Advanced Calculus AB—Semester A
We know our calculus; it says you + me = us.
- Course Length: 18 weeks
- Course Type: AP
- Category:
- Math
- High School
- College Prep
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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.
Technology Requirements
A graphing calculator is going to be a must for this course. Many of our activities require graphing calculators and come with explicit instructions on how to use the graphing calculator to complete the activity. The instructions are geared towards the TI-83 and TI-84.
Required Skills
The only skills required here are a good background in high school math covering algebra and geometry. Some knowledge of pre-calculus will definitely come in handy, but isn't a requirement.
Unit Breakdown
1 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.
2 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.
3 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.
4 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.
5 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.
Recommended prerequisites:
Sample Lesson - Introduction
Lesson 4.05: 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 4.4.05: 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.