**AP Calculus AB**

**Syllabus**

An outline of topics to be covered in AP Calculus AB is provided within this syllabus. This course is designed to provide students with instruction and a learning experience equivalent to a college course in single variable calculus.

Major areas to be taught include *Functions, Graphs, and Limits*; *Derivatives*; *Integrals (both indefinite and definite*); *Logarithmic, Exponential, and Other Transcendental Functions*; and *Applications of Integration*. Problems and their solutions will be approached from a graphical, numerical, and analytical perspective, with emphasis placed on how these representations are related. Students will be taught to explain and support their solutions both verbally and in writing. Throughout the course, students will concentrate on the importance of communication in their approach to understanding the problems, working through their solutions, and being able to present their solutions and rationale to classmates.

Graphing calculators will be used by the students to help solve problems and interpret their results and conclusions. Each student will be assigned a TI-83 Plus TI-84 Plus calculator for use both inside and outside the classroom. Although use of the calculator will be emphasized throughout the course, students will be required to work without calculators on specific assignments and assessments, where appropriate.

The textbook to be used for this course is *Calculus of a Single Variable*, Seventh Edition, by Ron Larson, Robert P. Hostetler, and Bruce H. Edwards. Each student will be issued a textbook for use throughout the course. Where necessary, supplemental materials from other sources will be provided to ensure that all required topics are included in the course.

Students will be encouraged to work in groups both inside and outside of class. Group work will be assigned during some class periods, and students will also be encouraged to work together outside of class to share ideas and approaches to problem-solving, and also to prepare for tests.

Throughout the course, assignments will be given to help review for the AP Exam. These assignments will include the type of multiple choice and free response questions students can expect to see on the AP Exam. These assignments, along with the unit tests that will be given throughout the course, will better prepare students for being successful on the AP Exam by requiring students to support their answers and communicate their thought process in writing.

**Students will be required to attend study sessions on four occasions throughout the course. These sessions will include students from other schools, and are generally held on Saturdays. These are designed to address specific topics that will be helpful in taking the AP exam at the end of the school year. In addition, tutoring sessions will be available here at school at least once per week, after school, at a time to be determined.**

An outline for AP Calculus AB is provided below. Chapter and section numbers from the textbook are included within the outline. Topics may not necessarily be presented in the order listed.

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**AP Calculus AB Outline**

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*Topic One: Limits and Their Properties*

Major assignments to be included within each topic include daily homework problems, writing assignments in which the student explains and communicates clearly the solutions to selected problems, group activitities involving use of graphing calculators, quizzes (vocabulary, rules, and formulas), and tests.* *** **

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*1.1 * *Linear Rates of Change*

*1.2 Finding Limits Graphically and Numerically* – understand and use the formal definition of limits

*1.3 Evaluating Limits Analytically* – use properties of limits, develop strategies for finding limits

*1.4 Continuity and One-Sided Limits* – determine the continuity of a function at a point and on an open interval; determine one-sided limits and continuity on a closed interval; understand and use the Intermediate Value Theorem

*1.5 Infinite Limits* – from left and right; find & sketch vertical asymptotes of the graph of a function

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*Topic Two: Differentiation*** **

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*2.1 The Derivative and the Tangent Line Problem* – find the slope of the tangent line to a curve at a point; understand the relationship between differentiability and continuity; use the limit definition to find a function’s derivative

*2.2 Basic Differentiation Rules and Rates of Change* – find derivatives using the Constant Rule, Power Rule, Constant Multiple Rule, and Sum and Difference Rules; find derivatives of trigonometric functions; use derivatives to find rates of change

*2.3 Product and Quotient Rules & Higher-Order Derivatives*

*2.4 The Chain Rule* – find the derivative of composite functions and trigonometric functions using the Chain Rule

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*2.5 Implicit Differentiation* – understand the difference between functions written in implicit form and explicit form; use implicit differentiation to find derivatives of functions

*2.6 Related Rates* – find rates that are related & use related rates to solve real-world problems** **

*3.1 Extrema on an Interval* – find extrema on open and closed intervals

*3.2 Rolle’s Theorem & the Mean Value Theorem* – understand and be able to use each theorem

*3.5 Limits at Infinity* – determine finite and infinite limits; determine horizontal asymptotes (if any) of the graph of a function

*7.7 Indeterminate Forms & L’Hopital’s Rule* – recognize limits that produce indeterminate forms; apply L’Hopital’s Rule to evaluate limits

*3.3 Increasing and Decreasing Functions & the First Derivative Test* – determine intervals on which a function increases or decreases; use First Derivative Test to find relative extrema of functions

*3.4 Concavity & the Second Derivative Test* – determine intervals over which a function is concave upward or concave downward; find points of inflection; use the Second Derivative Test to find relative extrema of functions

*3.6 Summary of Curve Sketching* – analyze and sketch the graphs of functions

*3.7 Optimization Problems* – solve real-world minimum and maximum problems

*3.9 Differentials* – understand tangent line approximations; calculate differentials using differential formulas

*Topic Three: Integration*

*4.1 Antiderivatives & Indefinite Integration* – write the general solution of a differential equation using integration; use basic integration rules to find antiderivatives; find particular solutions to differential equations

*4.2 Area* – use sigma notation to write and evaluate sums; approximate the area of a plane region; find the area of a plane region using limits

*4.3 Riemann Sums & Definite Integrals* – understand the definition of Riemann sums; evaluate definite integrals using limits and properties of definite integrals** **

*4.4 Fundamental Theorem of Calculus* – use to evaluate definite integrals; use the Mean Value Theorem for Integrals to find the average value of a function

*4.5 Integration by Substitution* – use to find indefinite integrals

*4.6 Numerical Integration* – approximate definite integrals using the Trapezoidal Rule & Simpson’s Rule

*7.2 Integration by Parts* – use to find antiderivatives of functions

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*Topic Four: Logarithmic, Exponential, and Other Transcendental Functions*

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*5.1 Natural Logarithmic Function: Differentiation* – use properties of natural logarithms to solve problems; understand the definition of the number e; find derivatives involving the logarithmic function

*5.2 Natural Logarithmic Function: Integration* – use the Log Rule to integrate a rational function; integrate trigonometric functions

*5.3 Inverse Functions* – verify inverse functions; determine whether a function has an inverse; find derivatives of inverse functions

*5.4 Exponential Functions: Differentiation & Integration* – recognize the natural exponential function, *e*, as the inverse of the natural logarithmic function, *ln x*; differentiate and integrate natural exponential functions; use properties of natural exponential functions

*5.5 Bases Other Than e and Applications* – differentiate and integrate exponential functions with bases other than e

*5.6 Differential Equations: Growth and Decay* – use separation of variables to solve simple differential equations; use exponential functions to model growth and decay in applied problems

*5.7 Differential Equations: Separation of Variables* – use initial conditions to find particular solutions of differential equations; recognize and solve differential equations that can be solved by separation of variables

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*Topic Five: Applications of Integration*

*6.1 Area of a Region Between Two Curves* – find the area of a region between two curves, and between intersecting curves, using integration; describe integration as an accumulation process

*6.2 Volume: The Disk Method* – find the volumes of solids of revolution using the disk method and using the washer method; find the volume of a solid with known cross sections

*6.3 Volume: The Shell Method* – find the volumes of solids of revolution using the shell method

*Appendix A & Supplemental Sources: Slope Fields* – use slope fields to sketch solutions of a differential equation

**Textbook:** Larson, Ron, Robert P. Hostetler, and Bruce H. Edwards. *Calculus of a Single Variable*. Seventh Edition. Boston: Houghton Mifflin Company, 2002.