Calculus 1

Purpose of Course  showclose

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, 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 = X+ 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.

This course provides students the opportunity to earn actual college credit. It has been reviewed and recommended for 4 credit hours by The National College Credit Recommendation Service (NCCRS). While credit is not guaranteed at all schools, we have partnered with a number of schools who have expressed their willingness to accept transfer of credits earned through Saylor. You can read more about our NCCRS program here.

National College Credit Recommendation Service

Course Information  showclose

Welcome to MA005: Calculus I. General information about this course and its requirements can be found below.
Course Designer: Lenny Tevlin
Primary Resources: While this course comprises a range of different free, online materials, the primary source used for this course is:
The chapters and sections of the original text have been reorganized and carefully aligned with the course subunits so that while solving any mathematical problem, the theory you might need to solve the problem is only a page or two away. The best way to proceed through the course is to read the assigned section in the order it is presented. You may download each assigned reading as you work through each subunit. If you prefer to download the entire text for the course, remember to refer to the reading titles, rather than the section or page numbers as these have been revised for the purpose of this course. Be advised that, depending upon your Internet speed, the file can take a couple of minutes to download, as it contains 329 pages and more than 300 megabytes of file size.
Requirements for Completion: In order to complete this course, you will need to work through each unit and all of its assigned materials. Pay special attention to units 1 and 2, as these lay the groundwork for understanding the more advanced, exploratory material presented in the latter units. You will also need to complete problem sets in each unit and the final exam.
Note that you will only receive an official grade on your final exam. However, in order to adequately prepare for this exam, you will need to work through the problems presented for solution.
In order to pass this course, you will need to earn a 70% or higher on the final exam. Your score on the exam will be tabulated as soon as you complete it. If you do not pass the exam, you may take it again.
Time Commitment: We recommend that you dedicate approximately 2 or 3 hours of work every weeknight and 6 or 7 hours each weekend if you expect to perform highly in this course.
This course should take you a total of approximately 130.75 hours to complete. Each unit includes a time advisory that lists the amount of time you are expected to spend on each subunit. These should help you plan your time accordingly. It may be useful to take a look at these time advisories, determine how much time you have over the next few weeks to complete each unit, and then set goals for yourself. For example, Unit 1 should take you 7.75 hours. Perhaps you can sit down with your calendar and decide to complete Subunit 1.1 and Subunit 1.2 (a total of 2.5 hours) on Monday night; Subunit 1.3 and Subunit 1.4 (a total of 3.5 hours) on Tuesday night; Subunit 1.5 and the unit assessment (a total of 1.75 hours) on Wednesday night; and so forth.
Tips/Suggestions: Calculus takes time. Most people who fail a calculus course do so because they are unwilling, or unable, to devote the necessary time to the course.
  • Do not skip topics. The understanding of calculus is typically sequential. It is very difficult to understand one topic after lightly skipping over a preceding topic. 
  • Test yourself. You are testing yourself when you follow the procedure of always solving a problem independently BEFORE looking at a solution of the same.
  • Work on details. Focus on the parts you missed. Determine what you did not understand before moving on.
  • While taking the final exam, you are welcome to make use of this Formula Sheet (PDF).

This course has been developed through a partnership with the Washington State Board for Community and Technical Colleges. Unless otherwise noted, all materials are licensed under a Creative Commons Attribution 3.0 Unported License. The Saylor Foundation has modified some materials created by the Washington State Board for Community and Technical Colleges in order to best serve our users.

Khan Academy  
This course features a number of Khan Academy™ videos. Khan Academy™ has a library of over 3,000 videos covering a range of topics (math, physics, chemistry, finance, history and more), plus over 300 practice exercises. All Khan Academy™ materials are available for free at

A version of this course is also available in iTunes U.
Preview the course in your browser or view all our iTunes U courses.  

Learning Outcomes  showclose

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.

Course Requirements  showclose

In order to take this course, you must:

√   have access to a computer;

√   have continuous broadband Internet access;

√   have the ability/permission to install plug-ins or software (e.g., Adobe Reader or Flash);

√   have the ability to download and save files and documents to a computer;

√   have the ability to open Microsoft files and documents (.doc, .ppt, .xls, etc.);

√   have competency in the English language;

√   have read the Saylor Student Handbook; and

√   have completed MA004, or the equivalent course in Intermediate College Algebra.

Unit Outline show close