AP Calculus BC 2014 – 2015
Mr. Cella
Lower Moreland High School
Course Overview
The AP Calculus AB course covers all those topics usually included in a first course in the
calculus, including the appropriate ones from analytic geometry. One of the early topics, limits,
is introduced from an intuitive approach and a more formal approach. The student is soon
introduced to the derivative, techniques for finding derivatives, simple curve sketching, the
solution of maximum and minimum problems, and the determination of velocity and
acceleration.
The integral is introduced through approximation methods, the Fundamental Theorem of
Calculus is developed, and the techniques of integration as well as applications of the definite
integral are studied. The integral is also presented as the limit of Riemann sums, and the student
learns how to approximate an integral numerically with techniques that can be supported with
technology.
Those students completing the AP Calculus AB curriculum as a junior, then take the AP
Calculus BC curriculum as a senior. Review and extensions are incorporated, as well as complete
units on conic sections, polar coordinates, parametric and vector applications, arc length and
surface area, techniques of integration, differential equations, and sequences and series.
Course Objectives
These goals have been adopted as established by the AP College Board and delineated at
apcentral.collegeboard.com.
- Students should be able to work with functions represented in a variety of way: graphical,
numerical, analytical, or verbal. They should understand the connections among these
representations.
- Students should understand the meaning of the derivative in terms of a rate of change and local
linear approximation and should be able to use derivatives to solve a variety of problems.
- Students should understand the meaning of the definite integral both as a limit of Riemann
sums and as the net accumulation of change and should be able to use integrals to solve a
variety of problems.
- Students should understand the relationship between the derivative and the definite integral as
expressed in both parts of the Fundamental Theorem of Calculus.
- Students should be able to communicate mathematics both orally and in well-written sentences
and should be able to explain solutions to problems.
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- Students should be able to model a written description of a physical situation with a function, a
differential equation, or an integral.
- Students should be able to use technology to help solve problems, experiment, interpret results,
and verify conclusions.
- Students should be able to determine the reasonableness of solutions, including sign, size,
relative accuracy and units of measurement.
- Students should develop an appreciation of calculus as a coherent body of knowledge and as a
human accomplishment.
Need Help?
If you need assistance for any reason (absent from class, confused, missing Mr. Conrad’s class,
etc.), schedule and appointment to meet me after school in advance. I am available MondayThursday until 3:15. If you are struggling with the material, I expect to see you there!
AP Calculus BC Course Outline
Extensions to the AP Calculus AB curriculum are noted via bold print.
Additional units in the AP Calculus BC curriculum are indicated via bold print.
Unit 1: Precalculus Review (Review as needed)
A. Lines
1. Slope as rate of change
2. Parallel and perpendicular lines
3. Equations of lines
B. Coordinate geometry applications
C. Functions and graphs
1. Functions
2. Domain and range
3. Families of functions
4. Piecewise functions
5. Composition of functions
D. Exponential and logarithmic functions
1. Exponential growth and decay
2. Inverse functions
3. Logarithmic functions
4. Properties of logarithms
E. Trigonometric functions
1. Graphs of basic trigonometric functions
a. Domain and range
b. Transformations
c. Inverse trigonometric functions
2. Applications
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Unit 2: Limits and Continuity (Review as needed)
A. An intuitive understanding of the limiting process
B. Estimating limits from graphs or tables of data
C. Limits at a Point
1. Algebraic properties of limits
2. Two-sided limits
3. One-sided limits
D. Limits involving Infinity
1. Asymptotic behavior
2. End behavior
3. Algebraic properties of limits at infinity
4. Visualizing limits
E. Continuity as a property of functions
1. Continuous functions
2. Discontinuous functions
a. Removable discontinuity
b. Jump discontinuity
c. Infinite discontinuity
3. Intermediate Value Theorem
4. Extreme Value Theorem
Unit 3: The Derivative (1 week)
A. Concept of the derivative
1. Derivative presented graphically, numerically, and analytically
2. Derivative interpreted as an instantaneous rate of change
3. Derivative defined as the limit of the difference quotient
4. Relationship between differentiability and continuity
B. Derivative at a point
1. Slope of a curve at a point
2. Tangent and normal lines
C. Derivative as a function
1. Corresponding characteristics of f , f ', and f "
2. Relationships between graph and derivative
D. Differentiation Rules
1. Derivative of polynomials
2. Derivative of the power function
3. Derivative of exponential functions
4. Product and quotient rules
5. Derivative of trigonometric functions
6. Chain Rule for composite functions
7. Derivative of inverse trigonometric functions
8. Derivative of hyperbolic functions
9. Implicit Differentiation
10. Derivative of logarithmic functions
E. Linear approximations and differentials
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Unit 4: Applications of Differentiation (2 weeks)
A. Graphing and the Analysis of Curves
1. Critical values
2. Increasing/decreasing
3. Relative and absolute extrema
4. Inflection points
5. Concavity
6. Tests for extrema
a. Closed interval test
b. First-derivative test
c. Second-derivative test
B. Using the derivative
1. The Mean Value Theorem
2. Rolle’s Theorem
C. Rectilinear motion
D. Projectile motion
E. Modeling rates of change
F. Optimization problems
G. Related rates problems
Unit 5: Integrals (2 weeks)
A. Riemann sums to represent area under a curve
1. Left-endpoint rectangular approximation method
2. Right-endpoint rectangular approximation method
3. Midpoint rectangular approximation method
4. Trapezoidal approximation method
5. Area as a limit of a Riemann sum
6. Simpson’s Rule
B. Integral of a rate of change to give accumulated change
C. The Fundamental Theorem of Calculus
D. Definite integrals
E. Indefinite integrals
F. The Average Value Theorem
G. Techniques of integration
1. Integration by inspection
2. Reversing the chain rule
3. Change of variable
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Unit 6: Techniques of Integration (5 weeks)
A. Integration by parts
B. Integrals of trigonometric functions
1. Powers of sine and cosine
2. Powers of tangent, cotangent, secant, and cosecant
C. Integration by trigonometric substitutions
D. Integration by partial fractions
1. Linear factors
2. Repeated linear factors
3. Quadratic factors
4. Repeated quadratic factors
5. Improper fractions
E. Integration of rational functions of sine and cosine
F. Miscellaneous substitutions
Unit 7: Applications of Integration (2 weeks)
A. Area between curves
B. Rectilinear motion
C. Projectile motion
D. Volume
1. Disks
2. Washers
3. Cylindrical shells
4. Known cross-sections
E. Arc length
F. Surface area
G. Work
Unit 8: Geometry in the Plane (5 weeks)
A. Conic sections in rectangular coordinates
B. Parametrically defined curves
1. Differentiation of a parametrized curve
2. Arc length of a parametrized curve
3. Conic sections defined parametrically
C. Vectors in the plane
1. The derived vector of a parametrized curve
2. Vector velocity and acceleration
D. Polar coordinates
1. Graphs of equations in polar coordinates
2. Differentiation of a polar curve
3. Area in polar coordinates
4. Arc length in polar coordinates
5. Conic sections defined in polar coordinates
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Unit 9: Indeterminate Forms (2 weeks)
A. The indeterminate form 0/0
1. L’Hôpital’s rule
2. Cauchy’s Mean-Value theorem
B. Other indeterminate forms
C. Improper integrals with infinite limits of integration
1. Convergent
2. Divergent
D. Other improper integrals
Unit 10: Infinite Series (4 weeks)
A. Sequences and their limits
B. Monotonic and bounded sequences
C. Infinite series of constant terms
1. Partial sums
2. Harmonic series
3. Geometric series
4. Telescoping series
5. Convergence and divergence
D. Infinite series of positive terms
1. Comparison Test
2. Limit Comparison Test
3. The p-series
E. The Integral Test
F. Alternating series
1. Alternating Series Test
2. Remainder
G. Absolute and conditional convergence
1. Definitions
2. Ratio Test
3. Root Test
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Unit 11: Power Series (3 weeks)
A. Convergence
1. Interval of convergence
2. Radius of convergence
B. Differentiation of power series
C. Integration of power series
D. Functions defined by power series
E. Taylor polynomial
1. Graphical demonstration of convergence
2. LaGrange error term
F. Maclaurin series
1. sin x and cos x
2. ex and ln x
1
1
3. 1−𝑥 and 1+𝑥
G. The Binomial series
Unit 12: An Introduction to Differential Equations (1 week)
A. Slope fields
B. Euler’s Method
C. Separable equations
D. Exponential growth and decay
E. The logistic model
Unit 13: Differential Equations (3 weeks)
A. First-Order Linear Differential Equations
B. Linear Differential Operators
C. Homogeneous Differential Equations
D. Nonhomogeneous Differential Equations
Primary Textbook
Calculus: Early Transcendentals – 5th edition
James Stewart
Thomson Higher Education
In my opinion, calculus is a fascinating discipline that blends mathematical theory with real
world application. When a student asks the dreaded question “When am I going to use this in
life”, calculus is my answer. With this discipline, fields such as engineering, physics, medicine,
and economics are viewed in a new light. I am looking forward to sharing this journey with you
this year, and I hope you enjoy it just as much as I do!
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