AP CALCULUS AB
ROOM: B 1.15, ground floor, blue pod
This topic outline incorporates the syllabus of the College Board for this course. Material is presented using the “Rule of Four” where problems are presented and solved 1) graphically 2) numerically 3) algebraically and 4) verbally and in writing. Students are engaged in authentic applications involving limits and continuity, derivatives, integrals, and transcendental functions. Graphing calculators are required for this course as mandated by the College Board. Students should be encouraged to talk about the mathematics of change in calculus, to use the language and symbols of calculus to communicate, and to discuss problems and methods of solution.
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.
-Intuitive understanding of continuity. (Close domain values lead to close range values.)
-Understanding continuity in terms of limits.
-Geometric understanding of graphs of continuous functions (Intermediate Value Theorem and Extreme Value Theorem).
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.
-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.
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.
-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.
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.
-Antiderivatives follow directly form derivatives of basic functions.
-Antiderivatives by substitution of variables (including change of limits for definite integrals).
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.
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.
TEXTBOOKS: Roland E. Larson and Robert P. Hostetler; Calculus of a Single Variable; Houghton Mifflin Company; 1998. Sixth edition.
Finney, Demana, Waits and Kennedy; Calculus: Graphical, Numerical Algebraic;
Pearson Prentice Hall; 2003.
SCOPE AND SEQUENCE:
Fall
Semester
Chapter P Algebraic Preparation for Calculus 7 days
Chapter 1 Limits and Their Properties 4 days
Chapter 2 Differentiation 10 days
Chapter 3 Applications of Differentiation 10 days
Chapter 4 Integration 10 days
Chapter 4 Integration 6 days
Chapter 5 Differentiation and Integration of Trig/Log/ Exp Functions 10 days
Chapter 6 Applications of Integration 10 days
AP Calculus Practice Exams 9 days
Chapter 7 Integration Techniques and L’Hospitals Rule 6 days
CONTINOUS SCHOOL
IMPROVEMENT:
AFNORTH
International High School’s Continous 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.
Calculus students will write four or five
essays explaining the meaning of and uses of derivatives and integrals in
solving problems. These essays will be scored based on a rubric involving
content, student understanding and use 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:
-Boardwork 5%
-Homework 10%
-Tests and Quizes 60%
-Final Exam (AP Calculus Exam) 25%
After the AP exam, the following topics will be taught: Integration by Parts, Partial Fractions and Indeterminate Forms and L’Hospitals Rule.
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. Homework assignments for the entire year are passed out the first week of class. You will often work homework problems on the board and explain your solution to the class. If an unplanned absence occurs, get class notes from another student and work on the assignment. Homework checks will be given on problems assigned or they will be collected.
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 by appointment after school or during seminar.