##### Course Description

This course provides a detailed introduction of functions, graphs, limits, continuity, and derivatives, and 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 (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 is comprised of the following units:

• Unit 1: Preview and Review (Optional)
• 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:

• 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.

Throughout this course, you'll also see related learning outcomes identified in each unit. You can use the learning outcomes to help organize your learning and gauge your progress.

##### Course Materials

The primary learning materials for this course are readings, lectures, video tutorials, and other resources.

All course materials are free to access, and can be found through the links provided in each unit and subunit of the course. Pay close attention to the notes that accompany these course materials, as they will instruct you as to what specifically to read or watch at a given point in the course, and help you to understand how these individual materials fit into the course as a whole. You can also access a list all of the materials used in this course by clicking on Resources in the course's "Activities" menu.

##### Evaluation and Minimum Passing Score

Only the final exam is considered when awarding you a grade for this course. 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 on your first attempt, you may take it again as many times as needed, following a 7-day waiting period between each attempt. Once you have successfully passed the final exam you will be awarded a free Saylor Certificate of Completion.

There are also 9 problem set quizzes in this course. These are intended to help you to gauge how well you are learning and do not factor into your final course grade. You may retake all of these as many times as needed to feel that you have an understanding of the concepts and material covered. You can locate a full list of these sorts of assessments by clicking on Quizzes in the course’s “Activities” menu.

##### Earning College Credit

This course is eligible for college credit via Saylor’s Direct Credit Program. If you are seeking to earn college credit, you must opt to take and pass the Saylor Direct Credit final exam. That exam will be password protected and require the presence of a proctor. Upon passing that final exam you will receive a Proctor Verified Course Certificate, and will be eligible to earn an Official Transcript. For more information about applying for college credit review the “Guide: College Credit Opportunities”. Be sure to check the section on proctoring for details (fees, technical requirements, etc.)

Note: There is a 14-day waiting period between attempts of the Direct Credit final exam. There is no imposed wait period between attempting the non-credit certificate-bearing exam and the credit exam. Some credit exams have a maximum number of attempts allowed, which will be detailed on the exam’s instructions page.

##### Tips for Success

MA005: Calculus I is a self-paced course in which you the learner determines when you will start and when you will complete the course. There is no instructor or predetermined schedule to follow. While learning styles can vary considerably and any particular student will take more or less time to learn or read, we estimate that the "average" student will take 122 hours to complete this course (129 hours if also completing the optional review unit). We recommend that you work through the course at a pace that is comfortable for you and allows you to make regular (daily, or at least weekly) progress. It's a good idea to also schedule your study time in advance and try as best as you can to stick to that schedule.

Learning new material can be challenging, so below we've compiled a few suggested study strategies to help you succeed:

• Pay special attention to Unit 1, as it will lay the groundwork for understanding the more advanced, explanatory material presented in the latter units.
• Take notes on the various terms, practices, and theories as you read. This can help you differentiate and contextualize concepts and later provide you with a refresher as you study.
• As you progress through the materials, take time to test yourself on what you have retained and how well you understand the concepts. The process of reflection is important for creating a memory of the materials you learn; it will increase the probability that you ultimately retain the information.
• Although you may work through this course completely independently, you may find it helpful to connect with other Saylor students through the discussion forums. You may access the discussion forums at https://discourse.saylor.org.

##### Technical Requirements

This course is delivered fully online. You will be required to have access to a computer or web-capable mobile device and have consistent access to the internet to either view or download the necessary course resources and to attempt any auto-graded course assessments and the final exam.

• To access the full course including assessments and the final exam, you will need to be logged into your Saylor Academy account and enrolled in the course. If you do not already have an account, you may create one, free of charge, here. Although you can access some course resources without being logged into your account, it’s advised that you log in to maximize your course experience. For example, some of the accessibility and progress tracking features are only available when you are logged in.
• Occasionally, Flash may be required to run certain multimedia and/or interactive applications in the course. Should you be prompted to enable Flash, click the option to allow or follow these instructions for enabling Flash on your computer or laptop.
• If you plan to attempt the optional credit recommended final exam that accompanies this course, then you will also need access to a webcam enabled computer. A webcam is needed so that our remote proctoring service can verify your identity, which will allow Saylor Academy to issue an official transcript to schools on your behalf.

For additional technical guidance check out Saylor’s tech-FAQ and the Moodle LMS tutorial.

##### Fees

There is no cost to access and enroll in this course. All required course resources linked throughout the course, including textbooks, videos, webpages, activities, etc are accessible for no charge. This course also contains a free final exam and course completion certificate.

This course does contain an optional final exam that will provide students an opportunity to earn college credit. Access to the exam itself is free, though it does require the use of a proctoring service for identity verification purposes. The cost for proctoring is \$25 per session.