APS
DISTRICT HIGH SCHOOL MATHEMATICS CURRICULUM FRAMEWORK
Course Title: AP Honors Calculus BC
Course Number: 38085
Department: Mathematics
ADS Number: 20595133
Prerequisites: Sucessful completion of AP Honors Calculus AB
Length of Course: One Year
Credit/PRI Area: .50 per Semester /Mathematics
Grade Level(s): 9 - 12
Important Notes: This calculus course consists of a full high school academic year of work that is comparable to calculus courses in colleges and universities.
The student is expected to seek college credit, college placement, or both from the college of his/her choice. The student is required to use a graphing calculator
and must have access to the graphing calculator at school and at home. The student must agree to participate in at least one organized mathematics competition
held in the district.
COURSE DESCRIPTION: AP Calculus BC extends the concepts of limits, differentiation, and integration studied in AP Calculus AB to include sequences, infinite series, and
parametric, polar and vector valued functions. More attention is given to techniques of integration using parts, trignometric substitution, partial fractions, and improper integrals.
Optional topics include vector analysis, developed to study lines, planes and surfaces in 3-dimensional space and/or advanced problem solving techniques used in major
mathematics contests. The student develops ideas from analysis throughout AP Calculus BC. The course prepares the student for the BC level Advanced Placement Exam in
Calculus and as a result includes an extensive review of relevant topics.
By the time the student takes this course, he/she has already met and exceeded the standards set forth by the National Council of Teachers of Mathematics (NCTM), the State of
New Mexico Mathematics Standards, and the Albuquerque Public Schools (APS) District Mathematics Standards. Therefore, the performance standards represented in this course
can be aligned to the strands “Topics for Further Study” recommended by the State Department of New Mexico.
AP HONORS CALCULUS BC
17.1.12
Albuquerque Public Schools 03/04
STRATEGIES:
The “Illustrations” column in the Program of Studies provides exemplars of the performance standards, strategies, and best practices suggested by mathematics teachers in the
Albuquerque Public Schools (APS).
ASSESSMENTS:
Assessments may include: authentic and performance-based assessment, cooperative learning, teacher observations, checklists, test and exams, formal and informal writing, small
group and full class discussions, oral and multimedia presentations, individual and group projects, demonstrations, and portfolios. Assessments are based on appropriate rubrics.
SUGGESTED TEXTBOOKS AND INSTRUCTIONAL MATERIALS:
 Current state adopted mathematics textbooks
 Supplemental materials
 Graphing calculators
 Computer software
 Released AP Calculus Exams and rubrics
 Released UNM/PNM Math contests and solutions
 Released MAA contests and solutions
SUGGESTED TITLES/AUTHORS WEB SITES:
 apcentral.collegeboard.com
 Calculus with Analytic Geometry, 7th edition; Larson, Hostetler, Edwards, Hyde
 Calculus – Graphical, Numerical, Algebraic – Finney, Demana, Waits, Kennedy
 Calculus – Concepts and Applications – Paul Foerester
 Calculus – Single Variable – Hughes-Hallett, Gleason, et al.
 Calculus – Concepts and Contexts – James Stewart
 Calculus from Graphical, Numerical, and Symbolic Points of View – Ostebee, Aaorn
 Solutions: AP Calculus Problems – Broadwin, Lenchner
 MAA Notes
Approved by HSCA:
AP HONORS CALCULUS BC
17.2.12
Albuquerque Public Schools 03/04
STRAND I: GLOBAL MATHEMATICS PROCESSES
CONTENT STANDARD: The student understands and uses mathematical processes.
BENCHMARK: The student uses problem solving, reasoning and proof, communication, connections, and representations as appropriate in all mathematical experiences.
GRADE
9 - 12
PERFORMANCE STANDARDS
ILLUSTRATIONS
NOTE: Illustrations include suggested activities for attaining each
performance standard. A check for (√) refers to a key feature to look for
while assessing student performance.
1.
Applies the “rule of four” (i.e., represents mathematics graphically,
symbolically, verbally, numerically) (APS – all of Strand I).
2. Prepares mathematically for future careers (APS – I. 14).
3.
Uses graphing technology throughout the curriculum (APS – I.9;
NM – IC.2).
4.
Uses reasoning and problem-solving strategies to solve new problems
[APS – I.3; NM – IIA (5-7)].
5.
Makes connections among mathematical concepts (APS – I.12;
NM – IA.6).
6.
Recognizes when to use previously learned strategies to solve new
problems (APS – I.2; NM – IC.1, IID.2).
7.
Identifies how seemingly different mathematical situations may be
essentially the same (e.g., The intersection of two lines is the same as the
solution to a system of linear equations) (APS – I.13; NM – IA.7,
IA.10).
AP HONORS CALCULUS BC
17.3.12
1 – 7. The student studies how drug dosage is established by researching nasal
allergy sprays. Based on the research he/she determines recommended dose
rates for a “new” nasal allergy spray. The student creates data, analyzes the
data and prepares a brief report including tables and graphs to support
answers. Based on research the student:
 Determines an amount of his/her “new” drug in the person’s body
necessary for the drug to be effective but safe (i.e., 50 µg to 80 µg).
 Gives the amount of drug remaining in the body over a specific time
period (i.e., 80% of what it was at the beginning of a 6 hour period).
 Determines the amount of drug delivered by each puff. (i.e., 7 µg).
 Writes a geometric series for the amount of drug remaining in the
person’s system after n puffs assuming a person uses one puff of the
spray each given time period.
 States when the amount in the person’s system exceeds the given
amount necessary to be effective.
 Determines if the amount would ever exceed the amount necessary to be
safe.
 Examines the effects of using two, three or four puffs each time period.
In each case determines when first effective and when, if ever, beyond
safe levels.
 Gives evidence to determine if several puffs of the spray could be taken
twice a day and have the levels stay effective and safe.
 Gives evidence to determine if several puffs of the spray could be taken
once a day and have the levels stay effective and safe.
 Reports on possible effects to the patient if not taken as prescribed.
 completion of all required components
 use of technology
 problem solving strategies
Albuquerque Public Schools 03/04
GRADE
9 - 12
PERFORMANCE STANDARDS
ILLUSTRATIONS



AP HONORS CALCULUS BC
17.4.12
effective communication
connections
multiple representations
Albuquerque Public Schools 03/04
STRAND II: FUNCTIONS, GRAPHS, AND LIMITS
CONTENT STANDARD: The students uses geometric, numeric and analytic information about curves.
BENCHMARK: The student uses calculus to predict and explain global and local behavior of functions, planar curves and sequences.
GRADE
9 - 12
PERFORMANCE STANDARDS
ILLUSTRATIONS
1.
Takes limits using l’hopital’s rule.
1.
The student recognizes seven indeterminate forms and can manipulate
expressions into ratios to apply l’hopitals rule. The student is given
opportunities throughout the course to solve problems by using l’hopital’s
rule in determining limits and convergence of improper integrals and series.
 comprehension
 accuracy
2.
Uses the limit as the number of terms of a sequence increases without
bound to determine convergence/divergence and boundedness.
2. The student creates both a table of values and a graph for a variety of
sequences as n increases, investigates patterns of possible outcomes and
order of magnitude, and formulates definitions for convergence, divergence,
and bounded.
 representations of data
 reasonable definitions
 identification of patterns
 conceptual understanding
3.
Computes the limit of the sequence of partial sums of an infinite series.
3. The student determines if a sum exists for an infinite series. If the sum can
not be given specifically, the student uses limits to bound the sum.
 calculations
 accuracy
4.
Recognizes and/or constructs and interprets graphs of planar curves
given in parametric, polar, and vector form.
4. The student creates and investigates (i.e., able to find dy/dx, rates of change)
planar curves including those given in parametric form, polar form, and vector
form.
OR
The student converts equations to parametric form and from parametric
form.
 correct interpretations
 accuracy
 elimination of parameters
AP HONORS CALCULUS BC
17.5.12
Albuquerque Public Schools 03/04
GRADE
9 - 12
PERFORMANCE STANDARDS
ILLUSTRATIONS



5.
Generates and interprets with and without the aid of a graphing calculator
graphs of differential equations.
AP HONORS CALCULUS BC
17.6.12
use of appropriate parameters
equivalence
ability to find dy/dx (i.e., rate of change)
5. As the student enters the room each is given a card with either a slope field
graph or a differential equation. He/She must then find the person who has
his/her match. Each pair then explains what attributes of the graph and of the
equation were used to determine why they belong together.
 team work, collaboration
 accuracy
 effective communication
Albuquerque Public Schools 03/04
STRAND III: DERIVATIVES
CONTENT STANDARD: The student uses differentiation in theoretical and real-world applications.
BENCHMARK: The student performs analysis of planar curves, differentiaties as a problem solving method, uses differentiation to create a Taylor series, and interprets
differential equations via slope fields.
GRADE
9 - 12
PERFORMANCE STANDARDS
ILLUSTRATIONS
1.
Determines the derivative of parametric, polar and vector functions.
2.
Calculates speed given parametric equations.
3.
Compares and contrasts velocity and acceleration vectors.
1 – 3. Working alone or in small groups, the student solves and presents
released exam problems taken from AP Calculus BC.
From 2002 # 3:
The position of a roller coaster car at time t seconds where 0 ≤t≤ 18 seconds is
given parametrically by x(t) = 10t + 4 sin t and y(t) = 920-t)sin t + cos t – 1,
where x and y are measured in meters.
The student does each of the following. In each case he/she writes at least one
complete sentence explaining or justifying his/her findings.




4.
Creates Taylor series using repeated differentiation of a given function.
5.
Manipulates infinite series using differentiation.
AP HONORS CALCULUS BC
17.7.12
Graph this situation using appropriate range and domain.
Find the slope of the path at time t = 2.
Find the acceleration vector of the car at the time when the car’s
horizontal position is x = 140.
Find the time t at which the car is at its maximum height, and find the
speed, in m/sec, of the car at this time.
 accuracy
 effective presentation, communication
 team work (if working in groups)
 completion of all components
4, 5. The student generates the Taylor series for f(x) = sin x using Taylor’s
Theorem and then uses his/her series to give a Taylor series for f(x) = cos x.
 accurate generation of series
 methodology
Albuquerque Public Schools 03/04
GRADE
9 - 12
PERFORMANCE STANDARDS
ILLUSTRATIONS
6.
Sketches solution curves on a slope field.
7.
Uses Euler’s method to calculate a numerical solution to a differential
equation.
AP HONORS CALCULUS BC
17.8.12
6 – 7. From AP Calculus BC #5, 2002 the student uses a graphing calculator able
to generate a slope field to do the following:
 Generate the slope field for the differential equation dy = 2y – 4x
dx
 Sketches the solution curve that passes through the point (0, 1) and
(0, -1).
 Lets f be the function that satisfies the given differential equation with
the initial condition f(0) = 1.. Use Euler’s method, starting at x = 0 with
a step size of 0.1, , to approximate f(0.2).
 use of appropriate technology
 representation
 accuracy
 prescribed methods
Albuquerque Public Schools 03/04
STRAND IV: INTEGRALS
CONTENT STANDARD: The student interprets and applies integrals in theoretical and real-world situations.
BENCHMARK: The student integrates using advanced techniques of integration, integrates to manipulate Taylor series, evaluates improper integrals, solves logistic
differential equations, and applies integration methods to find areas of regions bounded by parametric and polar curves, and calculates length of a curve and
distance traveled by a particle along a line and/or curve given by parametric or polar equations.
GRADE
9 - 12
PERFORMANCE STANDARDS
ILLUSTRATIONS
1.
Determines antiderivatives by parts, trigonometric substitution, and
partial fractions.
1.
The student integrates a variety of problems correctly choosing effective
and efficient methods (e.g., trigonometric substitution, partial fractions)
and tells what criteria of the integrand was used to select one method over
any other method.
 accuracy
 justification
 strategies
2.
Evaluates improper integrals.
2.
The student uses improper integrals in complex problems. From AP
Calculus BC #5, 2001, the student evaluates
function satisfying
f 1  4 and lim
x 


3.
Uses antidifferentiation to manipulate a Taylor series.
4.
Calculates arc length of a curve given by parametric or polar equations.
5.
Finds area of a region bounded by a planar curve given in parametric,
polar, or vector form.
AP HONORS CALCULUS BC
17.9.12

 3xf x dx if
1
f is a
f x   3xf x  , for all real numbers x , with
f x   0 .
accuracy
calculations
3.
Given an infinite power series for all x in the interval of convergence, the
student writes the first three nonzero terms and the general term for an
infinite series determined from integrating the given series.
 accuracy
4.
The student determines correct limits of integration so that a curve is
traced out only once on an interval of integration and applies the
appropriate arc length formula.
 correct use of formulas
 accuracy
5.
The student draws and/or constructs three interlocking circles given by
r  2a cos , r  2asin  , and r  a and calculates the area of each
of the seven regions inside one or more of the three interlocking circles.
 representations
Albuquerque Public Schools 03/04
GRADE
9 - 12
PERFORMANCE STANDARDS
ILLUSTRATIONS

correct calculations
6.
Solves logistic differential equations and uses them in modeling.
6.
The student separates and integrates analytically equations of the general
 P 
L  P 
dP
dP , and
form
 kP1   kP
, sketches a graph of
 L 
 L 
dt
dt
describes the meaning of the shape of the solution curves, where P is
increasing, decreasing, meaning of inflection point, asymptotes, etc. The
student uses this information to solve population problems.
 accuracy
 representations
 effective communication
 connections
7.
Integrates to calculate work done by a variable force.
7.
The student calculates work done in problems such as emptying/filling a
tank, compressing/stretching a spring, lifting a chain with or without any
weight attached (e. g., ignoring friction, air resistance).
 accuracy
 connections to real-world situations
AP HONORS CALCULUS BC
17.10.12
Albuquerque Public Schools 03/04
STRAND V: POLYNOMIAL APPROXIMATIONS AND SERIES
CONTENT STANDARD: The student represents and analyzes functions using infintite series.
BENCHMARK: The student investigates the concept of series, explores convergence or divergence of a sequence and of a series, determines error and/or can bound a sum of
an infinite series if possible, represents elementary functions as intinite power series, and can manipulate one series to create another for a composite
function.
GRADE
9 - 12
PERFORMANCE STANDARDS
ILLUSTRATIONS
1.
Defines a series as a sequence of partial sums and convergence as the
limit of the sequence of partial sums.
1. The student creates both a table of values of the sequence of partial sums and
a graph of the harmonic series as well as the natural log (ln(x)) function,
compares and contrasts the two and writes their reflections on convergence
of an infinite series. The student answers the question: What does “Find the
sum of an ∞ series.” mean?
 representation of data
 graphical representations
 effective communication
2.
Investigates and uses 10 tests of convergence/divergence.
(a) Geometric Series Test
(b) Telescoping Series
(c) Nth Term Test
(d) P-series Test
(e) Improper Integral Test
(f) Direct Comparison Test
(g) Limit Comparison Test
(h) Ratio Test
(i) Root Test
(j) Alternating Series Test
(i) Absolutely Convergent
(ii) Conditionally Convergent
2. The student is given 10 series such that each test can be used. The student,
working in pairs, determines convergence/divergence of each series using all
10 tests, once per problem.
 team work, collaboration
 completion of all required components
 accuracy
 comprehension
3.
Uses tests of convergence/divergence to find the values of x that
determine radius of convergence and interval of convergence for power
series.
3, 6. From AP Calculus BC #2, 1996; the student finds values of
the Maclaurin series for f x  given by
x for which
x x2 x3
xn


 ... 
 ... converges. If gx   xf x , the
2! 3! 4!
n  1!
student writes the Maclaurin series for g x  showing the first three nonzero
terms and the general term, writes g x  in terms of a familiar function
1
AP HONORS CALCULUS BC
17.11.12
Albuquerque Public Schools 03/04
GRADE
9 - 12
PERFORMANCE STANDARDS
ILLUSTRATIONS
without using series, and then, writes f x  in term of the same familiar
function.
 correct evaluations
 completion of all required components
 representation
 recognition of patterns
4.
Determines error and/or calculates possible bounds for sums of infinite
series.
 Alternating series error
 Improper integral error
 Lagrange error for Taylor polynomials
4, 6. From AP Calculus BC#5, 1994; the student finds the interval of
convergence of the power series for f x  about x  0 if f x   e
.
If g is the function given by the sum of the first four nonzero terms of the
2 x 2
f x  about x  0 , the student shows
f x  gx  0.02 for -0.6 ≤ x ≤ 0.6.
power series for


5.
Graphically demonstrates approximating functions using Taylor
polynomials centered at x  a .
6.
Uses Maclaurin series and formal manipulation of the series to solve
x
problems for elementary functions (i.e., e , sin x , cos x , and 1 ).
1 x
AP HONORS CALCULUS BC
17.12.12
accuracy
method
5. As a class activity, the student discusses how to generate and graph
polynomial functions that look like sin x and observes patterns as fine
tuning of the polynomials evolves.
 individual participation
 effective communication
 recognition of patterns
 applications
Albuquerque Public Schools 03/04