Calculus can be thought of as the mathematics of CHANGE. Because everything in the world is changing, calculus helps us track those changes. Calculus is among the most important and useful developments of human thought.
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, where t 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.
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.
Upon successful completion of this course, you will be able to:
- calculate or estimate limits of functions given by formulas, graphs, or tables by using properties of limits and L’Hopital’s Rule;
- state whether a function given by a graph or formula is continuous or differentiable at a given point or on a given interval, and justify the answer;
- calculate average and instantaneous rates of change in context, and state the meaning and units of the derivative for functions given graphically;
- calculate derivatives of polynomial, rational, and common transcendental functions, compositions thereof, and implicitly defined functions;
- apply the ideas and techniques of derivatives to solve maximum and minimum problems and related rate problems, and calculate slopes and rates for functions given as parametric equations;
- find extreme values of modeling functions given by formulas or graphs;
- predict, construct, and interpret the shapes of graphs;
- solve equations using Newton’s method;
- find linear approximations to functions using differentials;
- restate in words the meanings of the solutions to applied problems, attaching the appropriate units to an answer;
- state which parts of a mathematical statement are assumptions, such as hypotheses, and which parts are conclusions;
- find antiderivatives by changing variables and using tables; and
- calculate definite integrals.
