Welcome to an Introduction to Calculus. This course is intended to bring you up to speed quickly with ideas and techniques in calculus, which are commonly used throughout science and engineering, and also in many applications in business and economics. The course is organized into five modules, supported by videos, accompanying notes, formative practice exercises that develop mastery of the concepts, and graded exercises that verify your level of achievement. The first two modules discuss precalculus, setting the stage and helping you develop a repertoire of important and useful functions. A function is a process or rule that finds connections or relationships between different physical quantities or measurements represented in this course always by real numbers. The precalculus provides all the necessary background and foundations for the rest of the course. Precalculus can be difficult and challenging, especially if you're not used to the ideas, notational terminology. It might seem sometimes that I'm speaking in a foreign language. Nevertheless, I hope you'll be able to engage with the exercises, find them interesting and useful in developing familiarity and fluency with the key ideas, as this will set you up for a smooth transition into the remaining modules. In learning mathematics, there's no substitute for practice. The best way to develop focus and practice is to attempt the exercises, and they've been carefully designed to support your learning. The third and fourth modules introduce limits and derivatives. Limits are a precise mathematical tool for connecting finite objects, which we typically think of as represented by real numbers with intriguing processes that play with notions of infinity, going on forever. Have you ever stood in the middle of a road, gazed into the distance and seeing the sides of the road converge to a point on the horizon? How can that be? The sides of the road remain the same distance apart always, yet they seem to meet at infinity. The image on your retina, a single dot where they appear to meet, is an example of a limit point. When limits are combined with ratios or rates of changes, particular types of fractions, then we obtain derivatives, and the theory of differential calculus. Techniques of differential calculus lead to some of the most profound applications and insights. These techniques led to an explosion of progress and activity in science from their discovery in the 17th century. Are you intrigued by space travel, by the possibility of traveling to the stars and beyond? In the 17th century, Isaac Newton imagined just that. In a feat of imagination and creative thinking unmatched until Einstein in the 20th century, Newton worked out the escape velocity of a rocket. By pure reasoning, he figured out that in order for a rocket to escape the gravitational field of the earth, without any further forces acting, it would have to move at 11 kilometers per second. His calculation involves the theory of calculus, that he'd only just invented, and estimates about the radius of the earth from ancient astronomy. By the time you reach the end of this course, you'll be able to reconstruct Newton's thought processes, but you will need the last module. The fifth and final module introduces the integral calculus, which again uses limiting processes but now applied to areas. The fundamental theorem of calculus, discovered in the 17th century independently by Newton and Leibniz, reveals the most remarkable connection between areas defined by curves, and slopes of tangent lines to curves. Why might that be useful? Well, this connection leads naturally and surprisingly into theories of differential equations and their solutions, which are ubiquitous throughout science. We hope you enjoy this course. Calculus is a miracle of invention and one of the great pinnacles of the imagination of the human mind. We hope that you don't get tired from being surprised at every turn, and that you obtain great benefit from the videos, and the practice and challenges provided by the many exercises. So, let's begin.
