AP Calculus AB

Course Overview

AP Calculus AB is designed to follow the topic outline in the AP Calculus Course Description provided by the College Board. The primary objective of this course is to provide a fundamental understanding of Differentiable and Integral Calculus. The course teaches the concepts of calculus by drawing upon graphical, algebraic, numeric and verbal representations of functions. Communication of calculus is highly stressed throughout the course. On a daily basis, students are expected to verbally communicate methods and solutions in class discussions and in small group situations. Students are also expected to explain solutions in written sentences on both homework problems and on tests.

Course Outline and Pacing Guide

The following is a guide to topics and time spent for lecture, instructional labs and group work. Time of review and assessment is also included. The course is taught in a traditional 7-period day, where class times are 51 minutes 4 days/week and 41 minutes 1 day/week. Although class time is divided among lecture and class discussion and lab and group work, over 50% of the time, students work in small groups with the teacher acting as facilitator. The pacing goal is to complete the AP curriculum with at least 20 instructional days remaining before the AP exam. This time is spent reviewing and preparing for the AP exam.

Student Evaluation

AP Calculus grades are based upon the following: Tests ~50% In-class Labs and Group work ~30% Homework ~20% Tests are divided into two sections, a Calculator and a Non-Calculator section. Each of these sections is approximately half multiple-choice and half free-response. Free response questions are multi-part, require students to explain solutions in written sentences and are graded on a rubric that ranges from 5 to 9 points. Homework is collected on a weekly basis. Students are encouraged to work out homework and in-class problems with the assistance of others, but each student must submit his/her own solutions. Many solutions require written explanations to solutions.

Technology Resources

Students must use a College Board approved graphing calculator. Students are required to use calculators to find:

Zeros of a function

Points of intersection of functions

Derivative of a function at a given point

The value of a definite integral

Accompanying Textbook

Calculus: Graphical, Numerical, Algebraic (3rd

Edition)

Finney, Demana, Waits, Kennedy

Book Correlation

Topics

~Time

Related Assignments

(Problem numbers in bold print are

categorized “Writing to Learn”

problems in the text and require a

written explanation)

Chapter 1: Prerequisites for Calculus

1.3, 1.5, 1.6 Functions Review

Functions

Domain and Range

Viewing and Interpreting Graphs

Exponential & Logarithmic

Inverses

Trigonometric

Odd and Even

Inverse Trigonometric

4 days

p. 26: 3, 6, 9, 12, 13-18, 23, 30, 39, 41-46 p. 44: 1-6, 17-20, 33-42, 52-57 p. 52: 1, 3-8, 17-22, 25-30, 39-44

Chapter 2: Limits and Continuity

2.1 (A) Rates of Change And Limit

Definition of Limit

Limit as ax

One- and Two-side limits

2 days p. 66: 13-23, 38-44 even, 45-49

2.1 (B) Squeeze Theorem

Proving 1sin

lim0

using

geometry and Squeeze Theorem

1 day p. 66: 24-28, 52, 53, 57, 58, 67-70

2.2 Limits Involving Infinity

Limits as x

End Behavior Models

2 days p. 76: 3, 6, 9-12, 33-38, 54, 55, 68

2.3 Continuity

Continuity at a Point

Continuous Functions

Intermediate Value theorem

3 days p.84: 3-36 mult. of 3, 41-44, 47, 49, 51, 56-59

2.4 Rates of Change and Tangent Lines

Average Rates of Change

Average vs. Instantaneous Rates of Change

Tangent to a Curve

Slope of a Curve – Difference Quotient

Normal to a Curve

4 days p.92: 2-10 even, 11-16, 18, 29-32, 37-40

Review and Test 2 days

Chapter 3: Derivatives

3.1 Derivative of a Function

Def. Of Derivative

Notation

Relating Graphs of f and f’

Graphing Derivative from Data

5 days p. 105: 1, 2, 6, 9, 12, 13-17, 21, 22, 23, 26, 35

3.2 Differentiability

How f’(a) Might Fail to Exist

Local Linearity

Numeric Derivative on a Calculator

Differentiability Implies Continuity

IVT for Derivatives

3 days p. 114: 2-10 even, 11-15, 31, 33, 38, 44, 45

3.3 Rules for Differentiation

Power Rule

Constant Multiple Rule

Sums and Differences

Products and Quotients

Second and Higher Order Derivatives

6 days p. 124: 3-48 mult. of 3, 55-57 Lab: Given simultaneous graphs of f, f’ and f”; identify each.

3.4 Velocity and Other Rates of Change

Instantaneous Rates of Change

Motion along a Line – Position, Velocity and Displacement

3 days p. 135: 1, 2, 5, 9-12, 19, 22, 32, 36

Review and Test 3 days

3.5 Derivatives of Trigonometric Functions

Derivatives of Sine and Cosine developed visually

Derivatives of Sine and Cosine functions Developed graphically Using Graphing Calculator’s Ability to Graph f and f’ simultaneously

Derivatives of tan, sec, cot and csc Developed Using Rules of Differentiation

4 days p. 146: 2-10 even, 23-28, 37, 46-48, 51 Lab: Given graph of a period of sine (cosine) function, use tangent line slopes to visually produce its derivative.

3.6 Chain Rule

Review of Composite Functions

Chain Rule

Repeated Use of Chain Rule

Apply Chain Rule to Tabular Functions

3days p. 153: 3-39 mult. of 3, 56, 68, 73, 74

3.7 Implicit Differentiation

Implicitly Defined Functions

Tangent and Normal Lines

Higher Order Derivatives Using Implicit Differentiation

3 days p. 162: 3-39 mult. of 3, 46, 49, 50, 55, 56 Lab: Explicit diff of conic sections compared to implicit method

3.8 Derivatives of Inverse Trig Functions

xdx

d 1sin Developed

Geometrically

xdx

d 1tan Developed

Geometrically

Justification of

)(

1)(

1

1

xffxf

Geometrically

3 days p. 170: 3-27 mult. of 3, 28-31, 40

3.9 Derivatives of Exponential and Logarithmic Functions

Derivative of xe

Derivative of ln x

Derivative of xa

Derivative of xalog

Logarithmic Differentiation

5 days 3-27 mult. of 3, 30-50 even, 51, 52, 53, 56

Review and Test 3 days

Chapter 4: Applications of Derivatives

4.1 Extreme Values of Functions

Locating Critical Points

Absolute Extreme Values

Local Extreme Values

Finding Extreme Values

3 days p. 193: 5-13, 18, 23, 24, 26, 37, 39, 40, 43-47, 50, 51

4.2 Mean Value Theorem

MVT

Physical Interpretations

Increasing and Decreasing Functions

3 days p. 202: 1-13, 20, 21, 28, 39-42, 46, 51, 52

4.3 Connecting f’ and f” with the Graph of f

First derivative Test

Second Derivative Test

Concavity

Points of Inflection

4 days p. 215: 4, 5, 9, 12, 16, 22, 23, 24, 28, 37, 39-42, 43-46, 49

Review and Test 2 days

4.4 Modeling and Optimization

Real-world Optimization problems 3 days p.226: 2, 4, 5, 7, 10, 11, 12, 16, 20, 21, 30, 36, 39, 41, 42, 43, 44, 47, 48 Lab: Optimizing fuel consumption as a function of its velocity

4.5 Linearization Linear Approximation

Estimating Change

2 days p. 242: 1-4, 11, 12, 31, 33, 45

4.6 Related rates Related Rate Problems

Solution Strategies

4 days p. 251: 1, 2, 11, 13, 14, 16, 17, 19, 21, 27, 28, 31, 33, 42

Review and Test 3 days

Chapter 5: The Definite Integral

5.1 Estimating with Finite Sums

Distance Traveled Given Velocity

Rectangular Approximation Method Given Discrete Data

Left-Hand, Right-Hand and Midpoint Sums

2 days p.270: 2, 5, 6, 15, 17, 19, 23, 30 Lab: Compare endpoint/midpoint methods to produce better estimates of area under a curve.

5.2 Definite Integrals Riemann Sums

Terminology and Notation of Integration

Definite Integral and Area

Integrals on a Calculator – through graph screen and by using num integration function

Discontinuous Integrable Functions

5 days p. 282: 8-28 even, 29-36, 41-54, 57

5.3 Definite Integrals and Antiderivatives

Properties of Definite Integrals

Average Value of a Function – Geometrically and Algebraically

MVT For Definite Integrals

Connecting Differentiable and Integral Calculus

6 days p. 290: 1, 3, 7, 8, 11-36, 39, 40, 41, 47, 48

Review and Test 2 days

5.4 Fundamental theorem of Calculus

FTC, Part I

Graphing x

a

dttf )(

FTC, Part II

Area Connection

Graphically Analyzing Antiderivatives

5 days p. 302: 3-18, 27-60 mult. of 3, 64 Lab: Volume of a right circular cone

5.5 Trapezoidal Rule Trapezoidal Approximations

Comparing Geometric Approximations

3 days p. 312: 2-12 even, 19, 20, 22, 34, 36

Review and Test 3 days

Chapter 6: Differential Equations and Mathematical Modeling

6.1 Slope Fields and Euler’s Method

Defining and Solving Differential Equations

Constructing Slope Fields by Hand

Matching Slope Fields with Differential Equations

3 days p. 327: 3-24 mult. of 3, 25-28, 32-40, 42, 45, 48-48, 49, 50, 51, 52 Lab: reading slope fields

6.2 Antidifferentiation by Substitution

Evaluating an Indefinite Integral

Producing Antiderivatives from Derivatives of Basic Functions

Antiderivatives by Substitution of Variables

Change of Limits in U-Substitution

3 days p. 337: 2-12 even, 21-34, 40, 41, 53-66, 73, 74, 77, 79

6.4 Exponential Growth and Decay

Separable Differential Equations

Solving Differential Equation for a Given Initial Condition

Exponential Growth and Decay Review

Newton’s Law of Cooling and Other Applications

4 days p. 357: 1-10, 22, 23, 24, 25, 28, 31, 32, 40-44, 51, 52

Review and Test 2 days

Chapter 7: Applications of Definite Integrals

7.1 Integral as Net Change

Velocity to find displacement

Velocity to find total distance traveled

Solve Problems involving Linear Motion, Consumption over time, work and net change from data

3 days p. 389: 3-36 multiples of 3

7.2 Areas in the Plane Area between curves

Area enclosed by intersecting curves

Boundaries of Changing Functions

Integrating with respect to y

3 days p. 395: 2-16 even, 18-45 multiples of 3

7.3 Volumes Volume of known cross-section

Disks and Washers

Cavalieri’s Theorem

3 days p. 406: 3-42 multiples of 3, 44, 52, 66-68

Review and Test 3 days