Curriculum and Pacing Guide

Curriculum and Pacing Guide - Mathematics Honors Calculus (120230) and AP Calculus AB (1202310) (everyday block all year) Text: McDougal Littell Calculus of a Single Variable

Revised February 2007 Honors Calculus (120230) and AP Calculus AB (1202310) (everyday block all year)

I. AP Calculus AB Course Overview and Philosophy

In developing and using the topics of differential and integral calculus, students have the opportunity to solidify all their previous high school math experience, since practically all the major ideas from the prerequisite math courses are necessary for understanding. Consequently, we have high expectations which emphasize concepts reinforced with development and application rather than just procedure. Of course it is a goal that the successful student of Advanced Placement Calculus validate their success with a more that adequate performance on the AP Exam. However, beyond this, it is our desire that the successful student appreciates the high level of their mathematical success and at the same time come to realize that this accomplishment is not an end, but a beginning point for many other opportunities in various fields of study and work.

II. Course Outline for AP Calculus AB

The following is from the document, “Curriculum and Pacing Guide – Mathematics: Honors Calculus and AP Calculus AB (everyday block all year).” This document is relatively dynamic in that is regularly reviewed and updated as it should be. Although it may have more detailed than necessary, the columns in the pacing of each unit provide the general time line, sections of the text book, topics covered, major concepts, and suggested approaches and activities.

Curriculum and Pacing Guide - Mathematics Honors Calculus (120230) and AP Calculus AB (1202310) (everyday block all year)

Revised February 2007

This pacing guide is intended for use with the everyday block and will go all year. Students who successfully complete the 1st term will receive a credit for Honors Calculus. Students who successfully complete 1st and 2nd terms will receive one credit for Honors Calculus and one credit for AP Calculus AB and also take the AP Calculus AB Exam. Students who take AP Calculus in this manner cannot begin AP Calculus AB in the 2nd term. The intent is to cover the content of Calculus Honors and AP calculus AB without the intensity of completing the course on an all year alternating block. Honors Calculus (120230) and AP Calculus AB (1202310) (everyday block all year)

1. The beginning unit is a review of the concepts from algebra and pre-calculus concerning relations /functions will probably be necessary to some extent. These include linear, polynomial, rational, trigonometric, the generic functions, and piecewise functions. Topics relative to these are; slopes, domain and range, symmetry, intercepts, intersections, and the graphs of these relations/functions. The graphing calculator (TI-89) is used to reinforce these concepts, referring to the solving, table, and graphing capabilities of the calculator. Chapter P: Preparation for Calculus Block Section Topic Concepts Suggested Approaches Days (T) Technology, (V) Verbal, (G) Graphical, (N) Numerical 1 P.1 Graphs and models Symmetry, Intercepts, (T,G) Graphing Features of calculator, solving equations Introducing Families of Functions Video Tutorial - Learn how to Adjust the Graph Viewing Window 1 P.2 Linear Models and Slope, point-slope form, slope- (V) Rates of Change intercept form Move My Way--A CBR Analysis of Rates of Change X (orY) Marks the Spot 3 P.3* Functions and Their Elementary functions (T) Transformations of Functions Graph Domain and Range of Graphs Analysis of a Bouncing Ball Video Tutorial - Learn how to Evaluate Composite Functions Functions Horizontal and Vertical Transformations TI- Navigator, Activity Centre and Learncheck Transformation Graphing App 5 Appendix Precalculus Review Unit circle, solving trigonometric (V) D* functions, piece-wise functions Back and Forth---Analysis of Spring Motion The Trigonometric Functions and Their Inverse Func Special Ratios and the Unit Circle 1 Special Ratios and the Unit Circle 2 Special Ratios and the Unit Circle 3 Special Ratios and the Unit Circle 4 Graph It in Pieces: Piecewise-defined Functions 1 Test 11 Total Honors Calculus (120230) and AP Calculus AB (1202310) (everyday block all year)

2. The concepts of a limit, as the independent variable approaches a particular value, are addressed numerically, analytically, and graphically. This involves developing the ideas of the existence of a limit. The issues are, if the limit exist, what is it, and if it does not, why not? This necessitates the understanding of “broken graph” (addressed in detail later with continuity), oscillating, and asymptotic behaviors. Examination of limits, including one sided limits, is done using the various algebra techniques for the types of functions, the properties of limits and special techniques for rational trig related functions. Continuity and the Intermediate Value theorem and their applications are also part of this unit. Chapter 1: Limits and Their Properties Block Section Topic Concepts Suggested Approaches days (T) Technology, (V) Verbal, (G) Graphical, (N) Numerical 1 SUPP. Algebra Review Factoring perfect cubes, complex (V) verbal fractions, manipulating rational Absolute Value and Piecewise expressions, rationalizing, absolute Functions Solving Rational Expressions value Factoring trinomials and Expanding binomial x binomial 2 1.2 Finding Limits Graphically Limit (T) See www.calculus- and Numerically help.com (T) table features of calculator (T) trace features of calculator (V) p. 56/ 39-42, 9-52 Approaching Limits TI: Limits, Numerically Functions, Graphs, and Limits TI: Find a Delta For Any Given Epsilon 4 1.3 Evaluating Limits Indeterminate form, Squeeze (V) verbal Analytically Theorem The Derivative as a Limit 2 1.4 Continuity and One-Sided Continuity at a point, one-sided (T) See www.calculus- Limits limits, Intermediate Value Theorem help.com Is There a Limit to Which Side You Can Take? Graphical Consequences of Continuity 3 1.5 Infinite limits (may cover 3.5 Infinite limit, vertical asymptote (V) verbal after Section 1.5) To Infinity and Beyond! 1 Review (V) How Do I? Worksheets 1 Test 14 Total Honors Calculus (120230) and AP Calculus AB (1202310) (everyday block all year) 3. The derivative is developed in this unit which involves the geometric interpretation of the tangent line at a point, leading to the limit definition of the derivative of a function. The limit definition is used both to find derivatives at a point and to develop the basic derivative rules. The relationship between continuity and differentiability is examined with attention given to continuous functions that have points where the derivative does not exist. Basic derivative rules are used for first and second derivatives and also for implicit derivatives. Areas of application in this unit are; finding equations of tangent lines (and normal lines) at a point, beginning motion problems, and related rates. Chapter 2: Differentiation Block Days Section Topic Concepts Suggested Approaches (T) Technology, (V) Verbal, (G) Graphical, (N) Numerical 3 2.1 The derivative and the Tangent Derivative, differentiable, normal (V) p. 104/ 89-92 line problem line (V) p. 116/ 105 (T) http://www.ima.umn.edu/~arnold/graphics.html Seeing Is Believing TI: Writing Equations for Tangent and Normal Lines Graph of Functions and Their Derivatives 3 2.2 Differentiation rules and rates Instantaneous rate of change, (V) verbal of change. velocity, speed Average Rate of Change, Difference Quotients, and Approximate Instantaneous Rate of Change Derivatives (Trig Functions) 2 2.3 Product and Quotient rules Higher order derivatives (V) TI: Product Rule TI: The Product and Quotient Rules Local Linearity, Differentiability and Limits of Difference Quotients 3 2.4 Chain Rule Power rule (T) Calculator: graphs of functions and tangent lines TI: Derivatives of Composite Functions-Chain Rule TI: Logarithmic Differentiation 1 Test (V) verbal 2 2.5 Implicit Differentiation Explicit/implicit forms of (V) verbal functions Implicit Differentiation Implicit Differentiation TI: Implicit Differentiation on the TI-89 NUMB3RS - Season 3 - "Hardball" - Implicit Orbits 4 2.6 Related rates (V) verbal Related Rates Related Rates TI: Visualizing Related Rates Problems 1 Test 19 Total Honors Calculus (120230) and AP Calculus AB (1202310) (everyday block all year)

4. This unit continues the application of derivatives. Also three principle theorems are developed and used – the Extreme Value Theorem, Rolle’s Theorem, and the Mean Value Theorem. First and second derivatives are used to determine for a given function the critical values, intervals of increase and decrease, relative maxima and minima, points of inflection, and intervals concave up and concave down. These applications along with examination of limits at infinity for horizontal asymptotes and the previous study of asymptotic behaviors (vertical and slant) leads to curve sketching for the function. This application is done with and without graphing calculators. Included with this application is the examination of the relationships of the graphs of a function, the graph of its 1st derivative, and the graph of its 2nd derivative. The very useful and important derivative application of solving optimization problems are in this unit as well as linear approximations and differientials. ( Pacing on next page) Honors Calculus (120230) and AP Calculus AB (1202310) (everyday block all year) Chapter 3: Applications of Differentiation Block Section Topic Concepts Suggested Approaches Days (T) Technology, (V) Verbal, (G) Graphical, (N) Numerical 2 3.1 Extrema on an Critical numbers, absolute Functions and Their Extrema interval and relative extrema TI: Absolute Extrema 2 3.2 Rolle’s Theorem (T) Demonstrate Theorems using graphing calculator and Mean Value TI: Rolle's Theorem Theorem Average Rate of Change, Difference Quotients, and Approximate Instantaneous Rate of Change. 1 3.3 First Derivative Strictly monotonic (V) verbal (G) Graphical Test, increasing Graph of Functions and Their Derivatives and decreasing Graphing Relationships functions Projectile Motion 1 3.4 Concavity and the Point of inflection, (V) verbal (G) Graphical Second Derivative concave down, concave up Graphing Relationships Test Using Traffic Data to Reinforce Inflection Points in Calculus 1 3.5 Limits at infinity Horizontal asymptote (V) verbal (G)Graphical (T) Technology 2 3.6 Summary of curve Slant asymptote, (T) www.unitedstreaming.com : “applications of derivatives” sketching polynomial long division (G) “Getting at the concept” questions Relating Graphs of a Function To Derivatives 2 3.6 Examine relationships THE BIG DERIVATIVE PUZZLE: SUPP. between graphs of f, f’, & http://www.univie.ac.at/future.media/moe/tests/diff1/ablerkennen.html f” 1 Test (V) verbal 5 3.7 Optimization Primary/secondary (V) verbal Problems equations TI: Analyzing Optimization Problems TI: Optimization 2 3.9 Differentials differential forms, (V) verbal (G) Graphical propagated error, tangent line approximations 1 Review 1 Test 21 Total Honors Calculus (120230) and AP Calculus AB (1202310) (everyday block all year)

5. Integrals are begun in this unit through the introduction of the antiderivative (indefinite integral) which answers the question: What is a function whose derivative is given? This leads immediately to the basic integration rules. These are practiced. The notions of a family of curves and a general solution to a differential equation are introduced followed by the idea of a particular solution for an initial condition is developed and applied. This allows for more applications of motion problems. The definite integral is developed by first examining estimates of the areas of plane regions as sums of rectangles constructed by using a partitioning of an interval and the right, left, midpoint or any point of the partition. The definition of a definite integral can then be given as a limit to an infinite Riemann Sum, the exact area of the plane region. The Fundamental Theorem of Calculus is developed along with the Mean Value Theorem for Integrals leading to the Average Value of a function on an interval. The Second Fundamental Theorem is also given. The main applications here are areas of simple plane regions, accumulation problems, and average-value-of-a-function problems. More integration techniques are developed through pattern recognition, and substitution. This unit also includes estimation of plane regions by using trapezoidal approximation.

Chapter 4: Integration Block Days Section Topic Concepts Suggested Approaches (T) Technology, (V) Verbal, (G) Graphical, (N) Numerical 3 4.1 Antiderivative, Indefinite initial condition, general (V) p.252/ 89-94 Integration solution, differential Computing Indefinite Integrals equation Integration - Advanced Placement 1 4.2 Area Upper sum, lower sum, area (T, N) Graphing calculator: “sum(seq” approximation commands to find approximate areas using rectangles 3 4.3 Riemann Sums and Definite Definite integral defined as (V) verbal Integral limit of Riemann Sum Accumulation Functions Approximating Integrals with Riemann Sums 4 4.4 Fundamental Theorem of Mean value Theorem, (V) verbal Calculus Average value of a function, Riemann Sums and the Fundamental Theorem of Second Fundamental Calculus Theorem of calculus, Integral Fundamental Theorem of Calculus as accumulation function 5 4.5 Integration by Substitution General power rule. u- (V) verbal substitution Exploring derivatives using lists 1 4.6 Numerical Integration (Trapezoidal rule only) (V) verbal (G) Graphical (T) Technology Numerical Integration 1 Review Listen and Line Up! AP Calculus Study Cards 1 Test 19 Total Honors Calculus (120230) and AP Calculus AB (1202310) (everyday block all year)

Total instructional days in the pacing : 84 First term days: 90 This is the end of the 1st term. Students who successfully complete this term will be given a credit for Honors Calculus. It is intended that student continue to end of the year for the completion of AP calculus AB.

6. This is a substantial unit which addresses the calculus of transcendental functions – primarily logarithmic and exponential as well as the calculus of inverse functions. The integral definition of the natural logarithm function is used. Various derivative techniques are developed for natural logarithm functions and integral techniques for rational form functions, including trigonometric. Inverse functions are addressed with particular attention given to the natural exponential function as the inverse of the natural logarithm function. Solving logarithmic and exponential equations is reviewed. The derivative and integral of the natural exponential function are given. Logarithmic and exponential functions of any base are given along with their corresponding derivatives and integrals. This then gives the foundation for more application of differential equations and the examination of slope fields. Particular attention is given to differential equations in the form “ y’=ky” ( exponential growth and decay).

( Pacing on next page.)

Honors Calculus (120230) and AP Calculus AB (1202310) (everyday block all year)

Chapter 5: Logarithmic, Exponential, and other Transcendental Functions Block days Section Topic Concepts Suggested Approaches (T) Technology, (V) Verbal, (G) Graphical, (N) Numerical 3 5.1 Natural Logarithmic Function: Definition of e (V) verbal Differentiation Investigating the Derivatives of Some Common Functions What Is the Number "e"? 2 5.2 Natural Logarithmic Function: Log rule for integration (V) verbal (G) Graphical TI: Logarithmic Differentiation Integration 2 5.3 Inverse Function Derivative of inverse functions (V) verbal Derivative Inverse 1 Review (V) verbal 1 Test (V) verbal 3 5.4 Exponential Functions: (V) verbal Differentiation and Integration Using Derive 2 5.5 Bases other than e and (V) verbal application 5 5.6 Differential Equation: Growth Exponential Law of growth and (T, G) CBL Lab (Cooling) and decay decay, separation of variable, Exploring the Exponential Function Newton’s law of cooling Newton's Law Of Cooling and The Calculus Behind It Applications of Integrals 2 SUPP. Slope fields (G) www.apcentral.com : Teacher Resource: Slope fields Using Slope Fields Introduction to Slope Fields Differential Equations and Slope Fields - Activity 7 3 5.7 Differential Equation: Separation of variables, initial (V) p. 379/ 111-114 condition, particular solutions TI: Introduction to Solving Differential Equations 1 Review 1 Test 3 5.8 Inverse Trigonometric Range restrictions of inverse trig (V) verbal (G) Graphical Functions: Derivatives functions The Trigonometric Functions and Their Inverse Func 3 5.9 Inverse Trigonometric (V) Functions: Integration The Area Function Space Exploration AP: Next Generation Spacecraft - Orion Planting & Harvesting - An Application of the Definite Integral Integral Calculus 1 Review 1 Test 34 Total Honors Calculus (120230) and AP Calculus AB (1202310) (everyday block all year) 7. This unit involves the integral applications of finding areas and volumes. The definite Integral is used to find the areas of regions between curves using all types of functions. These are areas on an interval, areas between curves including curves with more than two intersections, also incorporating change of axis. The next application is volume. These are volumes of rotation using the disk, and washer method incorporating the change of axis. Volumes of three dimensional shapes with known cross sections completes this unit. Chapter 6: Applications of Integration Block Days Section Topic Concepts Suggested Approaches (T) Technology, (V) Verbal, (G) Graphical, (N) Numerical 3 6.1 Area of a Region between two Area of an accumulator (V) verbal curves Area Under the Curve TI: Area Between the Curves—The Game 6 6.2 Volume of Solids of Revolution Disk & Washer Methods (V) verbal The Region Between Two Curves 4 6.2 Solids of Known Cross-sections Integrating area to find (V) verbal volume 2 6.3 Volume: Shell method Horizontal/Vertical axis of (V) verbal revolution Optional 6.4 Arc length and surfaces of Smooth curve, (V) verbal revolution continuously differentiable Optional 6.5 Work (V) verbal Optional 6.6 Moments, Centers of Mass, and (V) verbal Centroids Exploring Hooke's Law Optional 6.7 Fluid pressure and fluid force (V) verbal Boyle's Law 1 Review 1 Test 17 Total

Total Instructional in the pacing: 51 Second term Days: 90

The remaining time in the 2nd term should be devoted to an over all review of the subject matter with the goal of preparing for the Advanced Placement AB exam. Depending on the scheduled dates for the exam this leaves about 25 to 30 days. Teachers should pick topics of interest from calculus such as moments and centers of mass and fluid pressure from chapter six for students to do after the AP exam. III. Major Text.

Larson, Ron, Robert P. Hostettler, and Bruce H. Edwards. Calculus with Analytic Geometry. 7th ad. Boston , New York: Houghton Mifflin, 2002

IV. Supplementary Materials

College Board AP Calculus Released Exams.

Getting Started with the Calculus Tools App