When comparing the AP Calculus AB and AP Calculus BC exams, you might wonder what additional topics you need to learn for the BC exam. First, note that the Calculus BC course includes all the topics covered in Calculus AB, plus some extra material. Generally, you can go straight to Calculus BC after completing a Pre-Calculus class. Below, we review the additional topics covered in the Calculus BC course and presented on the BC exam.
Extra BC Topics – Extensions of AB Units
The following units in BC Calculus are relatively small extensions of units seen in AB Calculus. Thus, these units are more accessible to all Calculus students.
Integrals
AP Calculus BC covers additional techniques of integration, namely:
- Integration by Parts
- Integrating the product of two functions, such as evaluating \( \int xe^x \, dx \).
- Integration by Partial Fractions
- Integrating a function of the form \( \frac{P(x)}{Q(x)} \), where \( P(x) \) and \( Q(x) \) are polynomials.
- Improper Integrals
- Evaluating integrals with bounds at infinity, such as \( \int_1^{\infty} \frac{1}{x^2} \,dx \), and evaluating definite integrals of functions with vertical asymptotes, such as \( \int_0^2 \frac{1}{x-1} \,dx \).
These topics deal with simplifying complicated integrals into ones that are easier to calculate. The only prerequisites for learning this material are the integration and differentiation rules taught in AP Calculus AB. As a result, students with a solid understanding of integration fundamentals shouldn’t have trouble with these topics.
Applications of Integration
AP Calculus BC covers one more application of integration:
- Finding Arc Length
- Finding the arc length of a curve is different from standard integration because it involves finding length, not area. The formula for arc length is more complex and can be difficult for students to remember. Namely, the arc length of a function \( y=f(x) \) from \( x=a \) to \( x=b \) is \( \int_a^b \sqrt{1+\left(\dfrac{dy}{dx}\right)^2} \,dx \).
This topic is conceptually similar to the topic of Riemann Sums covered in AP Calculus AB. As a result, students should be able to grasp this topic relatively quickly.
Differential Equations
AP Calculus BC adds extra topics to the Differential Equations unit, specifically:
- Euler’s Method
- Euler’s method is a method of approximating points on curves given their derivative. This technique involves the idea of linear approximation, using tangent lines over small intervals to approximate a function. Euler’s method only requires basic computation and differentiation and is generally straightforward.
- Restricted Growth + Logistic Models
- Restricted growth and logistic models are ways to model and analyze population growth. They are based on differential equations and are quite complex because of the number of parameters involved. Many students find them challenging to understand and apply.
It is important to keep in mind that the topics listed in this section are merely extensions of AP Calculus AB, and while harder, should not take too much time to learn.
Extra BC Topics – Completely Separate Units
The following units are completely separate from the AB curriculum, so a student will have to learn more new material and techniques relating to calculus. As a result, these topics may be more challenging.
Polar Coordinates, Vectors, Parametric Equations
This unit does not introduce new calculus formulas or concepts. Rather, it deals with thinking about the coordinate plane with a different approach.
- Polar Coordinates
- Polar coordinates plot points according to their magnitude and angle they make with the origin, rather than using the classic \( x \) and \( y \) rectangular coordinate system.
- Parametric Equations
- Parametric equations describe functions of the form \( y=f(x) \) in terms of \( x(t) \) and \( y(t) \), where \( t \) is a parameter, similar to a vector valued function.
- Vectors
- Instead of integrating and differentiating with respect to algebraic expressions, vectors involve vector valued functions of the form \( f(t) = \left< x(t), y(t) \right> \).
This unit will explore new ways of thinking about curves using the three new systems described above. It also uses other calculus concepts, such as integration and differentiation, and applies them in new ways.
Infinite Series
AP Calculus BC teaches another separate unit at the end of the curriculum called Infinite Series.
The focus of this unit revolves around whether an infinitely long series, such as \( \sum_{n=0}^{\infty} \frac{1}{n} \), converges (sum evaluates to a real number) or diverges (sum approaches infinity).
This unit explores various methods, including but not limited to:
- Convergence and Divergence
- P-Series
- Ratio, Direct Comparison, and Other Tests
- Alternating Series Test
- Power Series
This unit is often regarded as the most difficult to learn, mainly because of how conceptual it is. Rather than just focusing on the applications of formulas and rules, learning infinite series requires a comprehensive understanding of the concepts involved. While this unit might seem algebra-based on the outside, infinite series relies on several calculus concepts, including limits and integrals.
Because of its difficulty, students should rigorously practice a variety of problems within this unit to familiarize themselves with the different concepts and tricks that show up.
Studying BC Topics
These additional BC topics are available in the I Aced Calculus app. They are clearly marked in the Integrals, Applications of Integration, Differential Equations, Polar, and Series decks. The flashcards offer informative guides for learning the difficult material, and together with video explanations working out the precise details of every problem and solution, students will become more comfortable working through complicated calculus problems. There are also practice questions in the app to refine the knowledge and skills required to apply the material, each including individual hints and video solutions to guide the student through the learning experience. These features are especially useful for learning more challenging concepts, such as those found in the Infinite Series unit.
In addition to using the I Aced Calculus app, students can access the list of tutors, providing an alternative to those looking for a more interactive experience rather than purely self-studying.
