Welcome to MA005: Calculus I

Specific information about this course and its requirements can be found below. For more general information about taking Saylor Academy courses, including information about Community and Academic Codes of Conduct, please read the Student Handbook.

Course Description

Get a detailed introduction to functions, graphs, limits, continuity, and derivatives, and explore the relationship between derivatives and graphs.

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 \( Y = X_0+Rt \), where \( t \) is elapsed time and \( X_0 \) is the initial deposit. With compound interest, things get complicated for algebra, as the rate \( R \) is itself a function of time with \( Y = X_0 + R(t)t \). 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 (like physics, biology, social sciences, economics, 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 includes the following units:

Unit 1: Preview and Review. Unit 2: Functions, Graphs, Limits, and Continuity. Unit 3: Derivatives. Unit 4: Derivatives and Graphs. Unit 5: The Integral

Course Learning Outcomes

Upon successful completion of this course, you will be able to:

[1] Calculate or estimate limits of functions given by formulas, graphs, or tables by using properties of limits and L'Hopital's Rule; [2] 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; [3] Calculate average and instantaneous rates of change in context, and state the meaning and units of the derivative for functions given graphically; [4] Calculate derivatives of polynomial, rational, and common transcendental functions, compositions thereof, and implicitly defined functions; [5] 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; [6] Find extreme values of modeling functions given by formulas or graphs; [7] Predict, construct, and interpret the shapes of graphs; [8] Solve equations using Newton's method; [9] Find linear approximations to functions using differentials; [10] Restate in words the meanings of the solutions to applied problems, attaching the appropriate units to an answer; [11] State which parts of a mathematical statement are assumptions, such as hypotheses, and which parts are conclusions; [12] Find antiderivatives by changing variables and using tables; [13] Calculate definite integrals

Throughout this course, you will also see learning outcomes in each unit. You can use those learning outcomes to help organize your studies and gauge your progress.

Course Materials

This course's primary learning materials are articles, lectures, and videos.

All course materials are free to access and can be found in each unit of the course. Pay close attention to the notes that accompany these course materials, as they will tell you what to focus on in each resource and will help you understand how the learning materials fit into the course as a whole. You can also see a list of all the learning materials in this course at this link: Resources.

Evaluation and Minimum Passing Score

Only the final examination is considered when awarding you a grade for this course. To pass this course, you will need to earn 70% or higher on the final exam. The exam will be password-protected and requires a proctor.

Your score on the exam will be calculated as soon as you complete it. There is a 14-day waiting period between each attempt. You may only attempt the final exam a maximum of three times. Be sure to study in between each attempt! If you do not pass the exam after three attempts, you will not complete this course.

There is also a practice exam that you may take as many times as you want to help you prepare for the final exam. The course also contains end-of-unit assessments in this course. The end-of-unit assessments are designed to help you study and do not factor into your final course grade. You can take these as many times as you want to until you understand the concepts and material covered. You can see all of these assessments at this link: Quizzes.

Continuing Education Credits

The certificate earned by passing this self-paced course displays not only the program hours you completed, but also continuing education credits (CEUs) for documenting successful completion of courses that are designed to improve the knowledge and skills of working adults. Many industries value CEUs, and now your certificate reflects them clearly, and they may be used to support career advancement or to meet professional licensing standards. This course contains 4.5 CEUs.

Tips for Success

MA005: Calculus I is a self-paced course, meaning you can decide when to start and complete the course. We estimate the "average" student will take hours to complete. We recommend studying at a comfortable pace and scheduling your study time in advance.

Learning new material can be challenging, so here are a few study strategies to help you succeed:

  • Take notes on terms, practices, and theories. This helps you understand each concept in context and provides a refresher for later study.
  • Test yourself on what you remember and how well you understand the concepts. Reflecting on what you've learned improves long-term memory retention.

Technical Requirements

This course is delivered entirely online. You will need access to a computer or web-capable mobile device and consistent internet access to view or download resources and complete auto-graded assessments and the final exam.

To access the full course, including assessments and the final exam, log into your Saylor Academy account and enroll in the course. If you don’t have an account, you can create one for free here. Note that tracking progress and taking assessments require login.

For more details and guidance, please review our complete Technical Requirements and our student Help Center.

Optional Saylor Academy Mobile App

You can access all course features directly from your mobile browser, but if you have limited internet connectivity, the Saylor Academy mobile app provides an option to download course content for offline use. The app is available for iOS and Android devices.

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Fees

This course is entirely free to enroll in and access. All course materials, including textbooks, videos, webpages, and activities, are available at no charge. This course also contains a free final exam and course completion certificate.

Last modified: Friday, 1 August 2025, 3:27 PM