The emphasis in this course is on problems—doing calculations and story problems. To master problem solving one needs a tremendous amount of practice doing problems. The more problems you do the better you will be at doing them, as patterns will start to emerge in both the problems and in successful approaches to them. You will learn fastest and best if you devote some time to doing problems every day.
Typically the most difficult problems are story problems, since they require some effort before you can begin calculating. Here are some pointers for doing story problems:
1. Carefully read each problem twice before writing anything.
2. Assign letters to quantities that are described only in words; draw a diagram if appropriate.
3. Decide which letters are constants and which are variables. A letter stands for a constant if its value remains the same throughout the problem.
4. Using mathematical notation, write down what you know and then write down what you want to find.
5. Decide what category of problem it is (this might be obvious if the problem comes at the end of a particular chapter, but will not necessarily be so obvious if it comes on an exam covering several chapters).
6. Double check each step as you go along; don't wait until the end to check your work.
7. Use common sense; if an answer is out of the range of practical possibilities, then check your work to see where you went wrong.
Suggestions for Using This Text
1. Read the example problems carefully, filling in any steps that are left out (ask someone for help if you can't follow the solution to a worked example).
2. Later use the worked examples to study by covering the solutions, and seeing if you can solve the problems on your own.
3. Answers are given for most exercises; the availability of an answer is marked by "(answer)'' at the end of the exercise. Clicking on "answer'' will cause the answer to be displayed below the exercise. The answers should be used only as a final check on your work, not as a crutch. Keep in mind that sometimes an answer could be expressed in various ways that are algebraically equivalent, so don't assume that your answer is wrong just because it doesn't have exactly the same form as the given answer.
4. Although you should learn how to do the various important calculations involved in studying calculus by hand, it can be helpful to use a computer program for some computations, to check your work, or to determine if two answers are really the same, or to do tedious calculations that you could do by hand, but that are not the main focus of a particular problem. One program to help with this is called Sage; other computer algebra systems are Mathematica and Maple. Throughout the book there are some Sage "cells'' that allow you to use Sage right in the textbook, in your browser. To explain and summarize much of what you'll need to know about Sage, you can also work through the tutorial in chapter 19. If you are new to calculus, much of what you'll see there won't make sense, so you should plan on coming back to the chapter as you learn more about calculus.