AP Calculus AB and BC

Catalogue Description: This course is intended to give students an understanding of the concepts of calculus that they can apply to solve real-world problems.This course includes but are not limited to the following topics: characteristics and equations of conics, techniques of differentiation for algebraic and trigonometric functions, introduction to the Fundamental Theorem of Calculus, techniques and applications of integration, polar coordinates, and infinite series.

Students the Course is Expected to Serve:

1. Students who are preparing for the AP Calculus AB and BC Exam in order to obtain university credit.
2. Students who intend to major in Engineering or Physics and for those continuing in science programs at four-year institutions.

Course Objectives: Completion of the course will give students the skills needed for the level of study required at a university as well as provide an opportunity to gain college credits. It is assumed that all students entering the course are strong in Algebra and Trigonometry. Upon completion of the course content, students can choose to take the AP Calculus Exam in May and earn college credit with a passing score.

Calculus AB

1. Discuss the equations and characteristics of various conics.
2. Understand the concepts of a limit, continuity, and differentiability.
3. Apply the sum, product, quotient, and chain rules of differentiation.
4. Differentiate algebraic and trigonometric functions.
5. Apply the concepts of differential calculus to contextual (real-world) situations.
6. Understand the definition and basic properties of the Riemann sum.
7. Understand the concept of an antiderivative and its role in the Fundamental Theorem of Calculus.

Calculus BC

1. Differentiate inverse trigonometric, exponential, and logarithmic functions.
2. Differentiate and integrate in polar coordinates.
3. Apply the concepts of integral calculus to contextual (real-world) scenarios.
4. Apply various convergence tests to an infinite series.
5. Derive and apply the Taylor series of a function.
6. Apply various integration techniques, calculate improper integrals and numerically estimate definite integrals.

Student Learning Outcomes: In accordance with the AP Calculus AB/BC curriculum, upon satisfactory completion of the course, students will be able to:

Calculus AB

A. Estimate limits and derivatives graphically and by using tables of values.
B. Calculate limits of functions algebraically.
C. Calculate derivatives of functions using the definition of a derivative.
D. Identify points where a function fails to be continuous or differentiable.
E. Calculate derivatives of functions using the sum, product, quotient and chain rules.
F. Determine derivatives of functions using implicit differentiation.
G. Determine the equation of a tangent line to the graph of a function.
H. Approximate changes in a function using differentials.
I. Apply the Intermediate, Mean, and Extreme Value Theorems to a function defined on a closed and bounded interval.
J. Apply derivatives to problems involving optimization and related rates.
K. Analyze the behavior of functions and their graphs using first and second derivatives (e.g., determine local and absolute extrema, concavity, and inflection points).
L. Determine antiderivatives of functions.
M. Apply the concepts of first and second derivatives and antiderivatives to motion problems
N. Calculate a Riemann sum of a function on a closed interval.
O. Evaluate definite integrals by using the Fundamental Theorem of Calculus.

Calculus BC

A. Apply integration techniques such as partial fractions, trigonometric substitution, or use of integration tables.
B. Estimate definite integrals using the Midpoint Rule, Trapezoidal Rule and Simpson’s Rule.
C. Apply L’Hospital’s Rule to calculate limits of functions.
D. Evaluate improper integrals.
E. Apply integration to computing the area between two curves and the volume of a solid.
F. Graph a curve, including the conics, using polar coordinates.
G. Differentiate equations in parametric and polar form.
H. Calculate the area of regions in polar form using integrals.
I. Determine the limit of a sequence.
J. Calculate the sum of a geometric series.
K. Determine the convergence or divergence of a series using the Integral Test, comparison tests, Alternate Series Test, and Ratio Test.
L. Determine the interval of convergence for a power series.
M. Determine the McLaurin and Taylor series representation of a function at a point.
N. Apply Taylor series to estimate function values and definite integrals.
O. Calculate integrals using substitution and integration by parts methods.

Topical Outline:

1. Limits and Continuity
2. Differentiation
3. Applications of Differentiation
4. Integration
5. Applications of Integration
6. Techniques of Integration
7. Further Applications of Integration
8. Parametric Equations, Vectors, and Polar Coordinates (BC)
9. Infinite Sequences and Series (BC)

A more detailed breakdown of topics and curriculum online can be found here.

A Note on Graphing Calculators: The calculus AP exams consist of a multiple-choice and a free-response section, with each section including one part that requires use of a graphing calculator and one during which no electronic devices are permitted. While calculators cannot substitute for the necessary depth of understanding or provide any shortcuts where students are required to show their work, the tastemakers who develop the AP calculus exam recognize that a graphing calculator is an integral part of the course. Therefore students should become comfortable with their graphing calculators through regular use. Check the AP website at apcentral.collegeboard.com for more details on restrictions on calculators.