AP® Calculus AB



AP® Calculus AB

Syllabus

Course Overview

My major objectives in teaching AP® Calculus are to help my students appreciate the beauty and usefulness of Calculus and prepare them for success on the AP exam and in later college level mathematics courses.

An emphasis upon communicating mathematics is central to this course, both verbally through discussing the solutions to assigned problems in groups and also in writing by demonstrating complete solutions of problem to the class using the chalkboard, overhead projector, or Power Point presentations.

Technology is used to help the students explore and visualize the calculus concepts in a highly effective manner. Students are issued TI-89 calculators. The four required capabilities of a graphing calculator which follow are stressed:

1) Plotting the graph of a function within an arbitrary viewing window,

2) Finding the zeros of functions,

3) Numerically calculating the derivative of a function, and

4) Numerically calculating the value of a definite integral.

Functions are presented using the “rule of four”: graphically, numerically, analytically, and verbally. Through this approach, students are able to understand the connections between these representations and are able to grasp the “why” behind the major calculus concepts and thus apply these more easily to the solutions of multifaceted problems.

Course Outline

Limits and Their Properties (Chapter 1)

Time: 15 days

1. A Preview of Calculus

2. Finding Limits Graphically and Numerically

• Intuitive understanding of the limiting process

3. Evaluating Limits Analytically

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4. Continuity and One-Sided Limits

• Intuitive understanding of continuity

• Understanding continuity in terms of limits

• Geometric understanding of the graphs of continuous functions (Intermediate Value Theorem)

5. Infinite Limits

6. Limits at Infinity

• Understanding asymptotes in terms of graphical behavior

• Describing asymptotic behavior in terms of limits involving infinity

The review of precalculus concepts occurs throughout the course in conjunction with the introduction of related calculus concepts. For example, we review the graphs of basic families of functions with the introduction of the concept of local linearity and when explaining the observed local and global behavior of a function. Precalculus concepts are also reviewed daily with class warm-ups.

One of the activities in which the properties of the graphs of functions are reviewed involves the investigation of the relationship between the amount of liquid in various glass containers and the height of that liquid. Students collect data comparing the volume of water in increments of 100 ml. to the corresponding height of the liquid in centimeters and record the resulting data in a table. The data points are graphed and students compare the graphs obtained for the different glass containers with the equations of functions which have some of the same features. The students then explain in words what a graph can tell you about the shape of the matching glass container.

Differentiation (Chapter 2)

Time: 33 days

1) The Derivative and the Tangent Line Problem

• 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

• Slope of a curve at a point

• Tangent line to a curve at a point and local linear approximation

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• Instantaneous rate of change as the limit of average rate of change

• Approximate rate of change from graphs and tables of values

2) Derivative as a Function

• Corresponding characteristics of graphs of f, f’, and f”

• Equations involving derivatives.

3) Basic Differentiation Rules and Rates of Change

• Knowledge of derivatives of basic functions, including power and trigonometric functions

• Comparing relative magnitudes of functions and their rates of change

4) Product and Quotient Rules and Higher-Order Derivatives

• Basic rules for the derivative of sums, products, and quotients of functions

• Interpretation of the derivative as a rate of change in varied applied contexts, including velocity, speed, and acceleration

5) The Chain Rule

6) Implicit Differentiation

7) Related Rates

A student activity which is used involves sketching the graph of the derivative of a function. The graph of a function is displayed and each student is assigned a point on the graph in which the slope of the tangent line is determined. Each student then graphs the coordinate (x, f ‘(x)) which would lie on the graph of the function’s derivative. The graph of the derivative function f ‘ is found and the students write down and discuss verbally any relationships observed between the two graphs. Then the graphing calculator TI-89 program tanimate is used to display an animated sequence of tangent lines for several functions. The program also plots points representing the slopes of those tangent lines. By using this graphing calculator program, the students can verify the validity of the conjectures they wrote concerning the relationship between a function and its derivative graph.

Applications of Differentiation (Chapter 3)

Time: 30 days

1) Extrema on an Interval

2) Rolle’s Theorem and the Mean Value Theorem

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3) Increasing and Decreasing Functions and the First Derivative Test

• Relationship between the increasing and decreasing behavior of f and the sign of f’

4) Concavity and the Second Derivative Test

• Relationship between the concavity of f and the sign of f”

• Points of inflection as places where concavity changes

5) A Summary of Curve Sketching

• Prediction and explanation of observed local and global behavior of a function

• Corresponding characteristics of the graphs of f, f’, and f”

• Analysis of curves, including the notions of monotonicity and concavity

6) Optimization Problems

• Both absolute (global) and relative (local) extrema

7) Differentials

A student activity which illustrates optimization is investigating the problem of finding the size of an ice cream cone that will hold the largest amount of ice cream that is cut from a given circle. The students will model the problem by cutting a wedge from a circle, removing it and forming the remaining piece of the circle into a cone. The problem is to find the angle of the wedge that produces the cone with the greatest volume. The students will relate the formula for the volume of the cone to the angle of the wedge and determine a formula for the volume of the cone. Since the formula is complicated, the students will use their graphing calculators to graph the numerical derivative of the volume function, find its critical values, and determine the maximum volume.

Integration (Chapter 4)

Time: 22 days

1) Antiderivatives and Indefinite Integration

• Antiderivatives following directly from derivatives of basic functions

2) Area

• Finding the area of a region

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3) Riemann Sums and Definite Integrals

• Definite integral of the rate of change of a quantity over an interval interpreted as the change of the quantity over the interval: [pic]

• Basic properties of definite integrals (additivity and linearity)

• Using the integral of a rate of change to give accumulated change

• The average value of a function

• Distance traveled by a particle along a line

4) The 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 functions so defined

5) Integration by Substitution

• Antiderivatives by substitution of variables (including change of limits for definite integrals)

6) Numerical Integration

• Use of Riemann sums (using left, right, and midpoint evaluation points)

• Trapezoidal sums to approximate definite integrals of functions represented algebraically, graphically, and by tables of values

Average value of a function and its meaning is discovered in the lab titled, “It Averages Out in the End” from the book, A Watched Cup Never Cools, by Ellen Kamischke. The investigation is completed in pairs with students discussing their findings and reaching conclusions together. Functions are graphed with graphing calculators and the trace feature is used to complete a table of function values for given values of x. The students calculate the average of the function values and then compare it to the area of a region under the curve bounded by the given x-values. The process is repeated with different functions and questions are asked that relate the average value of a function to the average height of the function’s graph over an interval. The definition of average value of a function is discovered and then used to explore the mean distances of the planets in our solar system from the sun.

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Logarithmic, Exponential and Other Transcendental Functions (Chapter 5)

Time: 20 days

1) The Natural Logarithmic Function: Differentiation

2) The Natural Logarithmic Function: Integration

3) Inverse Functions

• Use of implicit differentiation to find the derivative of an inverse function

4) Exponential Functions: Differentiation and Integration

5) Bases Other Than e and Applications

6) Inverse Trigonometric Functions: Differentiation

7) Inverse Trigonometric Functions: Integration

An exploration using the graphing calculator will help students find the relationship between the derivatives of a function and its inverse. The students should graph the inverse functions[pic]. The graphing calculator will use its function of numerically calculating the derivative of a function to find the slope of f(x) at (1, 1), (2, 8) and (3, 27) and the slope of [pic] at (1, 1), (8, 2) and (27, 3). The students will discuss their observations and record their observations in writing.

Differential Equations (Chapter 6)

Time: 12 days

1) Slope Fields and Euler’s Method

• Geometric interpretation of differential equations via slope fields and the relationship between slope fields and solution curves for differential equations

2) Differential Equations: Growth and Decay

• Solving separable differential equations and using them in modeling (in particular, studying the equation [pic] and exponential growth)

3) Separation of Variables and the Logistic Equation

• Finding specific antiderivatives using initial conditions, including applications to motion along a line

To introduce slope fields, each student is assigned several coordinate points in a region of the Cartesian coordinate system surrounding the origin. For a given differential equation, each student computes the slope at the coordinate position and then goes to a

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graph board in front of the classroom, and draws a short line segment with the calculated slope and the coordinate point as the midpoint of the segment. It is important that all students accurately draw the line segments at a slant matching the calculated slope and compare the slopes that he/she draws to those drawn by other students. After all slopes are drawn, a discussion of observation by the students will occur.

Applications of Integration (Chapter 7)

Time: 20 days

1) Area of a Region Between Two Curves

2) Volume: The Disk Method (Includes disks, washers and volumes of solids with known cross sections)

3) Volume: The Shell Method

4) Basic Integration Rules

In order to visualize 3-dimensional solids with known cross-sections, students construct models in which the graphs of the equations that bound the base of the solid are sketched on a stiff board. A shape is chosen for the cross sections and at least 7 cross-sections are created to represent the cross-sectional slices perpendicular to the base. The cross sections are attached to the base. The students then set up and solve the integral that will be used to determine the volume of the constructed solid. Upon completion of the models, students will make a verbal presentation explaining the characteristics of their model and volume calculation to the class.

This schedule leaves about 4 weeks for review and practice prior to the AP Exam in May.

Student Evaluation

Quarter grades are computed using homework, performance assessments, quizzes, and exams. Each quarter grade represents 40% of the semester grade. The final exam represents the remaining 20% of the grade. Quizzes and exams are designed to reflect the AP Exam and contain both multiple choice and free-response questions. Each exam has two parts: one part

requiring graphing calculators and a second part not allowing graphing calculators.

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Free-response questions from past AP Exams are also assigned on a weekly basis. Students may discuss and work these problems together and come to me for extra help. Immediately prior to the AP Exam in May, the students take a complete released exam which is scored using the scoring guidelines provided by College Board.

Primary Textbook

Larson, Ron, Robert P. Hostetler, and Bruce H. Edwards. Calculus of a Single Variable. 8th ed. Boston: Houghton Mifflin, 2006.

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