Syllabus



  

 

Course Objectives

AP Calculus AB

Mr. Geiswhite

 

 

 

Pre-requisite: Successful completion of Honors Pre-calculus and teacher recommendation.

 

Primary Textbook: Calculus of a Single Variable, AP Edition, Ninth Edition; Larson, Edwards

 

Supplemental Textbooks: Applied Calculus, Third Edition; Hughes-Hallett, Gleason, Lock, Flath, et al.

                        Calculus, Graphical, Numerical, Algebraic, AP Edition, Fifth Edition; Finney, Demana, Waits, Kennedy, Bressoud

                        Calculus Concepts, Third Edition; Latorre, Kenelly, Fetta Reed, Harris, Carpenter

 

Course Overview: This advanced placement course consists of a full academic year of work in calculus and related topics comparable to courses in colleges and universities. This course begins with integral calculus and works through topics tested on the AP Calculus AB exam. I will be focusing on you being able to represent data graphically, numerically, algebraically, and in written form. Colleges may grant advanced placement and credit upon completion of the course and the Advanced Placement Examination taken in May. Students are expected to take the AP Calculus examination. (May 15, 2018)

 

Course Grading:

      - Pass/Fail Tests: Given at the beginning of the quarter. Worth 10% of your quarter grade.

                                    15 multiple choice questions; you may retake the same test twice.

 

- Manic Monday: Every other Monday you will have the option of 5 multiple choice AP Problems or 1 Free Response. Each assignment is worth 10 points.

 

- Homework: I will collect approximately 5 per quarter with notice. I will grade 5 problems worth 3 points each.

                              Due before the late bell rings or it’s worth half credit.

 

- Quizzes: You will have quizzes throughout the quarter worth approximately 30 points each.

 

- Tests: You will have a test at the end of each chapter/unit, approximately 3 per quarter.

                              You can do corrections in my room before or after school.

                              You may retake a test within three days of me returning your test.

 

- Free Response: Free Response questions will be part of each test.

                              Some will be part of the in-class test and some will be homework assignments.

                              FRQs assigned for homework will be due the next day before the late bell or it’s worth half credit.

 

 

Topics Covered and General Pacing Guide:

        This is a guide and days may be added or removed as I see fit throughout the year. My goal is to be finished the first week of April so we will have 3-4 weeks of AP review. I will have an AP review day on April 21, 2018 where we will take a full-length AP Calculus Exam.

 

        Chapter P: Preparation for Calculus (2 weeks)

            We will cover the following topics as needed after the Summer Review Packet Test.

            Graphs and Models

                        Sketch the graph of an equation

                        Find the intercepts of a graph

                        Test a graph for symmetry with respect to an axis and the origin

                        Find the points of intersection of two graphs

            Linear Models and Rates of Change

                        Find the slope of a line passing through two points

                        Write the equation of a line with a given point and slope

                        Interpret slope as a ratio or as a rate

                        Sketch the graph of a linear equation in slope-intercept form

                        Write equations of lines that are parallel or perpendicular to a given line

            Functions and Their Graphs

                        Use function notation to represent and evaluate a function

                        Find the domain and range of a function

                        Sketch the graph of a function

                        Identify different types of transformations of functions

                        Classify functions and recognize combinations of functions

            Trigonometric Functions, Identities, and Values

                        Evaluate trigonometric values

                        Solve trigonometric equations

                        Simplify trigonometric expressions using identities

                        Sketch the graph of trigonometric functions

 

        Chapter 1: Limits and Their Properties (2 weeks)

            Finding Limits Graphically and Numerically

                        Estimate a limit using a numerical or graphical approach

                        Learn different ways that a limit can fail to exist

                        Study and use a formal definition of limit

            Evaluating Limits Analytically

                        Evaluate a limit using properties of limits

                        Develop and use a strategy for finding limits

                        Evaluate a limit using dividing out and rationalizing techniques

                        Evaluate a limit using the Squeeze Theorem

            Continuity and One-Sided Limits

                        Determine continuity at a point and on an open interval

                        Determine one-sided limits and continuity on a closed interval

                        Use properties of continuity

                        Understand and use the Intermediate Value Theorem

            Infinite Limits

                        Determine infinite limits from the left and from the right

                        Find and sketch the vertical asymptotes of the graph of a function

 

 

            Limits at Infinity

                        Determine (finite) limits at infinity

                        Determine the horizontal asymptotes, if any, of the graph of a function

                        Determine infinite limits at infinity

 

        Chapter 2: Differentiation (4 weeks)

            The Derivative and the Tangent Line Problem

                        Find the slope of the tangent line to a curve at a point

                        Use the limit definition to find the derivative of a function

                        Understand the relationship between differentiability and continuity

            Basic Differentiation Rules and Rates of Change

                        Find the derivative of a function using the Constant Rule

                        Find the derivative of a function using the Power Rule

                        Find the derivative of a function using the Constant Multiple Rule

                        Find the derivative of a function using the Sum and Difference Rules

                        Find the derivatives of the sine function and of the cosine function

                        Use derivatives to find rates of change

            Product and Quotient Rules and Higher-Order Derivatives

                        Find the derivative of a function using the Product Rule

                        Find the derivative of a function using the Quotient Rule

                        Find the derivative of a trigonometric function

                        Find a higher-order derivative of a function

            The Chain Rule

                        Find the derivative of a composite function using the Chain Rule

                        Find the derivative of a function using the General Power Rule

                        Simplify the derivative of a function using algebra

                        Find the derivative of a trigonometric function using the Chain Rule

            Implicit Differentiation

                        Distinguish between functions written in implicit form and explicit form

                        Use implicit differentiation to find the derivative of a function

            Related Rates

                        Find a related rate

                        Use related rates to solve real-life problems

 

        Chapter 3: Applications of Differentiation (5 weeks)

            Sketching Derivative Graphs

                        Sketch a derivative graph from an original graph

                        Sketch an original graph from a derivative graph

                        Sketch a second Derivative given an original graph        

            Extrema on an Interval

                        Understand the definition of extrema of a function on an interval

                        Understand the definition of relative extrema of a function on an open interval

                        Find extrema on a closed interval

            Rolle’s Theorem and the Mean Value Theorem

                        Understand and use Rolle’s Theorem

                        Understand and use the Mean Value Theorem

            Increasing and Decreasing Functions and the First Derivative Test

                        Determine intervals on which a function is increasing or decreasing

                        Apply the First Derivative Test to find relative extrema of a function

            Concavity and the Second Derivative Test

                        Determine intervals on which a function is concave upward or concave downward

                        Find any points of inflection of the graph of a function

                        Apply the Second Derivative Test to find relative extrema of a function

 

            A Summary of Curve Sketching

                        Analyze and Sketch the graph of a function

            Optimization Problems

                        Solve applied minimum and maximum problems

            Differentials

                        Understand the concept of a tangent line approximation

                        Compare the value of the differential, dy, with the actual change in y,

                        Estimate a propagated error using a differential

                        Find the differential of a function using differentiation formulas

 

        Chapter 4: Integrals (4 weeks)

            Antiderivatives and Indefinite Integration

                        Write the general solution of a differential equation

                        Use indefinite integral notation for antiderivatives

                        Use basic integration rules to find antiderivatives

                        Find a particular solution of a differential equation

            Area

                        Use sigma notation to write and evaluate a sum

                        Understand the concept of area

                        Approximate the area of a plane region

                        Find the area of a plane region using limits

            Riemann Sums and Definite Integrals

                        Understand the definition of a Riemann sum

                        Evaluate a definite integral using limits

                        Evaluate a definite integral using properties of definite integrals

            The Fundamental Theorem of Calculus

                        Evaluate a definite integral using the Fundamental Theorem of Calculus

                        Understand and use the Mean Value Theorem for Integrals

                        Find the average value of a function over a closed interval

                        Understand and use the Second Fundamental Theorem of Calculus

                        Understand and use the Net Change Theorem

            Integration by Substitution

                        Use pattern recognition to find an indefinite integral

                        Use a change of variables to find an indefinite integral

                        Use the General Power Rule for Integration to find an indefinite integral

                        Use a change of variables to evaluate a definite integral

                        Evaluate a definite integral involving an even or odd function

            Numerical Integration

                        Approximate a definite integral using the Trapezoidal Rule

                        Approximate a definite integral using Simpson’s Rule

                        Analyze the approximate errors in the Trapezoidal Rule and Simpson’s Rule

 

        Chapter 5: Logarithmic, Exponential, and Other Transcendental Functions (4 weeks)

            The Natural Logarithmic Function: Differentiation

                        Develop and use properties of the natural logarithmic function

                        Understand the definition of the number e

                        Find derivatives of functions involving the natural logarithmic function

 

 

 

            The Natural Logarithmic Function: Integration

                        Use the Log Rule for Integration to integrate a rational function

                        Integrate trigonometric functions

            Inverse Functions

                        Verify that one function is the inverse function of another function

                        Determine whether a function has an inverse function

                        Find the derivative of an inverse function

            Exponential Functions: Differentiation and Integration

                        Develop properties of the natural exponential function

                        Differentiate natural exponential functions

                        Integrate natural exponential functions

            Bases Other Than e and Applications

                        Define exponential functions that have bases other than e

                        Differentiate and integrate exponential functions that have bases other than e

                        Use exponential functions to model compound interest and exponential growth

            Inverse Trigonometric Functions: Differentiation

                        Develop properties of the six inverse trigonometric functions

                        Differentiate an inverse trigonometric function

                        Review the basic differentiation rules for elementary functions

            Inverse Trigonometric Functions: Integration

                        Integrate functions whose antiderivatives involve inverse trigonometric functions

                        Use the method of completing the square to integrate a function

                        Review the basic integration rules involving elementary functions.

 

        Chapter 6: Differential Equations (2 weeks)

            Slope Fields

                        Use initial conditions to find particular solutions of differential equations

                        Use slope fields to approximate solutions of differential equations

            Differential Equation: Growth and Decay

                        Use separation of variables to solve a simple differential equation

                        Use exponential functions to model growth and decay in applied problems

            Separation of Variables and the Logistic Equation

                        Recognize and solve differential equations that can be solved by separation of variables

                        Recognize and solve homogeneous differential equations

                        Use differential equations to model and solve applied problems

                        Solve and analyze logistic differential equations

 

        Chapter 7: Application of Integration (3 weeks)

            Area of a Region Between Two Curves

                        Find the area of a region between two curves using integration

                        Find the area of a region between intersecting curves using integration

                        Describe integration as an accumulation process

            Volume: The Disk Method

                        Find the volume of a solid of revolution using the disk method

                        Find the volume of a solid of revolution using the washer method

                        Find the volume of a solid with known cross sections

 

 

 

 

        Chapter 8: Integration Techniques and L’Hopital’s Rule

            Basic Integration Rules

                        Review procedures for fitting an integrand to one of the basic integration rules

            Integration by Parts

                        Find an antiderivative using integration by parts

                        Use a tabular method to perform integration by parts

            L’Hopital’s Rule

                        Recognize limits that produce indeterminate forms

                        Apply L’Hopital’s Rule to evaluate a limit

 

 

        AP Exam Review (3-4 weeks)

                        Full AP Calculus Exam April 21, 2018

 

 

Final Exam

            Any student who does not take the AP Calculus Exam will be required to take a written final exam. All students who take the AP Calculus Exam will complete a project for their final.

 

Technology

You will need to know how to use a graphing calculator in order to complete some problems on the AP Exam. Every test you take in class will have a calculator and a non-calculator section as well as a Free Response question. The format is similar to the AP Exam so you are familiar with the setup.

            If you do not have a graphing calculator one will be issued to you for the year.

            I will be using TI-SmartView software to demonstrate how to use the graphing calculators.