# AP® Calculus BC—Semester A

## The learning is un*-limited.*

*This course has been approved by the College Board, which indicates that the syllabus "has demonstrated that it meets or exceeds the curricular expectations colleges and universities have for your subject." Please contact sales@shmoop.com if you would like to add this course to your official record of AP course offerings.*

*It has also 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.*

It's been said that preparation is the key to success. What's true in life is usually true in math just as well, so we wouldn't expect it things to be any different going into a course like Advanced Calculus BC.

Everything you've done in your high school math career has been leading up to this moment. After all that preparation, you're finally at the big dance. We're here to tell you, though, that calculus isn't the mysterious, impossible to understand subject many make out it be. Really, calculus is just a way of studying how functions change. If you could handle math up to this point, there's no reason you can't tackle a challenge like this, especially when we've got your back.

In this course, we'll cover everything related to limits and derivatives that you'll be expected to know for the AP Calculus BC exam. Here's a little more detailed run-down of what you can expect to see in this semester.

- Everything in calculus is a limit so we'll start there. We'll run through what a limit is, and a handful of techniques we can throw at 'em to see what they come out to be.
- Next up, we'll use limits to study continuity. You've seen plenty of continuous functions during your algebra and pre-calc days, but now we'll put everything in the context of limits.
- The last four units of this semester are all about derivatives. We'll start by using limits to relate them to average rates of change, and show what they can tell us about how a function is changing at a single point.
- Then it's on to the fun part: applications of derivatives. We'll use derivatives to construct pretty accurate graphs of functions and solve applied problems.

By the way, Advanced Calculus BC is a two-semester course. You're looking at Semester A, but if you want to check out Semester B, click here.

### Unit Breakdown

#### AP® Calculus BC—Semester A - Introduction to Limits

Ready to dive in? In the first unit, we'll introduce you to limits, and a handful of techniques for how to compute them. In the units to come, we'll start to see that everything in calculus boils down to a limit, so we'll really want to pay attention here. Really, when should we ever not pay attention in math class?

#### AP® Calculus BC—Semester A - Continuity

Isn't it great when a function's graph just runs smoothly through the coordinate plane without any weird breaks in it? Our dreams are filled will functions like these. This is more or less what continuity is all about, and in this unit we'll explain it in terms of limits, as well as uncover some neat properties of continuous functions.

#### AP® Calculus BC—Semester A - Introduction to Derivatives

Here's where we'll introduce a really special kind of limit that we'll be focusing on the rest of the semester, and will keep coming back to like day old Chinese food throughout the rest of the course. That special kind of limit is called a derivative, and it tells us how a function is changing at a single point.

#### AP® Calculus BC—Semester A - Computing Derivatives

Derivatives aren't much good if we don't actually have any reliable methods for computing them. Now that we know all about what a derivative is, and what it can do for us, we'll turn our attention to finding methods for actually computing derivatives. By the end of the unit, we should be able to find the derivative of any function we want.

#### AP® Calculus BC—Semester A - Analyzing Graphs with the Derivative

If there's ever a time to put derivatives to the test, this unit is it. We've already had plenty of experience graphing functions in other math classes, but derivatives can help us do it a lot better. Anything that can make us better at what we already know is definitely something worth learning.

#### AP® Calculus BC—Semester A - Applications of the Derivative

Pretty much everything in math can be used to solve real world problems and derivatives are no exception. Anytime we're dealing with a situation that involves change, there's a good chance there's a derivative or two loitering around. We'll also take a look at differential equations, which are equations involving a function and its derivatives. It'll be our job to find functions that make the equation true.

#### AP® Calculus BC—Semester A - Introduction to Integration

This unit introduces the next big idea in calculus: integration. We'll show you how to use integrals to find the area underneath a curve, and then how integration and differentiation are two sides of the same coin through the Fundamental Theorem of Calculus. When a theorem has a name as impressive as that, you know it's going to be important.

#### AP® Calculus BC—Semester A - Area, Volume, and Arc Length

When it comes to functions, we're used to taking in a point

*x,*and having our function spit out another point,*y.*That's not the only way to get the job done, though. Polar and parametric functions provide a different way of producing graphs in the coordinate plane. We'll show you how to work with these functions, as well as how to compute derivatives and integrals with them. Then we'll leave the coordinate plane and close out the unit by generating 3D solids, and using integrals to compute their volumes.#### AP® Calculus BC—Semester A - Further Applications of Integration

The applications of integration don't end at area, volume, and arc length. We can use integrals to find the average value of a function, as well as a function's net change over an interval. Then we'll revisit differential equations, and use integrals to solve differential equations we just couldn't crack before.

#### AP® Calculus BC—Semester A - Sequences and Series

You might remember that a sequence is just an infinite list of numbers. But what happens if we decide to add up all those numbers? The result is a series. We'll show you how to work with these guys, as well as how to tell if these infinite sums converge or diverge. It may seem counter intuitive, but it's totally possible to add up infinitely many numbers, and get a clean, finite answer.

#### AP® Calculus BC—Semester A - Power Series

This is where we link series with differentiation and integration to end the course with a bang. We'll show you how to take an infinitely differentiable function and represent it as an infinite series. You may be wondering why this is useful, but representing functions as series allows us to approximate function values and definite integrals, and make that estimate as precise as we want. We can't think of a better way to cap the course off.