Correspondence between Advanced Placement topic outline and course schedule

The following topic list, taken from the Advanced Placement Calculus BC course description, has been annotated to show the cross-references to specific sections in our textbook and/or supplemental activities or lectures. FDWK refers to the textbook Calculus: Graphical, Numerical, Algebraic by Ross L. Finney, Franklin D. Demana, Bert K. Waits, and Daniel Kennedy, published by Scott Foresman, 1999. Copies of this textbook are available on loan if you wish, but the Foerster text by itself should be adequate.

I. Functions, Graphs, and Limits

Analysis of graphs §§1-2, 3-1, and others

Limits of functions (including one-sided limits) §§1-5, 2-1, 2-2, 2-3, 2-5

Asymptotic and unbounded behavior §§2-5, 6-8

Continuity as a property of functions §§2-4, 4-6

IVT §2-6

EVT §2-6

Parametric, polar and vector functions

Parametric §§4-7, 4-8 [since implicit
relations can sometimes

be plotted parametrically]

Polar §8-9

Vector §10-7

II. Derivatives

Concept of the derivative

Deriv. presented graphically, num., analytically §§1-1, 1-2, 3-1, 3-2, 3-3

Deriv. interpreted as instantaneous rate of change §§1-1, 3-1, 4-8, 5-3, 7-4, 7-5

Deriv. defined as limit of difference quotient §§3-2, 3-3, 3-4

Relationship betw. differentiability and continuity §4-6

Derivative at a point

Slope of a curve at a point §§3-1, 3-5, 3-6, 4-7, 4-8, 7-4

Tangent line to a curve at a point and local lin. app. §§3-1, 3-2, 5-3, FDWK p. 107

Inst. rate of change as the limit of avg. rate of chg. §§3-2, 3-4, 3-7, 3-8, others

Approximate rate of change from graphs and tbls. §3-1, 3-3, 3-5, motion sensor
activities

Derivative as a function

Corresponding characteristics of graphs of f and f ' §§3-3, 3-4, 3-6, 3-8

Rel. betw. incr. and decr. beh. of f and sgn(f ') §§1-2, 3-3, 3-5, many others

MVT and its geometric consequences §§5-6, 5-7, 5-8

Equations involving derivatives, incl. verbal transl. Chapter 7, §10-4

Second derivatives

Corresp. characteristics of graphs of f, f ', and f '' §§3-5, 8-1, 8-2, 10-1, 10-2

Rel. betw. concavity of f and sgn(f '') §§8-2, 8-3, 10-6

Pts. of inflection as places where concav. changes §8-2

Applications of derivatives

Analysis of curves, incl. monotonicity and concav. §§3-3, 8-1, 8-2

Analysis of curves in parametric, polar, vector form

Parametric §§4-7, 4-8 [since implicit
relations can sometimes

be plotted parametrically]

Polar §8-9

Vector §10-7

Velocity and acceleration §§3-5, 10-2, 10-7

Optimization, both absolute and relative §§8-3, 10-5, 10-6

Modeling rates of change, incl. related rates §§7-1, 7-2, 7-3, 10-4

Use of implicit differentiation to find deriv. of inv. §§4-5 [in exercises], 4-8

Interp. of deriv. as a rate of change in applications §3-5, §4-1 activity,

Chapter 7, Chapter 10

Geometric interpretation of differential equations

...via slope fields §7-4

...rel. betw. slope fields and solution curves §§7-4, 7-5, 7-6,

Supplemental Lecture 3

Numerical solution of diff. eqs. using Euler’s Meth. §7-5, Supplemental Lecture 1

L’Hôpital’s Rule §§6-8, 9-10, 12-7

Computation of derivatives

Knowledge of derivatives of basic functions

Power functions §3-4

Exponential functions §6-5

Logarithmic functions §6-6

Trigonometric functions §§3-6, 3-8, 4-4

Inverse trigonometric functions §4-5

Hyperbolic and inverse hyperbolic functions §9-9 [post-exam topic]

Basic rules for deriv. of sums, products, quotients §§2-3, 3-2, 4-1, 4-2, 4-3

Chain rule and implicit differentiation §§3-7, 4-8

Derivatives of parametric, polar, and vector fcns.

Parametric §§4-7, 4-8 [since implicit
relations can sometimes

be plotted parametrically]

Polar §8-9

Vector §10-7

III. Integrals

Interpretations and properties of definite integrals

Definite integral as a limit of Riemann sums §§5-4, 5-5

Definite integral of the rate of change...(FTC1) §5-8

Basic properties of definite integrals §5-9

Applications of integrals

Integral of a rate to give accumulated change §§5-8, 5-9

Area of a plane region §8-4

Area of a plane region (polar) §8-9

Volume of a solid with known cross sections §8-5

Average value of a function §10-3

Distance traveled by a particle along a line §§10-1, 10-2

Length of a curve §8-7

Length of a curve (parametric) §8-7

Length of a curve (polar) §8-8

Other applications Chapter 11

Fundamental Theorem of Calculus (FTC)

Use of FTC to evaluate definite integrals §§5-8, 5-9, 5-10, others

Use of FTC to represent a particular antideriv. §6-3

Techniques of antidifferentiation

Antiderivatives following from basic functions Chapter 3, §5-2, Chapter 6

Antiderivatives by substitution of variables §6-9 and others

Antiderivatives by parts §§9-2, 9-3, 9-8

Antiderivatives by simple partial fractions §9-7

Improper integrals (as limits of def. integrals) §9-10

Applications of antidifferentiation

Finding specific antiderivs. using init. conds. Chapter 7, §10-2

Solving sep. diff. eqs., using them in modeling Chapter 7

Solving logistic diff. eqs., modeling with them §§7-4 and 7-5, Supplemental
Lecture 2

Numerical approximations to definite integrals

Riemann sums §§1-3, 5-5, 5-7

Trapezoidal sums §1-4 and in numerous
problems thereafter

IV. Polynomial Approximations and Series

Concept of series; convergence and divergence Chapter 12

Series of constants

Motivating examples, including dec. exp. Chapter 12

Geometric series with applications §12-2

The harmonic series §12-7

Alternating series with error bound §12-7

Terms of series as areas of rects., incl. int. test §12-7

The ratio test for convergence and divergence §12-6

Comparing series to test for conv. or div. §§12-6, 12-7

Taylor series

Taylor polynomial approximation with graphs §§12-3, 12-4, 12-5

Maclaurin series and the general Taylor series §12-5

Maclaurin series for e^x, sin x, cos x, and 1/(1-x) §12-5

Formal manipulation of Taylor series §12-5, Supplemental

Lecture 5