Barron’s AP Calculus, 14th Edition
Book Preface
This book is intended for students who are preparing to take either of the two Advanced Placement Examinations in Mathematics offered by the College Entrance Examination Board, and for their teachers. It is based on the May 2014 course description published by the College Board, and covers the topics listed there for both Calculus AB and Calculus BC.
Candidates who are planning to take the CLEP Examination on Calculus with Elementary Functions are referred to the section of this Introduction on that examination.
THE COURSES
Calculus AB and BC are both full-year courses in the calculus of functions of a single variable. Both courses emphasize:
(1) student understanding of concepts and applications of calculus over manipulation and memorization;
(2) developing the student’s ability to express functions, concepts, problems, and conclusions analytically, graphically, numerically, and verbally, and to understand how these are related; and
(3) using a graphing calculator as a tool for mathematical investigations and for problem-solving.
Both courses are intended for those students who have already studied college-preparatory mathematics: algebra, geometry, trigonometry, analytic geometry, and elementary functions (linear, polynomial, rational, exponential, logarithmic, trigonometric, inverse trigonometric, and piecewise). The AB topical course outline that follows can be covered in a full high-school academic year even if some time is allotted to studying elementary functions. The BC course assumes that students already have a thorough knowledge of all the topics noted above.
TOPICS THAT MAY BE TESTED ON THE CALCULUS AB EXAM
1.
Functions and Graphs
Rational, trigonometric, inverse trigonometric, exponential, and logarithmic functions.
2.
Limits and Continuity
Intuitive definitions; one-sided limits; functions becoming infinite; asymptotes and graphs; indeterminate limits of the form img or img using algebra; img; estimating limits using tables or graphs.
Definition of continuity (in terms of limits); kinds of discontinuities; theorems about continuous functions; Extreme Value and Intermediate Value Theorems.
3.
Differentiation
Definition of derivative as the limit of a difference quotient and as instantaneous rate of change; derivatives of power, exponential, logarithmic, trig and inverse trig functions; product, quotient, and chain rules; differentiability and continuity; estimating a derivative numerically and graphically; implicit differentiation; derivative of the inverse of a function; the Mean Value Theorem; recognizing a given limit as a derivative; L’Hôpital’s Rule.
4.
Applications of Derivatives
Rates of change; slope; critical points; average velocity; tangent line to a curve at a point and local linear approximation; increasing and decreasing functions; using the first and second derivatives for the following: local (relative) max or min, concavity, inflection points, curve sketching, global (absolute) max or min and optimization problems; relating a function and its derivatives graphically; motion along a line; related rates; differential equations and slope fields.
5.
The Definite Integral
Definite integral as the limit of a Riemann sum; area; definition of definite integral; properties of the definite integral; use of Riemann sums (left, right and midpoint evaluations) and trapezoidal sums to approximate a definite integral; estimating definite integrals from tables and graphs; comparing approximating sums; average value of a function; Fundamental Theorem of Calculus; graphing a function from its derivative; accumulated change as integral of rate of change.
6.
Integration
Antiderivatives and basic formulas; antiderivatives by substitution; applications of antiderivatives; separable differential equations; motion problems.
7.
Applications of Integration to Geometry
Area of a region, including between two curves; volume of a solid of known cross section, including a solid of revolution.
8.
Further Applications of Integration and Riemann Sums
Velocity and distance problems involving motion along a line; other applications involving the use of integrals of rates as net change or the use of integrals as accumulation functions; average value of a function over an interval.
9.
Differential Equations
Basic definitions; geometric interpretations using slope fields; solving first-order separable differential equations analytically; exponential growth and decay.
TOPICS THAT MAY BE TESTED ON THE CALCULUS BC EXAM
BC ONLY
Any of the topics listed above for the Calculus AB exam may be tested on the BC exam. The following additional topics are restricted to the BC exam.
1.
Functions and Graphs
Parametrically defined functions; polar functions; vector functions.
2.
Limits and Continuity
No additional topics.
3.
Differentiation
Derivatives of polar, vector, and parametrically defined functions; indeterminate forms.
4.
Applications of Derivatives
Tangents to parametrically defined curves; slopes of polar curves; analysis of curves defined parametrically or in polar or vector form.
5.
The Definite Integral
Integrals involving parametrically defined functions.
6.
Integration
By parts; by partial fractions (involving nonrepeating linear factors only); improper integrals.
7.
Applications of Integration to Geometry
Area of a region bounded by polar curves; arc length.
8.
Further Applications of Integration and Riemann Sums
Velocity and distance problems involving motion along a planar curve; velocity and acceleration vectors.
9.
Differential Equations
Euler’s method; applications of differential equations, including logistic growth.
10.
Sequences and Series
Definition of series as a sequence of partial sums and of its convergence as the limit of that sequence; harmonic, geometric, and p-series; integral, ratio, comparison and limit comparison tests for convergence; alternating series and error bound; power series, including interval and radius of convergence; absolute and conditional convergence; Taylor polynomials and graphs; finding a power series for a function; Maclaurin and Taylor series; Lagrange error bound for Taylor polynomials; computations using series.
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