AP
CALCULUS BC
TEACHER: Greg Houston
E-MAIL: greg_houston@eu.odedodea.edu
ROOM: B 1.15, ground floor, blue pod
Calculus BC incorporates everything in the AB course plus the objectives in italics or with an asterix. Material is presented using the “Rule of Four” where problems are presented and solved 1) graphicallyStudents 2) numerically 3) algebraically or 4)verbally or in writing. Students are expected to think on their feet, work problems on the board and present their solutions to the class. These include practical applications of integrals to model physical, biological, and economic situations.
COURSE GOALS/OBJECTIVES/STANDARDS: The topics cover
I.
FUNCTIONS, GRAPHS,
AND LIMITS
Analysis of Graphs. With the aid of technology, graphs of functions are often easy to produce. The emphasis is on the interplay between the geometric and analytic information and on the use of calculus both to predict and to explain the observed local and global behavior of a function.
Limits of functions (including one-sided limits).
-An intuitive understanding of the limiting process.
-Calculating limits using algebra.
-Estimating limits from graphs or tables of data.
Asymptotic and unbounded behavior.
-Understanding asymptotes in terms of graphical behavior.
-Describing asymptotic behavior in terms of limits involving infinity.
-Comparing relative magnitudes of functions and their rates of change. (For example, contrasting exponential growth, polynomial growth, and logarithmic growth.)
Continuity as a property of functions.
-An intuitive understanding of continuity. (Close values of the domain lead to close values of the range.)
-Understanding continuity in terms of limits.
-Geometric understanding of graphs of continuous functions (Intermediate Value Theorem and Extreme Value Theorem).
*Parametric, polar and vector
functions.
The analysis of planar curves
includes those given in parametric, polar and vector form.
Concept of the derivative.
-Derivative presented graphically, numerically, and analytically.
-Derivative interpreted as an instantaneous rate of change.
-Derivative defined as the limit of the difference quotient.
-Relationship between differentiability and continuity.
Derivative at a point.
-Slope of a curve at a point. Examples are emphasized, including points at which there are vertical tangents and points at which there are no tangents.
-Tangent line to a curve at a point and local linear approximation.
-Instantaneous rate of change as the limit of average rate of change.
-Approximate rate of change from graphs and tables of values.
Derivative as a function.
-Corresponding characteristics of graphs of f and f’.
-Relationship between the increasing and decreasing behavior of f and the sign of f’.
-The Mean Value Theorem and its geometric consequences.
-Equations involving derivatives. Verbal descriptions are translated into equations involving derivatives and vice versa.
Second derivatives.
-Corresponding characteristics of the graphs of f, f’, and f’’.
-Relationship between the concavity of f and the sign of f’’.
-Points of inflection as places where concavity changes.
Applications of Derivatives.
-Analysis of curves, including the notions of monotonicity and concavity.
+Analysis of planar curves given in parametric, polar and vector form. This includes velocity and acceleration vectors.
-Optimization, both absolute (global) and relative (local) extrema.
-Modeling rates of change, including related rates problems.
-Use of implicit differentiation to find the derivative of an inverse function.
-Interpretation of the derivative as a rate of change in varied applied contexts, including velocity, speed, and acceleration.
+Numerical solution of
differential equations using Euler’s method.
+L’Hospital’s Rule, including
its use in determining limits and convergence of improper integrals and series.
Computation of derivatives.
-Knowledge of derivatives of basic functions, including power, exponential, logarithmic, trigonometric, and inverse trigonometric functions.
-Basic rules for the derivative of sums, products, and quotients of functions.
-Chain rule and implicit differentiation.
+Derivatives of parametric,
polar, and vector functions.
-Computation of Riemann sums using left, right, and midpoint evaluation points.
-Definite integral as a limit of Riemann sums over equal subdivisions.
-Definite integral of the rate of change of a quantity.
-Basic properties of definite integrals. (Examples include additivity and linearity.)
Applications of Integrals. Appropriate integrals are used in a
variety of applications to model physical, biological or economic situations.
Although only a sampling of applications can be included in any specific
course, students should be able to adapt their knowledge and techniques to
solve other similar application problems. Whatever applications are chosen, the
emphasis is on using the integral of a rate of change to give accumulated
change or using the method of setting up an approximating Riemann sum and
representing its limit as a definite integral. To provide a common foundation,
specific applications should include finding the area of a region, the volume
of a solid with known cross sections, the average value of a function, and the
distance traveled by a particle along a line, and the length of a curve
(including a curve given in parametric form).
Fundamental Theorem of Calculus
-Use of the Fundamental Theorem to evaluate definite integrals.
-Use of the Fundamental Theorem to represent a particular antiderivative, and the analytical and graphical analysis of the functions so defined.
Techniques of
Antidifferentiation.
-Antiderivatives follow directly form derivatives of basic functions.
-Antiderivatives by substitution
of variables (including change of limits for definite integrals), parts, and
simple partial fractions (nonrepeating linear factors only).
Applications of antidifferentiation.
-Finding specific antiderivatives using initial conditions, including applications to motion along a line.
-Solving separable differential equations and using them in modeling. In particular, studying the equation y’ = kx and exponential growth.
+Solving logisctic differential
equations and using them in modeling.
Numerical approximations to definite integrals.
Use of Riemann and trapezoidal sums to approximate definite integrals of functions represented algebraically, graphically, and by tables of values.
*IV. POLYNOMIAL APPROXIMATIONS TO DEFINITE INTEGRALS
*Concept of series.
A series is
defined as a sequence of partial sums, and convergence is defined in terms of
the limit of the sequence of partial sums. Technology can be used to explore
convergence or divergence.
*Series of constants.
+Motivating examples, including
decimal expansion.
+The geometric series with
applications.
+The harmonic series.
+Alternate series with error
bound.
+Terms of series as areas of
rectangles and their relationship to improper integrals, including the integral
test and its use in testing the convergence of p-series.
+The ratio test for convergence
and divergence.
+Comparing series to test for
convergence or divergence.
*Taylor series.
+Taylor polynomial
approximation with graphical demonstration of convergence.
+Maclaurin series and the
general Taylor series centered at x = a.
+Maclaurin series for the
functions e^x, Sin x, Cos x and 1/(1-x).
+Formal manipulation of Taylor
series and shortcuts to computing Taylor series, including substitution,
differentiation, antidifferentiation, and the formation of new series from
known series.
+Functions defined by power
series.
+Radius and interval of
convergence of power series.
+Lagrange error bound for
Taylor polynomials.
TEXTBOOK: Roland E. Larson and Robert P. Hostetler; Calculus of a Single Variable; Houghton Mifflin Company; 1998. Sixth edition.
SCOPE AND SEQUENCE:
Fall
Semester
Chapter P Preparation for Calculus 4 days
Chapter 1 Limits and their Properties 4 days
Chapter 2 Differentiation 7 days
Chapter 3 Applications of Differentiation 7 days
Chapter 4 Integration 12 days
Chapter 8 Infinite Series 11 days
Chapter 5 Differentiation and Integration of Trig/Log/ Exp Functions 8 days
Chapter 6 Applications of Integration 8 days
Chapter 7 Integration Techniques and L’Hospitals Rule 5 days
Chapter 9 Conics, Parametric Equations, and Polar Coordinates 6 days
AP Calculus Practice Exams 6 days
CONTINOUS SCHOOL
IMPROVEMENT:
AFNORTH
International High School’s Continuous School Progress (CSP) goal is “all
students will
improve their written communication skills
across the curriculum.” The 6 + 1 Traits is the model
selected to improve school-wide writing in
all subject areas. The 6 + 1 Trait writing framework
is a powerful way to learn and use a common
language to refer to characteristics of writing as well
as establish a common vision of what strong
writing looks like. Teachers and students will use the
6 + 1 Trait model to identify areas of
strength and weakness as they continue to strive towards
continued writing improvement. Success of all
students requires that the 6 + 1 trait become a
consistent and integral component of each
course taught at AFNORTH International High School.
All tests (4 to 5 per semester) will contain at least one problem in which the student will be required
to write a paragraph detailing how they would solve and check that problem. Those problems will
be scored based on a rubric involving content, student understanding and use of one of the 6 + 1
traits.
Students will receive in class training and practice in writing the above
paragraphs.
COURSE ASSESSMENT:
Marks are cumulative and grades each semester will be based on:
-Chapter Tests 45%
-Projects/Assignments/Quizzes 20%
-Homework and Homework checks 10%
-Final Exam 25%
After the AP exam, Calculus topics of interest to the class will be covered.
HOMEWORK POLICY: The purpose of homework is to practice the skill learned that day and you can expect to have homework daily. It is YOUR responsibility to stay caught up and review your work regularly. You will often work homework problems on the board and explain your solution to the class. If an unplanned absence occurs, get the notes from another student and work on the assignment. Homework checks will be given on problems assigned.
MAKE-UP POLICY: Students are expected to get assignments PRIOR to planned absences. If your absence is 2 or more class periods, make an appointment to get help before you leave.
I am available during Seminar or during Academic Coaching on Wednesdays 1545-1730 when Activity busses run.