Mathematics 201-NYA-05
Calculus I
John Abbott College Science Program
Page 1
OBJECTIVES
STANDARDS
Statement of the Competency:
General Performance Criteria
Calculus I
Discipline:
Mathematics
Semester:
Fall 2008
Course Code: 201-NYA-05
Objectives: 00UN , 00UU
Ponderation: 3-2-3
Credits:
2.67
Prerequisite: Secondary V Mathematics 536 (or equivalent)
Your Teacher will give his/her Schedule and Availability
Students are strongly advised to seek help promptly from their instructor
if they encounter difficulties in the course.
To apply the methods of differential
calculus to the study of functions and
problem solving (00UN).
Appropriate use of concepts
Appropriate use of terminology
Representation of a situation as a function
Accurate graphical representation of a function
Correct choice and application of differential
techniques
Use of algebraic operations in conformity with
rules
Accuracy of calculations
Correct interpretation of results
Explanation of steps in problem-resolution
procedure
Elements of the Competency:
Specific Performance Criteria
Introduction:
Calculus I is the first of the required mathematics courses in the Science
Program. It is usually taken in the first semester. Calculus I introduces the
student to the limit process that is so vital to the development of differential
calculus. Since differential calculus is a basic tool in Physics, some of the
applications will be related to problems in Physics. To a lesser extent,
differential calculus can be applied to problems in Chemistry and Biology.
The primary purpose of the course is the attainment of objective 00UN ("To
apply the method of differential calculus to the study of functions and problem
solving"). To achieve this goal, this course must help the student understand
the following basic concepts: limits, continuity and derivatives involving realvalued functions (algebraic, trigonometric, exponential and logarithmic) of a
single variable.
Emphasis will be placed on clarity and rigour in reasoning and in the
application of methods. The student will learn to interpret the derivative both
as a mathematical tool and as a rate of change. The derivative will be used in
various contexts that include velocity, acceleration, curve sketching,
optimization and related rates. The basic concepts are illustrated by applying
them to various problems where their application helps arrive at a solution. In
this way the course encourages the student to apply learning acquired in one
context to problems arising in another. Towards the end of the course, the
student will be introduced to antiderivatives in order to help with the transition
from Calculus I to Calculus II.
Students will be encouraged to use a scientific graphing calculator; however,
only scientific non-graphical calculators will be permitted on tests and the final
exam. Students will also have access to the Mathematics Lab where suitable
mathematical software programs including MAPLE 6 are available for student
use. The course uses a standard college level Calculus textbook, chosen by
the Calculus I and Calculus II course committees.
1. To recognize and describe the
characteristics of a function expressed in
symbolic or graphic form.
2. To determine whether a function has a
limit. To determine whether a function is
continuous at a point or on an interval. To
determine whether a function is
differentiable at a point or on an interval.
3. To apply the rules and techniques of
differentiation.
4. To use the derivative and related
concepts to analyze the variations of a
function and to be able to graph it.
5. To solve optimization and rate-of-change
problems.
6. To apply basic rules and techniques of
integration.
Specific performance criteria for each of these
elements of the competency are shown below
with the corresponding intermediate learning
objectives. For the items in the list of learning
objectives it is understood that each is preceded
by:
"The student is expected to…".
Mathematics 201-NYA-05
Calculus I
Page 2
1.3 Graphing of functions
STANDARDS
OBJECTIVES
Specific Performance Criteria
Intermediate Learning Objectives
1. Functions
1.1 Recognition of functions
1.3.2 Graph piecewise defined functions whose
pieces are made up of the functions listed in
1.1.2.
1.3.3 Apply vertical and horizontal shifts and
reflections about the horizontal and vertical
axes, and any combination of these to the
functions listed in 1.1.2.
1.1.1 Decide whether a given relation is a function
from its graphical representation.
1.1.2 Recognize and name the following functions
from their symbolic representations:
1.4 Operations on functions
f ( x)  c
constant function
f ( x)  ax  b
f ( x)  ax  bx  c
2
f ( x)  x
1.3.1 Graph the functions listed in 1.1.2.
1.4.1 Perform addition, subtraction, multiplication,
division and composition of functions.
1.4.2 Divide two polynomial functions and express
the answer in the form
linear function
quadratic function
p( x)
r ( x)
.
 q ( x) 
d ( x)
d ( x)
absolute value function
f ( x)  x
x
f ( x)  a
f ( x)  log a x
f ( x)  sin x
f ( x)  cos x
f ( x)  tan x
f ( x)  csc x
f ( x)  sec x
f ( x)  cot x
square root function
1.4.3 Find the value of a function at a point in its
domain.
exponential function
1.4.4 Evaluate
sine function
quotient) for linear, quadratic and simple
rational functions.
1.5.1 Given an applied problem, decide which
function best represents the situation and
express the relationship using appropriate
notation.
cosine function
tangent function
1.5 Appropriate use of functions to
represent given situations
cosecant function
secant function
2.
Limits, Continuity & Derivatives
2.1 Determination of Limits
2.1.1
cotangent function
2.1.2
1.1.3 Recognize and name the following
function from its symbolic representation:
f ( x)  n x
2.1.3
nth root function
2.1.4
1.1.4 Recognize and name the functions listed in
1.1.2, from their graphical representations.
1.2 Finding domain, range and
intercepts
f ( x  h)  f ( x )
(the difference
h
logarithmic function
2.1.5
2.1.6
2.1.7
1.2.1 Find and state the domain of functions listed
in 1.1.2 from both their graphical and their
symbolic representations
1.2.2 Find and state the range of functions listed in
1.1.2 and 1.1.3 from both their graphical and
their symbolic representations.
1.2.3 Find and state the x and y intercepts, if they
exist, of functions listed in 1.1.2 from both
their graphical and their symbolic
representations.
2.2 Determination of whether a
function is continuous at a point or on
an interval
2.2.1
2.2.2
2.2.3
Give an intuitive description of the limit
of a function at a point.
Evaluate a limit of a function by viewing
the graph of the function.
Estimate a limit numerically by using
successive approximations (using a
table of values).
Evaluate a limit analytically by direct
substitution, factoring, rationalizing or
simplifying rational expressions.
Evaluate analytically limits at infinity.
Evaluate one-sided limits.
Recognize and evaluate infinite limits.
Define continuity of a function at a point;
that is, state the three conditions which
must be satisfied in order that a function
be continuous at a point.
Use the definition of continuity to
determine if a function is continuous at a
specific point.
Determine on which interval(s) a
function is continuous.
Mathematics 201-NYA-05
Calculus I
Page 3
3.2.2
2.3. Use of the limit definition of the
derivative
2.3.1
2.3.2
2.3.3
2.3.4
2.3.5
2.4 Use of the graph of a function to
2.4.1
determine whether a function is
differentiable at a point or on an
interval
3. Rules and Techniques of Differentiation
3.1 Recognition of the equivalence of
3.1.1
various derivative notations
Define the derivative of a function as
(i) the limit of a difference quotient
(ii) the slope of a tangent line and
(iii) the rate of change (in particular the
velocity function associated with a
position function).
Use the limit definition of the derivative
to determine the derivative of a
polynomial of degree 1, 2 or 3, square
root and simple rational functions.
Use the limit definition of the derivative
to determine the numerical value of the
derivative at a given point.
Use the limit definition of the derivative
to determine the slope of the tangent
line to a curve at a specific point.
Use the limit definition of the derivative
to determine the equation of the tangent
line to a curve at a specific point.
Determine if the derivative of a function
exists at a point or on an interval by
examining the graph of the function.
3.2.1
 
 
 
 
dy  d

,
f ( x) , D x y
dx  dx

3.2.3
3.2.4
Recognize when and how to use the
basic differentiation formulas:
d
d
c   0
(sin x)  cos x
dx
dx
d n
d
x  nx n 1
(cos x)   sin x
dx
dx
d x
d
e  ex
(tan x)  sec2 x
dx
dx
d
d
ln x   1
(sec x)  sec x tan x
dx
x
dx
d x
a  a x ln a
dx
d
(csc x)   csc x cot x
dx
d
d
log a x   1
(cot x)   csc 2 x
dx
x ln a
dx
 
d n
d
u  nu n 1u '
(sin u )  cos u u '
dx
dx
d u
d
e  e u u'
(cos u )   sin u u '
dx
dx
d
d
ln u   1 u '
(tan u )  sec 2 u u '
dx
u
dx
d u
a  a u ln a  u '
dx
d
(sec u )  sec u tan u u '
dx
d
log a u   1 u '
dx
u ln a
d
(csc u )   csc u cot u u '
dx
d
(cot u )   csc2 u u '
dx
Recognize different notations for the
derivative of y with respect to x.
y ' , f ' ( x),
3.2. Use of basic differentiation
formulas and rules and proof of
simple propositions
 
Recognize when and how to use the
following differentiation formulas derived
from the chain rule:
3.2.5
3.3 Determination of whether a function is
differentiable at a point or on an
interval
3.3.1
3.4 Use of differentiation rules to
perform implicit and logarithmic
differentiation
3.4.1
3.5 Evaluation and application of higher
order derivatives
3.5.1
3.5.2
3.6 Use of derivatives to find the slope of
a tangent (normal) line to a curve at a
point
3.6.1
3.4.2
3.6.2
3.6.3
Recognize when and how to use the
following rules: constant rule, power
rule, constant multiple rule, sum and
difference rule.
Recognize when and how to use the
product, quotient and chain rules.
Prove a selection of the rules in 3.2.1
using the limit definition of the derivative.
Determine whether a function is
differentiable at a specified point or on
an interval using graphical, numerical, or
analytical methods.
Recognize when and how to use implicit
differentiation.
Recognize when and how to use
logarithmic differentiation.
Find higher order derivatives.
Use higher-order derivatives to solve
position, velocity and acceleration
problems.
Use the differentiation rules listed in
3.2.1 and 3.2.2 to find the slope of the
tangent line to a curve at a point.
Use the differentiation rules listed in
3.2.1 and 3.2.2 to find the equation of
the tangent line to a curve at a point.
Use the differentiation rules listed in
3.2.1 and 3.2.2 to find the equation of
the normal line to a curve at a point.
Mathematics 201-NYA-05
4.
Calculus I
Graphing of Functions
4.1 Use of the derivative and related
concepts to analyze the
variations of a function and to
sketch a graph of the function
4.1.1
4.1.2
4.1.3
4.1.4
4.1.5
4.1.6
4.1.7
4.1.8
4.2 Demonstration of the ability to
understand abstract properties of
continuous and differentiable
functions, as illustrated by two
simple standard theorems.
4.2.1
4.2.2
4.2.3
4.2.4
5.
Find critical numbers.
Find intervals on which a function is
increasing and decreasing using the sign of
the first derivative.
Find relative and absolute extrema.
Use the first or second derivative test to
decide whether the critical points represent
relative maxima or relative minima.
Find inflection points.
Find intervals on which a function is concave
up or concave down using the sign of the
second derivative.
Use limits to find all vertical and horizontal
asymptotes.
Use 4.1.1 – 4.1.7 to graph polynomial,
rational, trigonometric, logarithmic and
exponential functions.
State the conditions necessary for Rolle’s
Theorem.
State the conclusion of Rolle’s Theorem.
State the conditions necessary for the
Mean Value Theorem.
State the conclusion of the Mean Value
Theorem.
Optimization and rate-of-change problems
5.1 Solution of optimization problems
5.1.1
Represent an optimization word problem in
functional form.
5.1.2
Determine the quantity, P , to be maximized
or minimized and identify the variables which
are involved.
Draw a diagram, if possible, to illustrate the
problem and list any other relationship(s)
between the variables.
5.1.3
P
5.1.4
Express
5.1.5
Find the derivative of the function for P
obtained in 5.1.4.
Find all the possible critical values by solving
5.1.6
5.1.7
5.1.8
Page 4
5.2 Solution of problems involving
related rates
5.2.1
5.2.2
5.2.3
5.2.4
5.2.5
5.2.6
5.2.7
6.
Integration
6.1 Evaluation of the Indefinite
Integral
6.1.1
6.1.2
6.1.3
6.1.4
6.2 Evaluation of the Definite
Integral
6.2.1
6.2.2
6.2.3
6.2.4
6.3 Solution of Simple Differential
Equations
6.3.1
Give the definition of the Indefinite Integral as
an antiderivative.
Express the basic differentiation formulas
listed in 3.2.1 as antidifferentiation formulas.
Recognize when and how to use the constant
multiple rule and the sum and difference rule
in the evaluation of integrals.
Use the antidifferentiation formulas from 6.1.2
and the rules in 6.1.3 to evaluate indefinite
integrals.
State the definition of the definite integral.
State the Fundamental Theorem of Calculus.
Find the definite integral of functions
described in 6.1.
Use the Fundamental Theorem of Calculus to
find the area of a region under a curve on a
closed interval.
Find a general solution to a differential
equation of the form
y ' , f ' ( x),
as a function of one variable.
the equation P' 0 .
Test the critical value(s) and interval
endpoints for absolute maximum or minimum.
Interpret (explain) the results found in the
optimization problem.
Represent a word problem involving
related rates in functional form.
Identify the variables and rates in the
problem.
Draw a diagram, if possible, to illustrate the
problem.
Determine the equation relating the
variables.
Differentiate the equation in 5.2.4 with
respect to time, t .
Solve the equation in 5.2.5 for the required
rate.
Interpret (explain) the results found in the
problem involving related rates.
6.3.2
dy  d

,
f ( x) , D x y
dx  dx

Find a particular solution to a differential
equation of the form y '  f ( x ) given an initial
condition y ( a )  b .
COURSE CONTENT with suggested exercises
The exercises listed below should help you practice and learn the material taught in this
course; they form a good basis for homework. Your teacher may supplement this list
during the semester. Regular work done as the course progresses should make it easier
for you to master the course.
1.1 – Functions and their Representations: 1, 2, 5-8, 23-40
1.2 – A Catalogue of Essential Functions: 15-19, 35-42, 45-48
1.3 – The Limit of a Function: 1-18
1.4 – Calculating Limits: 3-24, 29-38, 48-52
1.5 – Continuity: 3-6, 10-16, 25-33, 37-40
1.6 – Limits Involving Infinity: 1-8,10a-b,13-31, 33-34,36c
Mathematics 201-NYA-05
Calculus I
2.1 – Derivatives and Rates of Change: [Instructor’s Discretion]
2.2 – The Derivative as Function: 3-14, 17-23, 27-30, 39-41
2.3 – Basic Differentiation Formulae: 1-26, 29-41, 43-48, 52-54
2.4 – The Product and Quotient Rules: 1-30, 31a, 32a
2.5 – The Chain Rule: 7-40, 47-48, 51, 57-58
2.6 – Implicit Differentiation: 3-22,32,38,41
2.7 – Related Rates: 1-38
3.1 – Exponential Functions: 7-14
3.2 – Inverse Functions and Logarithms: 21-26, 41-54, 59-64
3.3 – Derivatives of Logarithmic Functions: 1-58, 61-66
4.1 – Maximum & Minimum Values: 1-48
4.2 – The Mean Value Theorem: [Instructor’s Discretion]
4.3 – Derivatives and the Shapes of Graphs: 1-42
4.4 – Curve Sketching: 1-36
4.5 – Optimization Problems: 1-22, 32, 34, 45-46, 48, 50
4.7 – Antiderivatives: 1-13, 15-21, 23-29, 33-36
5.1 – Areas & Distances: 1-5, 15, 16
5.2 – The Definite Integral: 1-7, 19-23, 29-41
5.3 – Evaluating Definite Integrals: 1-18, 21-30
5.4 – The Fundamental Theorem of Calculus: 3-14
REQUIRED TEXT
SINGLE VARIABLE ESSENTIAL CALCULUS: Early Transcendentals ,
by James Stewart - (Brooks/Cole) about $75.
TEACHING METHODS
This course will be 75 hours, meeting three times a week for a total of 5 hours a week.
Most teachers of this course rely mainly on the lecture method, although most also
employ at least one of the following techniques as well: question-and-answer sessions,
labs, problem-solving periods and class discussions, Generally, each class session starts
with a question period of previous topics, then new material is introduced, followed by
worked examples. No marks are deducted for absenteeism (however, see below). Failure
to keep pace with the lectures results in a cumulative inability to cope with the material,
and a failure in the course. A student will generally succeed or fail depending on how
many problems have been attempted and solved successfully. It is entirely the student's
responsibility to complete suggested homework assignments as soon as possible
following the lecture, as the material will be fresher in his/her mind. This also allows the
student the maximum benefit from any discussion of the homework (which usually occurs
in the following class). The answers to a selected number of problems can be found in the
back of the text. Each teacher will provide supplementary notes and problems as he/she
sees fit.
MATH WEBSITE: http://www.johnabbott.qc.ca
Go to Student Zone, Departments, Mathematics
MATH LAB:
The Math Lab is located in H-203 and is open from 11:00 to 16:00 for borrowing course
material or using the computers and printers for math assignments.
Page 5
MATH HELP CENTRE:
There is usually a teacher available for individual help (see posted schedule)
The Math Help Centre is located in H-203 from 9 to 11:00 and in H-222 after 11:00.
MATH TUTORING CENTRE:
Peer tutoring is available in the Math Lab. The schedule is usually posted around the 4th
week of each semester on the Math Lab bulletin board.
LEARNING CENTRE
The Learning Centre, located in H-117A , offers student skills classes and individual
tutoring.
DEPARTMENTAL ATTENDANCE POLICY
Regular attendance is expected. Missing six classes is grounds for automatic failure in
this course. Many of the failures in this course are due to students missing classes.
EVALUATION PLAN
( a) A student's Final Grade is a combination of the Class Mark and the mark on the Final
Exam. The class mark will be 75% (three to five tests) and 25% at you instructor’s
discretion (more tests, quizzes or assignments). The specifics of the Class Mark will be
given by your instructor in the last page or during the first week of classes. Every effort is
made to ensure equivalence between the various sections of the course.
The Final Exam is set by the Course Committee (which consists of all instructors currently
teaching this course), and is marked by each individual instructor.
( b) The Final Grade will be whichever is the better of:
50% Class Mark and 50% Final Exam Mark
OR
25% Class Mark and 75% Final Exam Mark
A student CHOOSING NOT TO WRITE the Final Exam will receive a failing grade of 50% or
their class mark, whichever is less.
COURSE COSTS
In addition to the cost of the text listed above, your instructor might recommend you
acquire an inexpensive scientific calculator ($10-$25). No graphing or programmable
calculators are allowed during tests or the final exam.
COLLEGE POLICY ON CHEATING AND PLAGIARISM
Cheating and plagiarism are unacceptable to John Abbott College. Students are expected
to conduct themselves accordingly and must be responsible for all their actions. For more
information on Cheating and Plagiarism, students should consult the Institutional Policy
on the Evaluation of Student Achievement (IPESA), which is reprinted in the College
Calendar or Student Agenda.
COLLEGE POLICY ON MID-SEMESTER ASSESSMENT
Students in their first and second semester have the right to feedback on basic skills in
the first weeks of the semester so that they can seek extra help if necessary.
Notice to Students on GRADE REVIEW
It is the responsibility of students to keep all assessed material for at least one month
past the grade review deadline in the event that they would want to request a grade
review.
Students can learn more about their rights and responsibilities by reading the IPESA.