# AP Calculus BC Syllabus

```AP Calculus BC Syllabus
PHILOSOPHY:
My main objective in teaching AP Calculus BC is to provide students with a rich and
thorough understanding of calculus that will enable them to succeed at higher levels of
mathematics. Students will be working in cooperative learning groups throughout the
school year. Within these groups students will be asked to discover the mathematical
concepts and relationships numerically, graphically, and algebraically. A graphing
calculator (TI 89) will be used to help students understand these mathematical ideas.
Students will also be asked to communicate their discoveries using the proper
mathematical vocabulary.
TIMELINE:
I. Students are to complete a summer packet prior to the opening of school. The
packet consists of problems from chapter one of Hughes – Hallet’s 4th edition and
Larson’s 7th edition.
II. Review (4 weeks)
A. Linear, quadratic, exponential, logarithmic, and sinusoidal functions
B. Compositions and Inverses
C. Parametric Equations
ACTIVITY: I use the old College Board Pacesetter activity “Ships in
the Fog” to introduce the concept of parametric equations.
Students are asked to use the TI-89 graphing calculator to explore
the concept of parametric equations numerically, graphically, and
algebraically.
D. Sequences and Series
E. Polar
II. LIMITS and CONTINUITY (2 weeks)
A. Limits at a point
1. Properties of limits
2. Two sided
3. One-sided
B. Limits involving infinity
1. Asymptotic behavior
2. Properties of limits
3. Visualizing limits
C. Continuity
1. Continuous
2. Discontinuous functions
ACTIVITY: I use the old College Board Pacesetter activity “Rational Functions” to
introduce the concept of limits and continuity. Students are asked to graph various
rational functions. The TI 89 calculator is used to verify their graphs as well as help
interpret the concepts of limits and continuity. The students are then asked to report
their findings in a short essay. The TI 89 helps students visualize the concepts of limits
and continuity both numerically (table of values) and graphically.
III. DERIVATIVES (5 weeks)
A. Definition
1. Secant line
2. Tangent line
B. Using the definition to find a derivative
1. Algebraic interpretation
2. Geometric interpretation
ACTIVITY: We play a derivative game. Each student is handed a card at random. The
card either contains a graph or a function. The card is also labeled either FUNCTION or
DERIVATIVE. The students are then to find their correct partners. I then use these
pairings as my cooperative learning groups.
C. Rates of change – To help students understand the concept of derivatives,
I use a CBL experiment. Students toss a ball in the air and examine the
height versus time graph that is generated. Students then use a quadratic
equation to fit the data and determine how high the ball went and how
long the ball was in the air. Students can then calculate the average
velocity over the time interval. They are then asked to determine the
velocity at exactly 0.5 seconds. We can then use the CBL to graph the
acceleration. Great classroom discussions take place during and after this
activity.
1. Average rate of change (slopes of secant lines)
2. Instantaneous rate of change (slopes of tangent lines)
D. Mean Value Theorem (MVT)
ACTIVITY: The Door Problem—Instantaneous Rate of Change of a Function (taken
from Calculus Concepts and Applications Instructor’s Resource Book by Paul A.
Foerster) This activity helps students further understand the relationship between
average and instantaneous rates of change.
E. The second derivative
F. Continuity and differentiability
IV. SHORTCUTS TO DIFFERENTIATION (4 weeks)
A. The power rule and polynomial functions
B. The exponential and logarithmic functions
C. Product and quotient rules
D. The chain rule
E. Inverses and Trigonometric functions
F. Related rates
G. Implicit differentiation
H. Local linearity, linear approximations, and Newton’s Method
ACTIVITY: “Investigating the Accuracy of the Tangent Line Approximation” (ancillary
materials from Hughes-Hallet)
V. USING THE DERIVATIVE (4 weeks)
A. First and second derivative tests
B. Curve sketching and family of curves – The TI 89 calculator is used to
help verify results graphically.
C. Optimization
ACTIVITY: “Packaging: The Optimal Form” (Larson, 7th ed., p.158)
MIDTERM: I review with the students for the midterm examination. The midterm
consists of: 15 M/C without the calculator, 10 M/C with the calculator, and 2 Free
Response (no calculator). Approximately 5 class periods prior to the exam, students are
tested. I give them old AP free response that covers limits and derivatives as well as
M/C from various sources.
VI. INTEGRATION (5 weeks)
A. Riemann Sums, Trapezoidal Rule and Simpson’s Rule
B. The Definite Integral
C. Interpretation of the definite integral
D. Fundamental Theorem of Calculus
ACTIVITY: “The Wankel Rotary Engine and Area” (Larson, 7th ed., p.240)
PROJECT: Students are to take a ride in a car with a parent. They are to record the
odometer reading on the car prior to the trip and after the trip is completed. During the
trip they are to use the speedometer and record the car’s speed every minute. The
students are then to use their data to create a speed versus time graph. From their
graph, students are to use integration techniques to calculate the distance traveled. The
students then compare their results with the actual distance traveled.
E. Antiderivatives and the Substitution Method
F. Integration by parts, trigonometric substitution, and partial fractions
G. Equations of Motion
H. The Second Fundamental Theorem of Calculus
I. Area between Curves
J. Volumes
1. Disc Method
2. Washer Method
3. Cylindrical Shells Method
4. Known Cross-sections
a. Squares
b. Semicircles
c. Equilateral Triangles
d. Isosceles Right Triangle
VII. DIFFERENTIAL EQUATIONS (3 weeks)
A. Separation of Variables
B. Slope Fields – I use a TI -89 graphing calculator to project a grid on the
chalkboard. Each student is then assigned a point and comes to the
board and draws the appropriate segment based on the differential
equation we are working with. The class then completes the slope field.
C. Euler’s Method
VIII. PARAMETRIC, POLAR, and VECTOR FUNCTIONS (3 weeks)
A. Length of parametrically defined curves
B. Vectors and vector-valued functions
C. Calculus of vector functions
D. Calculus of polar functions, including slope, length, and area
IX. SEQUENCES (2 weeks)
A. Arithmetic, harmonic, alternating harmonic, and geometric
B. Convergence and divergence
C. Bounded, monotonic, oscillating sequences
D. Limit properties of sequences
E. L’H&Ocirc;pital’s Rule and indeterminate forms
F. Relative rates of growth of functions
G. Improper integrals and the comparison test
X. SERIES (4 weeks)
ACTIVITY: An Investigation into the Accuracy of Polynomial Approximations to
Transcendental Functions
A. Geometric
B. Power - Term by term differentiation and integration
C. Taylor’s /Maclaurin
D. Tests of convergence/divergence
The month prior to the AP exam, we review. The students are administered the
1997, 1998, 2003, and released practice exams in their entirety. The ’97 exam is giving
as a Spring Break assignment. The ’98, ’03, and released practice exams are
administered on Saturdays. All exams are timed and graded like an actual AP exam.
ASSESSMENTS:
 Do Now: Students are administered a two question M/C quiz for the first 10
minutes of class every day beginning in October
 Exams are administered after each unit
 2 to 3 projects are assigned throughout the course
 Students are required to keep a Calculus Journal. Once every two weeks
students are giving a “Calculus Prompt.” They must respond to this prompt using
the appropriate calculus to support their solution.
Textbooks:
Hughes-Hallett, Deborah, et al. Calculus Single Variable. 4th ed. New York: Wiley &amp;
Sons 2005.
Larson, R., Hostetler, R., and Edwards, B. Calculus with Analytic Geometry. 7th ed.
Boston: Houghton Mifflin 2002.
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