Course Overview

My main objective in teaching AP Calculus is to enable students to appreciate the beauty of calculus and receive a strong foundation that will give them the tools to succeed in future mathematics courses, as well succeed on the AP exam.    

Textbook(s):

Finney, Demana, Waits and Kennedy. Calculus—Graphical, Numerical, Algebraic. Third edition. Pearson, Prentice Hall, 2007.
This textbook will be our primary resource.

Schwartz, Stu.  AP Calculus AB. www.mastermathmentor.com   2005
Selected sections of this online textbook will be used throughout the course.

Technology Requirement

The use of a graphing calculator is a required part of this course. I will use the Texas Instruments TI-Smartview software emulating the 84 Plus graphing calculator in class regularly. Although not required, I have issued students their own TI-84 plus graphing calculator for use both inside and outside of class. I recommend the TI-84 or the TI-89.  I will have a classroom set of TI-84 Plus calculators, and some are available for use in the learning center.

If students choose to use any other calculator they may do so, as long as it can perform the four required functionalities of graphing technology :

• Finding a root
• Sketching a function in a specified window
• Approximating the derivative at a point using numerical methods
• Approximating the value of a definite integral using numerical methods

In my class, it is expected that students use graphing calculators daily to explore, discover, and reinforce the concepts of calculus.  Students will be required to make connections between the graphs of functions and their analysis, and draw conclusions about the behavior of functions.  We will use the calculator in a variety of ways such as:

• Conducting explorations
• Graphing functions within arbitrary windows
• Solving equations numerically
• Analyzing and Interpreting Results
• Justifying and explaining results of graphs and equations

Example 1–

As an introduction to the AP Calc course at the beginning of the year we looked at a distance versus time graph, graphed the derivative of the graph, found the area under the curve of graph (anti-derivative) and drew conclusions about how all three were related. We then calculated instantaneous rates of change using the concept of average rate of change as the distance between two points approached 0. We looked at functions like \(f (x) =\frac{sin (x) }{ x} \)as x approaches 0. We used the WINDOW and ZOOM features to zero in on the behavior of the graph as x approaches from both sides. We used the Y1(input) function and STO feature to approximate the instantaneous rate of change. We used the CALC 6: dy/dx feature to calculate the instantaneous rate of change at x = 0. We used the MATH 8:nDeriv( feature to calculate the instantaneous rate of change at x = 0. We compared all answers and discussed similarities and differences.   

Example 2 –

In the limit unit we derive the \(lim=(\frac{sinx}{x})=1\)geometrically using the sandwich theorem. Prior to that proof, we use the graphing calculators to show howlies between the functions y = 1 and y = cosx and how they all approach 1 as x approaches 0 from both sides. We also use the graphing calculator to look at the limit as x approaches infinity of functions like to conclude what our limit will be and then attempt to find the limit algebraically using substitution methods.

 

A Balanced Approach

Current mathematical education emphasizes a “Rule of Four.” There are a variety of ways to approach and solve problems. The four branches of the problem-solving tree of mathematics are:

• Numerical analysis (where data points are known, but not an equation)
• Graphical analysis (where a graph is known, but again, not an equation)
• Analytic/algebraic analysis (traditional equation and variable manipulation)
• Verbal/written methods of representing problems (classic story problems as well as written justification of one’s thinking in solving a problem - such as on our state assessment)

These four methods will be stressed throughout the course.

---

Course Outline

Introduction to Calculus (1 week)

                A.  An intuitive approach to calculus will be developed using the classic motion problem viewed from several approaches.

                        1. Finding speed from the position (the derivative)

                        2. Finding distanced traveled from the speed (the integral)

                        3. Understanding the relationship between the two processes (Fundamental Theorem of

                             Calculus)

 

Unit 1: Limits and Continuity (3 - 4 weeks)

            A. Rates of Change

                        1. Average Speed

                        2. Instantaneous Speed

            B. Limits at a Point

                        1. 1-sided Limits

                        2. 2-sided Limits

                        3. Sandwich Theorem

            C. Limits involving infinity

                        1. Asymptotic behavior (horizontal and vertical)

                        2. End behavior models

                        3. Properties of limits (algebraic analysis)

                        4. Visualizing limits (graphic analysis)           

            D. Continuity

                        1. Continuity at a point

                        2. Continuous functions

                        3. Discontinuous functions

                                    a. Removable discontinuity (0/0 form)

                                    b. Jump discontinuity (We look at y = int(x).)

                                    c. Infinite discontinuity

            E. Rates of Change and Tangent Lines

                        1. Average rate of change

                        2. Tangent line to a curve

                        3. Slope of a curve (algebraically and graphically)

                        4. Normal line to a curve (algebraically and graphically)

                        5. Instantaneous rate of change

 

Unit 2: The Derivative (5 - 6 weeks)

            A. Derivative of a Function

                        1. Definition of the derivative (difference quotient)

                        2. Derivative at a Point

                        3. Relationships between the graphs of f and f’

                        4. Graphing a derivative from data

                        5. One-sided derivatives

            B. Differentiability

                        1. Cases where f’(x) might fail to exist

                        2. Local linearity

                        3. Derivatives on the calculator (Numerical derivatives using NDERIV)

                        4. Symmetric difference quotient

                        5. Relationship between differentiability and continuity

                        6. Intermediate Value Theorem for Derivatives

            C. Rules for Differentiation

                        1. Constant, Power, Sum, Difference, Product, Quotient Rules

                        2. Higher order derivatives

            D.  Derivatives of trigonometric functions

            E. Chain Rule

            F. Implicit Differentiation

                        1. Differential method

                        2. y’ method

            G. Derivatives of inverse trigonometric functions           

            H. Derivatives of Exponential and Logarithmic Functions

 

Unit 3: Applications of the Derivative (5 - 6 weeks)

            A. Extreme Values

                        1. Relative Extrema

                        2. Absolute Extrema

                        3. Extreme Value Theorem

                        4. Definition of a critical point           

            B. Implications of the Derivative

                        1. Rolle’s Theorem

                        2. Mean Value Theorem

                        3. Increasing and decreasing functions

            C. Connecting f’ and f’’ with the graph of f(x)

                        1. First derivative test for relative max/min

                        2. Second derivative

                                    a. Concavity

                                    b. Inflection points

                                    c. Second derivative test for relative max/min 

            D. Optimization problems

            E. Linearization models

                        1. Local linearization

                        2. Tangent line approximation

                        3. Differentials

            F. Particle motion, Position, velocity, acceleration, and jerk           

            G. Economics

                        1. Marginal cost

                        2. Marginal revenue

                        3. Marginal profit

            H. Related Rates

 

MIDTERM EXAM

 

Unit 4: The Definite Integral (3–4 weeks)

            A. Approximating areas

                        1. Riemann sums

                                    a. Left sums

                                    b. Right sums

                                    c. Midpoint sums

                                    d. Trapezoidal sums

                        2. Definite integrals

            B. Properties of Definite Integrals

                        1. Power rule

                        2. Mean value theorem for definite integrals

            C. The Fundamental Theorem of Calculus

                        1. Part 1

                        2. Part 2

 

Unit 5: Differential Equations and Mathematical Modeling (4 weeks)

            A. Slope Fields

            B. Antiderivatives

                        1. Indefinite integrals

                        2. Power formulas

                        3. Trigonometric formulas

                        4. Exponential and Logarithmic formulas

            C. Separable Differential Equations

                        1. Growth and decay

                        2. Slope fields (Resources from the AP Calculus website are liberally used.)

                        3. General differential equations

                        4. Newton’s law of cooling

            D. Logistic Growth

 

Unit 6: Applications of Definite Integrals (3 weeks)

            A. Integral as net change

                        1. Calculating distance traveled (particle motion)

                        2. Consumption over time

                        3. Net change from data

            B. Area between curves

                        1. Area between a curve and an axis

                                    a. Integrating with respect to x

                                    b. Integrating with respect to y

                        2. Area between intersecting curves

                                    a. Integrating with respect to x

                                    b. Integrating with respect to y

            C. Calculating volume

                        1. Cross sections

                        2. Disc method

                        3. Shell method

 

Unit 7: Review/Test Preparation (Approximately 2– 4weeks)

            A. Multiple-choice practice (Items from past exams—1997, 1998, and 2003

                 are used as well as items from review books I’ve purchased over the years.)

                        1. Test taking strategies are emphasized

                        2. Individual and group practice are both used

            B. Free-response practice (Released items from the AP Central website are used liberally.)

                        1. Rubrics are reviewed so students see the need for complete answers

                        2. Students collaborate to formulate team responses

                        3. Individually written responses are crafted. Attention to full explanations is emphasized.

 

Unit 8: After the exam…

            A. Review of the AP exam and try to identify points that students felt confident with or points that

                 students felt they needed more work with.  This will help strengthen those points before students

                 go to college, as well as help me adjust the course for next year.

            B. Students will be required to do and present projects designed to incorporate this year’s learning in

                 applied ways.  Projects can be of the students own choosing with teacher approval or chosen from

                 one of the following sources.

 

MAA Notes. Resources for Calculus Collection. , D.C.: MAA, 1993.

Volume 3: Applications of Calculus, edited by Philip Straffin. MAA Notes Number 29.

Volume 4: Problems for Student Investigation, edited by Michael B. Jackson and John R. Ramsay. MAA Notes Number 30.

Volume 5: Readings for Calculus, edited by Underwood Dudley. MAA Notes Number 31.

This course uses the A-List book of Knowledge SAT 2nd Edition and SAT/ACT practice tests throughout the semester to expose students to curriculum, strategies, and techniques when taking the SAT/ACT exam.

Course Overview

My main objective in teaching AP Calculus is to enable students to appreciate the beauty of calculus and receive a strong foundation that will give them the tools to succeed in future mathematics courses, as well succeed on the AP exam.    

Textbook(s):

Finney, Demana, Waits and Kennedy. Calculus—Graphical, Numerical, Algebraic. Third edition. Pearson, Prentice Hall, 2007.
This textbook will be our primary resource.

Schwartz, Stu.  AP Calculus AB. www.mastermathmentor.com   2005
Selected sections of this online textbook will be used throughout the course.

Technology Requirement

The use of a graphing calculator is a required part of this course. I will use the Texas Instruments TI-Smartview software emulating the 84 Plus graphing calculator in class regularly. Although not required, I have issued students their own TI-84 plus graphing calculator for use both inside and outside of class. I recommend the TI-84 or the TI-89.  I will have a classroom set of TI-84 Plus calculators, and some are available for use in the learning center.

If students choose to use any other calculator they may do so, as long as it can perform the four required functionalities of graphing technology :

• Finding a root
• Sketching a function in a specified window
• Approximating the derivative at a point using numerical methods
• Approximating the value of a definite integral using numerical methods

In my class, it is expected that students use graphing calculators daily to explore, discover, and reinforce the concepts of calculus.  Students will be required to make connections between the graphs of functions and their analysis, and draw conclusions about the behavior of functions.  We will use the calculator in a variety of ways such as:

• Conducting explorations
• Graphing functions within arbitrary windows
• Solving equations numerically
• Analyzing and Interpreting Results
• Justifying and explaining results of graphs and equations

Example 1–

As an introduction to the AP Calc course at the beginning of the year we looked at a distance versus time graph, graphed the derivative of the graph, found the area under the curve of graph (anti-derivative) and drew conclusions about how all three were related. We then calculated instantaneous rates of change using the concept of average rate of change as the distance between two points approached 0. We looked at functions like \(f (x) =\frac{sin (x) }{ x} \)as x approaches 0. We used the WINDOW and ZOOM features to zero in on the behavior of the graph as x approaches from both sides. We used the Y1(input) function and STO feature to approximate the instantaneous rate of change. We used the CALC 6: dy/dx feature to calculate the instantaneous rate of change at x = 0. We used the MATH 8:nDeriv( feature to calculate the instantaneous rate of change at x = 0. We compared all answers and discussed similarities and differences.   

Example 2 –

In the limit unit we derive the \(lim=(\frac{sinx}{x})=1\)geometrically using the sandwich theorem. Prior to that proof, we use the graphing calculators to show howlies between the functions y = 1 and y = cosx and how they all approach 1 as x approaches 0 from both sides. We also use the graphing calculator to look at the limit as x approaches infinity of functions like to conclude what our limit will be and then attempt to find the limit algebraically using substitution methods.

 

A Balanced Approach

Current mathematical education emphasizes a “Rule of Four.” There are a variety of ways to approach and solve problems. The four branches of the problem-solving tree of mathematics are:

• Numerical analysis (where data points are known, but not an equation)
• Graphical analysis (where a graph is known, but again, not an equation)
• Analytic/algebraic analysis (traditional equation and variable manipulation)
• Verbal/written methods of representing problems (classic story problems as well as written justification of one’s thinking in solving a problem - such as on our state assessment)

These four methods will be stressed throughout the course.

---

Course Outline

Introduction to Calculus (1 week)

                A.  An intuitive approach to calculus will be developed using the classic motion problem viewed from several approaches.

                        1. Finding speed from the position (the derivative)

                        2. Finding distanced traveled from the speed (the integral)

                        3. Understanding the relationship between the two processes (Fundamental Theorem of

                             Calculus)

 

Unit 1: Limits and Continuity (3 - 4 weeks)

            A. Rates of Change

                        1. Average Speed

                        2. Instantaneous Speed

            B. Limits at a Point

                        1. 1-sided Limits

                        2. 2-sided Limits

                        3. Sandwich Theorem

            C. Limits involving infinity

                        1. Asymptotic behavior (horizontal and vertical)

                        2. End behavior models

                        3. Properties of limits (algebraic analysis)

                        4. Visualizing limits (graphic analysis)           

            D. Continuity

                        1. Continuity at a point

                        2. Continuous functions

                        3. Discontinuous functions

                                    a. Removable discontinuity (0/0 form)

                                    b. Jump discontinuity (We look at y = int(x).)

                                    c. Infinite discontinuity

            E. Rates of Change and Tangent Lines

                        1. Average rate of change

                        2. Tangent line to a curve

                        3. Slope of a curve (algebraically and graphically)

                        4. Normal line to a curve (algebraically and graphically)

                        5. Instantaneous rate of change

 

Unit 2: The Derivative (5 - 6 weeks)

            A. Derivative of a Function

                        1. Definition of the derivative (difference quotient)

                        2. Derivative at a Point

                        3. Relationships between the graphs of f and f’

                        4. Graphing a derivative from data

                        5. One-sided derivatives

            B. Differentiability

                        1. Cases where f’(x) might fail to exist

                        2. Local linearity

                        3. Derivatives on the calculator (Numerical derivatives using NDERIV)

                        4. Symmetric difference quotient

                        5. Relationship between differentiability and continuity

                        6. Intermediate Value Theorem for Derivatives

            C. Rules for Differentiation

                        1. Constant, Power, Sum, Difference, Product, Quotient Rules

                        2. Higher order derivatives

            D.  Derivatives of trigonometric functions

            E. Chain Rule

            F. Implicit Differentiation

                        1. Differential method

                        2. y’ method

            G. Derivatives of inverse trigonometric functions           

            H. Derivatives of Exponential and Logarithmic Functions

 

Unit 3: Applications of the Derivative (5 - 6 weeks)

            A. Extreme Values

                        1. Relative Extrema

                        2. Absolute Extrema

                        3. Extreme Value Theorem

                        4. Definition of a critical point           

            B. Implications of the Derivative

                        1. Rolle’s Theorem

                        2. Mean Value Theorem

                        3. Increasing and decreasing functions

            C. Connecting f’ and f’’ with the graph of f(x)

                        1. First derivative test for relative max/min

                        2. Second derivative

                                    a. Concavity

                                    b. Inflection points

                                    c. Second derivative test for relative max/min 

            D. Optimization problems

            E. Linearization models

                        1. Local linearization

                        2. Tangent line approximation

                        3. Differentials

            F. Particle motion, Position, velocity, acceleration, and jerk           

            G. Economics

                        1. Marginal cost

                        2. Marginal revenue

                        3. Marginal profit

            H. Related Rates

 

MIDTERM EXAM

 

Unit 4: The Definite Integral (3–4 weeks)

            A. Approximating areas

                        1. Riemann sums

                                    a. Left sums

                                    b. Right sums

                                    c. Midpoint sums

                                    d. Trapezoidal sums

                        2. Definite integrals

            B. Properties of Definite Integrals

                        1. Power rule

                        2. Mean value theorem for definite integrals

            C. The Fundamental Theorem of Calculus

                        1. Part 1

                        2. Part 2

 

Unit 5: Differential Equations and Mathematical Modeling (4 weeks)

            A. Slope Fields

            B. Antiderivatives

                        1. Indefinite integrals

                        2. Power formulas

                        3. Trigonometric formulas

                        4. Exponential and Logarithmic formulas

            C. Separable Differential Equations

                        1. Growth and decay

                        2. Slope fields (Resources from the AP Calculus website are liberally used.)

                        3. General differential equations

                        4. Newton’s law of cooling

            D. Logistic Growth

 

Unit 6: Applications of Definite Integrals (3 weeks)

            A. Integral as net change

                        1. Calculating distance traveled (particle motion)

                        2. Consumption over time

                        3. Net change from data

            B. Area between curves

                        1. Area between a curve and an axis

                                    a. Integrating with respect to x

                                    b. Integrating with respect to y

                        2. Area between intersecting curves

                                    a. Integrating with respect to x

                                    b. Integrating with respect to y

            C. Calculating volume

                        1. Cross sections

                        2. Disc method

                        3. Shell method

 

Unit 7: Review/Test Preparation (Approximately 2– 4weeks)

            A. Multiple-choice practice (Items from past exams—1997, 1998, and 2003

                 are used as well as items from review books I’ve purchased over the years.)

                        1. Test taking strategies are emphasized

                        2. Individual and group practice are both used

            B. Free-response practice (Released items from the AP Central website are used liberally.)

                        1. Rubrics are reviewed so students see the need for complete answers

                        2. Students collaborate to formulate team responses

                        3. Individually written responses are crafted. Attention to full explanations is emphasized.

 

Unit 8: After the exam…

            A. Review of the AP exam and try to identify points that students felt confident with or points that

                 students felt they needed more work with.  This will help strengthen those points before students

                 go to college, as well as help me adjust the course for next year.

            B. Students will be required to do and present projects designed to incorporate this year’s learning in

                 applied ways.  Projects can be of the students own choosing with teacher approval or chosen from

                 one of the following sources.

 

MAA Notes. Resources for Calculus Collection. , D.C.: MAA, 1993.

Volume 3: Applications of Calculus, edited by Philip Straffin. MAA Notes Number 29.

Volume 4: Problems for Student Investigation, edited by Michael B. Jackson and John R. Ramsay. MAA Notes Number 30.

Volume 5: Readings for Calculus, edited by Underwood Dudley. MAA Notes Number 31.

Course Overview

Unit #1 – Trig Proofs and Trig Equations (13 days)

• Simplifying Trig Expressions (1 day)
  • Derive three trig identities
  • Simplify trig expressions using trig identity substitution, equivalent trig expressions, and algebra
• Proving Trig Identities (1 day)
• Sum and Difference of Two Angles Formulas for Trig Functions (2 days)
• Double Angles Formulas for Trig Functions (1 day)
• Half Angle Formulas for Trig Functions (.5 day)
• Solving Trig Equations (5.5 days)
  • Solving Linear Trig Equations
  • Solving Quadratic Trig Equations
  • Solving Trig Equations using the quadratic formula
  • Solving Trig Equations using Substitution
  • Solving Trig Equations using “u-substitution”

For Do Now questions during this unit, we will begin to spiral in the concept of evaluating inverse trig functions and looking at their graphs. 

Allow 1 day for review and 1 day to test.

Unit #2 – Trig Applications (9 days)

• Law of Sines (2.5 days)
  • Include angle of elevation and depression word problems
  • Ambiguous Case
• Law of Cosines (1 day)
• Area of a Triangle – (include law of sines and cosines within problems) (1.5 days)
• Force Word Problems (1.5 days)

Allow 1.5 days for review and 1 day to test.

Unit #3 – Graphing Rational Functions (11 days)

• Review of Even and Odd Functions (1 days)
  • Algebraic and graphical arguments including, but not limited to polynomial functions, rational functions, and trig functions. 
• Synthetic Division (1 day)
  • Cover linear factors with coefficients other than 1
• Graphing Rational Functions (8 days)
  • Finding vertical and horizontal asymptotes algebraically
  • Oblique and curvilinear asymptotes (not so much emphasis on curvilinear)
  • Holes
  • Transforming rational functions
  • Make sure rational function range in difficulty level (ie., include rational functions whose polynomials are of a degree than 2 where understanding of factoring cubes, factoring by grouping, and factor using synthetic division can be incorporated)

Allow 1 day to test for this unit.

Unit #4 – Limits of Polynomial, Rational and Piecewise Functions (13 days)

• Limits Graphically (2 days)
  • Definition of a limit (left/right hand limits)
  • Limits as x approaches a number including, but not limited to rational and piecewise functions, as well as functions with holes, step functions, fractional equations with absolute values etc…
  • Approximating limits without graphing the function (ie., plugging in values close to the “a” value and anticipating the value of the limit)
• Limits Algebraically (5 days)
  • Limit as x approaches the infinities (2/5 days)
    • Rational functions (powers equal, top/bottom heavy)
    • Polynomial functions (end behavior and leading coefficients)
    • Fractional equations with radicals
    • Expressions with radical (indeterminate case of )
  • Limit as x approaches the value a (3/5 days)
    • Polynomial/Continuous Functions (at x = a)
    • (indeterminate) form where a common factor needs to be removed
    • form where limit either DNE or is one of the infinities
    • One-sided limits
• Continuity (3 days)
  • Definition of Continuity
  • Types of Continuity
    • Removable
    • Essential Finite
    • Essential Infinite
    • If removable, redefining the function to make it continuous (continuous extension)
  • Determining whether functions (primarily piecewise functions and rational functions) are continuous and if not, stating where the definition of continuity fails
  • Finding the value of a constant(s) so that a function is continuous

Allow 2 days for review and 1 day to test.

*** This next unit is only for the honors level if time permits.

Unit #5 – Limits of Trig, ln, e^x, and Inverse Trig Functions (12 days)

• Trig limits (4 days)
  • Squeeze Theorem to derive 
    
    
  • Mixed problems involving all of the above limits
  • Intuitive approach to limits involving trig functions that don’t involve the above theorems
• Oscillating trig functions
• Non-ratio trig functions
• Trig functions with angles that are ratios
• Trig limits using substitution
• Introduction to the idea of L’Hopital’s Rule (0.5 days)
• Limits involving e^x and lnx (1.5 days)
• Limits comparing rates of change for growth functions (.5 day)
• Limit Theorems (1 day)
• Continuity and Continuity Theorems (1.5 days)
• Intermediate Value Theorem (1 day)

Allow 1 day for review and 1 day to test for this unit.

Unit #6 – Derivatives (11 days)

• Definition of a derivative (2 days)
• Power Rule for derivatives (1.5 days)
• Derivatives as they relate to increasing/decreasing/constant functions (.5 day)
• Chain Rule (1 day)
• Product Rule (1 day)
• Quotient Rule (2 days)

Allow 2 days to review and 1 day to test for this unit.

Try to allow for 2 – 3 days of review for the final exam in this course.

Common Core Algebra 2 Year 1 of 1.5 – Course Outline (Revised March 2019)

Unit #1 – Review of Algebraic Ideas (Important Topics from Common Core Algebra I and Common Core Geometry Relating to Algebra) – 15 days

Summary: Includes but not limited to: Basic Review of Operations with Radicals, Review of Basic Terms and Vocabulary, Solving Linear Equations, Brief Exponent Review, Operations with Polynomials and Basic Calculator Work Using the TI-83Plus Graphing Calculator. 

Specifics:

 

-          Review of Operations with Radicals (2.5 days)

o    Adding, Subtracting, Multiplying, Dividing, Rationalizing, Simplifying (including square roots and nth roots)

-          Review of Properties (1 day)

o    Properties of Real Numbers (Commutative, Associative, Distributive)

o    Properties of Equalities (Addition and Multiplication)

o    Properties of Inequalities (If time permits)

-          Review of Linear Equations (2 days)

o    Analyzing linear equations to determine the number of real solutions (unique, identity, and inconsistent cases)

o    Solving linear equations by recognizing applications of properties of real numbers and equalities

o    Solving multi-step linear equations

o    Solving fractional linear equations with emphasis on extraneous solutions (Cross multiply and Common Denominator technique)

-          Review of Laws of Exponents with Monomial Expressions (2 days)

o    All power rules (multiplying, dividing, zero exponents, negative exponents, fractions raised to exponents, factors raised to an exponent, powers raised to an exponents, etc…)

o    Simplifying monomial expressions (be sure to include expressions with variable bases and numerical bases)

-          Review of Multiplying Polynomials (2 days)

o    Use an algebraic approach and a geometric approach (area array) to multiply binomials times binomials, binomials times trinomials, trinomials time trinomials, polynomials squared time polynomials, monomials times polynomials, etc…

o    Determining the algebraic equivalence of expressions after multiplying polynomials and showing their equivalence by substituting values into the expressions and comparing.

o    Application problems involving multiplying polynomials

-          Review of Evaluating Expressions (1.5 days)

o    Review of GEMDAS

o    Understanding how to use the variable and storage keys on the calculator to check evaluated expressions

o    Introduction of a rational expression and how to use the fraction key, absolute value and square root function of a calculator and parentheses on a calculator to check the evaluated function. 

-          Review of the Exploration of other Features on the Graphing Calculator (2 days)

o    Using the calculator to find zeros of a function (CALC: ZERO, TABLE)

o    Associating a zero with its appropriate factor to write equations of a function

o    Other functions to review:  y=, TABLE, VALUE, adjusting TBL SETTINGS, function notation on the home screen, WINDOW

o    Briefly discuss Location Principle

o    Relative and Absolute MINIMUM and MAXIMUM Points

o    INTERSECT

 

Allow 1 day for review and 1 day to test on this unit.

 

Unit #2 – Intro to Functions (24 days)

Summary:  Includes but not limited to: Graphs of Functions and Important Ideas with the Graph of a Function (intervals of Increase/Decrease, Zero(s), y-intercepts, Maximum and Minimum (Absolute and Relative)), Definition of a Function, Function Notation, Evaluating Functions, Domain and Range of Functions, Composition of Functions, One to One Functions and Inverse Functions

Specifics:

-          Intro to Vocabulary and Definition of a Function (2 days)

o    Independent, Dependent, Domain, Range, Codomain, Input, Output, x, y

o    Relative vs. Function using multiple representations like mappings, equations, tables, graphs, charts, sets, etc…

o    Function Notation

o    Standard Form vs. Vertex Form for Functions

o    Parent Functions

-          Review of Completing the square with and without leading coefficients other than 1 (1 day)

-          Review of 9 Parent Functions from Algebra 1 and Geometry and Respective Behaviors of those functions: (2.5 days)

o    Linear Function

o    Quadratic Function

o    Absolute Value Function

o    Square Root Function

o    Exponential Growth/Decay

o    Cubic Function

o    Cube Root Function

o    Step Function (Greatest Integer Function)

o    Circle (not a function, but good to review it here)

-          Transforming Parent Functions (2.5 days)

o    Vertical and Horizontal Shifts

o    Vertical Compressions and Stretches

o    Reflections over the x-axis

-          Evaluating Functions Algebraically using Function Notation (1 day)

o    Given Input, find output (evaluating functions)

o    Given Output, find input

o    Understanding that f(a) = b is another way to write the point (a, b)

o    Given a restricted domain, find range.

-          Evaluating Functions using Function Notation  (1 day)

o    Using Graphs

o    Using Tables

o    Using Charts

-          Finding the Domain of the following Functions: (1.5 days)

o    Polynomial, rational, square root, fractional equations with radicals in the denominator (linear only), exponential, absolute value, circles (not functions)

-          Finding the Range of Functions: (1.5 days)

o    Linear, Exponential, Absolute Value, Square Root, Quadratic (standard and vertex form), circles

o    Mixed Review of Domain and Range Questions with restricted domains and ranges, charts, tables, etc…

-          Operations with Functions (Add, subtract, Multiply, and Divide) (0.5 days)

-          Composition of Functions (1.5 days)

o    Algebraically

o    Graphically

o    Using Tables/Charts

-          One-to-One (1-1) Functions (1 day)

o    Horizontal Line Test, identifying 1-1 functions given equations, graphs, charts, etc…, restricting domains of functions, like quadratics, in order to make them 1-1

-          Inverse Functions (3 days)

o    Notation

o    Finding inverses algebraically and graphically

-          Key Features of Functions (Algebraically and Using Other Methods Necessary, like the graphing calculator) (3 days)

o    Intercepts

o    Zeros, Roots, Solutions, Answers, and x-intercepts

o    Increasing, Decreasing, and Constant Functions

o    Positive, Negative intervals of a function

o    Relative Extrema

o    Absolute Extrema

o    End Behavior

 

Allow 1 day to review and 1 day to test for this unit.

 

Unit #3 – Linear Functions (16 days)

Summary:  Includes but not limited to: Direct Variation, Rate of Change, Various Forms of a Linear Equation, Modeling using linear functions, Inverses of Linear functions, Piecewise Equations/Graphing involving Linear Functions and Systems of Linear Equations (focus on Three Equations and Three Unknowns)

Specifics: 

-          Review of Terminology and Parts of a Linear Function (1.5 days)

o    Slope (rate of change), y-intercept

o    Slope-Intercept form, Point-Slope form

o    Graphing using the Table Method

o    Graphing using the Slope-Intercept and Point-Slope method

-          Vertical and Horizontal Lines (0.5 days)

-          Parallel and Perpendicular Lines (1 days)

-          Writing the equation of lines (1 day)

-          Average Rate of Change (0.5 days)

-          Linear Modeling (2 days)

-          Direct Variation and Indirect Variation (3 days)

o    Identifying Constant of Variation

o    Identifying graphs of DV and IDV (linear and rectangular hyperbolas)

o    Application problems and Charts for DV and IDV

-          Systems of Linear Equations (2 x 2) (1 day)

o    Algebraically using Elimination Method and Substitution Method

o    Graphically

-          3 x 3 Systems of Linear Equations Algebraically (1.5 days)

-          Piecewise Functions (2 days)

 

Allow 1 day for Review and 1 day for Test on this unit.

 

Unit # 4 – Exponential Functions (20.5 days)

Summary:  Includes but not limited to: Exponents (integer and fractional), Review of the Rules of Exponents, Graph of Exponential Functions, Writing the Equation of an Exponential Function given the Base and Initial Value, Solving Exponential Equations with Common Bases, Modeling Exponential Functions (Growth and Decay)

Specifics:

-          Rearranging Expressions Involving Exponents (1.5 days)

o    Rewrite expressions as a single power

o    Rewrite a single power as multiple powers

o    Evaluate expressions involving exponents

o    Review of exponent laws

o    Simplifying exponential expressions review (including scientific notation)

-          Exponent Work with Function Notation, Manipulating Expressions involving Powers using Laws of Exponents (1 day)

-          Fractional Exponents (2 days)

o    Converting radical expression to an expression with a fractional exponent

o    Converting a fractional exponent to a radical expression

o    Simplifying expressions involving radicals and fractional exponents

-          Solving Equations with Fractional Exponents (1 day)

-          Solving Exponential Equations with the same base and with an uncommon base (1 day)

-          Graphing Exponential Functions (1 day)

o    Making an appropriate table of values

o    Understanding how to identify the equation of an asymptote

o    Identifying intercepts, intervals of increase/decrease

o    Exponential Growth/Decay

o    Relationship between equations that are reflections of each other over the y-axis and x-axis

o    Graphing exponential functions with a restricted domain

o    Transforming exponential functions

-          Writing the equation of exponential functions: (2 days)

o    Given “a” and a point

o    Given “b” and a point

o    Given two points

o    Given a table

-          Introduction to Exponential Word Problems (1 day)

o    Identifying initial amount, growth/decay factor, growth/decay rate (as a non-% and a %)

o    General form(s) for exponential word problems

o    Creating and Solving Exponential Word Problems (2.5 days)

§   Find output given input

§   Find input given output

§   Write a model that best represents a given situation

§   Interpret the behavior of a given exponential model

§   Exponential Word Problems involving Money (Financial), Compounding “n” times a year

-          Nominal vs. Effective Yearly Interest Rates (1.5 days)

-          Calculating the % of Growth/Decay within Various Periods of Time (Mindful Manipulations) (2 days)

-          Exponential Word Problems using base “e” (2 days)

o    Compounding Continuously

o    Naturally Occurring Situations and Half-Life Problems

 

Allow 1 day for Review and 1 day for Testing for this unit.

 

      Allow 3 days for Midterm Review for this course leading up to the exam.

 

Unit # 5 – Logarithmic Functions (19 days)

Summary:  Definition of a Logarithm, Graphs of Logarithmic Functions, Rules of Logarithms, Solving Exponential Equations Using Logarithms, the Number e and the Natural Logarithm and Applications of Exponential Equations and Logarithmic Equations

Specifics:

-          Introduction to Logs (2 days)

o    Understanding the inverse relationship between an exponential function and a logarithmic function

o    Domain and Range of log functions (and their restrictions)

o    Converting from exponential form to log form and vice versa

o    Common log

o    Inverse Property for Logs

o    Evaluating log expressions

o    Estimating between which two consecutive integers a log expression lies

-          Solving Simple Log Equations (1 day)

-          Graphing Log Functions and Other Features of Log Graphs like: (2 days)

o    Domain, Range, Intervals of Increase/Decrease, Intervals of Positive/Negative, Intercepts, Asymptotes, End Behavior

o    Finding an appropriate table of values for log graphs with knowledge of transformation shifts

o    Finding intercepts, domain, and range of log graphs algebraically

o    Graphing log graphs with restricted domains

o    Finding the average rate of change over an interval of a log graph, finding ∆y given an interval of x

-          Log Properties for Multiplication, Division, Powers (2 days)

o    Log Identity and Log Inverse Property

o    Proving log identities using log properties only

o    Expanding/Condensing log expressions

o    Evaluating log expressions using log properties

-          Manipulating Log Expressions (2 days)

-          Solving Log Equations (3 days)

o    Simple log equations (derive the change of base formula)

o    Log equations with logs on one side where you have to use log condensing properties

o    Log equations with logs on both sides where you have to get in the form log_ba = log_bc and then drop the logs

-          Log Application Word Problems (1 day)

o    Word problems given a log equation

o    Word problems where you have to use logs to solve them

-          Introduction to the Natural Log (ln) (2 days)

o    Ln rules and properties

o    Mixed problems involving ln like, expanding/condensing expressions, evaluating expressions, manipulating expressions, etc…

o    Ln application word problems

-          Newton’s Law of Cooling (2 days)

 

Allow 1 day of review and 1 day for testing for this unit.

 

Unit # 6 – Sequences and Series (13 days)

Summary:  Includes but not limited to: Definition a Sequence, Arithmetic Sequences, Geometric Sequences, Sigma Notation (Summation Notation), Definition of a Series, Arithmetic Series, Geometric Series and Applications of Series

Specifics:

-          Review of Sequences and Intro to Sigma Notation (2 days)

o    What is a sequence?  What is the domain/range of a sequence?  What types of sequences are there?  Recursive vs. Explicit formulas?  What is a series?

o    Introduction to sigma notation and evaluating expressions that are in sigma notation

-          Converting a series to Sigma Notation (1 day)

-          Generate sequences given explicit and recursive definitions (identify patterns where applicable) (1 day)

-          Given a sequence, write an explicit or recursive formula (0.5 days)

-          Arithmetic Sequences (1.5 days)

o    Identify common difference, d

o    Derive and apply explicit formula

o    Write recursive definitions

o    Find a₁, d, and a_n using a system of equations

o    Other types of problems relating to arithmetic sequences

-          Geometric Sequences (1.5 days)

o    Identify common ratio, r

o    Derive and apply explicit formula

o    Write recursive definitions

o    Find a₁, r, and a_n using a system of equations

o    Other types of problems relating to geometric sequences

-          Arithmetic and Geometric Series (2.5 days)

o    Derive and apply the formulas for arithmetic and geometric series.

o    Find the nth partial sum of a series

-          Mortgage Unit as it applies to Geometric Series (2 days)

o    Calculating the Amount Owed on a Mortgage

o    Calculating the Number of Payments needed to pay off a loan

o    Calculating the Monthly Payment

o    Finding the monthly interest rate given the yearly interest rate

o    Being able to solve word problems given equations involving the above unknowns

 

Allow 1 day for review and 1 day to test for this unit.

 

Unit #7 – Quadratic Functions (17 days)

Summary:  Includes but not limited to: Review of a Quadratic Equation (in one variable) and its Graph, Factoring (GCF, Factoring Completely, Difference of Perfect Squares (Conjugate Factors), Perfect Square Trinomials, Factoring Trinomials, Factoring by Grouping, Zero Product Property, Quadratic Inequalities in One Variable, Completing the Square, Vertex-Form of a Parabola, Horizontal and Vertical Shifts of a Parabola, Modeling with Quadratic Functions

Specifics:

-          Factoring of All Types (5.5 days)

o    GCF

o    Difference of Perfect Squares

o    Sum and Difference of Perfect Cubes

o    Trinomials with leading coefficients of 1 and coefficients other than 1

o    Factor by grouping

o    Factor using substitution

o    Factoring Completely

-          Solving Quadratic Equations (2.5 days)

o    Solving equations in standard form using Zero-Product Property Method (by Factoring)

o    Solving equations in vertex form using Square Root Method

o    Getting equations in vertex form by completing the square (then using the Square Root Method)

o    Fractional Equations with and without extraneous solutions

-          Solving Equations of a Higher Degree (1 day)

-          Quadratic Word Problems of All Types: (2 days)

o    Area, Perimeter, Ratio, Consecutive Integer, Volume, Systems, Proportions

-          Quadratic Inequalities (2 days)

o    Solving Quadratic Inequalities Algebraically

-          Modeling Quadratic Word Problems (2 days)

o    Building models for projectile motion in feet and meters

o    Appropriate domain questions

o    Intercept questions

o    Vertex/Turning Point Questions

o    Identifying the meaning of coefficients in a quadratic projectile model in standard form

o    Identifying the meaning of numbers in a quadratic projectile model in vertex form

 

Allow 1 day for review and 1 day to test for this unit.

Unit #8 – Locus of a Circle and Parabola (7 days)

Summary:  Distance Formula, Equations of a Circle (Center-Radius and Expanded Form) and the Definition of a Parabolas as a Locus of Points Equidistant from a Point (Focus) and a Line (Directrix)

Specifics:

-          Definition of a Circle as a locus of points (1 day)

o    Review of distance formula, midpoint formula, slope formula, Pythagorean theorem, center-radius form, slope/intercept and point/slope form of a line, and points of tangency

o    Writing the equation of a circle in center-radius form given information that can find the center and radius of a circle

-          Convert from standard form to Center-Radius Form by completing the square twice (1 day)

-          Systems Involving Circles Algebraically and Graphically (1 day)

-          Definition of a Parabola as a Locus of Points (2 days)

o    Define and identify focus and directrix

o    Derive focus-directrix form of a parabola

o    Mixed set of problems involving focus-directrix form

§   Find focus, vertex, directrix, p

§   Write focus-directrix form given information, convert focus-directrix form to standard or vertex form

§   Derive the focus-directrix equation of a parabola using the distance formula

 

Allow 1 day for review and 1 day for testing for this unit.

 

Unit #9 – Transformations of Functions (6 days)

Summary:  Includes, but not limited to: Vertical and Horizontal Shifts of Various Functions, Reflections of a Parabola, Vertical and Horizontal Dilations of a Function (Stretching and Compressing) and Even & Odd Functions

Specifics:

-          Transforming Functions (focus on answering questions without necessarily the use of a graph to read off of) (4 days)

o    Vertical and Horizontal Shifts

o    Reflections over the x and y axes

o    Vertical Stretches and Compressions

o    Horizontal Stretches and Compressions

-          Even and Odd Functions (1 day)

o    Algebraic Definition of an Even and Odd Function

o    Graphically Representation of Even and Odd Functions

o    Completing Tables given Even and Odd Functions

o    Evaluating expressions given Even and Odd Functions

 

Allow 1 day for a quiz/test on this unit.  No review built in.

 

Unit #10 – Radicals and the Quadratic Formula, Complex Numbers (10 days)

Summary:  Includes but not limited to: Domain, Range and Graphs of Square Root Functions, Solving Square Root Equations (extraneous roots), revisit Fractional Exponents as a means of discussing roots other than square roots and Quadratic Formula, Definition of an Imaginary Number, Powers of I, Definition of Complex Number, Operations with Complex Numbers, Definition and Meaning of the Discriminant and Solving Quadratic Equations with Complex Roots

Specifics:

-          Domain of Radical Equations (0.5 days)

o    Graphical representation

o    Algebraic approach (include quadratic and higher degree equations under the radical)

-          Solving Equations Involving Radicals (1 day)

o    Understanding inverse operations eliminate radicals

o    Solving equations with one or two radical expressions within them and extraneous solutions

o    Solving systems involving radical equations

-          Introduction to “i” and the Complex Number System (2.5 days)

o    Visual Representation of the Set of Complex Numbers

o    Definition of “i” (Complex number vs. Purely Imaginary or Purely Real Number)

o    Cyclic Nature of “i”

o    Powers of “i”

o    Operations involving “i”

o    Simplifying radical expressions involving “i” and/or negative radicands

-          Properties of Complex Numbers (1.5 days)

o    Conjugates (the product of conjugates)

o    Multiplicative and Additive Inverse

o    Rationalizing expressions with monomial and polynomial imaginary terms in the denominator

-          Solving Equations with Complex Roots (1 day)

o    Quadratic Formula

o    Completing the Square (Square Root Method)

o    Systems

-          Discriminant Work (1.5 days)

o    Review the four cases for the discriminant

o    Understand where it comes from and how it relates to the roots of an equation

o    Find the value(s) of k such that….type questions

 

Allow 1 day for review and 1 day for testing on this unit.

This course uses the A-List book of Knowledge SAT 2nd Edition and SAT/ACT practice tests throughout the semester to expose students to curriculum, strategies, and techniques when taking the SAT/ACT exam.

Course Overview

My main objective in teaching AP Calculus is to enable students to appreciate the beauty of calculus and receive a strong foundation that will give them the tools to succeed in future mathematics courses, as well succeed on the AP exam.    

Textbook(s):

Finney, Demana, Waits and Kennedy. Calculus—Graphical, Numerical, Algebraic. Third edition. Pearson, Prentice Hall, 2007.
This textbook will be our primary resource.

Schwartz, Stu.  AP Calculus AB. www.mastermathmentor.com   2005
Selected sections of this online textbook will be used throughout the course.

Technology Requirement

The use of a graphing calculator is a required part of this course. I will use the Texas Instruments TI-Smartview software emulating the 84 Plus graphing calculator in class regularly. Although not required, I have issued students their own TI-84 plus graphing calculator for use both inside and outside of class. I recommend the TI-84 or the TI-89.  I will have a classroom set of TI-84 Plus calculators, and some are available for use in the learning center.

If students choose to use any other calculator they may do so, as long as it can perform the four required functionalities of graphing technology :

• Finding a root
• Sketching a function in a specified window
• Approximating the derivative at a point using numerical methods
• Approximating the value of a definite integral using numerical methods

In my class, it is expected that students use graphing calculators daily to explore, discover, and reinforce the concepts of calculus.  Students will be required to make connections between the graphs of functions and their analysis, and draw conclusions about the behavior of functions.  We will use the calculator in a variety of ways such as:

• Conducting explorations
• Graphing functions within arbitrary windows
• Solving equations numerically
• Analyzing and Interpreting Results
• Justifying and explaining results of graphs and equations

Example 1–

As an introduction to the AP Calc course at the beginning of the year we looked at a distance versus time graph, graphed the derivative of the graph, found the area under the curve of graph (anti-derivative) and drew conclusions about how all three were related. We then calculated instantaneous rates of change using the concept of average rate of change as the distance between two points approached 0. We looked at functions like \(f (x) =\frac{sin (x) }{ x} \)as x approaches 0. We used the WINDOW and ZOOM features to zero in on the behavior of the graph as x approaches from both sides. We used the Y1(input) function and STO feature to approximate the instantaneous rate of change. We used the CALC 6: dy/dx feature to calculate the instantaneous rate of change at x = 0. We used the MATH 8:nDeriv( feature to calculate the instantaneous rate of change at x = 0. We compared all answers and discussed similarities and differences.   

Example 2 –

In the limit unit we derive the \(lim=(\frac{sinx}{x})=1\)geometrically using the sandwich theorem. Prior to that proof, we use the graphing calculators to show howlies between the functions y = 1 and y = cosx and how they all approach 1 as x approaches 0 from both sides. We also use the graphing calculator to look at the limit as x approaches infinity of functions like to conclude what our limit will be and then attempt to find the limit algebraically using substitution methods.

 

A Balanced Approach

Current mathematical education emphasizes a “Rule of Four.” There are a variety of ways to approach and solve problems. The four branches of the problem-solving tree of mathematics are:

• Numerical analysis (where data points are known, but not an equation)
• Graphical analysis (where a graph is known, but again, not an equation)
• Analytic/algebraic analysis (traditional equation and variable manipulation)
• Verbal/written methods of representing problems (classic story problems as well as written justification of one’s thinking in solving a problem - such as on our state assessment)

These four methods will be stressed throughout the course.

---

Course Outline

Introduction to Calculus (1 week)

                A.  An intuitive approach to calculus will be developed using the classic motion problem viewed from several approaches.

                        1. Finding speed from the position (the derivative)

                        2. Finding distanced traveled from the speed (the integral)

                        3. Understanding the relationship between the two processes (Fundamental Theorem of

                             Calculus)

 

Unit 1: Limits and Continuity (3 - 4 weeks)

            A. Rates of Change

                        1. Average Speed

                        2. Instantaneous Speed

            B. Limits at a Point

                        1. 1-sided Limits

                        2. 2-sided Limits

                        3. Sandwich Theorem

            C. Limits involving infinity

                        1. Asymptotic behavior (horizontal and vertical)

                        2. End behavior models

                        3. Properties of limits (algebraic analysis)

                        4. Visualizing limits (graphic analysis)           

            D. Continuity

                        1. Continuity at a point

                        2. Continuous functions

                        3. Discontinuous functions

                                    a. Removable discontinuity (0/0 form)

                                    b. Jump discontinuity (We look at y = int(x).)

                                    c. Infinite discontinuity

            E. Rates of Change and Tangent Lines

                        1. Average rate of change

                        2. Tangent line to a curve

                        3. Slope of a curve (algebraically and graphically)

                        4. Normal line to a curve (algebraically and graphically)

                        5. Instantaneous rate of change

 

Unit 2: The Derivative (5 - 6 weeks)

            A. Derivative of a Function

                        1. Definition of the derivative (difference quotient)

                        2. Derivative at a Point

                        3. Relationships between the graphs of f and f’

                        4. Graphing a derivative from data

                        5. One-sided derivatives

            B. Differentiability

                        1. Cases where f’(x) might fail to exist

                        2. Local linearity

                        3. Derivatives on the calculator (Numerical derivatives using NDERIV)

                        4. Symmetric difference quotient

                        5. Relationship between differentiability and continuity

                        6. Intermediate Value Theorem for Derivatives

            C. Rules for Differentiation

                        1. Constant, Power, Sum, Difference, Product, Quotient Rules

                        2. Higher order derivatives

            D.  Derivatives of trigonometric functions

            E. Chain Rule

            F. Implicit Differentiation

                        1. Differential method

                        2. y’ method

            G. Derivatives of inverse trigonometric functions           

            H. Derivatives of Exponential and Logarithmic Functions

 

Unit 3: Applications of the Derivative (5 - 6 weeks)

            A. Extreme Values

                        1. Relative Extrema

                        2. Absolute Extrema

                        3. Extreme Value Theorem

                        4. Definition of a critical point           

            B. Implications of the Derivative

                        1. Rolle’s Theorem

                        2. Mean Value Theorem

                        3. Increasing and decreasing functions

            C. Connecting f’ and f’’ with the graph of f(x)

                        1. First derivative test for relative max/min

                        2. Second derivative

                                    a. Concavity

                                    b. Inflection points

                                    c. Second derivative test for relative max/min 

            D. Optimization problems

            E. Linearization models

                        1. Local linearization

                        2. Tangent line approximation

                        3. Differentials

            F. Particle motion, Position, velocity, acceleration, and jerk           

            G. Economics

                        1. Marginal cost

                        2. Marginal revenue

                        3. Marginal profit

            H. Related Rates

 

MIDTERM EXAM

 

Unit 4: The Definite Integral (3–4 weeks)

            A. Approximating areas

                        1. Riemann sums

                                    a. Left sums

                                    b. Right sums

                                    c. Midpoint sums

                                    d. Trapezoidal sums

                        2. Definite integrals

            B. Properties of Definite Integrals

                        1. Power rule

                        2. Mean value theorem for definite integrals

            C. The Fundamental Theorem of Calculus

                        1. Part 1

                        2. Part 2

 

Unit 5: Differential Equations and Mathematical Modeling (4 weeks)

            A. Slope Fields

            B. Antiderivatives

                        1. Indefinite integrals

                        2. Power formulas

                        3. Trigonometric formulas

                        4. Exponential and Logarithmic formulas

            C. Separable Differential Equations

                        1. Growth and decay

                        2. Slope fields (Resources from the AP Calculus website are liberally used.)

                        3. General differential equations

                        4. Newton’s law of cooling

            D. Logistic Growth

 

Unit 6: Applications of Definite Integrals (3 weeks)

            A. Integral as net change

                        1. Calculating distance traveled (particle motion)

                        2. Consumption over time

                        3. Net change from data

            B. Area between curves

                        1. Area between a curve and an axis

                                    a. Integrating with respect to x

                                    b. Integrating with respect to y

                        2. Area between intersecting curves

                                    a. Integrating with respect to x

                                    b. Integrating with respect to y

            C. Calculating volume

                        1. Cross sections

                        2. Disc method

                        3. Shell method

 

Unit 7: Review/Test Preparation (Approximately 2– 4weeks)

            A. Multiple-choice practice (Items from past exams—1997, 1998, and 2003

                 are used as well as items from review books I’ve purchased over the years.)

                        1. Test taking strategies are emphasized

                        2. Individual and group practice are both used

            B. Free-response practice (Released items from the AP Central website are used liberally.)

                        1. Rubrics are reviewed so students see the need for complete answers

                        2. Students collaborate to formulate team responses

                        3. Individually written responses are crafted. Attention to full explanations is emphasized.

 

Unit 8: After the exam…

            A. Review of the AP exam and try to identify points that students felt confident with or points that

                 students felt they needed more work with.  This will help strengthen those points before students

                 go to college, as well as help me adjust the course for next year.

            B. Students will be required to do and present projects designed to incorporate this year’s learning in

                 applied ways.  Projects can be of the students own choosing with teacher approval or chosen from

                 one of the following sources.

 

MAA Notes. Resources for Calculus Collection. , D.C.: MAA, 1993.

Volume 3: Applications of Calculus, edited by Philip Straffin. MAA Notes Number 29.

Volume 4: Problems for Student Investigation, edited by Michael B. Jackson and John R. Ramsay. MAA Notes Number 30.

Volume 5: Readings for Calculus, edited by Underwood Dudley. MAA Notes Number 31.

Course Overview

Unit #1 – Trig Proofs and Trig Equations (13 days)

• Simplifying Trig Expressions (1 day)
  • Derive three trig identities
  • Simplify trig expressions using trig identity substitution, equivalent trig expressions, and algebra
• Proving Trig Identities (1 day)
• Sum and Difference of Two Angles Formulas for Trig Functions (2 days)
• Double Angles Formulas for Trig Functions (1 day)
• Half Angle Formulas for Trig Functions (.5 day)
• Solving Trig Equations (5.5 days)
  • Solving Linear Trig Equations
  • Solving Quadratic Trig Equations
  • Solving Trig Equations using the quadratic formula
  • Solving Trig Equations using Substitution
  • Solving Trig Equations using “u-substitution”

For Do Now questions during this unit, we will begin to spiral in the concept of evaluating inverse trig functions and looking at their graphs. 

Allow 1 day for review and 1 day to test.

Unit #2 – Trig Applications (9 days)

• Law of Sines (2.5 days)
  • Include angle of elevation and depression word problems
  • Ambiguous Case
• Law of Cosines (1 day)
• Area of a Triangle – (include law of sines and cosines within problems) (1.5 days)
• Force Word Problems (1.5 days)

Allow 1.5 days for review and 1 day to test.

Unit #3 – Graphing Rational Functions (11 days)

• Review of Even and Odd Functions (1 days)
  • Algebraic and graphical arguments including, but not limited to polynomial functions, rational functions, and trig functions. 
• Synthetic Division (1 day)
  • Cover linear factors with coefficients other than 1
• Graphing Rational Functions (8 days)
  • Finding vertical and horizontal asymptotes algebraically
  • Oblique and curvilinear asymptotes (not so much emphasis on curvilinear)
  • Holes
  • Transforming rational functions
  • Make sure rational function range in difficulty level (ie., include rational functions whose polynomials are of a degree than 2 where understanding of factoring cubes, factoring by grouping, and factor using synthetic division can be incorporated)

Allow 1 day to test for this unit.

Unit #4 – Limits of Polynomial, Rational and Piecewise Functions (13 days)

• Limits Graphically (2 days)
  • Definition of a limit (left/right hand limits)
  • Limits as x approaches a number including, but not limited to rational and piecewise functions, as well as functions with holes, step functions, fractional equations with absolute values etc…
  • Approximating limits without graphing the function (ie., plugging in values close to the “a” value and anticipating the value of the limit)
• Limits Algebraically (5 days)
  • Limit as x approaches the infinities (2/5 days)
    • Rational functions (powers equal, top/bottom heavy)
    • Polynomial functions (end behavior and leading coefficients)
    • Fractional equations with radicals
    • Expressions with radical (indeterminate case of )
  • Limit as x approaches the value a (3/5 days)
    • Polynomial/Continuous Functions (at x = a)
    • (indeterminate) form where a common factor needs to be removed
    • form where limit either DNE or is one of the infinities
    • One-sided limits
• Continuity (3 days)
  • Definition of Continuity
  • Types of Continuity
    • Removable
    • Essential Finite
    • Essential Infinite
    • If removable, redefining the function to make it continuous (continuous extension)
  • Determining whether functions (primarily piecewise functions and rational functions) are continuous and if not, stating where the definition of continuity fails
  • Finding the value of a constant(s) so that a function is continuous

Allow 2 days for review and 1 day to test.

*** This next unit is only for the honors level if time permits.

Unit #5 – Limits of Trig, ln, e^x, and Inverse Trig Functions (12 days)

• Trig limits (4 days)
  • Squeeze Theorem to derive 
    
    
  • Mixed problems involving all of the above limits
  • Intuitive approach to limits involving trig functions that don’t involve the above theorems
• Oscillating trig functions
• Non-ratio trig functions
• Trig functions with angles that are ratios
• Trig limits using substitution
• Introduction to the idea of L’Hopital’s Rule (0.5 days)
• Limits involving e^x and lnx (1.5 days)
• Limits comparing rates of change for growth functions (.5 day)
• Limit Theorems (1 day)
• Continuity and Continuity Theorems (1.5 days)
• Intermediate Value Theorem (1 day)

Allow 1 day for review and 1 day to test for this unit.

Unit #6 – Derivatives (11 days)

• Definition of a derivative (2 days)
• Power Rule for derivatives (1.5 days)
• Derivatives as they relate to increasing/decreasing/constant functions (.5 day)
• Chain Rule (1 day)
• Product Rule (1 day)
• Quotient Rule (2 days)

Allow 2 days to review and 1 day to test for this unit.

Try to allow for 2 – 3 days of review for the final exam in this course.

Course Overview

Unit #11 – Rational Functions (Depending on time, we will cover a portion of this unit) (10 days if completely done)

Summary:  Includes but not limited to: Simplifying Rational Expressions, Operations with Rational Expressions, Complex Fractions

Specifics: 

• Simplifying Algebraic Fractions (1 day)
  • Mono over Mono
  • Mono over Poly
  • Poly over Mono
  • Poly over Poly
  • Different cases for cancelling binomial factors
• Multiplying and Dividing Algebraic Fractions (1 day)
  • Include Area and Volume word problems 
• Complex Fractions (1 day)
• Adding and Subtracting Algebraic Fractions (2 days)
  • Common Denominator and Uncommon Denominator
  • Monomial and Polynomial Expressions in the Denominator
• Solving Fractional Equations (1.5 days)
  • Cross-Multiplying
  • Finding the Least Common Denominator
• Solving Rational Inequalities (1.5 days)

Test and Review based on the amount of time left for the year and the amount of this unit that was covered.

Try to allow for 3 days of review for the final, if possible.

Unit #12 – Polynomial and Rational Functions (12 days)

Specifics:  Includes but not limited to: Power Functions, Graphs and Zeros of a Polynomial, Writing Polynomial Equations, Polynomial Identities, Simplifying Rational Expressions, Operations with Rational Expressions, Solving Rational Equations, Solving Rational Inequalities, Discussion of Rational Functions, Complex Fractions, Long Division of Polynomials and Remainder Theorem  

• Identities (1.5 days)
• Intro to Power Functions (1 day)
• Behavior of Polynomial Functions (1.5 days)
  • End Behavior
  • Even/Odd Functions
  • Multiplicity of Roots
• Writing the Equation of Polynomial Functions (2.5 days)
  • Given no graph (with and without complex roots)
  • Given graphs
  • Finding the exact equation of polynomial functions
•  Polynomial Long Division (1.5 days)
• Synthetic Division (1 day)
• Remainder Theorem (.5 day)
• Location Principle (.5 day)

Allow 1 day for review and 1 day for testing on this unit.

***NOTE: AT THIS POINT, THE HONORS CLASS DOES A DIFFERENT ORDER OF UNITS COMPARED TO THE NON-HONORS CLASS.  THE ORDER BELOW REPRESENTS THE SEQUENCE OF THE NON-HONORS CLASS.  THE ORDER OF UNITS FOR THE HONORS CLASS WILL BE AS FOLLOWS:  UNITS 15, 16, 17, 13, AND THEN 14.  BOTH COURSES WILL FINISH THE COURSE IN THE SAME AMOUNT OF TIME.

Unit #13 – Introduction to Trigonometric Functions (12 days)

Specifics: Includes but not limited to: Standard Position, the meaning of Positive/Negative Angle Measures, Coterminal Angles, Reference Angles, Radian and Degree Measures of an Angle, The Unit Circle, Definition of Sine and Cosine, Pythagorean Identity, and Evaluating Using Sine and Cosine. 

• Introduction to basic right triangle trigonometry, definition of a radian, the relationship between degree and radian measure, the relationship between degree and degree/minute/second measure  (1.5 day)
• Angle of Depression/Elevation (.5 day)
• Types of Angles (1 day)
  • Standard Position
  • Coterminal Angles
  • Quadrantal Angles
• Unit Circle Concepts (1.5 day)
  • Pythagorean Trig Identity
  • Relationship between sine, cosine, and tangent on a unit circle
  • Value of quadrantal angles on a unit circle
  • Sign of trigonometry functions
• Special right triangles and the value of sine, cosine, and tangent for 30, 45, and 60 degree angles (1 day)
• Introduction to reference angles and rewriting trig functions as functions of positive acute reference angles (1 day)
• Evaluating trig functions (2.5 days)
• Evaluating reciprocal trig functions (2 days)

Allow 1 day for review and 1 day for testing on this unit.

Unit #14 – Graphing Trigonometric Functions (10 days)

Specifics:  Graphs of Sine and Cosine, Frequency, Period and Amplitude of a Trig Graph, Transformations of Trig Graphs, Modeling Using Sinusoidal Graphs, the Tangent Function and Reciprocal Trig Functions.

• Graphing the parent function for sine, cosine, and tangent (1.5 days)
• Exploring transformational shifts (reflections, vertical shifts, amplitude and frequency) for sinusoidal curves (1 day)
• Practice graphing sinusoidal curves with transformations (2 days)
• Graphing trig functions with phase changes (horizontal shifts) (1 day)
• Trig application word problems (2 days)

Allow 1.5 days of review and 1 day to test for this unit.

Unit #15 – Probability (12 days)

Specifics:  Includes but not limited to: Basic Terminology, Definition of Probability, Theoretical and Empirical Probabilities, Definitions of Sets/Sample Spaces, Independent and Dependent Events, Adding and Multiplying Probabilities and Conditional Probabilities

• Intro to the Basics of Probability (1 day)
  • Discuss basics like make-up of a deck of cards, number of digits, even vs. odd, positive vs. negative, vowels vs. consonants, etc.
  • Union and Intersection with Venn Diagrams
  • Counting Principle
• Concepts of Probability (1 day)
  • Terminology (experiment, trial, outcome, event, sample space, set, elements, subset, compliment of a set)
  • Basic definition of probability
  • Probability of Certainty and Impossibility
  • Drawing and/or listing sample spaces
• Theoretical vs. Empirical Probability (.5 day)
• Analyzing relationships between sets that are united or mutually exclusive (disjoint) using Venn Diagrams and union/intersection/complement notation (.5 day)
• Probabilities of an event not happening (.25 day)
• Probability of events involving “or” (adding probabilities, include two-way frequency tables) (2.25 days)
• Conditional Probability (1.5 days)
• Independence Tests (1.5 days)
  • Conditional probability test
  • Multiplying probability test
• Multiplying Probabilities (1.5 days)

Allow for 1 day of review and 1 day to test for this unit.

Unit #16 – Scatterplots and Regressions (7 days)

Specifics:  Includes but not limited to:  Scatterplots, Correlation Coefficients, Regressions

• Basic concepts of scatterplots (linear, quadratic, exponential, logarithmic, power, and sinusoidal models) (.5 day)
• Linear Regressions (.5 day)
  • Positive/Negative Trends
  • Correlation Coefficient (no/weak/moderate/strong/perfect)
  • Correlation doesn’t imply causation
• Using the graphing calculator to find linear regressions (2 days)
  • Identifying and interpreting the correlation coefficient
  • Interpolation/Extrapolation
• Practice with other regressions (3 days)

There was no review for this unit.  There is 1 day to test.

Unit #17 – Statistics (19 days)

Specifics:  Includes but not limited to: Variability and Sampling, Population Parameters, Normal Distribution and Z=scores, Sample Means, Sample Proportions, Difference in Sample Means and Margin of Error

• Types of Variability (Observational, Natural, Induced, Sampling) (.5 day)
• Ways to collect data randomization (surveys, questionnaires, observational and experimental studies) and discussion of sampling methods that lead to un-biased/biased results (random number table, etc…) (.5 day)
• Population Parameters (1.5 days)
  • Measures of central tendency (mean, median, mode)
  • Measures of dispersion (range, IQR, variance, standard deviation)
  • Population proportion
• Ways of organizing data (stem-and-lead, box plot, dot plot, histogram) (1.5 day)
• Types of distributions and what measures of central tendency are best (skewed, symmetrical) (.5 day)
•  Normal Distributions (2.5 days)
  • 68-95-99 rule
  • Percentile rank
  • normalcdf function
• Z-scores and z-score table (2 days)
• Intro to Inferential Statistics (8 days)
  • Concept of a simulation
  • Population vs. sample statistics and notation
  • Sample Means and Sample Proportions
  • Difference in Sample Means
  • Distributions of Sample Means
  • Distributions of Sample Proportions
  • Margin of Error

Allow 1 day of review and 1 day to test.

 

 

This course uses the A-List book of Knowledge SAT 2nd Edition and SAT/ACT practice tests throughout the semester to expose students to curriculum, strategies, and techniques when taking the SAT/ACT exam.

Course Overview

My main objective in teaching AP Calculus is to enable students to appreciate the beauty of calculus and receive a strong foundation that will give them the tools to succeed in future mathematics courses, as well succeed on the AP exam.    

Textbook(s):

Finney, Demana, Waits and Kennedy. Calculus—Graphical, Numerical, Algebraic. Third edition. Pearson, Prentice Hall, 2007.
This textbook will be our primary resource.

Schwartz, Stu.  AP Calculus AB. www.mastermathmentor.com   2005
Selected sections of this online textbook will be used throughout the course.

Technology Requirement

The use of a graphing calculator is a required part of this course. I will use the Texas Instruments TI-Smartview software emulating the 84 Plus graphing calculator in class regularly. Although not required, I have issued students their own TI-84 plus graphing calculator for use both inside and outside of class. I recommend the TI-84 or the TI-89.  I will have a classroom set of TI-84 Plus calculators, and some are available for use in the learning center.

If students choose to use any other calculator they may do so, as long as it can perform the four required functionalities of graphing technology :

• Finding a root
• Sketching a function in a specified window
• Approximating the derivative at a point using numerical methods
• Approximating the value of a definite integral using numerical methods

In my class, it is expected that students use graphing calculators daily to explore, discover, and reinforce the concepts of calculus.  Students will be required to make connections between the graphs of functions and their analysis, and draw conclusions about the behavior of functions.  We will use the calculator in a variety of ways such as:

• Conducting explorations
• Graphing functions within arbitrary windows
• Solving equations numerically
• Analyzing and Interpreting Results
• Justifying and explaining results of graphs and equations

Example 1–

As an introduction to the AP Calc course at the beginning of the year we looked at a distance versus time graph, graphed the derivative of the graph, found the area under the curve of graph (anti-derivative) and drew conclusions about how all three were related. We then calculated instantaneous rates of change using the concept of average rate of change as the distance between two points approached 0. We looked at functions like \(f (x) =\frac{sin (x) }{ x} \)as x approaches 0. We used the WINDOW and ZOOM features to zero in on the behavior of the graph as x approaches from both sides. We used the Y1(input) function and STO feature to approximate the instantaneous rate of change. We used the CALC 6: dy/dx feature to calculate the instantaneous rate of change at x = 0. We used the MATH 8:nDeriv( feature to calculate the instantaneous rate of change at x = 0. We compared all answers and discussed similarities and differences.   

Example 2 –

In the limit unit we derive the \(lim=(\frac{sinx}{x})=1\)geometrically using the sandwich theorem. Prior to that proof, we use the graphing calculators to show howlies between the functions y = 1 and y = cosx and how they all approach 1 as x approaches 0 from both sides. We also use the graphing calculator to look at the limit as x approaches infinity of functions like to conclude what our limit will be and then attempt to find the limit algebraically using substitution methods.

 

A Balanced Approach

Current mathematical education emphasizes a “Rule of Four.” There are a variety of ways to approach and solve problems. The four branches of the problem-solving tree of mathematics are:

• Numerical analysis (where data points are known, but not an equation)
• Graphical analysis (where a graph is known, but again, not an equation)
• Analytic/algebraic analysis (traditional equation and variable manipulation)
• Verbal/written methods of representing problems (classic story problems as well as written justification of one’s thinking in solving a problem - such as on our state assessment)

These four methods will be stressed throughout the course.

---

Course Outline

Introduction to Calculus (1 week)

                A.  An intuitive approach to calculus will be developed using the classic motion problem viewed from several approaches.

                        1. Finding speed from the position (the derivative)

                        2. Finding distanced traveled from the speed (the integral)

                        3. Understanding the relationship between the two processes (Fundamental Theorem of

                             Calculus)

 

Unit 1: Limits and Continuity (3 - 4 weeks)

            A. Rates of Change

                        1. Average Speed

                        2. Instantaneous Speed

            B. Limits at a Point

                        1. 1-sided Limits

                        2. 2-sided Limits

                        3. Sandwich Theorem

            C. Limits involving infinity

                        1. Asymptotic behavior (horizontal and vertical)

                        2. End behavior models

                        3. Properties of limits (algebraic analysis)

                        4. Visualizing limits (graphic analysis)           

            D. Continuity

                        1. Continuity at a point

                        2. Continuous functions

                        3. Discontinuous functions

                                    a. Removable discontinuity (0/0 form)

                                    b. Jump discontinuity (We look at y = int(x).)

                                    c. Infinite discontinuity

            E. Rates of Change and Tangent Lines

                        1. Average rate of change

                        2. Tangent line to a curve

                        3. Slope of a curve (algebraically and graphically)

                        4. Normal line to a curve (algebraically and graphically)

                        5. Instantaneous rate of change

 

Unit 2: The Derivative (5 - 6 weeks)

            A. Derivative of a Function

                        1. Definition of the derivative (difference quotient)

                        2. Derivative at a Point

                        3. Relationships between the graphs of f and f’

                        4. Graphing a derivative from data

                        5. One-sided derivatives

            B. Differentiability

                        1. Cases where f’(x) might fail to exist

                        2. Local linearity

                        3. Derivatives on the calculator (Numerical derivatives using NDERIV)

                        4. Symmetric difference quotient

                        5. Relationship between differentiability and continuity

                        6. Intermediate Value Theorem for Derivatives

            C. Rules for Differentiation

                        1. Constant, Power, Sum, Difference, Product, Quotient Rules

                        2. Higher order derivatives

            D.  Derivatives of trigonometric functions

            E. Chain Rule

            F. Implicit Differentiation

                        1. Differential method

                        2. y’ method

            G. Derivatives of inverse trigonometric functions           

            H. Derivatives of Exponential and Logarithmic Functions

 

Unit 3: Applications of the Derivative (5 - 6 weeks)

            A. Extreme Values

                        1. Relative Extrema

                        2. Absolute Extrema

                        3. Extreme Value Theorem

                        4. Definition of a critical point           

            B. Implications of the Derivative

                        1. Rolle’s Theorem

                        2. Mean Value Theorem

                        3. Increasing and decreasing functions

            C. Connecting f’ and f’’ with the graph of f(x)

                        1. First derivative test for relative max/min

                        2. Second derivative

                                    a. Concavity

                                    b. Inflection points

                                    c. Second derivative test for relative max/min 

            D. Optimization problems

            E. Linearization models

                        1. Local linearization

                        2. Tangent line approximation

                        3. Differentials

            F. Particle motion, Position, velocity, acceleration, and jerk           

            G. Economics

                        1. Marginal cost

                        2. Marginal revenue

                        3. Marginal profit

            H. Related Rates

 

MIDTERM EXAM

 

Unit 4: The Definite Integral (3–4 weeks)

            A. Approximating areas

                        1. Riemann sums

                                    a. Left sums

                                    b. Right sums

                                    c. Midpoint sums

                                    d. Trapezoidal sums

                        2. Definite integrals

            B. Properties of Definite Integrals

                        1. Power rule

                        2. Mean value theorem for definite integrals

            C. The Fundamental Theorem of Calculus

                        1. Part 1

                        2. Part 2

 

Unit 5: Differential Equations and Mathematical Modeling (4 weeks)

            A. Slope Fields

            B. Antiderivatives

                        1. Indefinite integrals

                        2. Power formulas

                        3. Trigonometric formulas

                        4. Exponential and Logarithmic formulas

            C. Separable Differential Equations

                        1. Growth and decay

                        2. Slope fields (Resources from the AP Calculus website are liberally used.)

                        3. General differential equations

                        4. Newton’s law of cooling

            D. Logistic Growth

 

Unit 6: Applications of Definite Integrals (3 weeks)

            A. Integral as net change

                        1. Calculating distance traveled (particle motion)

                        2. Consumption over time

                        3. Net change from data

            B. Area between curves

                        1. Area between a curve and an axis

                                    a. Integrating with respect to x

                                    b. Integrating with respect to y

                        2. Area between intersecting curves

                                    a. Integrating with respect to x

                                    b. Integrating with respect to y

            C. Calculating volume

                        1. Cross sections

                        2. Disc method

                        3. Shell method

 

Unit 7: Review/Test Preparation (Approximately 2– 4weeks)

            A. Multiple-choice practice (Items from past exams—1997, 1998, and 2003

                 are used as well as items from review books I’ve purchased over the years.)

                        1. Test taking strategies are emphasized

                        2. Individual and group practice are both used

            B. Free-response practice (Released items from the AP Central website are used liberally.)

                        1. Rubrics are reviewed so students see the need for complete answers

                        2. Students collaborate to formulate team responses

                        3. Individually written responses are crafted. Attention to full explanations is emphasized.

 

Unit 8: After the exam…

            A. Review of the AP exam and try to identify points that students felt confident with or points that

                 students felt they needed more work with.  This will help strengthen those points before students

                 go to college, as well as help me adjust the course for next year.

            B. Students will be required to do and present projects designed to incorporate this year’s learning in

                 applied ways.  Projects can be of the students own choosing with teacher approval or chosen from

                 one of the following sources.

 

MAA Notes. Resources for Calculus Collection. , D.C.: MAA, 1993.

Volume 3: Applications of Calculus, edited by Philip Straffin. MAA Notes Number 29.

Volume 4: Problems for Student Investigation, edited by Michael B. Jackson and John R. Ramsay. MAA Notes Number 30.

Volume 5: Readings for Calculus, edited by Underwood Dudley. MAA Notes Number 31.

Course Overview

Unit #1 – Trig Proofs and Trig Equations (13 days)

• Simplifying Trig Expressions (1 day)
  • Derive three trig identities
  • Simplify trig expressions using trig identity substitution, equivalent trig expressions, and algebra
• Proving Trig Identities (1 day)
• Sum and Difference of Two Angles Formulas for Trig Functions (2 days)
• Double Angles Formulas for Trig Functions (1 day)
• Half Angle Formulas for Trig Functions (.5 day)
• Solving Trig Equations (5.5 days)
  • Solving Linear Trig Equations
  • Solving Quadratic Trig Equations
  • Solving Trig Equations using the quadratic formula
  • Solving Trig Equations using Substitution
  • Solving Trig Equations using “u-substitution”

For Do Now questions during this unit, we will begin to spiral in the concept of evaluating inverse trig functions and looking at their graphs. 

Allow 1 day for review and 1 day to test.

Unit #2 – Trig Applications (9 days)

• Law of Sines (2.5 days)
  • Include angle of elevation and depression word problems
  • Ambiguous Case
• Law of Cosines (1 day)
• Area of a Triangle – (include law of sines and cosines within problems) (1.5 days)
• Force Word Problems (1.5 days)

Allow 1.5 days for review and 1 day to test.

Unit #3 – Graphing Rational Functions (11 days)

• Review of Even and Odd Functions (1 days)
  • Algebraic and graphical arguments including, but not limited to polynomial functions, rational functions, and trig functions. 
• Synthetic Division (1 day)
  • Cover linear factors with coefficients other than 1
• Graphing Rational Functions (8 days)
  • Finding vertical and horizontal asymptotes algebraically
  • Oblique and curvilinear asymptotes (not so much emphasis on curvilinear)
  • Holes
  • Transforming rational functions
  • Make sure rational function range in difficulty level (ie., include rational functions whose polynomials are of a degree than 2 where understanding of factoring cubes, factoring by grouping, and factor using synthetic division can be incorporated)

Allow 1 day to test for this unit.

Unit #4 – Limits of Polynomial, Rational and Piecewise Functions (13 days)

• Limits Graphically (2 days)
  • Definition of a limit (left/right hand limits)
  • Limits as x approaches a number including, but not limited to rational and piecewise functions, as well as functions with holes, step functions, fractional equations with absolute values etc…
  • Approximating limits without graphing the function (ie., plugging in values close to the “a” value and anticipating the value of the limit)
• Limits Algebraically (5 days)
  • Limit as x approaches the infinities (2/5 days)
    • Rational functions (powers equal, top/bottom heavy)
    • Polynomial functions (end behavior and leading coefficients)
    • Fractional equations with radicals
    • Expressions with radical (indeterminate case of )
  • Limit as x approaches the value a (3/5 days)
    • Polynomial/Continuous Functions (at x = a)
    • (indeterminate) form where a common factor needs to be removed
    • form where limit either DNE or is one of the infinities
    • One-sided limits
• Continuity (3 days)
  • Definition of Continuity
  • Types of Continuity
    • Removable
    • Essential Finite
    • Essential Infinite
    • If removable, redefining the function to make it continuous (continuous extension)
  • Determining whether functions (primarily piecewise functions and rational functions) are continuous and if not, stating where the definition of continuity fails
  • Finding the value of a constant(s) so that a function is continuous

Allow 2 days for review and 1 day to test.

*** This next unit is only for the honors level if time permits.

Unit #5 – Limits of Trig, ln, e^x, and Inverse Trig Functions (12 days)

• Trig limits (4 days)
  • Squeeze Theorem to derive 
    
    
  • Mixed problems involving all of the above limits
  • Intuitive approach to limits involving trig functions that don’t involve the above theorems
• Oscillating trig functions
• Non-ratio trig functions
• Trig functions with angles that are ratios
• Trig limits using substitution
• Introduction to the idea of L’Hopital’s Rule (0.5 days)
• Limits involving e^x and lnx (1.5 days)
• Limits comparing rates of change for growth functions (.5 day)
• Limit Theorems (1 day)
• Continuity and Continuity Theorems (1.5 days)
• Intermediate Value Theorem (1 day)

Allow 1 day for review and 1 day to test for this unit.

Unit #6 – Derivatives (11 days)

• Definition of a derivative (2 days)
• Power Rule for derivatives (1.5 days)
• Derivatives as they relate to increasing/decreasing/constant functions (.5 day)
• Chain Rule (1 day)
• Product Rule (1 day)
• Quotient Rule (2 days)

Allow 2 days to review and 1 day to test for this unit.

Try to allow for 2 – 3 days of review for the final exam in this course.

Course Overview

Unit #1 – Trig Proofs and Trig Equations (13 days)

• Simplifying Trig Expressions (1 day)
  • Derive three trig identities
  • Simplify trig expressions using trig identity substitution, equivalent trig expressions, and algebra
• Proving Trig Identities (1 day)
• Sum and Difference of Two Angles Formulas for Trig Functions (2 days)
• Double Angles Formulas for Trig Functions (1 day)
• Half Angle Formulas for Trig Functions (.5 day)
• Solving Trig Equations (5.5 days)
  • Solving Linear Trig Equations
  • Solving Quadratic Trig Equations
  • Solving Trig Equations using the quadratic formula
  • Solving Trig Equations using Substitution
  • Solving Trig Equations using “u-substitution”

For Do Now questions during this unit, we will begin to spiral in the concept of evaluating inverse trig functions and looking at their graphs. 

Allow 1 day for review and 1 day to test.

Unit #2 – Trig Applications (9 days)

• Law of Sines (2.5 days)
  • Include angle of elevation and depression word problems
  • Ambiguous Case
• Law of Cosines (1 day)
• Area of a Triangle – (include law of sines and cosines within problems) (1.5 days)
• Force Word Problems (1.5 days)

Allow 1.5 days for review and 1 day to test.

Unit #3 – Graphing Rational Functions (11 days)

• Review of Even and Odd Functions (1 days)
  • Algebraic and graphical arguments including, but not limited to polynomial functions, rational functions, and trig functions. 
• Synthetic Division (1 day)
  • Cover linear factors with coefficients other than 1
• Graphing Rational Functions (8 days)
  • Finding vertical and horizontal asymptotes algebraically
  • Oblique and curvilinear asymptotes (not so much emphasis on curvilinear)
  • Holes
  • Transforming rational functions
  • Make sure rational function range in difficulty level (ie., include rational functions whose polynomials are of a degree than 2 where understanding of factoring cubes, factoring by grouping, and factor using synthetic division can be incorporated)

Allow 1 day to test for this unit.

Unit #4 – Limits of Polynomial, Rational and Piecewise Functions (13 days)

• Limits Graphically (2 days)
  • Definition of a limit (left/right hand limits)
  • Limits as x approaches a number including, but not limited to rational and piecewise functions, as well as functions with holes, step functions, fractional equations with absolute values etc…
  • Approximating limits without graphing the function (ie., plugging in values close to the “a” value and anticipating the value of the limit)
• Limits Algebraically (5 days)
  • Limit as x approaches the infinities (2/5 days)
    • Rational functions (powers equal, top/bottom heavy)
    • Polynomial functions (end behavior and leading coefficients)
    • Fractional equations with radicals
    • Expressions with radical (indeterminate case of )
  • Limit as x approaches the value a (3/5 days)
    • Polynomial/Continuous Functions (at x = a)
    • (indeterminate) form where a common factor needs to be removed
    • form where limit either DNE or is one of the infinities
    • One-sided limits
• Continuity (3 days)
  • Definition of Continuity
  • Types of Continuity
    • Removable
    • Essential Finite
    • Essential Infinite
    • If removable, redefining the function to make it continuous (continuous extension)
  • Determining whether functions (primarily piecewise functions and rational functions) are continuous and if not, stating where the definition of continuity fails
  • Finding the value of a constant(s) so that a function is continuous

Allow 2 days for review and 1 day to test.

*** This next unit is only for the honors level if time permits.

Unit #5 – Limits of Trig, ln, e^x, and Inverse Trig Functions (12 days)

• Trig limits (4 days)
  • Squeeze Theorem to derive 
    
    
  • Mixed problems involving all of the above limits
  • Intuitive approach to limits involving trig functions that don’t involve the above theorems
• Oscillating trig functions
• Non-ratio trig functions
• Trig functions with angles that are ratios
• Trig limits using substitution
• Introduction to the idea of L’Hopital’s Rule (0.5 days)
• Limits involving e^x and lnx (1.5 days)
• Limits comparing rates of change for growth functions (.5 day)
• Limit Theorems (1 day)
• Continuity and Continuity Theorems (1.5 days)
• Intermediate Value Theorem (1 day)

Allow 1 day for review and 1 day to test for this unit.

Unit #6 – Derivatives (11 days)

• Definition of a derivative (2 days)
• Power Rule for derivatives (1.5 days)
• Derivatives as they relate to increasing/decreasing/constant functions (.5 day)
• Chain Rule (1 day)
• Product Rule (1 day)
• Quotient Rule (2 days)

Allow 2 days to review and 1 day to test for this unit.

Try to allow for 2 – 3 days of review for the final exam in this course.

This course uses the A-List book of Knowledge SAT 2nd Edition and SAT/ACT practice tests throughout the semester to expose students to curriculum, strategies, and techniques when taking the SAT/ACT exam.

Course Overview

My main objective in teaching AP Calculus is to enable students to appreciate the beauty of calculus and receive a strong foundation that will give them the tools to succeed in future mathematics courses, as well succeed on the AP exam.    

Textbook(s):

Finney, Demana, Waits and Kennedy. Calculus—Graphical, Numerical, Algebraic. Third edition. Pearson, Prentice Hall, 2007.
This textbook will be our primary resource.

Schwartz, Stu.  AP Calculus AB. www.mastermathmentor.com   2005
Selected sections of this online textbook will be used throughout the course.

Technology Requirement

The use of a graphing calculator is a required part of this course. I will use the Texas Instruments TI-Smartview software emulating the 84 Plus graphing calculator in class regularly. Although not required, I have issued students their own TI-84 plus graphing calculator for use both inside and outside of class. I recommend the TI-84 or the TI-89.  I will have a classroom set of TI-84 Plus calculators, and some are available for use in the learning center.

If students choose to use any other calculator they may do so, as long as it can perform the four required functionalities of graphing technology :

• Finding a root
• Sketching a function in a specified window
• Approximating the derivative at a point using numerical methods
• Approximating the value of a definite integral using numerical methods

In my class, it is expected that students use graphing calculators daily to explore, discover, and reinforce the concepts of calculus.  Students will be required to make connections between the graphs of functions and their analysis, and draw conclusions about the behavior of functions.  We will use the calculator in a variety of ways such as:

• Conducting explorations
• Graphing functions within arbitrary windows
• Solving equations numerically
• Analyzing and Interpreting Results
• Justifying and explaining results of graphs and equations

Example 1–

As an introduction to the AP Calc course at the beginning of the year we looked at a distance versus time graph, graphed the derivative of the graph, found the area under the curve of graph (anti-derivative) and drew conclusions about how all three were related. We then calculated instantaneous rates of change using the concept of average rate of change as the distance between two points approached 0. We looked at functions like \(f (x) =\frac{sin (x) }{ x} \)as x approaches 0. We used the WINDOW and ZOOM features to zero in on the behavior of the graph as x approaches from both sides. We used the Y1(input) function and STO feature to approximate the instantaneous rate of change. We used the CALC 6: dy/dx feature to calculate the instantaneous rate of change at x = 0. We used the MATH 8:nDeriv( feature to calculate the instantaneous rate of change at x = 0. We compared all answers and discussed similarities and differences.   

Example 2 –

In the limit unit we derive the \(lim=(\frac{sinx}{x})=1\)geometrically using the sandwich theorem. Prior to that proof, we use the graphing calculators to show howlies between the functions y = 1 and y = cosx and how they all approach 1 as x approaches 0 from both sides. We also use the graphing calculator to look at the limit as x approaches infinity of functions like to conclude what our limit will be and then attempt to find the limit algebraically using substitution methods.

 

A Balanced Approach

Current mathematical education emphasizes a “Rule of Four.” There are a variety of ways to approach and solve problems. The four branches of the problem-solving tree of mathematics are:

• Numerical analysis (where data points are known, but not an equation)
• Graphical analysis (where a graph is known, but again, not an equation)
• Analytic/algebraic analysis (traditional equation and variable manipulation)
• Verbal/written methods of representing problems (classic story problems as well as written justification of one’s thinking in solving a problem - such as on our state assessment)

These four methods will be stressed throughout the course.

---

Course Outline

Introduction to Calculus (1 week)

                A.  An intuitive approach to calculus will be developed using the classic motion problem viewed from several approaches.

                        1. Finding speed from the position (the derivative)

                        2. Finding distanced traveled from the speed (the integral)

                        3. Understanding the relationship between the two processes (Fundamental Theorem of

                             Calculus)

 

Unit 1: Limits and Continuity (3 - 4 weeks)

            A. Rates of Change

                        1. Average Speed

                        2. Instantaneous Speed

            B. Limits at a Point

                        1. 1-sided Limits

                        2. 2-sided Limits

                        3. Sandwich Theorem

            C. Limits involving infinity

                        1. Asymptotic behavior (horizontal and vertical)

                        2. End behavior models

                        3. Properties of limits (algebraic analysis)

                        4. Visualizing limits (graphic analysis)           

            D. Continuity

                        1. Continuity at a point

                        2. Continuous functions

                        3. Discontinuous functions

                                    a. Removable discontinuity (0/0 form)

                                    b. Jump discontinuity (We look at y = int(x).)

                                    c. Infinite discontinuity

            E. Rates of Change and Tangent Lines

                        1. Average rate of change

                        2. Tangent line to a curve

                        3. Slope of a curve (algebraically and graphically)

                        4. Normal line to a curve (algebraically and graphically)

                        5. Instantaneous rate of change

 

Unit 2: The Derivative (5 - 6 weeks)

            A. Derivative of a Function

                        1. Definition of the derivative (difference quotient)

                        2. Derivative at a Point

                        3. Relationships between the graphs of f and f’

                        4. Graphing a derivative from data

                        5. One-sided derivatives

            B. Differentiability

                        1. Cases where f’(x) might fail to exist

                        2. Local linearity

                        3. Derivatives on the calculator (Numerical derivatives using NDERIV)

                        4. Symmetric difference quotient

                        5. Relationship between differentiability and continuity

                        6. Intermediate Value Theorem for Derivatives

            C. Rules for Differentiation

                        1. Constant, Power, Sum, Difference, Product, Quotient Rules

                        2. Higher order derivatives

            D.  Derivatives of trigonometric functions

            E. Chain Rule

            F. Implicit Differentiation

                        1. Differential method

                        2. y’ method

            G. Derivatives of inverse trigonometric functions           

            H. Derivatives of Exponential and Logarithmic Functions

 

Unit 3: Applications of the Derivative (5 - 6 weeks)

            A. Extreme Values

                        1. Relative Extrema

                        2. Absolute Extrema

                        3. Extreme Value Theorem

                        4. Definition of a critical point           

            B. Implications of the Derivative

                        1. Rolle’s Theorem

                        2. Mean Value Theorem

                        3. Increasing and decreasing functions

            C. Connecting f’ and f’’ with the graph of f(x)

                        1. First derivative test for relative max/min

                        2. Second derivative

                                    a. Concavity

                                    b. Inflection points

                                    c. Second derivative test for relative max/min 

            D. Optimization problems

            E. Linearization models

                        1. Local linearization

                        2. Tangent line approximation

                        3. Differentials

            F. Particle motion, Position, velocity, acceleration, and jerk           

            G. Economics

                        1. Marginal cost

                        2. Marginal revenue

                        3. Marginal profit

            H. Related Rates

 

MIDTERM EXAM

 

Unit 4: The Definite Integral (3–4 weeks)

            A. Approximating areas

                        1. Riemann sums

                                    a. Left sums

                                    b. Right sums

                                    c. Midpoint sums

                                    d. Trapezoidal sums

                        2. Definite integrals

            B. Properties of Definite Integrals

                        1. Power rule

                        2. Mean value theorem for definite integrals

            C. The Fundamental Theorem of Calculus

                        1. Part 1

                        2. Part 2

 

Unit 5: Differential Equations and Mathematical Modeling (4 weeks)

            A. Slope Fields

            B. Antiderivatives

                        1. Indefinite integrals

                        2. Power formulas

                        3. Trigonometric formulas

                        4. Exponential and Logarithmic formulas

            C. Separable Differential Equations

                        1. Growth and decay

                        2. Slope fields (Resources from the AP Calculus website are liberally used.)

                        3. General differential equations

                        4. Newton’s law of cooling

            D. Logistic Growth

 

Unit 6: Applications of Definite Integrals (3 weeks)

            A. Integral as net change

                        1. Calculating distance traveled (particle motion)

                        2. Consumption over time

                        3. Net change from data

            B. Area between curves

                        1. Area between a curve and an axis

                                    a. Integrating with respect to x

                                    b. Integrating with respect to y

                        2. Area between intersecting curves

                                    a. Integrating with respect to x

                                    b. Integrating with respect to y

            C. Calculating volume

                        1. Cross sections

                        2. Disc method

                        3. Shell method

 

Unit 7: Review/Test Preparation (Approximately 2– 4weeks)

            A. Multiple-choice practice (Items from past exams—1997, 1998, and 2003

                 are used as well as items from review books I’ve purchased over the years.)

                        1. Test taking strategies are emphasized

                        2. Individual and group practice are both used

            B. Free-response practice (Released items from the AP Central website are used liberally.)

                        1. Rubrics are reviewed so students see the need for complete answers

                        2. Students collaborate to formulate team responses

                        3. Individually written responses are crafted. Attention to full explanations is emphasized.

 

Unit 8: After the exam…

            A. Review of the AP exam and try to identify points that students felt confident with or points that

                 students felt they needed more work with.  This will help strengthen those points before students

                 go to college, as well as help me adjust the course for next year.

            B. Students will be required to do and present projects designed to incorporate this year’s learning in

                 applied ways.  Projects can be of the students own choosing with teacher approval or chosen from

                 one of the following sources.

 

MAA Notes. Resources for Calculus Collection. , D.C.: MAA, 1993.

Volume 3: Applications of Calculus, edited by Philip Straffin. MAA Notes Number 29.

Volume 4: Problems for Student Investigation, edited by Michael B. Jackson and John R. Ramsay. MAA Notes Number 30.

Volume 5: Readings for Calculus, edited by Underwood Dudley. MAA Notes Number 31.

Course Overview

Unit #1 – Trig Proofs and Trig Equations (13 days)

• Simplifying Trig Expressions (1 day)
  • Derive three trig identities
  • Simplify trig expressions using trig identity substitution, equivalent trig expressions, and algebra
• Proving Trig Identities (1 day)
• Sum and Difference of Two Angles Formulas for Trig Functions (2 days)
• Double Angles Formulas for Trig Functions (1 day)
• Half Angle Formulas for Trig Functions (.5 day)
• Solving Trig Equations (5.5 days)
  • Solving Linear Trig Equations
  • Solving Quadratic Trig Equations
  • Solving Trig Equations using the quadratic formula
  • Solving Trig Equations using Substitution
  • Solving Trig Equations using “u-substitution”

For Do Now questions during this unit, we will begin to spiral in the concept of evaluating inverse trig functions and looking at their graphs. 

Allow 1 day for review and 1 day to test.

Unit #2 – Trig Applications (9 days)

• Law of Sines (2.5 days)
  • Include angle of elevation and depression word problems
  • Ambiguous Case
• Law of Cosines (1 day)
• Area of a Triangle – (include law of sines and cosines within problems) (1.5 days)
• Force Word Problems (1.5 days)

Allow 1.5 days for review and 1 day to test.

Unit #3 – Graphing Rational Functions (11 days)

• Review of Even and Odd Functions (1 days)
  • Algebraic and graphical arguments including, but not limited to polynomial functions, rational functions, and trig functions. 
• Synthetic Division (1 day)
  • Cover linear factors with coefficients other than 1
• Graphing Rational Functions (8 days)
  • Finding vertical and horizontal asymptotes algebraically
  • Oblique and curvilinear asymptotes (not so much emphasis on curvilinear)
  • Holes
  • Transforming rational functions
  • Make sure rational function range in difficulty level (ie., include rational functions whose polynomials are of a degree than 2 where understanding of factoring cubes, factoring by grouping, and factor using synthetic division can be incorporated)

Allow 1 day to test for this unit.

Unit #4 – Limits of Polynomial, Rational and Piecewise Functions (13 days)

• Limits Graphically (2 days)
  • Definition of a limit (left/right hand limits)
  • Limits as x approaches a number including, but not limited to rational and piecewise functions, as well as functions with holes, step functions, fractional equations with absolute values etc…
  • Approximating limits without graphing the function (ie., plugging in values close to the “a” value and anticipating the value of the limit)
• Limits Algebraically (5 days)
  • Limit as x approaches the infinities (2/5 days)
    • Rational functions (powers equal, top/bottom heavy)
    • Polynomial functions (end behavior and leading coefficients)
    • Fractional equations with radicals
    • Expressions with radical (indeterminate case of )
  • Limit as x approaches the value a (3/5 days)
    • Polynomial/Continuous Functions (at x = a)
    • (indeterminate) form where a common factor needs to be removed
    • form where limit either DNE or is one of the infinities
    • One-sided limits
• Continuity (3 days)
  • Definition of Continuity
  • Types of Continuity
    • Removable
    • Essential Finite
    • Essential Infinite
    • If removable, redefining the function to make it continuous (continuous extension)
  • Determining whether functions (primarily piecewise functions and rational functions) are continuous and if not, stating where the definition of continuity fails
  • Finding the value of a constant(s) so that a function is continuous

Allow 2 days for review and 1 day to test.

*** This next unit is only for the honors level if time permits.

Unit #5 – Limits of Trig, ln, e^x, and Inverse Trig Functions (12 days)

• Trig limits (4 days)
  • Squeeze Theorem to derive 
    
    
  • Mixed problems involving all of the above limits
  • Intuitive approach to limits involving trig functions that don’t involve the above theorems
• Oscillating trig functions
• Non-ratio trig functions
• Trig functions with angles that are ratios
• Trig limits using substitution
• Introduction to the idea of L’Hopital’s Rule (0.5 days)
• Limits involving e^x and lnx (1.5 days)
• Limits comparing rates of change for growth functions (.5 day)
• Limit Theorems (1 day)
• Continuity and Continuity Theorems (1.5 days)
• Intermediate Value Theorem (1 day)

Allow 1 day for review and 1 day to test for this unit.

Unit #6 – Derivatives (11 days)

• Definition of a derivative (2 days)
• Power Rule for derivatives (1.5 days)
• Derivatives as they relate to increasing/decreasing/constant functions (.5 day)
• Chain Rule (1 day)
• Product Rule (1 day)
• Quotient Rule (2 days)

Allow 2 days to review and 1 day to test for this unit.

Try to allow for 2 – 3 days of review for the final exam in this course.

This course uses the A-List book of Knowledge SAT 2nd Edition and SAT/ACT practice tests throughout the semester to expose students to curriculum, strategies, and techniques when taking the SAT/ACT exam.