Calculus 1Saylor Academy
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- Course Introduction
Calculus can be thought of as the mathematics of CHANGE. Because everything in the world is changing, calculus helps us track those changes. Algebra, by contrast, can be thought of as dealing with a large set of numbers that are inherently CONSTANT. Solving an algebra problem, like y = 2x + 5, merely produces a pairing of two predetermined numbers, although an infinite set of pairs. Algebra is even useful in rate problems, such as calculating how the money in your savings account increases because of the interest rate R, such as Y = X0+Rt, wheret is elapsed time and X0 is the initial deposit. With compound interest, things get complicated for algebra, as the rate R is itself a function of time with Y = X0 + R(t)t. Now we have a rate of change which itself is changing. Calculus came to the rescue, as Isaac Newton introduced the world to mathematics specifically designed to handle those things that change.
Calculus is among the most important and useful developments of human thought. Even though it is over 300 years old, it is still considered the beginning and cornerstone of modern mathematics. It is a wonderful, beautiful, and useful set of ideas and techniques. You will see the fundamental ideas of this course over and over again in future courses in mathematics as well as in all of the sciences (e.g., physical, biological, social, economic, and engineering). However, calculus is an intellectual step up from your previous mathematics courses. Many of the ideas you will gain in this course are more carefully defined and have both a functional and a graphical meaning. Some of the algorithms are quite complicated, and in many cases, you will need to make a decision as to which appropriate algorithm to use. Calculus offers a huge variety of applications and many of them will be saved for courses you might take in the future.
This course is divided into five learning sections, or units, plus a reference section, or appendix. The course begins with a unit that provides a review of algebra specifically designed to help and prepare you for the study of calculus. The second unit discusses functions, graphs, limits, and continuity. Understanding limits could not be more important, as that topic really begins the study of calculus. The third unit introduces and explains derivatives. With derivatives, we are now ready to handle all of those things that change mentioned above. The fourth unit makes visual sense of derivatives by discussing derivatives and graphs. The fifth unit introduces and explains antiderivatives and definite integrals. Finally, the reference section provides a large collection of reference facts, geometry, and trigonometry that will assist you in solving calculus problems long after the course is over.Page: 1
- Unit 1: Preview and Review
While a first course in calculus can strike you as something new to learn, it is not comparable to learning a foreign language where everything seems different. Calculus still depends on most of the things you learned in algebra, and the true genius of Isaac Newton was to realize that he could get answers for this something new by relying on simple and known things like graphs, geometry, and algebra. There is a need to review those concepts in this unit, where a graph can reinforce the adage that a picture is worth one thousand words. This unit starts right off with one of the most important steps in mastering problem solving: Have a clear and precise statement of what the problem really is about.Page: 1Quiz: 1
- Unit 2: Functions, Graphs, Limits, and Continuity
The concepts of continuity and the meaning of a limit form the foundation for all of calculus. Not only must you understand both of these concepts individually, but you must understand how they relate to each other. They are a kind of Siamese twins in calculus problems, as we always hope they show up together.Page: 1Quiz: 1
A student taking a calculus course during a winter term came up with the best analogy that I have ever heard for tying these concepts together: The weather was raining ice - the kind of weather in which no human being in his right mind would be driving a car. When he stepped out on the front porch to see whether the ice-rain had stopped, he could not believe his eyes when he saw the headlights of an automobile heading down his road, which ended in a dead end at a brick house. When the car hit the brakes, the student's intuitive mind concluded that at the rate at which the velocity was decreasing (assuming continuity), there was no way the car could stop in time and it would hit the house (the limiting value). Oops. He forgot that there was a gravel stretch at the end of the road and the car stopped many feet from the brick house. The gravel represented a discontinuity in his calculations, so his limiting value was not correct.
- Unit 3: Derivatives
In this unit, we start to see calculus become more visible when abstract ideas such as a derivative and a limit appear as parts of slopes, lines, and curves. Then, there are circles, ellipses, and parabolas that are even more geometric, so what was previously an abstract concept can now be something we can see. Nothing makes calculus more tangible than to recognize that the first derivative of an automobile's position is its velocity and the second derivative of that position is its acceleration. We are at the very point that started Isaac Newton on his quest to master this mathematics, what we now call calculus, when he recognized that the second derivative was precisely what he needed to formulate his Second Law of Motion F = MA, where Fis the force on any object, Mis its mass, and A is the second derivative of its position. Thus, he could connect all the variables of a moving object mathematically, including its acceleration, velocity, and position, and he could explain what really makes motion happen.Page: 1Quiz: 1
- Unit 4: Derivatives and Graphs
A visual person should find this unit extremely helpful in understanding the concepts of calculus, as a major emphasis in this unit is to display those concepts graphically. That allows us to see what, so far, we could only imagine. Graphs help us to visualize ideas that are hard enough to conceptualize - like limits going to infinity but still having a finite meaning, or asymptotes - lines that approach each other but never quite get there.Page: 1Quiz: 1
Graphs can also be used in a kind of reverse by displaying something for which we should take another mathematical look. It is hard enough to imagine a limit going to infinity, and therefore never quite getting there, but the graph can tell us that it has a finite value, when it finally does get there, so we had better take a serious look at it mathematically.
- Unit 5: The Integral
While previous units dealt with differential calculus, this unit starts the study of integral calculus. As you may recall, differential calculus began with the development of the intuition behind the notion of a tangent line. Integral calculus begins with understanding the intuition behind the notion of an area.In fact, we will be able to extend the notion of the area and apply these more general areas to a variety of problems. This will allow us to unify differential and integral calculus through the Fundamental Theorem of Calculus. Historically, this theorem marked the beginning of modern mathematics and is extremely important in all applications.Page: 1Quiz: 1
- Optional Course Evaluation Survey
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- Final ExamQuizzes: 2
- Unit 6: Appendix
By reviewing and having access to this unit, you will have an invaluable list of references to assist you in solving future calculus problems after this course has ended. It is a standard experience, when solving calculus problems on your own, to react to the new problem with the following: "We did not solve that kind of problem in the course.” Ah, but we did, in that the new problem is often a combination, or composition, of two problem types that were covered.
The course could not cover all possible trigonometric functions you will encounter. If you encounter a need for the derivative of tan(x),it is sufficient to recall that tan(x) = sin(x)/cos(x)and that sine and cosine were covered. You can eventually become so good at this that future calculus problems can almost seem to be little more than plugging into formulas.
Engineering students, who have to take several courses that involve the use of calculus, are noted for having a Table of Integrals on their hip wherever they go, such as this one posted on Wikipedia.*