you will launch an Acrobat Reader pdf file that will display the text on-line. You can open the link for the Lecture you are working on or for any other topic if there is something you have forgotten and want to review, or if you want to peek ahead. The textbook material will open in a separate window so that you can leave it open on your desktop while you are going through the rest of the lecture.
- Today's Homework: The link here will take you to the WeBWorK login page. If you do not have a password, you can sign in as a practice user with practice1 as both the user name and password. If "practice1" is in use, try practice2, or practice3, etc. until you find one that is not in use. After logging in, you will see a page that has the option to "Begin Problem Sets." After selecting it, you will see a list of homework sets that are listed by day. For example, the homework set "Setm3f02day04" corresponds to the homework for COD Lecture 4. Similarly, "Setm3f02day13" is the homework for COD Lecture 13. Ignore the fact that the sets all say "CLOSED;" this just means that the answers are available, and that WeBWorK will not retain your answer once you log off. However, by choosing a problem set and "getting a problem" you may input your answers and WeBWorK will tell you if you are right or wrong. You could also get a copy of the problem set (in pdf).
You may not want to do the homework until you have looked at some Examples and tried the on-line Quiz. This will be true if you are studying the material for the first time. In that case, you should look at the Examples, take the Quiz, and review the Textbook material as necessary before turning to the homework exercises. For those students who have seen the material before, there is nothing wrong with going directly to the homework. You can always come back later to the examples and quiz as needed.
- Quiz: The quiz consists of a set of problems that come up in their own window. You can work out the answers on paper and check them by looking at the COD answers on-line.
- Examples: Here you can find examples of problems and their step-by-step solutions on-line.
- Applets: These carry out little computer programs that illustrate a calculus idea or provide a tool to implement a calculus procedure. Look at and experiment with them. They are meant to help you learn the material better, or to function much like advanced calculators. Don't be shy. Have fun with them.
- Videos: Click on a video to see a calculus problem being worked out by real people. Be on the lookout for Dartmouth graduate students, faculty, and undergraduates. So far, there have been no lucrative film contracts in the offing, but who knows? How long did it take Matt Damon to be discovered?
- Lecture-page sidebar: The green sidebar on the left contains links to several resources. First, there is a link to the Math 3 Course Home Page. Math 3 is the Dartmouth course for which COD is the on-line version. The Math 3 Home Page talks about all of the issues relevant to a student taking the course on campus in that particular term. We thought you might be interested. The next link on the COD sidebar gives the Syllabus for that offering of Math 3. The day-by-day syllabus corresponds to the layout of the Lectures in COD. Be careful, though, and don't be misled. If you are studying calculus on your own it may take more than one day to master the material of a given lecture: having a real live teacher does make a difference!
There is also a link to past Math 3 exams (covering Lectures 1-10, 11-19; and the course final). This link is called Practice Exams and should be used as such to test your understanding of blocks of the COD Lectures. There is also a Textbook Home Page link to the home page of the book Principles of Calculus Modeling: An Interactive Approach by Donald Kreider and Dwight Lahr. Here you will find section-pages that give links to calculus resources on the World Wide Web. And finally, there is a link to the Post a Comment page that we discuss below.
What do you do if you have a problem? If you are studying calculus in a classroom setting, you can talk to other students or to the teacher. However, if you are studying on your own, you can Post a Comment using the bottom link on the green sidebar of a lecture-page. There you can communicate with other students who also are studying calculus on-line. You could either state your problem and ask for help, or you may find that someone has already answered that or a similar question. You may even find questions that you can answer for other students. We encourage you to look at the Post a Comment section of COD on a regular basis.