calculus 1 - edison · calculus 1 3 statement of purpose calculus 1 follows the course description...
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PUBLIC SCHOOLS OF EDISON TOWNSHIP
DIVISION OF CURRICULUM AND INSTRUCTION
CALCULUS 1
Length of Course: Term
Elective/Required: Elective
School: High Schools
Student Eligibility: Grade 12
Credit Value: 5 Credits
Date Approved: 11/22/10
CALCULUS 1
TABLE OF CONTENTS
Statement of Purpose ------------------------------------------------------------------- 1
Course Objectives ----------------------------------------------------------------------- 2
Suggested Time Table ------------------------------------------------------------------ 3
Course Content ---------------------------------------------------------------------------- 4
Career Related Lessons ---------------------------------------------------------------- 14
Basic Text / References ---------------------------------------------------------------- 15
Course Requirements ------------------------------------------------------------------- 16
Essential Instructional Behavior (Draft 14) ------------------------------------------ 17
Modifications will be made to accommodate IEP mandates for classified
students.
CALCULUS 1 3
STATEMENT OF PURPOSE
Calculus 1 follows the course description of the College Entrance Examination
Board for the Level AB Advanced Placement Calculus examination. It combines
the study of elementary functions and topics in differential and integral calculus.
Students who choose to take the level AB Advanced Placement Calculus
examination and are evaluated as "extremely well-qualified" can receive up to a
full year of credit in college calculus.
Students enrolled in this course must have completed the study of trigonometry
or can be concurrently taking Trigonometry/Analytical Geometry.
This curriculum guide was revised and updated by: Lynn Harris - EHS
Bruce Ralli - JPSHS
Coordinated by: Don Jobbins, Supervisor of Mathematics
Vincent Ciraulo, Supervisor of Mathematics
CALCULUS 1 4
CALCULUS 1 6
CALCULUS 1 7
COURSE OBJECTIVES
The student will demonstrate proficiency in:
1. recognizing and using the terminology and symbols which are related to
Calculus.
2. identifying and applying the properties of functions and their graphs.
3. determining the limit of a function as a variable approaches a given value.
4. determining the derivative of a function.
5. solving problems involving applications of the derivative.
6. determining the definite integral of a function.
7. solving problems involving applications of the integral.
8. becoming familiar with the capabilities of the graphing calculator.
CALCULUS 1 7
STATEMENT OF PURPOSE
Calculus 1 follows the course description of the College Entrance Examination
Board for the Level AB Advanced Placement Calculus examination. It combines
the study of elementary functions and topics in differential and integral calculus.
Students who choose to take the level AB Advanced Placement Calculus
examination and are evaluated as “extremely well-qualified” can receive up to a
full year of credit in college calculus.
Students enrolled in this course must have completed the study of trigonometry
or can be concurrently taking Trigonometry/Analytical Geometry.
This curriculum guide was revised and updated by: Lynn Harris – EHS
Bruce Ralli – JPSHS
Coordinated by: Jessica Lewis, Supervisor of Mathematics
Vincent Ciraulo, Supervisor of Mathematics
CALCULUS 1 7
Suggested Time Schedule
UNIT # Class Periods
1 Prerequisites for Calculus -------------------------------------- 16
2 Limits and Continuity --------------------------------------------- 14
3 Derivatives --------------------------------------------------------- 25
4 Application of the Derivative ----------------------------------- 35
5a Integration ---------------------------------------------------------- 16
5b Integration ---------------------------------------------------------- 25
6 Applications of the Definite Integral ------------------------ 16
7 Calculus of Transcendental Function ------------------------ 23
8 Testing (HSPA, PSAT), Final Exam Review 10
Total = 180
*Midterm Exam Grade will be determined by selecting 4 of the 5 established
Performance Assessments:
1. Derivatives Rules
2. Implicit Differentiation
3. Related Rates Problem
4. 12 Steps of Graphing Ratl Functions
5. Optimization Problem
CALCULUS 1
5
UNIT 1: Prerequisites for Calculus
Enduring Understanding: To gain an understanding of the prerequisite for Calculus.
Mastery Objectives
Materials
Strategies
Notes To Teacher
Reference To
Standards
The student will be able to:
1. be able to use and understand the
following:
a. slope
b. equations of lines
c. absolute value
d. distance formula
e. domain and range
f. even and odd functions
g. composite functions
h. piece-wise functions
i. greatest-integer function
j. zeros of a function
k. inverse functions
2. graph basic functions with shifts,
reflections, sketches and shrinks
3. solve equations and inequalities of
functions
4. find inverses of functions
5. use basic trigonometric functions
and identities
Basic text -
Chapter 1
pp. 1-104
Graphing
calculator
Supplemental
worksheets on
the basic
graphs
1. Use graphs to stress
properties of
functions:
a. domain
b. range
c. symmetry
d. asymptotes
2. Use graphing
calculators to verify
assumptions about
graphical
transformations and
nature of functions.
3. Establish a list of
basic functions that
students should be
able to graph without
a calculator.
4. Since many students
are dabbling in trip
and calculus, the
teacher may omit the
review of trip and use
trig functions until
marking period 2.
1. Stress the meaning of slope,
as this will be the basis for the
derivative.
2. Stress how a composition of
functions is affected by
domain restrictions.
3. Stress the use of proper math
terminology, vocabulary and
symbols.
*Quiz on basic graphs
*Test at end of unit
4.1.12.A.1
4.1.12.B.1
4.1.12.B.2
4.1.12.B.3
4.2.12.A.3
4.1.12.C.1
4.3.12.B.1
4.3.12.B.2
4.3.12.B.4
4.3.12.C.1
4.5.A.1
4.5.A.2
4.5.A.3
4.5.A.4
4.5.C.1
4.5.C.2
4.5.C.3
4.5.C.6
4.5.E.1
4.5.E.2
4.5.F.3
CALCULUS 1
6
UNIT 2: LIMITS AND CONTINUITY
Enduring Understanding: To gain an understanding of limits and continuity
Mastery Objectives
Materials
Strategies
Notes To Teacher
Reference To
Standards
The student will be able to:
1. understand the concept of a limit.
2. find a limit algebraically.
3. estimate a limit graphically.
4. use limit notation correctly.
5. understand the definition and
properties of a continuous function.
6. understand and be able to test for
continunity at a point.
7. explain the Sandwich Theorem
graphically and numerically.
8. understand limits approaching
infinity.
9. find vertical and horizontal
asymptotes using limits.
Basic text -
Chapter 2
pp. 105-166
Graphing
calculator
1. Use graphical
representation to
emphasize the
difference between a
limit and the value of
a function.
2. Use a piece-wise
function to discuss
left-handed and right-
handed limits and
continuity.
3. Use practical
examples, such as
roads, to give
examples of limits
(Grove Ave. by New
Dover Rd. does not
connect to same pt. -
thus, limit does not
exist).
4. Use basic list of
functions to discuss
which are continuous
and which are not.
1. Students have difficulty with
distinguishing
LIM f(x) and f(a).
x a
2. Need to give examples of
where, if both the left and
right-hand limits are not
equal, then the limit at the
point does not exist.
3. Derive LIM SINx = 1
x o x
geometrically and using the
Sandwich Theorem.
*Quiz on graphical, Algebraic and
Trig limits
*Test on limits, Asymptotes,
continuity
4.1.12.A.1
4.1.12.A.2
4.1.12.B.1
4.1.12.B.2
4.1.12.B.4
4.1.12.C.1
4.3.12.A.2
4.3.12.A.3
4.3.12.B.1
4.3.12.B.2
4.3.12.B.4
4.3.12.C.1
4.3.12.D.1
4.3.12.D.2
4.5.A.1
4.5.A.2
4.5.A.3
4.5.A.4
4.5.B.1
4.5.B.4
4.5.C.1
4.5.C.2
4.5.C.3
4.5.C.4
4.5.C.6
4.5.D.1
4.5.D.2
4.5.D.3
4.5.D.4
4.5.E.1
4.5.E.2
4.5.F.3
4.5.F.4
CALCULUS 1
7
UNIT 3: DERIVATIVES
Enduring Understanding: To gain an understanding of derivatives.
Mastery Objectives
Materials
Strategies
Notes To Teacher
Reference To
Standards
The student will be able to:
1. understand the graphical
interpretation of the derivative and
the algebraic definition
f1(x) = LIM f(x+h) - f(x)
h o h
2. apply the definition to finding the
derivatives of functions.
3. explain the relationship between
differentiability and continuity.
4. understand and apply the following
methods for taking derivatives of
functions:
a. Power rule for positive integer
powers of x
b. Constant multiple rule
c. Sum and difference rule
d. Product rule
e. Quotient rule
f. Power rule for negative integer
powers of x
g. Power rule for fractional
exponents
5. find velocity, speed, acceleration
and other rates of change.
Basic text -
Chapter 3
pp. 167-257
Graphing
calculator
1. Use f'(x) = LIM F(x + x) - f(x)
x o x
algebraically find the
first derivative.
2. Use the graph of f(x)
to sketch f'(x).
3. Graph f(x), f'(x), f"(x)
simultaneously on the
graphing calculator.
4. Work with the chain
rule as separate
functions, within the
composition of
functions.
5. When covering
product rule, also
extend to the
"extended product
rule" for products of
more than 2
functions.
1. Review binomial expansion
(Pascal's Triangle) for
expansion of higher order
polynomials before using
definition of the derivative.
2. Make sure students
understand the quotient rule,
and do not take the derivative
of the numerator and
denominator separately.
3. Students usually have
most difficulty with the
chain rule - make sure
they can use this in the
case of parametric
equations and in a
composition of functions.
*Quiz on Basic Rules (Power,
Product, Quotient)
*MP Cumulative Quarterly
*Quiz on Choem Rule,
*Test on all derivative methods
4.1.12.B.1
4.1.12.B.2
4.1.12.B.4
4.1.12.C.1
4.2.12.A.3
4.3.12.A.2
4.3.12.A.3
4.3.12.B.1
4.3.12.B.2
4.3.12.B.4
4.3.12.C.1
4.3.12.D.1
4.3.12.D.1
4.4.12.C.4
4.5.A.1
4.5.A.2
4.5.A.3
4.5.A.4
4.5.A.4
4.5.A.5
4.5.B.1
4.5.B.2
4.5.B.4
4.5.C.1
4.5.C.2
4.5.C.3
4.5.C.4
4.5.C.6
4.5.D.1
4.5.D.2
4.5.D.3
CALCULUS 1
8
UNIT 3: DERIVATIVES - CONTINUED
Enduring Understanding: To gain an understanding of derivatives.
Mastery Objectives
Materials
Strategies
Notes To Teacher
Reference To
Standards
The student will be able to:
6. find derivatives of trig functions.
7. use the chain rule
8. use implicit differentiation to find
f’(x).
9. illustrate the difference between df
and ∆f and use df to approximate
∆f.
4.5.D.4
4.5.E.1
4.5.E.2
4.5.E.3
4.5.F.3
4.5.F.4
CALCULUS 1
9
UNIT 4: APPLICATION OF THE DERIVATIVE
Enduring Understanding: To use the concept of differentiation as it applies to real-world phenomenon and other subject areas.
Mastery Objectives
Materials
Strategies
Notes To Teacher
Reference To
Standards
The students will be able to:
1. explain how to find critical points
and extreme values of a function.
2. state and apply the Mean Value
Theorem.
3. sketch a graph of f(x) using:
a. increasing/decreasing
b. relative maximum/minimum
c. concavity
d. inflection points
e. asymptotes
4. sketch a graph of f(x) from a graph
of f’(x).
5. use calculus to solve optimization
problems.
6. sketch rational functions.
7. solve related rates of change
problems.
8. find antiderivatives.
9. solve initial value problems.
10. understand mathematical modeling.
Basic text
Chapter 4
pp. 259-346
Graphing
calculator
1. Use graphing
calculator to graph
f(x), f'(x) and f"(x) in
same viewing
windows.
2. When covering curve
sketching, use the 9-
step process:
a. find domain/range
b. all roots
c. y-intercept
d. all asymptotes
e. local max/min pts.
f. intervals
increasing/
decreasing
g. inflection points
h. intervals concave
up/down
i. extreme values
(what happens at
or - )
3. When covering
max/min word
problems, students
should link the
algebraic solution to
the graph of the
function.
1. Work with physics teacher on
initial value problems
involving position, velocity
and acceleration.
2. See "Career Applications"
section in this guide for
marginal cost/revenue and
physics applications.
*Quiz on related rates, bases
graphing
*Test on Linearization, rel rates,
graphing
*Quiz on Optimization
*MP II Quarterly Exam
4.1.12.A.1
4.1.12.B.1
4.1.12.B.2
4.1.12.B.4
4.1.12.C.1
4.2.12.A.1
4.2.12.E.2
4.3.12.A.2
4.3.12.B.1
4.3.12.B.2
4.3.12.B.3
4.3.12.C.1
4.3.12.D.1
4.3.12.D.2
4.3.12.D.3
4.5.A.1
4.5.A.2
4.5.A.3
4.5.A.4
4.5.A.5
4.5.A.6
4.5.B.1
4.5.B.2
4.5.B.3
4.5.B.4
4.5.C.1
4.5.C.2
4.5.C.3
4.5.C.4
4.5.C.6
4.5.D.1
4.5.D.2
CALCULUS 1
10
UNIT 4: APPLICATION OF THE DERIVATIVE (CONT.)
Enduring Understanding: To use the concept of differentiation as it applies to real-world phenomenon and other subject areas.
Mastery Objectives
Materials
Strategies
Notes To Teacher
Reference To
Standards
4.5.D.4
4.5.E.1
4.5.E.2
4.5.E.3
CALCULUS 1
11
UNIT 5A: INTEGRATION
Enduring Understanding: To understand the process of integration as it relates to finding the area under a curve.
Mastery Objectives
Materials
Strategies
Notes To Teacher
Reference To
Standards
The student will be able to:
1. approximate the area under a curve
of a nonnegative continuous
function by using a rectangular
approximation method (RAM)
2. express a definite integral as a limit
of a Riemann sum.
3. express the area under a curve as a
definite integral.
4. approximate the value of a definite
integral using a numerical
integration procedure as provided in
a graphing calculator.
5. establish and apply the algebraic
rules of definite integrals.
6. find the average value of a function.
7. understand and apply the Mean
Value Theorem for definite
integrals.
8. use the properties of definite
integrals.
Basic text
Chapter 5.1-
5.3
pp. 347-379
Graphing
calculator
Overhead
projector
1. Use the classical
approach of the early
Greeks to illustrate
the approximation of
area of regions
bounded by curves.
(pp. 347-349).
2. Introduce the built-in
numerical integrator
function for the
graphing calculator to
evaluate definite
integrals.
3. Relate the algebraic
rules for definite
integrals to the
concept of area under
a curve.
4. Illustrate the definition
of the average value
of a function by
relating it to the
familiar idea of a finite
set of numbers (p.
375).
1. The concept of finding the
area under a curve is the
underlying principle of
integration as it pertains to
the limit of Riemann Sums.
2. For small sub-intervals,
student should be able to
calculate the approximate
area under a curve by hand.
3. Emphasize that the definite
integral will represent the area
between the curve and the
horizontal axis only for
functions that are completely
positive-valued on the interval
in question. In the case that
the function is negative-
valued the definite integral will
have a negative value and is
negated to represent area.
Consequently, for functions
that are both positive-and
negative-valued on an
interval, the definite integral
will yield the “net-area”, which
is a signed number.
4.1.12.A.1
4.1.12.B.1
4.1.12.C.1
4.2.12.A.1
4.2.12.A.3
4.2.12.A.4
4.2.12.B.4
4.2.12.C.1
4.2.12.D.1
4.2.12.E.2
4.3.12.A.1
4.3.12.A.2
4.3.12.B.1
4.3.12.B.2
4.3.12.B.4
4.3.12.D.1
4.3.12.D.2
4.3.12.A.2
4.3.12.B.1
4.3.12.B.2
4.3.12.B.4
4.3.12.D.1
4.3.12.D.2
4.5.A.1
4.5.A.2
4.5.A.3
4.5.A.4
4.5.A.5
4.5.A.6
4.5.B.1
4.5.B.2
4.5.B.4
CALCULUS 1
12
UNIT 5A: INTEGRATION - Continued
Enduring Understanding: To understand the process of integration as it relates to finding the area under a curve.
Mastery Objectives
Materials
Strategies
Notes To Teacher
Reference To
Standards
4. The MVT can be illustrated
(for positive-valued functions)
by relating the area under the
curve to that of a rectangle
with a base equal to the
length of the interval and a
height equal to the average
value of the function (9. 375)
*Quiz on Basic Integrals
Initial Value Problem
*Test on 1st half of chapter
4.5.C.1
4.5.C.2
4.5.C.3
4.5.C.4
4.5.C.6
4.5.D.1
4.5.D.2
4.5.D.3
4.5.D.4
4.5.D.5
4.5.E.1
4.5.E.2
4.5.E.3
4.5.F.4
CALCULUS 1
13
UNIT 5B: INTEGRATION
Enduring Understanding: To establish the connection between differential and integral calculus.
Mastery Objectives
Materials
Strategies
Notes To Teacher
Reference To
Standards
The student will be able to:
1. understand the concept of defining
functions using integrals.
2. establish and apply the
Fundamental Theorem of Integral
Calculus (both parts).
3. compute definite integrals using the
Fundamental Theorem of Integral
Calculus.
4. apply the formulas and algebraic
rule for indefinite integrals.
5. use indefinite integrals to solve
initial value problems.
6. find antiderivatives and evaluate
definite integrals using the
substitution method.
7. approximate the value of a definite
integral using numerical methods:
a. Trapezoidal Rule
b. Simpson’s Rule
Basic Text
Chapter 5.4-
5.7
pp. 380-432
Graphing
calculator
Overhead
projector
1. Use the graphing
calculator to illustrate
the first part of the
Fundamental
Theorem (as in
Examples and
Exploration 2, p. 383).
1. Be sure to emphasize the
significance of the constant of
integration and the family of
solutions.
2. When teaching integration by
substitution of variables, be
sure that students re-
substitute (or with the definite
integral, be sure that students
change the bounds).
3. A derivation of the
Trapezoidal Rule often helps
students remember and
understand the formula.
*Quiz on FTIL, Area Under Curve,
Properties
*Test on 2nd
half of chapter
*MP III Quarterly Exam
*See
Standards for
5A above –
same for 5B
3
CALCULUS 1
14
UNIT 6: APPLICATIONS OF DEFINITE INTEGRALS
Enduring Understanding: To model a given physical situation using a definite integral.
Mastery Objectives
Materials
Strategies
Notes To Teacher
Reference To
Standards
The student will be able to:
1. find the area between two curves.
2. find the volume of a solid with
known cross sections (“slicing”
method).
3. find the volume of a solid of
revolution using:
a. disk method
b. washer method
c. shell method (optional)
4. find the length of a curve.
(The following topics are optional)
5. find the surface area of a solid.
6. model other physical, social, or
economic situations by adaptation
of the knowledge and techniques
utilized in similar problems.
Basic text
Chapter 6
pp. 433-513
Graphing
calculator
Overhead
projector
1. Emphasize the basic
concepts behind
these integral
applications.
2. Many interesting
problems cannot be
integrated without the
use of a graphing
calculator. The setup
of the integral should
be emphasized.
3. Do not encourage
students to memorize
formulas, but instead
to be able to model to
the situation.
1. Sections 6.4 – 6.9 are
optional.
*Quiz on area, volumes of
rotation
*Test on Dist/Disp, Area, Volume,
Length
4.1.12.B.1
4.1.12.B.2
4.1.12.B.4
4.1.12.C.1
4.2.12.A.1
4.2.12.B.1
4.2.12.B.2
4.2.12.E.2
4.3.12.A.3
4.3.12.B.3
4.3.12.C.1
4.3.12.D.1
4.5.A.1
4.5.A.2
4.5.A.3
4.5.A.4
4.5.A.5
4.5.C.1
4.5.C.3
4.5.C.4
4.5.C.6
4.5.D.1
4.5.D.2
4.5.E.1
4.5.E.2
4.5.E.3
4.5.E.4
CALCULUS 1
15
UNIT 7: CALCULUS OF TRANSCENDENTAL FUNCTIONS
Enduring Understanding: To be able to find derivatives and integrals of
Mastery Objectives
Materials
Strategies
Notes To Teacher
Reference To
Standards
The student will be able to:
1. graph the following:
a. y = lnx
b. y = ex
2. find the derivatives and
antiderivatives of :
a. lnu
b. eu
c. au
d. logau
e. inverse trig functions
3. use log differentiation.
4. solve law of exponential change
problems.
5. apply L’Hôpital’s Rule.
6. use integration by parts to integrate.
Basic text
Chapter 7
pp. 519-595
Chapter 8.1-
8.2
pp. 613-627
1. Review the properties
of logs and natural
logs with the students
before doing log
differentiations.
2. Show the “support
box” on p. 536.
“Show graphically that
ex and its constant
multiples are their
own derivatives.”
1. This unit can be broken down
to 2 parts:
a. All derivatives
b. All integrals
2. Stress to students that when
using L’Hôpital’s Rule, the
derivatives of the numerator
and denominator are taken
separately. Do not use the
quotient rule.
*Quiz on properties on logs,
inverse
*Test on Deriv/Integ of Logs, Exp,
Inv Trig Exponential
Growth/Decay
*Quiz on Integ by Parts, L’Hopital
Rule
*Mini-Final Exam
4.1.12.A.1
4.1.12.B.1
4.1.12.B.2
4.1.12.B.4
4.2.12.E.1
4.2.12.E.2
4.3.12.A.2
4.3.12.B.1
4.3.12.B.2
4.3.12.B.4
4.3.12.C.1
4.3.12.D.1
4.3.12.D.2
4.3.12.D.3
4.5.A.1
4.5.A.2
4.5.A.3
4.5.A.4
4.5.A.6
4.5.B.1
4.5.B.2
4.5.B.4
4.5.C.1
4.5.C.3
4.5.C.4
4.5.C.5
4.5.C.6
4.5.D.1
4.5.D.2
4.5.D.6
4.5.E.1
CALCULUS 1
16
UNIT 7: CALCULUS OF TRANSCENDENTAL FUNCTIONS (CONT.)
Enduring Understanding: To be able to find derivatives and integrals of
Mastery Objectives
Materials
Strategies
Notes To Teacher
Reference To
Standards
4.5.E.2
4.5.E.3
4.5.F.3
4.5.F.4
CALCULUS 1
17
CAREER LESSONS FOR CALCULUS 1
LESSON #1
Objective: To gain an understanding of how Calculus concepts such as rates of
change and derivatives are used in economics.
Procedure: Investigate problems in economics such as the marginal cost of
production and marginal revenue from sales. Also, show how profits are
maximized and costs of inventory are minimized.
(Basic Text: pp. 206-208, 302-307)
LESSON #2
Objective: To gain an understanding of how Calculus concepts such as related rates
of change, differentials, and linearization of functions is used in the
medical profession.
Procedure: Investigate problems in medicine concerning the clogging of arteries,
measuring cardiac output, the best branching angle for arterial flow.
(Basic Text: pp. 247-251, 323-326, 425)
LESSON #3
Objective: To gain an understanding of how Calculus concepts of solutions of
differential equations (both exact and numerical approximations) are used
in electrical/electronic engineering).
Procedure: Investigate the flow of current in basic circuits, more complicated RL-
circuits (both open and closed) and the discharge of capcitors.
(Basic Text and Thomas/Finney, 8th
Edition: pp. 438, 1057-1061)
LESSON #4
Objective: To gain understanding of how techniques of differential and integral
Calculus is used in the chemical and biological professions.
Procedure: Investigate problems such as the rate of chemical reactions and control of
them, Delesse’s Rule, transfer of heat, growth and decay curves,
logistical growth.
(Basic Text: pp. 295, 511, 563 and 647)
CALCULUS 1
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BASIC TEXTS/REFERENCES/RESOURCES
BASIC TEXTS:
Advanced Placement Course Description, Calculus, College Entrance Examination
Board, New York, New York, 1997.
Finney, Ross L., Thomas, George R., Demana, Franklin D., Wairs, Bert K., Calculus -
Graphical, Numerical, Algebraic, Addison-Wesley Pub. Co., Reading, MA, 1995.
REFERENCES:
Thomas, George B., Finney, Ross L. Calculus and Analytic Geometry, 7th Edition,
Addison-Wesley Pub., Co., Reading, Mass. 1989.
Anton, Howard. Calculus with Analytic Geometry, 2nd
Edition, John Wiley and Sons,
1984.
Larson and Hostetler, Calculus with Analytic Geometry, 3rd Edition, D.C. Heath and
Company, 1986.
Thomas, George B and Finney, Ross L. Calculus and Analytical Geometry, Eighth
Edition, Addison-Wesley Pub. Co., Reading, MA, 1992.
Foerster, Paul A., Calculus - Concepts and Applications, Key Curriculum Press,
Berkeley, CA, 1998.
ADDITIONAL RESOURCES:
"Calc Master" Studyware Computer Program for MacIntosh
Finney, Ross L.; Thomas, George B; Demana, Franklin D.; Waits, Bert K.
a. Teachers' Guide, Calculus - Graphical, Numerical, Algebraic, 1995.
b. Quizzes - Calculus - Graphical, Numerical, Algebraic, 1995.
c. Technology Resources Manual for Calculus TI Graphing Calculators, 1995.
Addison-Wesley Pub. Co, Reading, MA.
T1-83 Graphing Calculator Guidebook, Texas Instruments Incorporated, 1996.
CD-Rom, Multimedia Calculus I, Pro One Software, Lax Cruces, NM, 1996. (Windows
95, Window 3.1 or 3.11)
CALCULUS 1
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PUBLIC SCHOOLS OF EDISON TOWNSHIP
DIVISION OF CURRICULUM AND INSTRUCTION
COURSE REQUIREMENTS
CALCULUS
Grades: 12 Length of Course: Term
I. Course Content - This course will consist of the following units of study:
A. Elements Functions: algebraic, trigonometric, exponential and logarithmic
functions, properties, limits, continuity, theorems, graphs
B. Differential Calculus: derivatives of functions, implicit differentiation, Mean
Value Theorem, applications, curve sketching approximations, extreme value
problems, logarithmic differentiation, velocity, acceleration, rates of change
C. Integral Calculus: antiderivative, applications, integration, definite integral,
approximations, properties, theorems, mean value, area, volume
(Additionally, career-related topics and information will be presented/reviewed.)
II. Course Requirements - To complete this course successfully, students will be
required to demonstrate a satisfactory (or higher) level of proficiency in:
A. recognizing and using terminology and symbols which relate to Calculus.
B. identifying and applying properties of functions.
C. determining the limit of a function as a variable approaches a given value.
D. determining the derivative of a function.
E. solving problems involving applications of the derivative.
F. determining the definite integral of a function.
G. solving problems involving applications of the integral.
III. Evaluation Process - Throughout the length of this course, students will be
evaluated on the basis of: A. Test/quizzes
B. Homework assignments
C. Class participation
D. A notebook
Note: Both midterm and final examinations will be administered. The
midterm grade is composed of 4 performance assessment tasks.
9/99, 8/05
CALCULUS 1
27
CALCULUS 1 28
Public Schools of Edison Township
Divisions of Curriculum and Instruction
Draft 14
Essential Instructional Behaviors
Edison’s Essential Instructional Behaviors are a collaboratively developed statement of effective
teaching from pre-school through Grade 12. This statement of instructional expectations is
intended as a framework and overall guide for teachers, supervisors, and administrators; its use as
an observation checklist is inappropriate.
1. Planning which Sets the Stage for Learning and Assessment
Does the planning show evidence of:
a. units and lessons directly related to learner needs, the written curriculum, the New Jersey Core Content Curriculum Standards (NJCCCS), and the Cumulative Progress Indicators (CPI)?
b. measurable objectives that are based on diagnosis of learner needs and readiness levels and reflective of the written curriculum, the NJCCCS, and the CPI?
c. lesson design sequenced to make meaningful connections to overarching concepts and essential questions? d. provision for effective use of available materials, technology and outside resources? e. accurate knowledge of subject matter? f. multiple means of formative and summative assessment, including performance assessment, that are
authentic in nature and realistically measure learner understanding? g. differentiation of instructional content, processes and/or products reflecting differences in learner interests,
readiness levels, and learning styles? h. provision for classroom furniture and physical resources to be arranged in a way that supports student
interaction, lesson objectives, and learning activities?
2. Observed Learner Behavior that Leads to Student Achievement
Does the lesson show evidence of:
a. learners actively engaged throughout the lesson in on-task learning activities? b. learners engaged in authentic learning activities that support reading such as read alouds, guided reading,
and independent reading utilizing active reading strategies to deepen comprehension (for example inferencing, predicting, analyzing, and critiquing)?
c. learners engaged in authentic learning activities that promote writing such as journals, learning logs, creative pieces, letters, charts, notes, graphic organizers and research reports that connect to and extend learning in the content area?
d. learners engaged in authentic learning activities that promote listening, speaking, viewing skills and strategies to understand and interpret audio and visual media?
e. learners engaged in a variety of grouping strategies including individual conferences with the teacher, learning partners, cooperative learning structures, and whole-class discussion?
f. learners actively processing the lesson content through closure activities throughout the lesson? g. learners connecting lesson content to their prior knowledge, interests, and personal lives? h. learners demonstrating increasingly complex levels of understanding as evidenced through their growing
perspective, empathy, and self-knowledge as they relate to the academic content? i. learners developing their own voice and increasing independence and responsibility for their learning? j. learners receiving appropriate modifications and accommodations to support their learning?
CALCULUS 1 29
3. Reflective Teaching which Informs Instruction and Lesson Design
Does the instruction show evidence of: a. differentiation to meet the needs of all learners, including those with Individualized Education Plans? b. modification of content, strategies, materials and assessment based on the interest and immediate needs of
students during the lesson? c. formative assessment of the learning before, during, and after the lesson, to provide timely feedback to
learners and adjust instruction accordingly? d. the use of formative assessment by both teacher and student to make decisions about what actions to take to
promote further learning? e. use of strategies for concept building including inductive learning, discovery-learning and inquiry activities? f. use of prior knowledge to build background information through such strategies as anticipatory set,
K-W-L, and prediction brainstorms? g. deliberate teacher modeling of effective thinking and learning strategies during the lesson? h. understanding of current research on how the brain takes in and processes information and how that
information can be used to enhance instruction? i. awareness of the preferred informational processing strategies of learners who are technologically
sophisticated and the use of appropriate strategies to engage them and assist their learning? j. activities that address the visual, auditory, and kinesthetic learning modalities of learners? k. use of questioning strategies that promote discussion, problem solving, and higher levels of thinking? l. use of graphic organizers and hands-on manipulatives? m. creation of an environment which is learner-centered, content rich, and reflective of learner efforts in which
children feel free to take risks and learn by trial and error? n. development of a climate of mutual respect in the classroom, one that is considerate of and addresses
differences in culture, race, gender, and readiness levels? o. transmission of proactive rules and routines which students have internalized and effective use of
relationship-preserving desists when students break rules or fail to follow procedures?
4. Responsibilities and Characteristics which Help Define the Profession
Does the teacher show evidence of: a. continuing the pursuit of knowledge of subject matter and current research on effective practices in teaching
and learning, particularly as they tie into changes in culture and technology? b. maintaining accurate records and completing forms/reports in a timely manner? c. communicating with parents about their child’s progress and the instructional process? d. treating learners with care, fairness, and respect? e. working collaboratively and cooperatively with colleagues and other school personnel? f. presenting a professional demeanor?
MQ/jlm
7/2009