Q&A
 

MT360A  Calculus 



General Information
Facilitator:
Wayne Cook
Home: (970)2299282
Cell: (970)6909976
Email:
wcook@regis.edu or
ww_cook@msn.com
Term:
Summer 8W2 2006
Meeting Times:
Starting Monday, 3 July 2006  6:00 PM until 10:00 PM
Grading:
Grade 
Range 
Grade 
Range 
A 
93%100% 
D+ 
67%69% 
A 
90%92 % 
D 
63%66% 
B+ 
87%89 % 
D 
60%62% 
C+ 
77%79% 
F 
0%59% 
C 
73%76% 


C 
70%72% 


5% lost for every week a homework is
late (unless other arrangements are made), no homework can be turned in past last night of class.
Course Description
MT
360A. CALCULUS I (4).
Treats standard topics of single variable calculus including limits,
continuity, derivatives, applications of derivatives, and elements of
integration.
Class Schedule
Please read the
assigned chapters in each book and complete each homework assignment
before the scheduled class. We will cover key elements
of each assignment and any areas on which you might have questions.
However, we will not be able to cover all of the topics presented in the
textbooks, so please read them well so that you may have a better
understanding of the material. Historically, there has been
a significant amount of work associated with this class, so please plan
to spend time out of class preparing for each week.
Please also see
the General Notes at the bottom of this web page.
Class Date 
Learning Topic 
Sections Covered 
H/W
Assignments 
3 July 
#1 Review: Functions and Trigonometry
#2 Limits and Continuity 
Sections
1.1 – 1.4
Appendix
E
Section
2.1 
To Be Added  suggestion, go through the problems and look
for those problems which may be a challenge. Do at
least ten of these. 
10 July 
#2 Limits and Continuity
#3 Derivatives 
Sections
2.2  2.5
Sections
3.1 & 3.2 

17 July 
#4 Evaluating Derivatives 
Sections
3.3 & 3.4 

24 July 
Quiz # 1
(Topics 1  4)
#5 Chain Rule & Implicit Differentiation 
Sections
3.5 & 3.6 

31 July 
#6 Related Rates & Differentials
#7 Exponential & Logarithmic Functions
#8 Inverse Trigonometric Functions 
Sections
3.7 & 3.8
Sections
4.1 – 4.3
Section
4.4 

7 Aug 
#9 Derivatives in Analysis of Functions
#10 Applications of Derivatives
#11 Indefinite Integrals 
Section
5.1  5.2
Section
5.4  5.8
Section
6.2  6.3 

14 Aug 
Quiz # 2
(Topics 5  10)
#12 The Area Problem & Definite Integrals
Review 
Sections
6.1, 6.4,

6.6


21 Aug 
Final
Exam
(Topics 1 – 12) 
ALL 





Class Outline (taken from Regis outline for class)
General Notes
Calculus is dependent on all of the mathematics you have been
able to perform before this class. Calculus requires a slightly different
mindset that the mathematics you have had before, but if you use the logic you
have gained from the other mathematics classes, you should be able to do just
fine. Do not fight thinking in a different way, just enjoy the class and
"go with the flow." ;) You will do well.
Questions and Answers
As I receive questions, I will create a file that will contain
answers to these questions.
Homework Status
You can check your homework here (eventually).
Learning
Topic #1: Review: Functions and Trigonometry
This
topic includes a review of functions and concepts from college trigonometry.
Topic
Outcomes
After
completing this topic, students will be able to:

Find the slope of a line, given two points.

Find the slope of a line, given a linear function.

Simplify rational functions.

Given two functions, use function composition to create new functions.

Find trigonometric ratios for given right triangles.

Evaluate trigonometric ratios for angles that are not in the first quadrant.

Manipulate trigonometric functions.

Apply basic trigonometric identities.

Use
a calculator to evaluate and graph the following types of functions:

The facilitator will
show examples of evaluating limits graphically, algebraically, and
intuitively.

Polynomial functions

Rational functions

Exponential functions

Trigonometric functions
 Solve trigonometric equations.
Discussion
Questions

What are several types
of change that are important to understand?

What are the positive
and negative aspects of representing functions graphically?

What is the difference
between discrete and continuous models and when do you think this makes a
difference?

Describe
a procedure that could be used to solve a right triangle.
Suggested Classroom Activities

The facilitator and the
students will discuss the idea of a function and how it relates to its
inverse (graphically and analytically).

The class will evaluate
functions.

The class will combine
functions by addition, subtraction, multiplication, division, and
composition.

The facilitator will
demonstrate several useful features of a TI 83 Plus in graphing functions.

The class will use
calculators to analyze functions numerically and graphically.

The class will review
the six trigonometric functions.

The class will work on
applications of right triangle.

The facilitator will
derive basic trigonometric identities and their applications.

The class will prove
some basic trigonometric identities and apply the same in solving
trigonometric equations.
Suggested Enrichment Activities

Investigate
rational functions using a graphing calculator, paying particular attention
to local extreme values, asymptotes, and other discontinuities.
Learning Topic #2: Limits and
Continuity
This
topic includes the historical rationale for the concept of a limit, and explores
limits numerically, graphically and formally. This definition is then applied
to explain the concept of continuity.
Topic
Outcomes
Upon
completion of this topic, students will be able to:

Provide a definition of
a limit.

Interpret the existence
of a limit graphically.

State algebraic and
graphical examples of the existence or nonexistence of a limit.

Find limits
numerically, graphically, and algebraically.

Use the definition of
continuity to determine the intervals where a function is continuous.
Discussion Questions

Graphically, how would you recognize the points or intervals where limits do
not exist?

What algebraic techniques help us locate points of discontinuity?

Which functions are continuous everywhere?
Suggested Classroom Activities

The facilitator will
lead a discussion of the concept of a limit and some applications.

The facilitator will
show examples of evaluating limits graphically, algebraically, and
intuitively

The facilitator will
graphically illustrate the difference between continuous and discontinuous
functions

The class will analyze
the definition of continuity at a point and at an interval. The class will
then use this definition to determine continuity of some algebraic and
trigonometric functions.
Suggested Enrichment Activities
Learning Topic #3:
Derivatives
This
topic includes introduction of the derivative as the limit of a difference
quotient, statement of the formal definition of the derivative.
Topic Outcomes
Upon completion of this topic, students will be able to:

Find
the slope of the secant line for two points from the graph of a function.

Express the slope of a secant line as a difference quotient.

Find
the equation of the secant line for two points from the graph of a function.

Find
the slope of the tangent line at one point of the graph of a function.

Find
the equation of the tangent line for one point of the graph of a function.

Find
the instantaneous rate of change of a function for a given xvalue.

Determine graphically whether a function is locally linear.

State the definition of instantaneous rate of change of a function for a given
xvalue.

State the limit definition for the derivative function.

Use the limit
definition to find the derivative function for polynomials and radical
functions.

Estimate the graph of
a derivative function from a given function.
Discussion
Questions

Why is the concept
of a limit necessary to understand derivatives?

What is the
difference between instantaneous and average velocity?

What is a secant
line?

Explain the relationship between the
slope of a tangent line and derivative.
Suggested Classroom Activities

The facilitator and
the class will develop the connection between limits, slopes of lines, velocity,
and derivative.

The class will
evaluate limit of the difference quotient and relate this to the value of the
derivative.

The class will
analyze graphically analyze points where the derivative is not defined.

The facilitator will
demonstrate methods of determining points of nondifferentiability.
Suggested Enrichment Activities
Learning Topic
#4: Techniques of Differentiation
This topic focuses on the
algebraic methods of differentiation
Topic Outcomes:
Upon completion of this topic, students will be able to:

Apply the appropriate derivative formula.

Identify problems that require the use of the Chain Rule.

Apply the Chain Rule correctly.

Find the derivative of trigonometric functions.

Find the derivative of logarithmic functions.

Apply the method of logarithmic differentiation.

Find the derivative of exponential functions.

Find higher order derivatives.
Discussion Questions

When is the power
rule used?

Can quotients be
expressed as products?

What is the product
rule?

What is the quotient
rule?

How is the derivative
of the sum/difference of functions evaluated?

What is logarithmic
differentiation?

How do you separate a
composition of functions into its component functions?

How is the product
rule different (in application and form) from the chain rule?
Suggested Classroom Activities

The facilitator will
demonstrate the application of the product rule, quotient rule, and
sum/difference rule to evaluate the derivative of a function.

The facilitator will
demonstrate the application of the derivative formulas for logarithmic and
exponential functions.

The facilitator will
present an example of the application of the chain rule.

The class will
discuss how the values of the derivative will apply to velocity, acceleration,
and slope of tangent lines.

The class will work
on a set of problems involving finding derivatives.

The class will
discuss several derivative problems and determine which method is appropriate in
solving the problems.
Suggested Enrichment Activities
Learning Topic #5: More Derivatives & Applications
This topic includes evaluating the
derivative of functions expressed implicitly, evaluating derivatives of inverse
trigonometric functions, and solving problems on applications of derivatives.
Topic Outcomes
Upon completion of this topic,
students will be able to:

Derive the
formula for the derivative of inverse trigonometric
functions.

Apply the
appropriate derivative formula for inverse
trigonometric
functions together with previously established rules and formulas to
evaluate the derivative of a function.

Apply the
method of implicit differentiation to evaluate the
derivative
of a function.

Solve related
rate problems.

Determine if a
function is increasing or decreasing over an
interval.

Determine if
the graph of a function is concave up or concave
down over an interval.

Determine
relative extrema.
Discussion Questions

How is the derivative
of the inverse of a trigonometric function derived?

What information
about a function is derived from the first derivative?

What information
about a function is derived from the second derivative?

When is implicit
differentiation applied?

What are related rate
problems?

What is a relative
extremum?

Describe a procedure
that will determine the relative minimum and relative maximum of a function.
Suggested Classroom Activities

The facilitator will
explain the interpretation of rates of change as a derivative with respect to
time.

The class will solve
related rate problems.

The facilitator will
demonstrate how derivatives can be used to approximate nonlinear functions by
simpler linear functions.

The facilitator will
illustrate the process of implicit differentiation.

The class will
discuss several derivative problems and determine which method is appropriate in
solving the problems.

The class will practice using the method of implicit differentiation.

The facilitator will illustrate the process of analyzing a function by using
its first and second derivative.

The class will work on problems that will determine the relative extrema of a
function.
Suggested Enrichment Activities
·
Use CAS to verify solutions to chain rule
and implicit differentiation problems.
Learning Topic
#6: Applications of Differentiation
This topic includes
absolute extreme values of functions on a closed interval and its applications,
rectilinear motion, differential, and Newton’s method.
Topic Outcomes:
After completing this topic, the student should
be able to:

Find
the absolute extremes for a function on a closed interval graphically.

Find
the absolute extremes for a function on a closed interval algebraically.

Apply a fourstep process to set up and solve applied optimization problems.

Given a position function, produce a time vs. position graph.

Given a position function, determine a velocity function and an acceleration
function.

Given a position function, find the extreme positions for a specified time
interval.

Define the differential.

Use
differentials to estimate the value of a function near a known value.

Iterate a function.

Use
a formula to determine the intersection of a tangent line to a function with the
xaxis.

Use Newton’s method
to estimate the root of a function.

Show graphically a
situation where Newton’s method would fail.
Discussion Questions

What is a
differential?

What is error
propagation?

What are some applications of differentials?

Explain the concept behind Newton’s Method.

How is the velocity function determined from the position function?

Explain how the absolute extrema is determined.
Suggested Classroom Activities

The facilitator will illustrate the
process of finding extreme values of a function.

The class will solve a problem on finding
the maximum and minimum value of a function.

The facilitator will demonstrate the use of derivatives to solve optimization
problems.

The facilitator will
introduce the concepts of related rates and differentials.

The class will apply
the concepts above to related rate problems.

The class will apply
differentials to solving problems involving error propagation.

The class will apply
Newton’s method to solve an equation.

The class will use derivatives to solve rectilinear motion problems.
Learning Topic #7:
Antidifferentiation
This topic includes
antiderivatives and the idea of area as a limit of a summation.
Topic Outcomes:
After completing
this topic, the student should be able to:

Sketch a graphical illustration of the mean value theorem.

Find
a family of antiderivatives for a power.

Find
a family of antiderivatives for sin (x), cos (x), sec^{ 2 }(x), and
sec(x) tan (x).

Find
a family of antiderivatives for exponential (base e) function.

Solve initial value problems involving functions with simple antiderivatives.

Given a linear,
quadratic, or sinusoidal velocity function and an initial position, find a
position function.

Find antiderivatives
of functions that are constant multiples, sums, or differences of other
functions with known antiderivatives.

Represent sums using sigma notation.

Manipulate sums expressed in sigma notation algebraically.

Evaluate sums using a calculator.

Estimate area between the graph of a function and the xaxis using rectangles.

State the definition
of a Riemann sum.

Define areas as
limits of Riemann sums.
Discussion
Questions

What is an antiderivative?

Explain how the velocity function relate to the velocity function.

What is a sigma notation?

What is the syntax for using the calculator to evaluate the sum of finite number
of terms?
Suggested Classroom Activities

The
facilitator explains the Mean Value Theorem.

Students work on problems involving the Mean
Value Theorem.

The facilitator leads a review of the
application of the derivative formulas and
use the results to find an antiderivative for each function.

The
facilitator leads a discussion of the sigma notation and the symbols used.

The students work on evaluating sums
manually and with the use of calculators.

The facilitator leads
activities on approximating areas using the leftendpoint, rightendpoint, and
midpoint methods.

The facilitator
explains the relationship between the approximate area under a curve and the
Riemann Sum.
Learning Topic
#8: Definite Integrals
This topic includes evaluating definite integrals
using the Fundamental Theorem of Calculus, the Second Fundamental Theorem of
Calculus, and the Mean Value Theorem.
Topic Outcomes:
After completing this topic, the student should
be able to:

Relate Riemann Sum to Definite Integral

Evaluate definite integrals using formulas
from geometry.

Evaluate definite integrals using the
Fundamental Theorem of Calculus Part I.

Use the TI83 calculator to find an
approximate value of a definite integral.

Evaluate the derivative of an integral
function by using the Fundamental Theorem of Calculus Part II.

Evaluate definite integrals by
usubstitution.

Demonstrate the relationship between the area
under a curve and the definite integral.
Discussion Questions

How does the Riemann Sum relate to the
definite integral?

How is a definite integral evaluated?

How does the definite integral relate to the
area under a curve?
Suggested Classroom Activities

The facilitator shows an example on how the
definite integral can be evaluated using an area formula from geometry.

The facilitator explains the First
Fundamental Theorem of Calculus.

The class works on problems applying the
First Fundamental Theorem of Calculus.

The facilitator demonstrate evaluating the
application of the Second Fundamental Theorem of Calculus.

The class works on evaluating definite
integrals by usubstitution.
Resources
Textbooks
and Study Guides

Ayres, Frank and Elliott Mendleson; Schaum’s Outline of Theory and
Problems of Differential and Integral Calculus, McGrawHill 1990; ISBN:
0070026629

Passow, Eli; Schaum’s Outline of Theory and Problems of Understanding
Calculus Concepts, McGrawHill 1996; ISBN: 0070487383

Thompson, Silvanus P., Calculus Made Easy, 3^{rd} Edition,
St. Martins Press, 1988, ISBN: 0312114109
Software
Student versions of each of the following packages are available through the
bookstore. The academic price may apply to students. One of these packages may
be useful for learning and experimenting, but check with your instructor first
to determine if one may be preferable over the others. In each case, you will
have to learn some programming syntax, but it will facilitate the learning
process and will help with the application of the calculus to later classes and
work situations.
·
Mathcad
·
Matlab
·
Mathematica
Internet
The
following sites contain tutoring information and may be searched as an aid to
learning.
This site was last updated
07/03/06
