AP® Calculus BC—Semester A
The learning is un-limited.
- Course Length: 18 weeks
- Course Type: AP
- Category:
- Math
- High School
- College Prep
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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.
Technology Requirements
Seeing as this is an online course and all, a computer with internet access will be a must. A graphing calculator is required for many of the activities in this course as well. The instructions given in the activities on how to use many calculator functions apply most directly to calculators in the TI-83 and TI-84 series.
Required Skills
The only skills required here are a good background in high school math covering algebra and geometry. Ideally, you will have knowledge of pre-calculus, but it isn't required.
Unit Breakdown
1 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?
2 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.
3 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.
4 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.
5 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.
6 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.
Recommended prerequisites:
Sample Lesson - Introduction
Lesson 3.05: Secant Lines and the Difference Quotient
Everyone loves a good party. What's not to love? There's always good food and drink, good music, and hopefully great entertainment. We always keep our fingers crossed hoping someone had the good sense to hire a magician. Better yet, a mathemagician. That way we'll have learning and entertainment bundled into one package. Once again, what's not to love?
But what really makes a party special is the people we get to share the experience with. It doesn't matter what band we're seeing if we don't have anyone to dance with, and what's the point of hiring a mathemagician if we don't have anyone to make fun of him with? Every party seems to have that one person who gets things going and makes it a memorable experience for everyone. They show up with the best appetizers, tell the best stories, and are typically a pretty good dancer. They're called the life of the party for a reason.
When it comes to calculus, we can't throw a party without limits. They just have to be there. We can't really do rates of change any justice without them. In fact, during Isaac Newton's college years he was often quoted as saying, "we can't get this party started until a limit or two gets here." It's kind of what gave him the inspiration to create limits to begin with. True story.
Sample Lesson - Reading
Reading 3.3.05: Let's Get It Started in Here
Let's get this party started. We just extended an invitation to limits and they RSVP'd yes.
When we first introduced the idea of an instantaneous rate of change, we were sort of taking limits of a bunch of average rates of change. We used tables, though, as a way of getting an approximation of what the instantaneous rate of change would be. If we actually take a limit, we'll be able to get an exact number for the instantaneous rate of change. Wouldn't that be nice?
Here's the setup. Take a function f, and a point a. Now if we pick some other number h we can look at the average rate of change of the function on the interval [a, a + h]. We already know how this looks. It's given by the expression
and represents the slope of the line connecting the points (a, f(a)) and (a + h, f(a + h)). This expression is also called a difference quotient. Here's where limits join the party. If we choose smaller and smaller values of h we'll get closer to what the actual instantaneous rate of change is. So really, we're just taking the limit as h goes to 0 and hoping everything works out.
or
Either limit works and will give the same value for the rate of change.
Notice though, that if we plug in zero for h in the denominator, we're always going to end up dividing by zero. So every time we're evaluating this limit we're going to end up with an indeterminate form. Luckily, we've already learned a whole host of ways to maneuver around indeterminate forms and get a nice answer for our limits.
But before we wrap this lesson up, it's time for some new terminology. "Instantaneous rate of change" is just way too wordy. The word we'll be using from here on out is derivative. Seeing as the title of this unit is "Introduction to Derivatives" we knew that word was going to creep into our vocabulary at some point. For the sake of convenience, we usually write the derivative of a function f at some point a as f '(a). To sum it up,
or
Seeing as how the derivative is a rate of change and is a limit of a bunch of slopes of secant lines, we should expect the derivative to be the slope of something as well. The question is what? How can we have slope at a single point?
The answer lies in the secant lines. As h gets closer and closer to zero, the secant lines connecting the points (a, f(a)) and (a + h, f(a + h)) get closer and closer to a line we call the tangent line. The tangent line is the line that comes in and just barely touches the graph of our function at (a, f(a)), usually without crossing it.
Check out this reading for an overview of what tangent lines might look like.
Alright, party's over. Time for everybody to clear out.
Recap
We can find the exact value of the instantaneous rate of change of a function at a point using limits. The limit 
represents the instantaneous rate of change of a function at a point, a. This limit is also called the derivative of f at a and is written as f '(a).
The derivative represents the slope of the tangent line at the point (a, f(a)). The tangent line is what the secant lines between the points (a, f(a)) and (a + h, f(a + h)) approach as h approaches zero in the limit.
Sample Lesson - Activity
- Course Length: 18 weeks
- Course Type: AP
- Category:
- Math
- High School
- College Prep
Schools and Districts: We offer customized programs that won't break the bank. Get a quote.