AP Calculus AB Course Syllabus Frenchtown High School Course Design and Purpose This course is designed to provide instruction and an opportunity for student work on each topic listed in the AP Course Description. The primary goal of this course is to link the study of Limits, Derivatives, and Integrals in such a way that students gain a conceptual understanding of Calculus. Through a conceptual approach students will have the ability to apply their knowledge to application based problems and develop their ability to communicate findings. The secondary goal of this course is to prepare students to pass the AP Calculus AB exam in the spring of the year. Technology Integration Technology is an important part of the world we work and live in. To that end graphing calculators and computer technology will be integrated into the course in order to further students conceptual understanding of Calculus. Students will become proficient using the Ti- 84 family of calculators to interpret results and make connections within calculus topics. Calculators will also be used to analyze data and make predictions based on the observed results regarding the rate that data changes at various times. Students will be required to have this technology; however those who do not personally own calculators will have the ability to check calculators out for the entire school year. Computer applets and software programs will be used in conjunction with the classroom SmartBoard to further students conceptual understanding of key calculus topics. Teaching Strategies The course will begin with a review of Pre-Calculus with a particular emphasis placed on review of functions and various representations of functions. This will also present an opportunity for students to review and strengthen their use of graphing calculator capabilities and furthering their ability to interpret the meaning of the information obtained from their calculator. As the course progresses into the exploration of limits, derivatives, and integrals, students will be expected to communicate and demonstrate their understanding of these concepts both verbally and in writing; demonstrating the usage of numerical, graphical, and analytical information to present a sound knowledge of calculus and an ability to apply concepts in multiple situations will be a key component of this course. Students will regularly present their work to the class in order to develop communication skills and enhance their clarity of understanding and allow for the instructor to check for conceptual understanding. The expectation is that frequent student presentation and participation will
develop a community of calculus learners who are comfortable presenting their work and who understand the expectation of communicating their ideas is paramount to their success in this course and on the AP exam. This class will also use calculator based activities to introduce new concepts and help students make connections with old concepts. Students will be expected to communicate their findings in written form, following proper rules of grammar and sentence construction. Lessons will be a mixture of lecture, student investigation, and discussion. The use of activities will be integrated to further student understanding of the key components of the course. Students will be expected to clearly communicate their findings from these activities in a written format. Included is a course calendar for the each of the 90 minute block periods. After the AP exam students will be placed in teams and will be required to explore an additional topics (primarily from the Calculus BC topic list) and present this information to the class. Assessment Throughout the year students will be assigned homework problems to ensure development of a sound foundation in calculus. In addition, students will be given quizzes and exams that are comprised of multiple choice and free response questions. The course will frequently use released AP Calculus AB exam items as daily exercises as well as exam and quiz questions. Proper answer methods for free response questions will be taught and expected for full credit. Students will take a mid-year exam comprised entirely of selected released items from both the multiple choice and the free response sections of the AP Calculus AB exam. Two weeks prior to the spring AP exam students will take a practice exam comprised of the most recently released AP Calculus AB exam. Students will be strongly encouraged to take the exam in the spring, those who do will be exempt from a year-end final. Those who do not will be administered a released exam as their final, with their corresponding through 5 score being related to an appropriate letter grade matching the grading scale of Frenchtown High School. Primary Textbook and Resources Calculus, Ron Larson, Robert Hostetler, Bruce Edwards, Seventh Edition. Houghton Mifflin Company, 00 A Watched Cup Never Cools; Ellen Kamischke, Key Curriculum Press, 999 Additional resources will be integrated as needed to enhance student learning.
The following is a course calendar delineating the progression of the topics covered in this course. Each course day is considered one 90 minute class. CHAPTER : LIMITS AND THEIR PROPERTIES Understanding of what calculus is and what types of problems it can be used to solve A Preview of Calculus Using a graphing calculator to find a limit graphically Finding Limits Graphically and Numerically Evaluating Limits Analytically Continuity and One-Sided Limits Infinite Limits Using a graphing calculator and tables to find a limit Understanding when a limit fails to exist and demonstrating this idea using a graphing calculator to show a function does not approach the same value from the right and the left Algebraic properties of limits Limits of trigonometric functions Substitution method to find limits Dividing and rationalizing technique Definition of continuity Properties of continuity Intermediate Value Theorem Relationship between infinite limits and asymptotes Using graphing calculator to explore graphs as they approach vertical asymptotes from both the right and left side
CHAPTER : DIFFERENTIATION The Derivative and Tangent Line Problem Basic Differentiation Rules and Rates of Change The Product Rule The Quotient Rule Higher Order Derivatives The Chain Rule Implicit Differentiation Related Rates Tangent Line Problem Limit definition of a derivative Using the zoom feature of a calculator to explore local linearity Differentiation and continuity Where f may fail to exist Basic differentiation rules Trigonometric differentiation rules Derivatives as rates of change (velocity and acceleration) Higher order derivatives Using the product rule The relation between a function and its derivative Using the quotient rule Using the calculator to discover the relation between the graph of function and its derivative graph Using the calculator to find numeric derivatives Second derivatives and beyond Relationship to velocity and acceleration Analyzing the relationship between the graphs of a function and the first and second derivative The general power rule and the chain rule Trigonometric functions and the chain rule Guidelines for using implicit differentiation Finding tangent lines to relations using implicit differentiation and relating their meaning Finding the second derivative implicitly Strategies for setting up related rates problems Using implicit differentiation to solve related rate problems Communication of results of related rate problems using complete sentences in the explanation
CHAPTER 3: APPLICATION OF DIFFERENTIATION Extrema on an Interval Rolle s Theorem and Mean Value Theorem Increasing and Decreasing Functions Concavity and the Second Derivative Limits and Infinity Optimization Differentials Value of the derivative at a relative extrema Extreme value theorem and continuity Non-differentiable locations and the relation to extrema Illustration and application of Rolle s Theorem Illustration and application of Mean Value Theorem Using graphing calculators to explore applications of Rolle s and Mean Value Theorems First derivative test and it s relation to the graphs of a function Organizing data using tables Verification of the first derivative test using graphing calculator Using the second derivative to test concavity of functions Finding points of inflection Using the graphing calculator to find zeros, maximums, and minimums while relating them to points of inflection Connecting the graphs of f and f with the graph of f Limits as they relate to horizontal asymptotes of functions Evaluating limits at infinity Limits involving trigonometric functions Overall summary of curve sketching and comparison to creating graphs with graphing calculators Maximization problems Minimization problems Application of graphing technology into problem solving and expressing results in clearly written sentences Relationship to change in x and y values Local linearity and approximations using a calculator to explore the concept Calculating differentials 3 3
CHAPTER 4: INTEGRATION Antiderivatives Indefinite Integrals Riemann Sums Fundamental Theorem of Calculus The second Fundamental Theorem of Calculus Integration by Substitution Numeric Integration Definition of antiderivatives Basic rules of integration Initial conditions and particular solutions Application of basic integration rules Connection to differential equations Partitions of a region to find area Connection to limits of a function Relationship between continuity and integration Evaluating a definite integral Using FTC to find area Mean value as it relates to area Average Value of a function Applications of the nd FTC Application problems relating to FTC Using the graphing calculator to find the area under a curve Change of variables Recognition of patterns for use of substitution General power rule and integration Change of variables and definite integrals Trapezoid Rule Simpson s Rule 3
CHAPTER 5: LOGARITHMIC, EXPONENTIAL, AND OTHER TRANSENDENTAL FUNCTIONS Differentiation of Natural Logarithms Integration of Natural Logarithms Inverse Functions Differentiation of Exponential Functions Integration of Exponential Functions Differentiation of Inverse Trigonometric Functions Integration of inverse trigonometric functions Definition of natural logarithm function Logarithmic properties Derivatives of natural logarithms Log rule for integration Integrals of 6 basic trigonometric functions Derivatives of inverse functions Continuity and differentiability of inverse functions Graphical analyses of inverse functions Derivatives of exponential functions Applications with bases other than e Modeling and solving with graphing calculator Integrals of exponential functions Applications with bases other than e Modeling and solving with graphing calculator Evaluating inverse trigonometric functions Finding the derivative of inverse trigonometric functions Integrating with inverse trigonometric functions
CHAPTER 5.5: DIFFERENTIALS Constructing slope fields manually and on graphing calculators Slope Fields Interpreting slope fields Growth and Decay Separation of Variables Using slope fields and calculators to estimate solutions Solving differential equations Applications with differential equations Separation of variables techniques Particular solutions Applications involving separation of variables CHAPTER 6: APPLICATIONS OF INTEGRATION Area of a Region Between Two Curves Volume: Disk Method Volume: Washer Method Calculating the area between two curves Intersecting regions created by two curves Using graphing calculators to find points of intersection Horizontal and vertical representations Revolution around multiple axis Using technology for visual recognition Applications and expression of solutions in writing Removal of Internal areas to create hollowed shapes Using technology for visual recognition Applications and expression of solutions in writing 3
REVIEW FOR AP EXAM Review for AP Exam Multiple choice problems Free response problems Connections between major themes of calculus Practice exam 0 Review will be an ongoing process throughout the year with frequent opportunities to work released exam items. The major themes of Limits, Derivatives, and Integrals will be linked throughout the course, often using your previous knowledge to interpret information and solve problems. The schedule above has built in time for assessment and flexibility in teaching. In addition, there are 4 additional days that can be used as needed for further instruction of an idea or for integration of lab experiments that are not already incorporated into the syllabus.