# A.P. Calculus (12) (ELCA)

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**AP Calculus AB**

# Scope and SequenceEdit

This scope and sequence has been officially authorized by the College Board for use in my Calculus AB classroom and will provide the student with AP status.

The following is an outline of the topics covered in AP Calculus AB. An approximate number of days are allotted to the topics since the actual time varies from year to year depending upon the richness of class discussions that are generated.

## Teaching StrategiesEdit

Students use graphing calculators on a routine basis to explore, discover, and re-enforce the concepts of calculus. Students may use the graphing calculators on some, but not all assessments. It is stressed to students that they must know how to use a graphing calculator to do the following four required features: finding a root, sketching a function in a specified window, numerically calculating the derivative of a function at a value, and numerically calculating the value of a definite integral. Students are taught by the rule of four: analytically, graphically, numerically, and verbally. Communication in this course is important, as students are expected to give written justification for many of the processes used throughout the course. Students do best when they have an understanding of the conceptual nature of calculus, therefore stressing the reasons why behind the major ideas is essential. Quizzes are given routinely and students can expect a minimum of two tests every six weeks. AP questions and resources from previous years are used extensively throughout the course.

## ScheduleEdit

- Functions, Graphs, and Limits (approx. 3 weeks)
- Analysis of Graphs
- How to read a graph and graph a function (Pre-Calculus Skill)

- Limits
- Intuitive feel of limit using a graphing calculator for exploration
- Limit Notation
- One-sided and Two-sided limits
- Calculate limits algebraically
- Estimate limits from a graph or table
- The importance of limits to calculus

- Asymptotes (vertical, horizontal, slant, and other)
- Understand and identify vertical and horizontal asymptotes graphically
- (Pre-Calculus Skill)
- Justify asymptotes with calculus notation and definitions
- How does one identify crossing over asymptotes in calculus
- Infinite Limits
- Limits at infinity

- Continuity
- Test continuity of a function in terms of limits and graphs
- Intermediate Value Theorem and Extreme Value Theorem

- Analysis of Graphs
- Differential Calculus (Five Weeks)
- Definition of Derivative
- Discussion on the relationship between tangent lines and secant lines
- During this time we take a curve and point of tangency and use multiple delta x’s close by and calculate the secant slopes and watch them become the tangent value.

- Find derivative using limit of the difference quotient
- Know the relationship between differentiability and continuity

- Discussion on the relationship between tangent lines and secant lines
- Derivative at a Point
- Power Rule, Product and Quotient Rules, Chain Rule
- Slope of a curve at a point, Tangent and Normal lines to a curve at a point
- Local linear approximation and differentials to estimate the tangent lines to a curve at a point
- Instantaneous rate of change of a function from a graph or table of values
- Instantaneous rates of change versus average rates of change from graphs or tables
- Discussion on vertical tangent lines, cusps, and corners along with differentiability
- Higher-Order Derivatives and Implicit Differentiation
- Locations of vertical and horizontal tangent lines to curves

- Derivative of a Function
- Relate graph of function and derivative graph
- Rolle’s Theorem and Mean Value Theorem for Derivatives
- Identification of critical numbers
- Relationship between sign of derivative and increasing/decreasing nature of function
- Find relative and absolute maxima and minima

- Second Derivative
- Relate graph of function, first derivative graph, and second derivative graph
- Relationship between concavity and sign of second derivative
- Find points of inflection

- Definition of Derivative
- Applications of Derivatives (Six Weeks)
- Curve Sketching
- Sketch a curve using first and second derivatives, analyze critical points
- Use points of inflections and concavity
- Use asymptotes
- Use symmetry

- Optimization and Related Rates
- Solve optimization problems and related rates problems

- More applications of Derivatives
- Solve Rectilinear Motion problems
- L’Hopital’s Rule
- Newton’s Method

- Curve Sketching
- Integral Calculus (Five Weeks)
- Riemann Sums
- Find the sum of a region using left, right, midpoint evaluations, and Trapezoidal Rule
- During this time students sketch a region bounded by a nonnegative curve and use the regular partitions to fill the space as well as try to get as close to the actual value using any size and any number of rectangles (or other figures as well). I share with them the actual answer for comparison.

- Summation formulas through the cubes
- Using summation and limit process to generate an exact value for a sum
- Find a sum geometrically
- Discussion on a sum versus area

- Find the sum of a region using left, right, midpoint evaluations, and Trapezoidal Rule
- Fundamental Theorem of Calculus
- First and Second Fundamental Theorem of Calculus
- Find the derivative of a function defined by an integral
- Discussion on the power this allows us
- Definite Integral Properties including additivity, even, odd, change of limits

- Techniques of Antidifferentiation
- Discussion on the difference between the definite integral and indefinite integral
- Integrate using Power Rule and U-Substitution
- Find particular antiderivatives using initial conditions
- Integration by Parts

- Some Applications of Integral Calculus
- Mean Value Theorem for Integrals
- Average value of a function
- Rectilinear Motion

- Total distance versus displacement and the use of the integral
- Average velocity using the integral as compared to change in position divided by change in time
- Slope Fields

- Visual interpretation of a differential equation
- The TI-89 and presenter are used to observe various slope fields and particular solutions on the calculator. We discuss asymptotic behavior, how to sketch a slope field efficiently by hand, etc.
- Discuss how to justify increasing/decreasing behavior as well as max/min and concavity at a point using the differential equation
- Discussion of isoclines, nullclines, and equilibrium

- Riemann Sums
- Transcendental Functions (Five Weeks)
- Differentiation and Integration of the Natural Logarithmic Function
- Definition of natural log in terms of an integral function
- Use slope field on TI-89 to tap into visual aspect
- Discussion on domain and necessity of absolute value
- Completion of the list of derivatives of all of the trig functions

- Differentiation and Integration of the Exponential Functions
- Base e
- Bases other than e
- Applications

- Differential Equations
- Slope Fields as visual interpretations of the differential equations
- Separation of Variables
- Growth and Decay Applications

- Differentiation and Integration of Inverse Trigonometric Functions
- Derivatives of Inverse Functions
- From original function
- Implicitly
- From a table using formula

- Differentiation and Integration of the Natural Logarithmic Function
- Applications of Integration (Three Weeks)
- Area of a Region with finite boundaries
- Volume
- Find the volume of a solid with a known cross-section
- Find the volume of a solid of revolution by the Disc and Washer Method as well as the Shell Method

- More on the Integral as an accumulator

This schedule leaves 3 weeks per semester for flexibility with teaching, learning, and review of material. At numerous times during the course students are required to communicate verbally through written explanation as well as orally in class. Students are also given a 50-60 question take-home test each semester where they are encouraged to do it individually and then get together in a group and discuss the answers. This is a very effective tool used to increase their communication skills.

## TextbookEdit

### Primary TextEdit

Larson, Roland. E.; Hostetler, Robert P., Edward, Bruce. H. (2002) Calculus of a single variable (7th Ed.) Houghton Mifflin Company, Boston, MA

### SupplementsEdit

During the topics of slope fields and rectilinear motion I have to supplement the textbook heavily. I do so with worksheets of my own creation which are written specifically for the way I want to develop those topics and include links to other concepts as well.

# SyllabusEdit

## ObjectiveEdit

To provide students with the academic rigor and college-level experience that is associated with an Advanced Placement course and to help students perform well on the AP EXAM while bringing glory to GOD.

## Help ClassEdit

Q/A time: Tuesday morning 7:30-7:55am

## Grading PolicyEdit

AP EXAM DATE: May 6, 2009

- Tests 55%
- Daily Grade 30%
- Final Exam 15%

## Classroom ExpectationsEdit

- Be prompt
- Be prepared
- Be positive
- Be polite
- NO FOOD OR DRINK IN THE CLASSROOM
- NO CELLPHONE OR IPODS,ETC..

## Materials NeededEdit

- Graphing Calculator TI-83+, TI-84
- Notebook (3-ring binder with pockets)
- Paper
- Pen/Pencil

- The textbook is utilized as a resource tool but other rich AP resources will be used. I will refer to potential content on the AP EXAM through the use of past AP EXAMS as we progress through the concepts. The approach to problem-solving will involve the “rule of four” numerically, graphically, analytically, and verbally. Tests will vary in format and students will be expected to approach problems in various ways with/without the aid of a graphing calculator. There will be a minimum of two tests within a 6-week period.

## Learn WhyEdit

It is imperative that students make the effort to understand the “why” behind the concepts because the AP rigor requires deep thinking and understanding. There will be times when rote skill is necessary but an emphasis on the conceptual nature of calculus will help the student comprehend the material thoroughly enough for college courses and the AP EXAM

# Conclusion and DisclaimerEdit

Thank you for the opportunity to teach you and if I can be of any assistance please don’t hesitate to ask. However, it will be most helpful to form a study group within the classroom.

Disclaimer: The scope and sequence and syllabus are subject to change at the instructor’s discretion.

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