### 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:

• 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 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

The primary learning materials for this course 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 to 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 by clicking on Resources in the navigation bar.

### 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 calculated as soon as you complete it. If you do not pass the exam on your first try, you may take it again as many times as you want, with a 7-day waiting period between each attempt. Once you have successfully passed the final exam you will be awarded a free Course Completion Certificate.

There are also problem sets and other assessments in this course. These 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 until you understand the concepts and material covered. You can see all of these assessments by clicking on Quizzes in the course's navigation bar.

### Earning College Credit

This course is eligible for college credit via Saylor Academy's Direct Credit Program. If you want to earn college credit, you must take and pass the Direct Credit final exam. That exam will be password protected and requires a proctor. If you pass the Direct Credit exam, you will receive a Proctor Verified Course Certificate and be eligible to earn an official transcript. For more information about applying for college credit, review the guide to college credit opportunities. Be sure to check the section on proctoring for details like fees and technical requirements.

There is a 14-day waiting period between attempts of the Direct Credit final exam. There is no waiting period between attempts for the not-for-credit exam and the Direct Credit exam. You may only attempt each Direct Credit final exam a maximum of 3 times. Be sure to study in between each attempt!

### Tips for Success

MA005: Calculus I is a self-paced course, which means that you can decide when you will start and when you will complete the course. There is no instructor or an assigned schedule to follow. We estimate that the "average" student will take 45 hours to complete this course. We recommend that you work through the course at a pace that is comfortable for you and allows you to make regular 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 we've compiled a few study strategies to help you succeed:

• Take notes on the various terms, practices, and theories that you come across. This can help you put each concept into context, and will create a refresher that you can use as you study later on.
• As you work through the materials, take some time to test yourself on what you remember and how well you understand the concepts. Reflecting on what you've learned is important for your long-term memory, and will make you more likely to retain information over time.

### Technical Requirements

This course is delivered entirely 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 for free here. Although you can access some of the course without logging in to your account, you should log in to maximize your course experience. For example, you cannot take assessments or track your progress unless you are logged in.
• If you plan to attempt the optional Direct Credit final exam, then you will also need access to a webcam. This lets our remote proctoring service verify your identity, which is required to issue an official transcript to schools on your behalf.