Welcome to calculus. I'm Professor Ghrist. We are about to begin the course Calculus in a single variable. Welcome to calculus. I'm Professor Ghrist and for the next 13 weeks I'll be your calculus professor. Calculus is a wonderful subject. One of the loftiest achievements of human thought with thousands of years in the making. Whether you are at the beginning of your studies, or whether you've come back to deepen your understanding, my course will give you a novel experience. This course is built on the main objects of calculus, functions, limits, derivatives, and integrals, both continuous and discreet. You'll learn how to compute with these objects, but you'll also learn what they mean. And how they are useful in the engineering, biological, social, and physical sciences. Are you ready? Let's go! You may be wondering, what do you need in order to be successful in this course? Well, there are several things that you do not need. You don't need a big calculus book, you don't need a fancy calculator, you don't need any money, this is a free course, but there is one thing that you do need a lot of, and that is time. Mathematics is difficult and it takes time to work through the homework assignments, to think about what you are doing. This is a hard course and you're going to need time and perseverance to get through it. You're also going to need prerequisites. It's assumed that you know the basics, such as algebra. You need to be very familiar with how exponents work, how polynomials are factored. Sometimes we'll be doing some of the basic algebra off-screen and it's going to be up to you to fill those steps in. You're going to have some background in basic geometry. Knowing about things like curves and circles, volumes, areas of basic shapes. You're going to want to make sure that you've seen some trigonometry before. We'll be reviewing things like sine, cosine and tangent, but you will need to have some prior exposure, likewise, in pre-calculus. It's assumed that you've seen the exponential function, e to the x, and the natural logarithm, l n of x. We'll review this a little bit, but you're going to want to make sure that it's not your first time seeing that. And even though this is a calculus course, it is not your first calculus course. I'm going to assume that you've seen some of the basics such as differentiation or integration of polynomials and exponential functions before. You need to know, or at least have seen, a definition of a derivative, maybe in terms of slopes. And it will be helpful if you've seen a definition of an integral in terms of, say, an area under a curve. Now we're going to go through all of that material again and make your understanding deeper and clearer. But, if it's your first time seeing this, then this might not be the course for you. Let's continue with an overview of the course. What you're going to see over the next 13 weeks. The course is broken into five chapters. The first chapter is on functions and beginning with a simple function, e to the x, we're going to reconsider functions from the perspective of series. Taylor series, to be precise. We'll learn a new asymptotic language for understanding growth. And then, in chapter two, we'll put that language and that intuition to work by reconsidering rates of change and our notion of differentiation. From there we'll turn in chapter three to the notion of an anti-derivitive. That is an integral. Motivated by applied problems in differential equations we will build up integrals both indefinite and then definite. In chapter 4, we'll take what we've learned about derivatives and integrals and put them to use considering applications in the physical, social, engineering and biological sciences. Finally, in chapter five we'll revisit everything that we have done in the course, rebuilding calculus in a discrete setting. A calculus four sequences. Your next step should be to take the diagnostic exam and see if you remember all of those prerequisites. Then, begin with lecture one. Watch the lecture, and then go to the homework assignments. There are two homework sets per lecture. One, a set of core or basic problems, and then a set of challenge problems that are optional, for those who want to go deeper. Now, when you get to the homeworks, you may or may not encounter some difficulty. If you do, we have several resources available. You can go to the Penn Calc Wiki, which operates something like a text for the course. But even better, you can go to the discussion forums. Help and be helped by other people. As you complete homework assignments, move on to the next lectures and repeat. When you get to the end of a chapter, then there will be a quiz. Those happen at particular times. You can see the schedule on the website for the course. Eventually, when we're done with all five chapters, we'll get to the final exam and, if you make it to the end, you will be done. It is by no means an easy path to get to the end of this course, it is a difficult subject and is only learned by means of hard work. You are going to have to work hard and persevere to get to the end. But I am confident that together, we can make it. It's a privilege for me to be your calculus professor this term, and I'm glad that you've chosen this course. I want you to make it to the end. Your next step should be to take the diagnostic exam, and make sure that this is the right place. Then, start with lesson one, and move lesson by lesson. Don't skip around a lot. This course has a flow. Calculus is like an epic story with main characters, grand themes, struggle, and eventual achievement. I want you to see that story. I want you to live that story. This course is going to be an odyssey, a long and difficult journey, but if you work hard, you'll make it to the end with something you can be really proud of. A mastery of calculus.