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A P CALC SYLLABUS

CalculusAB Syllabus

SwansboroHigh School

Instructor: Ed Walsh

Course Design and Philosophy

The course is designed to give students the skills and conceptual understandings necessary to be successful on the CollegeBoard’s®CalculusAB examination. Topics covered in the course are taken directly from the topic outline set forth by the CollegeBoard® and mandated by the North Carolina Standard Course of Study. In order to emphasize conceptual understanding of calculus a multi-faceted educational approach using graphical, numerical, analytical, and verbal instructional strategies is implemented. Additionally, connections between calculus and other subjects such as physics and chemistry allow students to appreciate the real world applications of concepts covered.

The class meets five days a week for 90 minutes and lasts one full semester. Students enrolled in this class are expected to take responsibility for their own learning. As such, the teacher acts more as a facilitator of learning than a presenter of information. Students must engage in class discussions and be willing to analyze information, interpret facts, communicate their reasoning, and discover expanded meanings. The primary goal of the course is for students to gain a conceptual understanding of the foundations of calculus. Therefore, rather than basing the course on skills memorization the focus is on developing students’ understandings of the underpinnings and purpose of calculus.

Teaching Strategies

Success in calculus requires that students have a strong algebra background. Students enrolled in calculus are required to take precalculus the preceding semester. This ensures students will have covered all prerequisite material immediately prior to embarking upon calculus. In the first weeks of the course we all work extensively on establishing guidelines related to study habits, use of graphing calculators, and student communication in both verbal and written form. Students are immediately made aware of the high performance expectations for the course and their responsibility with regards to obtaining assistance when they do not grasp certain concepts.

Calculus differs from many mathematics courses in that the emphasis is often more on the student’s ability to explain and defend their conclusions instead of obtaining a correct numeric value. To this end,considerable time and effort is devoted to having students practice giving verbal and written explanations of their reasoning processes and conceptual understandings. The limited amount of lecture in the class requires students to interact with the instructor and peers and develop the ability to communicate in the language of math. Cooperative learning is used extensively throughout the course.

Effective use of the graphing calculator allows students to focus on conceptual understanding and less on tedious or cumbersome tasks such as graph creation and numeric computation. All students have access to a TI-83 Plus or TI-84 Plus both at home and school. The calculators are particularly useful in completion of the labs implemented throughout the course. For example, calculator usage is an essential component of the discovery lab for instantaneous rate of change and the lab on falling objects.

Course Outline

Chapter P: Preparation for Calculus (4 days)

(Note: all students will have taken precalculus the preceding semester)

  1. Graphs and Models
  2. Intercepts
  3. Symmetry
  4. Points of intersection
  5. Modeling (Scatter plots/Regression equations)
  1. Linear Models and Rates of Change
  2. Slope as a rate of change
  3. Linear equations
  4. Parallel and perpendicular lines
  1. Functions and Their Graphs
  2. Functions
  3. Functional notation
  4. Domain and range
  5. Eight basic functions
  6. Transformations
  7. Combinations of Functions
  1. Fitting Models to Data
  2. Linear models
  3. Quadratic models
  4. Trigonometric models

Chapter 1: Limits and Their Properties (10 days)

  1. A Preview of Calculus
  2. What is calculus?
  3. The tangent line problem
  4. The area problem
  1. Finding Limits Graphically and Numerically
  2. Introduction to limits
  3. Limits that fail to exist
  4. Formal definition of a limit
  1. Evaluating Limits Analytically
  2. Properties of limits
  3. Direct substitution
  4. Strategies for finding limits
  1. Continuity and One-Sided Limits
  2. Continuity at a point
  3. Continuity of a function
  4. One-sided limits
  5. Properties of continuity
  6. Intermediate value theorem
  1. Infinite Limits
  2. Infinite limits
  3. Vertical asymptotes

Chapter 2: Differentiation

  1. The Derivative and the Tangent Line Problem
  2. The tangent line problem
  3. The derivative of a function
  4. Differentiability and continuity
  1. Basic Differentiation Rules and Rates of Change
  2. The power rule
  3. The constant multiple rule
  4. The sum and difference rules
  5. Derivatives of sine and cosine Functions
  6. Rates of change
  1. The Product and Quotient Rules and Higher-Order Derivatives
  2. The product rule
  3. The quotient rule
  4. Derivatives of trigonometric rules
  5. Higher-order derivatives
  1. The Chain Rule
  2. The chain rule
  3. The general power rule
  4. Simplifying derivatives
  5. Trigonometric functions and the chain rule
  1. Implicit Differentiation
  2. Implicit and explicit functions
  3. Implicit differentiation
  1. Related Rates
  2. Finding related rates
  3. Problem solving related rates

Chapter 3: Applications of Differentiation (15 days)

  1. Extrema on an Interval
  1. Extrema of a function
  2. Relative extrema and critical numbers
  3. Finding extrema on a closed interval
  1. Rolle’s Theorem and the Mean Value Theorem
  1. Rolle’s theorem
  2. The Mean Value theorem
  1. Increasing and Decreasing Functions and the First Derivative Test
  1. Increasing and decreasing functions
  2. The first derivative test
  1. Concavity and the Second Derivative Test
  1. Concavity
  2. The second derivative test
  1. Limits at Infinity
  1. Limits at infinity
  2. Horizontal asymptotes
  3. Infinite limits at infinity
  1. A Summary of Curve Sketching
  1. Analyzing the graph of a function
  1. Optimization Problems
  1. Applied minimum and maximum problems
  1. Newton’s Method
  1. Newton’s method
  2. Algebraic solutions of polynomial equations
  1. Differentials
  1. Linear approximations
  2. Differentials
  3. Error propagation
  4. Calculating differentials

Chapter 4: Integration (12 days)

  1. Antiderivatives and Indefinite Integration
  1. Antiderivatives
  2. Notation for antiderivatives
  3. Basic integration rules
  4. Initial conditions and particular solutions
  1. Area
  2. Sigma notation
  3. Area
  4. The area of a plane region
  5. Upper and lower sums
  1. Riemann Sums and Definite Integrals
  2. Riemann sums
  3. Definite integrals
  4. Properties of definite integrals
  1. The Fundamental Theorem of Calculus
  2. The fundamental theorem of calculus
  3. The mean value theorem for integrals
  4. Average value of a function
  5. The second fundamental theorem of calculus
  1. Integration by Substitution
  2. Pattern recognition
  3. Change of variables
  4. The general power rule for integration
  5. Change of variables for definite integrals
  1. Numerical Integration
  2. The trapezoidal rule
  3. Simpson’s rule

Chapter 5: Logarithmic, Exponential, and Other Transcendental Functions (15 days)

  1. The Natural Logarithmic Function: Differentiation
  2. The natural logarithmic function
  3. The number e
  4. The derivative of the natural logarithmic function
  1. The Natural Logarithmic Function: Integration
  2. Log rule for integration
  3. Integrals of trigonometric functions
  1. Inverse Functions
  2. Inverse functions
  3. Existence of an inverse function
  4. Derivative of an inverse function
  1. Exponential Functions: Differentiation and Integration
  2. The natural exponential function
  3. Derivatives of exponential functions
  4. Integrals of exponential functions
  1. Bases Other than e and Applications
  2. Bases other than e
  3. Differentiation and integration
  4. Applications of exponential functions
  1. Differential Equations: Growth and Decay
  2. Differential equations
  3. Growth and decay models
  1. Differential Equations: Separation of Variables
  2. General and particular solutions
  3. Separation of variables
  4. Applications
  1. Inverse Trigonometric Functions: Differentiation
  2. Inverse trigonometric functions
  3. Derivatives of inverse trigonometric functions
  4. Review of Basic Differentiation Rules
  1. Inverse Trigonometric Functions: Integration
  2. Integrals involving inverse trigonometric functions
  3. Completing the square
  4. Review of basic integration rules

Chapter 6: Applications of Integration (10 days)

  1. Area of a Region Between Two Curves
  2. Area between two curves
  3. Area between intersecting curves
  4. Integration as an accumulation process
  1. Volume: The Disk Method
  2. The disk method
  3. The washer method
  4. Solids with known cross sections
  1. Volume: The Shell Method
  2. The shell method
  3. Comparison of disk and shell methods
  1. Arc Length and Surfaces of Revolution
  2. Arc length
  3. Area of a surface of revolution

Chapter 7: Integration Techniques. L’Hopital’s Rule, and Improper Integrals (8days)

  1. Basic Integration Rules
  2. Fitting integrands to basic rules
  1. Integration by Parts
  2. Integration by parts
  3. Tabular method
  1. Trigonometric Substitution
  2. Indeterminate Forms and L’Hopital’s Rule
  3. Indeterminate forms
  4. L’Hopital’s Rule

Activities

  1. After learning basic derivation rules and discussing position, velocity, and acceleration students complete a lab on falling objects. The purpose of the lab is to have students take real-world data related to time, height, and velocity of a falling ball and make both graphical and conceptual connections. The lab involves creation of a mathematical model for the position function of an object in freefall based on data, a comparison of the derived model and the function s(t) = .5gt2+v0t+s0, and a calculation of earth’s gravity based on the data.
  2. Once students understand the graphical implications of the first and second derivatives they complete a project that requires them to take the graph of f(x) and sketch a feasible graph for both f’(x) and f’’(x). Then they take a graph of f’(x) and sketch f(x) and f’’(x). Written justifications incorporating the concepts of increasing/decreasing and concavity must accompany each graph.
  3. To emphasize the relationship between calculus and chemistry the students do a lab dealing with Boyle’s Law. Given a table of pressure and volume data students are able to use graphical analysis and differential calculus to explore the relationship between the pressure of a trapped gas and its container. To expand the activity students are also led to combine Charles’s Law with Boyle’s Law to see how temperature changes relate to pressure and volume.
  4. To make the study of optimization problems more meaningful students complete a project on finding optimal packaging for products. Starting with right circular containers students develop a function to describe the relationship between radius and surface area for a package of given volume. Students are then able to determine the optimal radius for packaging products with different volumes. The process is then repeated for packages with square bases. Students answer questions such as: “what advantages do rectangular containers have over cylinders?” and “why would a company make a container that does not use an optimal surface area?’.
  5. As fun opportunity for students to show their conceptual understanding of topics in calculus they complete a project at the end of the semester that requires them to write a children’s book explaining one or more calculus concepts in a way that an eight year old child could understand. The book does not need to explain the computational work; it just needs to make clear the concept. For example, the book may explain what a limit is but does not need to explain how to find a limit. Students are encouraged to be creative.

Major Text

Larson, R., Hostetler, R., & Edwards, B. Calculus of a Single Variable. 7th

ed.).Boston: Houghton Mifflin Company, 2002.