Course: MATH 550/555 AP Calculus
-Students will be able to work with functions represented in a variety of ways: graphically, numerically, analytically, or verbally.
-Students will come to understand the meaning of the derivative in terms of a rate of change and be able to use derivatives to solve a variety of problems.
-Students will come to understand the meaning of the definite integral as a limit of Riemann sums and as an accumulation of a rate of change and use integrals to solve a variety of problems.
-Students will come to understand the relationship between the derivative and the definite integral as expressed in both parts of the Fundamental Theorem of Calculus.
-Students will be able to communicate mathematics both orally and in well written sentences.
-Students will become able to model a written description of a physical situation with a function, a differential equation, or an integral.
-Students will be able to use technology to help solve problems.
-Students will develop an appreciation of Calculus as a coherent body of knowledge.
Calculus: Graphical, Numerical, Algebraic: AP Edition
By: Ross L. Finney, Franklin D. Demana, Bert K. Waits,
and Daniel Kennedy, Published by Pearson Prentice Hall, 2007
Calculators will play a major role in the day to day teaching, investigation and problem solving of Calculus. Students are
required to have access to a graphing calculator.
The recommended model is the TI-84 or TI-83 (Plus).
1. Analysis of Functions (14 Days)
A. Review of linear functions, trigonometric functions, exponential functions, piecewise functions and composite functions.
B. Limits of functions as X approaches both positive and negative infinity (end behavior of the function) and as X approaches a constant (vertical asymptotes versus holes in the graph).
C. Implications of continuity of a function including Intermediate Value Theorem.
2. Derivatives of Functions (30 Days)
A. The derivative presented as an instantaneous rate of change graphically, numerically and analytically.
B. The derivative defined as a difference quotient.
C. Use of the derivative to determine slope of curve at a point and to find the equation of tangent line to a curve.
D. Estimate rates of change from graphs and from tables of data.
E. Rules developed for finding derivatives of basic elementary functions including power, and trigonometric functions.
F. Rules for finding derivatives of sums, products, and quotients of functions.
G. The chain rule for finding derivatives of composite functions and implicit differentiation for functions not explicitly stated.
H. Derivatives of inverse trigonometric functions, exponential, and logarithmic functions.
3. Applications of the Derivative (30 Days)
A. Derivative used to find local and global extreme values of functions.
B. Relationship between sign of F'(x) and increasing or decreasing behavior of F(x).
C. Use second derivative, F'', to determine concavity and location of inflection points where concavity changes.
D. Connecting derivatives, F' and F'', to the graph of F.
E. Use of derivative to find solutions to optimization problems.
F. Linearity and Newton's Method.
G. Solving problems of related rates.
H. Modeling applications of rates of change, specifically position, velocity, and acceleration.
4. The Definite Integral (30 Days)
A. Computation of Riemann sums with finite approximations (rectangular approximations with right, midpoint, or left evaluation points).
B. Definite integral as limit of Riemann sum as number of partitions approaches infinity.
C. Finding integrals by signed areas and using the calculator.
D. Compute the average value of a function.
E. Fundamental Theorem of Calculus to evaluate definite integrals.
F. Fundamental Theorem of Calculus demonstrates relationship between Integration and Differentiation.
G. Trapezoidal approximation of integral and Simpson's rule.
H. Antiderivatives and slope fields.
I. Evaluating antiderivatives from basic rules of differentiation.
J. Antiderivatives by substitution of variables, including changing limits of definite integrals.
K. Integration by parts.
4. Applications of the Definite Integral (30 Days)
A. Solving the differential equation dy/dx =ky (exponential growth and decay, Newton's law of cooling).
B. Use of definite integral to find areas enclosed by curves.
C. Compute volumes of solids with known cross section.
D. Compute volumes of solids generated by rotating an area about an axis using disc or cylindrical shell methods.
E. Lengths of curves.
F. Definite integral to solve applications from science such as work or force problems.
5. L'Hopital's Rule, Special Situations or Applications (20 Days)
A. Evaluation of limits by L'Hopital's rule.
B. Evaluating integrals by manipulations such as long division, and the use of partial fractions.
C. Use of limits to evaluate improper integrals..
D. Use initial conditions to evaluate specific antiderivatives, particularly related to acceleration, velocity and position.
E. Modeling projectile motion.
Preparation For the AP Exam
Throughout the year, test questions and weekly bonus offerings are modeled after AP test questions. Students become very familiar with the format of the multiple choice and free response. The free response questions particularly become the subject of classroom discussion as students are asked to present their solutions with the requirement that they must "Justify" all answers.
We maintain a pace through the year that allows us to complete the designated topics two weeks prior to the exam date. We then spend the remaining time reviewing the material and working in small groups or with partners on sample tests taken from the workbook, Multiple Choice & Free Response Questions for the AP Calculus (AB) Examination by David Lederman, published by D & S Marketing Systems, Inc. The groups take turns explaining their solutions and methodology for the various problems as we make a little competition out of the review. By test day, the students are well acquainted with the format of the test.
My Philosophy of Calculus
Calculus is an approach to problem solving. The underlying philosophy of Calculus is, "No problem is too big if tackled with confidence, care, and taken in small steps." We reference that theme the first time we try to approximate the slope of a function at a specific point or when we try to find the area under a curve using rectangular approximations. The class becomes well versed in the concept, "If you want more accuracy, use smaller steps".
Applications from business, medicine, physics, and economics are frequently incorporated in our study. The text has many problems with real data from cited sources. Through all of this the students are using the same philosophy, the methodology of Calculus, to solve problems that may be presented analytically/algebraically, numerically, graphically, or verbally. They see the connection of the various modes as merely being different ways of modeling real life situations. Calculus is not a dry, procedural course with lots of "recipes" for working out meaningless equations. It is full of problem solving procedures that help explain the world around us.