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 gonna wanna 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.