Advanced Calculus AB—Semester B
It's an integral part of life.
- Course Length: 14 weeks
- Course Type: AP
- Category:
- College Prep
- Math
- High School
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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.
Ready for a course filled with more limits, derivatives, and integrals than anyone bargained for? Semester B of Shmoop's AP Calculus course picks up right where Semester A left off through one epic toss of the baton. This semester is where the material really comes to life, though.
We'll start by actually putting derivatives to use in constructing graphs and solving related rates and optimization problems, and then it's on to integration and its applications. By the end of the course, you'll be ready for almost anything calculus can throw at you, especially an AP exam.
Through readings, guided exercises, problem sets, and engaging activities, there won't be any excuse to not love calculus by the time we're done. Here's a preview of what's covered in this semester:
- We'll use derivatives to study properties of graphs and construct graphs of functions. If you thought you knew everything there was to know about graphing, well...you were wrong.
- We'll solve related rates and optimization problems. If you wondered how many lemons you should order to make sure you get as much profit out of a lemonade stand (who hasn't), we'll show you.
- Then we finish out the semester with two units on integration. We'll show you how to find the area under a curve, and how that process is a lot like differentiation in reverse.
In case it wasn't obvious, AP Calculus AB is a two-semester course. You're looking at Semester B, but Semester A can be found 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 required for many of the activities in this course. 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, as well as what was covered in Semester A. In so many words, a working knowledge of limits and derivatives will give you enough background to dive right in.
Unit Breakdown
6 Advanced Calculus AB—Semester B - 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.
7 Advanced Calculus AB—Semester B - 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.
8 Advanced Calculus AB—Semester B - 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.
9 Advanced Calculus AB—Semester B - 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.
Recommended prerequisites:
Sample Lesson - Introduction
Lesson : The Fundamental Theorem Of Calculus
As young kids, tying our shoes was a pretty daunting task. No matter how hard we tried, we just couldn't seem to remember all the motions. It was pretty stressful, really. Our parents even came up with little stories to help us remember each step. We seem to remember something about a rabbit going into its hole, but who knows, really.
Eventually, after a lot of practice we managed to tie everything together, in more ways than one. We finally saw how every cross of the laces came together to form a perfect bow. We didn't have to wear sneakers with Velcro straps anymore. It was a great day.
Right now, there's a lot going in this unit that all probably seems pretty unrelated. We went from sums, to definite integrals, and we just left off with antiderivatives. We spent all that time learning about derivatives, only to spend the last two lessons reversing the process, for reasons we didn't really bother to tell you. Well it's time for us to tie everything together in a way that's going to make a lot more sense.
Tighten those laces. With a lesson title like "The Fundamental Theorem of Calculus," we know we're in for a real treat.
Sample Lesson - Reading
Reading 8.: Tying It All Together
This is where it all comes together. Integration may be all about finding the area under a curve, but it's related to derivatives in more ways than we might have ever imagined. That's why the fundamental theorem of calculus is just so darn fundamental. It ties all the major ideas of calculus together in a pretty simple equation, that doesn't really leave much to the imagination.
So, we could spend all day hyping this theorem up, but we'd rather just get it over with and go for the big reveal.
Fundamental Theorem of Calculus: Take a function f, continuous on some closed interval, [a, b]. If F is an antiderivative of f, then 
.
The continuity assumption is super important. We've already showed how to find definite integrals with non-continuous functions, but in order to apply the fundamental theorem, we really need continuity.
If that wasn't the most amazing thing we've ever seen, then we definitely don't know what is. Let's just reiterate what that equation actually says. If we can find a function's antiderivative, all we have to do is take the difference between the value of the antiderivative at each endpoint of [a, b], and we'll have the area under the graph of the function. No more taking limits with Riemann sums, or any of that nonsense.
At least for right now, though, this theorem should seem a little bit like, well…magic really. It's almost a little too convenient. We promise it's true, though, and for an intuitive explanation as to why, go ahead and check out this reading on integrating the velocity function.
The proof of the Fundamental Theorem of Calculus actually relies on the Mean Value Theorem in a really cool way, but it's a little beyond the scope of what we're trying to show here. If you're really curious about it, though, go ahead and check out this video.
Recap
The Fundamental Theorem of Calculus provides a surprising link between differentiation and integration, and makes the problem of solving definite integrals a whole lot easier. The theorem states that if f is continuous on the interval [a, b], and F is an antiderivative of f, then 
.
Sample Lesson - Activity
Sample Lesson - Activity
- Course Length: 14 weeks
- Course Type: AP
- Category:
- College Prep
- Math
- High School
Schools and Districts: We offer customized programs that won't break the bank. Get a quote.