• Course Introduction

        • Time: 45 hours
        • Free Certificate
        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 that 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.

        • Course Syllabus

          First, read the course syllabus. Then, enroll in the course by clicking "Enroll me". Click Unit 1 to read its introduction and learning outcomes. You will then see the learning materials and instructions on how to use them.

        • Unit 1: Preview and Review

          While your first course in calculus can strike you as something new to learn, it is not comparable to learning a foreign language where everything seems different. Calculus still depends on most of the things you learned in algebra, and the true genius of Isaac Newton was to realize that he could get answers for this something new by relying on simple and known things like graphs, geometry, and algebra. There is a need to review those concepts in this unit, where a graph can reinforce the adage that a picture is worth one thousand words. This unit starts right off with one of the most important steps in mastering problem solving: Have a clear and precise statement of what the problem really is about.

          Completing this unit should take you approximately 6 hours.

        • Unit 2: Functions, Graphs, Limits, and Continuity

          The concepts of continuity and the meaning of a limit form the foundation for all of calculus. Not only must you understand both of these concepts individually, but you must understand how they relate to each other. They are "twins" in calculus problems: they usually show up together.

          A student taking a calculus course during a winter term came up with a great analogy for tying these concepts together: "The weather was raining ice – the kind of weather where no one should be driving a car. The student stepped out on his front porch to see whether the ice rain had stopped, and he couldn't believe his eyes: he saw headlights heading down the road that dead-ended at a brick house. When the car hit the brakes, the student intuited that, at the rate the car's velocity was decreasing (the continuity), there was no way it could stop in time without hitting the house (the limiting value). Oops! However, the student forgot that there was a gravel stretch at the end of the road, so the car stopped before hitting the brick house. The gravel represented a discontinuity in the student's calculations, so his limiting value was not correct."

          Completing this unit should take you approximately 7 hours.

        • Unit 3: Derivatives

          In this unit, we start to see calculus become more visible when abstract ideas such as a derivative and a limit appear as parts of slopes, lines, and curves. Circles, ellipses, and parabolas are even more geometric; what were abstract concepts are now something we can see. Nothing makes calculus more tangible than recognizing that the first derivative of an automobile's position is its velocity and the second derivative of that position is its acceleration. We are at the very point that started Isaac Newton on his quest to master this math that we now call calculus. He recognized that the second derivative was just what he needed to formulate his Second Law of Motion  F = MA , where  F is the force on any object,  M is its mass, and  A is the second derivative of its position. Thus, he could connect all the variables of a moving object mathematically, including its acceleration, velocity, and position, and he could explain what really makes motion happen.

          Completing this unit should take you approximately 13 hours.

        • Unit 4: Derivatives and Graphs

          If you are a visual person, you should find this unit very helpful for understanding the concepts of calculus. Displaying concepts graphically allows us to see what we have only been able to imagine so far. Graphs help us visualize ideas that are hard to conceptualize, like limits going to infinity but still being finite, or asymptote lines that approach each other but never quite get there. Graphs can also be used "in reverse" by causing us to question what we see, and causing us to take another look at the mathematics behind what we find.

          Completing this unit should take you approximately 9 hours.

        • Unit 5: The Integral

          While previous units dealt with differential calculus, this unit starts the study of integral calculus. As you may recall, differential calculus began with developing the intuition behind the notion of a tangent line. Integral calculus begins with understanding the intuition behind the idea of an area. We will be able to extend the notion of the area and apply these more general areas to various problems. This will allow us to unify differential and integral calculus through the Fundamental Theorem of calculus. Historically, this theorem marked the beginning of modern mathematics and is extremely important in all applications.

          Completing this unit should take you approximately 10 hours.

        • Study Guide

          This study guide will help you get ready for the final exam. It discusses the key topics in each unit, walks through the learning outcomes, and lists important vocabulary terms. It is not meant to replace the course materials!

        • Course Feedback Survey

          Please take a few minutes to give us feedback about this course. We appreciate your feedback, whether you completed the whole course or even just a few resources. Your feedback will help us make our courses better, and we use your feedback each time we make updates to our courses.

          If you come across any urgent problems, email contact@saylor.org.

        • Certificate Final Exam

          Take this exam if you want to earn a free Course Completion Certificate.

          To receive a free Course Completion Certificate, you will need to earn a grade of 70% or higher on this final exam. Your grade for the exam will be calculated as soon as you complete it. If you do not pass the exam on your first try, you can take it again as many times as you want, with a 7-day waiting period between each attempt.

          Once you pass this final exam, you will be awarded a free Course Completion Certificate.

        • Saylor Direct Credit

          Take this exam if you want to earn college credit for this course. This course is eligible for college credit through Saylor Academy's Saylor Direct Credit Program.

          The Saylor Direct Credit Final Exam requires a proctoring fee of $5. To pass this course and earn a Credly Badge and official transcript, you will need to earn a grade of 70% or higher on the Saylor Direct Credit Final Exam. Your grade for this exam will be calculated as soon as you complete it. If you do not pass the exam on your first try, you can take it again a maximum of 3 times, with a 14-day waiting period between each attempt.

          We are partnering with SmarterProctoring to help make the proctoring fee more affordable. We will be recording you, your screen, and the audio in your room during the exam. This is an automated proctoring service, but no decisions are automated; recordings are only viewed by our staff with the purpose of making sure it is you taking the exam and verifying any questions about exam integrity. We understand that there are challenges with learning at home - we won't invalidate your exam just because your child ran into the room!


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          Once you pass this final exam, you will be awarded a Credly Badge and can request an official transcript.