# Advanced Calculus AB—Semester A

## We know our calculus; it says you + me = us.

*This course has been granted a-g certification, which means it has met the rigorous iNACOL Standards for Quality Online Courses and will now be honored as part of the requirements for admission into the University of California system.*

Calculus has been known to spark fear in many a heart, but hey, it's only natural to fear what we don't understand. The only way to actually understand calculus is to well...learn it. We're happy to report calculus isn't some exotic, magical branch of math that only geniuses can understand; it's actually just a very natural extension of the algebra and geometry we've been learning all through high school. If we could handle that stuff, there's no reason we can't tackle this.

So what is calculus all about, anyway? It's nothing more than the study of change. If that sounds a bit vague, well, it should, so let's just say that without calculus we wouldn't have been able to put a man on the moon. Someone had to figure out how get a rocket up there efficiently, and the algebra and geometry we know so far just isn't enough. If that isn't enough motivation to get us interested, then we don't know what is.

Here's a sneak peek of what's to come:

- We'll start with a nice little review of pre-calculus. We'll run through all the different functions we should already know a thing or two about before we get to analyzing them with calculus.
- Then we can get the calculus party started. We'll go over what a limit is, which enables us to do pretty much everything we want to do in calculus.
- Then we'll use limits to study continuity and derivatives. Derivatives let us see how a function is changing a single point, something we just can't do with algebra alone.
- Once we get through the concepts of continuity and derivatives and have all the theory laid out, we'll need to take a moment to play catch-up and figure out how to actually compute derivatives. We'll run through the tricks of the trade so that there won't be a single derivative we can't conquer.

Calculus is tricky enough as it is, so that's why we've also packed every lesson with guided exercises, problem sets, and activities. We really left no stone unturned.

Oh, and a bit of disclaimer: This is a two-semester course. You're looking at Semester A, but Semester B is right on over here.

_{AP® is a trademark registered and/or owned by the College Board, which was not involved in the production of, and does not endorse, this product.}

### Unit Breakdown

#### Advanced Calculus AB—Semester A - Analysis of Graphs

This unit may have a fancy title, but all we're doing here is a little pre-calculus review. We wouldn't want to be unprepared before stepping barefoot into the murky of waters of calculus, so here we're just going to run through all the function types we've worked with in the past, and the tools we already have for analyzing them.

#### Advanced Calculus AB—Semester A - Introduction to Limits

This is where we'll start to get our toes wet with some calculus. A limit tells us what a function is doing as we let its input approach some number. We almost get to be fortune tellers by trying to predict where these unpredictable functions are headed. We'll start to see that everything in calculus is a limit of one form or another, too, so we'd better pay attention here.

#### Advanced Calculus AB—Semester A - Continuity

Every now and then we're blessed with a function whose graph just cruises through the coordinate plane without any breaks or interruptions. That's what continuity is all about: everything just flows seamlessly without a care in the world. We've seen plenty of continuous functions in past, but here is where we'll understand them in terms of limits.

#### Advanced Calculus AB—Semester A - Introduction to the Derivative

This unit is where we'll start to get to the real meat of the course. Hopefully it's juicy. Derivatives are just a special kind of limit that help us understand how a function is changing at a specific point. Try accomplishing that with that just algebra. We dare you.

#### Advanced Calculus AB—Semester A - Computing Derivatives

Now that we've got the gist of derivatives, it's time to learn how to actually compute them—they wouldn't be of much use otherwise. This unit finishes out the semester by running through every law, rule, and trick that will help us compute derivatives. By the time we're through, we'll be able to find the derivative of pretty much any function we want.

#### Advanced Calculus AB—Semester A - Analyzing Graphs with the Derivative

We'll kick the semester off by using everything we learned about derivatives last semester to study graphs of functions. The first and second derivatives can tell us a lot about how a graph is behaving, or more likely misbehaving.

#### Advanced Calculus AB—Semester A - Applications of the Derivative

What good would a derivative be if all we could use it for was studying graphs? If a derivative can be used to model change, it should follow we can use it to study situations that involve...change. We'll also get a sneak peak at differential equations, which are equations involving derivatives instead of your plain old

*x*and*y*.#### Advanced Calculus AB—Semester A - Introduction to Integration

The last major problem we need to tackle in this calculus course is how to find the area under a curve. That's more or less what integration is all about. The real surprise, though, is just how much integration and differentiation are related. That's where the Fundamental Theorems of Calculus come in. Anytime we throw "fundamental" in the name of a theorem, you know it must be important.

#### Advanced Calculus AB—Semester A - Applications of Integration

This is it; the final unit and culmination of the entire course. After a few basic applications we'll revisit differential equations and then we're going 3D. Why confine ourselves to finding the area of a 2D region when we can create shapely 3D solids and find their volumes instead? It's a great way to end the course.