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AP Calculus AB

Syllabus

Pre-Requisites

:

Students intending to take AP Calculus AB are expected to have

successfully completely 3 years of HS math and Pre-calculus either at their

home school or as a summer class at a local community college.

At their

home school, Pre-Calculus students will study families of functions with

emphasis on their behavior and modeling real world applications.

Students

will have used CBL’s in a lab setting as an introduction to certain functions

and will have fit curves to the data collected.

Written responses and

explanation are required for many of the activities and assessments

throughout the year.

Course Overview

:

The students begin their study of Calculus before the end of their

Accelerated Pre-Calculus course.

They are initially exposed to the concepts

of derivatives, integral, and limits from intuitive, graphical, and numerical

perspectives.

They continue on with Delta-epsilon definition of limit with

applications, and limits algebraically and graphically.

When they return in

September, we spend approximately one and a half weeks reviewing these

concepts, and learning the Intermediate and Extreme value theorems.

The

remainder of the year is devoted to exploring the concepts of AP Calculus

AB in more depth.

Students are encouraged to work in groups and to form

study groups outside of the school day.

Graphing Calculators

:

Each student either owns or is provided the use of a Graphing Calculator

from the TI-83 or 84 families of calculators.

By the time they enroll in

calculus, the students have been using their calculators for at least a year.

Their Pre-Calculus course refines and expands on their algebraic technique

and builds their skills in using the graphing calculator for graphing functions

and solving equations using multiple methods (intersect, zeros, solver).

These skills are reviewed throughout the AP Calculus course.

In addition,

students will be able to find numerical derivatives at a specified point, the

definite integral over a specified domain, and the equation of a tangent line

with their calculators.

Course Outline:

May to June of Accelerated Pre-Calculus

Unit 1:

Introduction to Calculus

•

Average rate of change to approximate instantaneous rate of change –

numerically and graphically from functions, graphs, and tables.

•

Definite Integral as area – counting squares on a graph and Trapezoid

Rule on functions and tables.

•

Limits of a function from graphs and tables

Unit 2:

Limits

•

Limits numerically

•

Limits graphically

•

Limits algebraically

•

Delta/Epsilon definition of limits

•

Limits involving infinity.

September to May of AP Calculus AB

Unit 3:

•

Review of topics from units 1 and 2

•

Definition of continuity

( )

( )

c

f

x

f

c

x

=

→

lim

•

Intermediate and Extreme Value Theorems

Unit 4:

•

Difference quotients- forward, backward, and symmetric

•

Numerical and Algebraic definitions of a derivative (limit of a

difference quotient emphasis on derivative as a slope)

•

Analysis of functions and their derivatives graphically (preliminary

comparison of behaviors of one to predict behavior of the other)

•

Development of power rule using algebraic definition of derivative

(limit of difference quotient, formalization of notation such as

y

′

and

(

)

x

f

′

)

•

Introduction of velocity and acceleration as

1

st

and 2

nd

derivatives of a

position function

Unit 4 (cont.)

•

Equations of tangent and normal lines algebraically and

graphically

Unit 5:

•

Chain rule to find derivative of composition of functions

•

Derivatives of Sine and Cosine

•

Squeeze Theorem and proof of

( )

1

sin

lim

0

=

→

x

x

x

•

Applications for sine and cosine and derivatives

•

Introduction of anti-derivatives and u-substitution

•

Introduction of additional derivative notation (

2

2

,

dx

y

d

y

dx

dy

y

=

′

′

=

′

)

Unit 6:

•

Proof of product rule

•

Quotient rule (without proof)

•

Development of derivatives of secant, tangent, cosecant, and

cotangent

Unit 7:

•

Inverse trigonometric functions and their derivatives

•

Differentiability, continuity, and limits

•

Implicit differentiation

Unit 8:

•

Review of anti-derivatives

•

Linear approximation

•

Indefinite Integral-formal definition

•

Riemann sums-lower, upper, left, right, and midpoint

•

Mean Value Theorem

•

Rolle’s Theorem

Unit 9:

•

Proof of 1

st

Fundamental Theorem of Calculus

•

Definite integrals and their properties

•

Application of definite integral

(as accumulation function)

•

Integrals using u-substitution (including change of bounds)

•

Simpson’s Rule

Unit 10:

•

Review of exponential modeling

•

Derivatives and integrals of natural logarithms

•

2

nd

Fundamental Theorem of Calculus

•

Derivatives of exponential functions and logarithmic

differentiation

Two week review using selected MC and FRQ questions from released

exams followed by a cumulative midterm exam.

Exam is modeled on the

format of the AP Calculus AB exam.

Unit 11:

•

Derivatives and Integrals involving

x

e

•

L’Hospital’s rule

Unit 12:

•

Separable differential equations

•

Applications involving differential equations

•

Slope fields

•

Euler’s method for solving non-separable differential equations

•

Predator/Prey models

Unit 13:

•

Analysis of functions and 1

st

and 2

nd

derivatives in order to graph a

function including review of behavior near vertical asymptotes,

behavior of functions as x approaches

∞

±

, and monotonicity.

•

1

st

derivative test and critical points.

•

2

nd

derivative test, points of inflection, and concavity

•

Applications involving optimization

Unit 14:

•

Area of plane regions

•

Volume of solids by disk/washer method

•

Volume of solids by cylindrical shells method

•

Volume of solids

of known cross sections

Unit 15:

•

Displacement, distance, velocity, and acceleration

•

Average value function

•

Related rates problems

Remaining time is used to review for the AP Calculus AB exam.

Student Activities:

Exploration

s:

Supplementary resource with primary textbook.

Students

complete selected calculator-active explorations introducing topics and

develop an intuitive sense of the topic prior to formal presentation.

Writing

:

In addition to writing explanations supporting their work, students are

required to keep a journal during the year.

Entries include responses to

prompts in the textbook allowing them to explain their understanding of the

current topics, reaction to math articles both print and internet based, and

responses to readings about calculus and the people involved in its

development.

Review activity

:

PBS Mathline “Bottles and Divers” activity

(

http://www.pbs.org/mathline

)

Students work in small groups on the Bungee Jumper Problem and the Dive

Problem.

Students review rates of change and use average rates of change to

approximate the instantaneous rate of change.

Students review the

techniques used to find the rates of change from tables, graphs and

functions.

They review methods to simplify calculations using their

graphing calculator.

Groups share their results and generalize their findings

as a class.

Trig Derivatives Activity

:

Students work in small groups to derive a formula for the derivative of

secant, tangent, cosecant, and cotangent using their knowledge of derivatives

(power rule, product and quotient rules, derivatives of sine and cosine) and

trig identities.

Each group finds one of the derivatives and the groups

present their results to the class on the whiteboard or chart paper.

Slope Field Activity

:

Students, working in small groups, are assigned a differential equation and

selected sample points.

They calculate the value of

dx

dy

at each point and plot

the slope field on grid chart paper.

After all groups have completed their

graph, the class makes generalizations about the slope fields.

The class will

then make predictions about the solution curves for the differential equations

based on the slope fields.

Volumes of Solids of Revolution

:

Classic Coke Bottle Activity

(disk method):

Students, working in groups of two or three, measure the circumference of a

classic coke bottle at evenly spaced intervals.

These values are entered into

the statistics editor of their calculators.

Other lists are used to generate the

corresponding radii and areas of the circles.

The students use their

knowledge of integrals to calculate the volume of the soda in the bottles.

The groups compare their volumes and the actual volume of the soda given

on the bottle.

They discuss the reasons for the differences between their

answers and the actual volume.

The students also create scatter plots of the

data and find a regression model for the bottle.

Calculus Cake Day

(washer and cylindrical shells methods):

Students calculate the volume of a bundt cake and an angel food cake using

equations modeling the cross section of the cake rotated about the y and x-

axes, respectively.

(The equations were found using regression techniques

on the pans the cakes were baked in.)

After the calculations are complete,

the students eat the models.

Student Assessment

:

Students grades are determined using a combination of quizzes, tests, daily

homework, spiral review assignments (Do These Quickly exercises in the

primary textbook), Graded Homework (prior to each test), and Journals.

Quizzes and tests are drawn from or model released AP exam questions.

MC questions are modified to short answer questions in the early part of the

year.

Graded homework in the latter part of the course consist of released

FRQ questions related to the current topic.

Primary Textbook:

Foerster, Paul A.,

CALCULUS: Concepts and Applications,

Key

Curriculum Press, Berkley, CA, 1998

Supplementary Textbooks:

Hughes-Hallett, et al, CALCULUS, John Wiley & Sons, Inc., New York,

1994

Anton, Howard, Calculus with Analytic Geometry, 5

th

Edition (Brief), John

Wiley & Sons, New York, 1995

Additional resources:

Print:

Explorations by Paul Foerster

Internet:

PBS Mathline: Bottles and Divers Activity

Hadley Math Page

AP Central

The Calculus Page

Surfin’ Sinefeld

Calculus on the Web

Calculus-Help.com

Whiteboard movies

Powerpoint presentations by Greg Kelly

The MacTutor History of Mathematics archive

PLUS magazine