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Course Title:
Introduction to Calculus (Level 3)
Length of Course:
One Year (5 credits)
Prerequisites:
Precalculus
Description:
This course presents a preliminary introduction to Calculus for students who will likely specialize in business, economics, management, life and
social sciences. Calculus plays an important role in these areas. It is the mathematics of change and we, of course, live in a constantly changing
world. The goal of this course is to equip students with the powerful tools of Calculus. At the foundation of mathematics is the idea of a function.
Functions express the way one variable quantity is related to another quantity. Calculus was invented to deal with the rate at which a quantity varies,
particularly if that rate does not stay constant. Clearly, this course needs to begin with a thorough review of functions, particularly the properties,
behavior and manipulation of the polynomial function. However, reviewing should be accomplished through the lens of calculus. This should be
engaging because it is a different, but a unifying way to look at all the functions. Beyond review, students will construct a firm understanding of the
derivative and its applications.
This course is designed to give students a proper balance between the mastery of skills and the comprehension of key concepts. This curriculum
will be guided by two principles. The first is the Rule of Three which requires that every topic be presented geometrically, numerically and
algebraically. The second guiding principle is the Way of Archimedes which states that formal definitions and procedures evolve from the
investigation of practical problems. Specifically, this curriculum provides many problems that are applications of the Social Sciences, Life Sciences
and Business arenas and are generally accepted as important. This course exceeds requirements specified in the CCSS and State Standards.
Evaluation:
Student performance will be measured using a variety of instructor-specific quizzes and chapter tests as well as departmental common
assessments, Midterm and Final Exam. Assessments will balance the degree to which required concepts and skills have been mastered.
Text:
Calculus: Concepts & Contexts 3rd Edition, James Stewart, Thomson Brooks/Cole 2005
Reference Texts:
Calculus, Deborah Hughes-Hallett, Andrew M. Gleason, et al, Wiley & Sons, 1994
Calculus: Graphical, Numerical, Algebraic, Ross Finney, Franklin Demana, et al, Pearson/Prentice Hall, 2007
Applied Calculus, Bernard Kolman & Charles G. Denlinger, 1989
Humongous Book of Calculus Problems, W. Michael Kelley, 2007
Introduction to Calculus
August 2011
1
Preliminary Unit: Review of Algebra
Learning objective
Content outline
1. Students will
distinguish and
apply number
systems and their
properties.
2. Students will
represent and
solve problems
involving linear
equations and
inequalities.
Concepts
1. Represent, identify and use equivalent
forms, and compare real number
systems- counting numbers, whole
numbers, integers, rational and
irrational numbers
2. Operations
3. Absolute value
4. Closed/open interval of real numbers
Skills
1. Correctly place a real number in the
appropriate number sets
2. Correctly order real numbers
3. Correctly graph open/closed intervals
of real numbers
4. Correctly identify subset relationships
among the real numbers
Concepts
1. Distinguish a linear function from every
other type of function
2. Select a form of a linear equation
based on the use/context/relationship
3. Correctly identify subset relationships
among the real numbers
Skills
1. Identify and solve linear functions and
inequalities
2. Write the solution to a linear inequality
using interval notation
3. Graph the solution to a linear inequality
on a number line or Cartesian plane
4. Solve absolute value equations.
Introduction to Calculus
Instructional Materials
Algebra 2 (Glencoe/McGraw Hill)
Sections1.1-1.2
SAT items
Teacher constructed materials
Assessment – sample questions
Key questions
What are counting numbers, whole numbers,
integers, rational and irrational numbers?
What are the ways in which these numbers
are noted (fractions, decimals, etc.)?
What operations can be performed on the
real numbers?
What is the definition of the absolute value of
a number?
What is a closed/open interval of real
numbers?
Concept check:
What is the reason we do not permit division
by zero?
Why must you use parenthesis when
squaring a negative number on a calculator?
Algebra 2 (Glencoe/McGraw Hill)
Sections1.3-1.6, Chapter 2
Functions, Statistics, and
Trigonometry
Chapter 2
SAT items
Teacher constructed materials
August 2011
Key questions
What's different between a line and every
other type of function you know? What is a
useful form of the equations of a line that we
didn't use often in the prior courses but that
expresses this
relationship?
Concept check:
T/F: A line is the only function with constant
slope.
How can a linear function be recognized from
its equation, table and graph?
Do all linear equations have solutions?
Explain.
2
Learning objective
3. Students will
represent and
solve problems
involving systems
of equations and
inequalities.
4. Students will
represent and
solve problems
involving
exponents and
radicals.
Content outline
Concept
1. Represent and solve system of linear
equations & inequalities
Instructional Materials
Algebra 2 (Glencoe/McGraw Hill)
Chapter 3
SAT items
Skills
2. Solve and graph system of linear
equations & inequalities
Concepts
1. Represent positive, negative, and
fractional exponential values.
2. Identify equivalent forms of exponential
values
3. Identify the domain of a radical function
from the given context
Teacher constructed materials
Algebra 2 (Glencoe/McGraw Hill)
Chapter 5
SAT items
Teacher constructed materials
Skills
1. Rewrite a rational expression using
positive exponents
2. Rewrite a radical expression using
rational exponents
3. Identify the domain of a radical function
4. Sketch a radical function
Introduction to Calculus
Assessment – sample questions
Do all systems of linear equations and
inequalities have solutions? Under what
circumstances would a system have no
solution?
Key questions
What is a radical? A radicand? An index
number?
What is meant by a negative exponent?
What is meant by a fractional exponent?
How does one calculate the domain of a
radical function?
Concept check:
Explain why the expression
simplified as
x
x 2 y should be
y rather than x y .
What is meant by: “You CANNOT take an
even root of a negative number in the real
number system?”
August 2011
3
Learning objective
5.Students will
represent and
solve problems
involving
polynomial
functions.
Content outline
Concepts
1. Parabolas represented in situations,
equations and graphs
2. Look at the parabola graphically
Consider slope using only
a. the graph
b. points
c. equations
3. Recognize the degree of a polynomial
4. Identify the operations performed on
polynomials
5. Explain what it mean to factor a
polynomial and demonstrate it
6. Identify, graph and solve problems
involving a quadratic
function/equation/inequality
7. Explain what is meant by the slope of a
curve
8. Explain how are the roots of a
polynomial equation are shown in the
sketch of the corresponding polynomial
function
Assessment – sample questions
Key questions
Algebra 2 (Glencoe/McGraw Hill)
What is a parabola? How can we talk
Chapter 6
about the slope of a parabola?
How can we look at the slope using
Functions, Statistics, and
only the graph, using only points and
Trigonometry
using only equations? What is the
Chapter 9
degree of a polynomial?
What are the operations performed on
Applied Calculus (Kolman & Denlinger) polynomials?
What does it mean to factor a
SAT items
polynomial and how is it done?
What is a quadratic
Teacher constructed materials
function/equation/inequality and how is
it graphed/solved?
What is meant by the slope of a curve?
How are the roots of a polynomial
equation shown in the sketch of the
corresponding polynomial function?
Instructional Materials
Concept check:
Are all polynomials factorable?
Students expand binomials such as
( x  3) 2 as x 2  9 . Is this correct and
Skills
Identify a polynomial function
Identify the degree of a polynomial
Factor polynomial functions
Solve polynomial equations
Graph quadratic functions
Solve quadratic equations
Use the Quadratic Formula
Solve quadratic inequalities
Sketch the tangent to a curve at a given
point.
Introduction to Calculus
how can you determine if this is?
Students solve x  9 as x  3 . Is
this correct? Explain.
How can one tell from looking at the
graph of a polynomial function that it is
increasing/decreasing?
Do all polynomial equations have real
roots? What would lack of real roots
imply about the sketch of the
corresponding polynomial function?
2
August 2011
4
Learning objective
6. Students will
represent and
solve problems
involving rational
functions.
Content outline
Concepts
1. Identify a rational function.
2. Distinguish rational functions from
polynomial functions.
3. Explain how one calculates the
domain of a rational function.
4. Determine the unique feature(s) that
would one expect to find in the graph
of a rational function.
Instructional Materials
Skills
1. Identify rational functions
2. Identify the domain of a rational
function
3. Sketch a rational function indicating
locations of holes/asymptotes
4. Solve a rational equation
SAT items
Algebra 2 (Glencoe/McGraw Hill)
Chapter 9
Functions, Statistics, and
Trigonometry
Chapter 4
Assessment – sample questions
Key questions
What is a rational function? How are
rational functions different from
polynomial functions? How does one
calculate the domain of a rational
function? What unique feature(s)
would one expect to find in the graph
of a rational function?
Applied Calculus (Kolman & Denlinger)
Concept check:
Introduction to Calculus
1. True/False:
Teacher constructed materials
August 2011
x
 1 . Explain.
x
2. What is meant by a divide by zero
error?
3. T/F: All rational functions have either
holes or asymptotes. Explain.
4. Do all rational equations have real
roots?
5. How can you tell if a function will
have a hole, an asymptote or
neither?
5
Unit One: Functions & Models
Learning objective
7. Using applications
and examples,
students will relate
and use four forms
to represent a
function.
Content outline
Concepts
1. Recognize a function from its graph,
algebraic representation, and table.
2. Recognize why we care about
whether a pairing of data is a function
3. Represent a function 4 ways.
4. Explain and demonstrate the Vertical
Line Test
5. Describe a piecewise function
6. Determine how we recognize and
note when a function has symmetry
7. Describe a function as increasing or
decreasing
Instructional Materials
Assessment – sample questions
Calculus: Concepts & Contexts
Section 1.1,1.3
12-3: Handouts and worksheets
Concept check:
What does it mean to say that f is a
function of x?
True or false: f (x ) the same
as f  x . Explain why.
Explain how a function is similar to
a machine. Be able to pair math
words with
Input, output and rule. Be able to
state the ‘golden’ rule of a function.
If f is a function and f(a) = f(b),
must a=b? Why or why not?
Skills
2. Given either the table or graph of a
function, be able to evaluate f(c), c a
constant.
3. Given the algebraic representation of
a function be able to evaluate f(x + h).
4. State the domain &range of a
function.
5. State the intervals over which a
function is increasing or decreasing.
6. Determine if the graph is that of a
function using the vertical line test.
7. Create a graph of a function given
anecdotal data, a table or an algebraic
equation.
8. Given either the graph or algebraic
representation of a function,
determine if it’s even or odd.
Introduction to Calculus
August 2011
6
Learning objective
Content outline
Concepts
1.Construct & relate mathematical models
2. Describe the use of a mathematical
model
3. Identify the essential functions from
which they will create models
8. Students will
create
mathematical
models for
essential
functions.
Instructional Materials
Assessment – sample questions
Calculus: Concepts & Contexts
Section 1.2
12-3: Handouts and worksheets
Concept check:
When can we be confident in the
predictions based upon modeling?
What is interpolation v. extrapolation?
What is the real world significance of
constants we determine in modeling?
(i.e. the slope and y-intercept in a
linear model, the base in an
exponential model)
Skills
1. Create the following kinds of models
from data: linear, polynomial, power,
rational, trigonometric, exponential,
logarithmic
2. Choose the most appropriate model
based upon the data
3. 3. Use technology to create a model
Introduction to Calculus
August 2011
7
Learning objective
9. Students will
transform
functions
algebraically and
graphically.
Students will use
the graphing
calculator to
efficiently aid in
their study of
functions
Introduction to Calculus
Content outline
Concepts
What are dilations and translations?
How are these accomplished graphically
and algebraically? What is composition
of functions? What is the domain of a
composed function? How are reflections
over the coordinate axes and y=x
accomplished graphically and
algebraically?
Skills
1. Draw the graph of f(x  c), f(cx), cf(x),
and f(x)  c, given the graph of f
Identify parent curves.
2. Evaluate f(g(c)) given rules, graphs or
tables for f and g
3. Find f(g(x)) given rules for f and g Find
the domain of f(g(x)) given domains for
f and g
4. Express a complicated function as the
composition of easier functions
With a TI-83+…
5. Identify a correct window in order to
show the portion of the function in
which we are interested
6. Correctly determine the intersection of
two functions
7. Create a customized table of values
for a function
8. Solve an equation using the TI83+
9. Use the regression feature of a TI83+
10.Use the features of a TI-83+calculator
to identify domain& range of a function
Instructional Materials
Assessment – sample questions
Calculus: Concepts & Contexts
Sections 1.3-1.4
Concept check:
What is the benefit of understanding
about transformations of functions?
What are the real world applications of
composition of functions?
If c > 0, why does f ( x  c) shift the
12-3: Handouts and worksheets
Section 2.1
Functions, Statistics, and
Trigonometry 2-1 (The Language of
Functions)
August 2011
graph of f (x ) to the right and not to
the left as one might expect?
Is the function f(g(x)) the same as
g(f(x))? Why or why not?
If f(x) is linear and g(x) is linear, must
f(g(x)) also be linear? Why or why
not?
8
Learning objective
Content outline
Concepts
1. Determine the shape of a graph of
10. Students will
an exponential function
represent and solve
2. Determine its domain and range
problems involving
exponential functions. 3. Describe e and when is it used
4. Determine factors that distinguish
the graph of decay v growth
5. Identify half life
Instructional Materials
Calculus: Concepts & Contexts
Section 1.5
Functions, Statistics, and
Trigonometry (FST)
3-1 (Using an Automatic Grapher)
Assessment – sample questions
Concept check:
What distinguishes an exponential
function from a linear function?
Why is e called the natural base?
Identify real world uses of exponential
functions.
State a real world question that has
Total = 90 * 3
x
as its answer.
Skills
1. Graph exponential functions, and
identify domain and range
2. Apply transformations on exponential
functions
3. Use an exponential function as one
function when composing functions
4. Express an exponential function
x
in y = c*a
5. Solve half life problems
Introduction to Calculus
August 2011
9
Learning objective
Content outline
Concepts
11. Students will
1. Define an inverse function.
represent and
2. Determine if all functions have
solve problems
inverse functions & how you can tell
involving inverse
3. Explain the horizontal like test.
functions and
4. Graphically construct inverses.
logarithmic
5. Define a one-to-one function
functions.
6. Algebraically construct inverses
7. Define a logarithmic function
8. Explain the relationship between an
exponential and logarithmic function
10. Define a natural log
11. Define the change of base formula
12. Describe the kinds of equations logs
allow someone to solve
Skills
1. Test if a given function is 1:1 and the
existence of an inverse function
2. Verify if two functions are inverses
3. Find the inverse of a 1:1 function
(algebraic, graphical, and numerical
method) and its corresponding
domain and range
4. Use a calculator to compute logs to
bases other than 10 and e?
5. Find equivalent forms for logarithmic
and exponential expressions.
6.Evaluate logarithms using properties
7. Graph logarithmic functions
8.Transform logarithmic functions
6. 9.Evaluate common &natural logs
10.Find the domain of a log function
Introduction to Calculus
Instructional Materials
Calculus: Concepts & Contexts
Section 1.6
Assessment – sample questions
Concept check:
1. How can a function that is not oneto-one, have an inverse function?
1
Functions, Statistics, and
2. In the expression f (x), is -1 an
Trigonometry
exponent?
3-2 (The Graph Translation Theorem)
3-4 (Symmetries of Graphs)
3-5 (The Graph Scale ChangeTheorem)
3-7 (Composition of Functions)
August 2011
10
Unit Two: Limits and Derivatives
Learning objective
12. Students will
solve tangent
and velocity
problems.
Content outline
Concepts
Average rate of change
instantaneous rate of change
Distinguish how they are they
different and how are they the
same?
Graphically explore the
correspondence of average and
instantaneous rate of change
Identify the units of rates of change
Identify the difference quotient
Instructional Materials
Assessment – sample questions
Calculus: Concepts & Contexts
Section 2.1
Key questions
What is average rate of change?
What is instantaneous rate of
change? How are they different and
how are they the same? Graphically,
what corresponds to average and
instantaneous rate of change? What
are the units of rates of change?
What is the difference quotient?
Concept check:
Will the average rate of change and
instantaneous rate of change of a
function be the same? Why or why
not?
Identify the significance of a
positive/negative rate of change.
Skills
Find the average and
instantaneous rates of change of
a function that is expressed
graphically and algebraically.
This should include proper units.
Introduction to Calculus
August 2011
11
Learning objective
13. Students will
determine the
limits of a
function.
Content outline
Concepts
What is a limit? What is appropriate
limit notation? What is a one sided
limit? Do all functions have limits?
What would a graph look like where a
limit either does not exist or is one
sided?
Instructional Materials
Calculus: Concepts & Contexts
Sections 2.2- 2.3
Assessment – sample questions
If lim f ( x)  L , must f(c) = L? Explain
x c
True/False: If f(x) =
x2  4
and g(x) =
x2
x + 2, then we can say that the
functions f and g are equal.
The statement: “Whether or not
lim f ( x ) exists , depends on how f(a)
xa
is defined” is true: Sometimes, always
or never
Explain the difference between these
two statements: f(a) = L and
Skills
Identify the limit of a function given its
graph
Identify the limit of a function given its
equation
Identify values at which the limit of a
function does not exist
Identify limits in a piecewise function
Sketch a function given information
about its limits
Use a graphing calculator to evaluate
limits of a function
Find limits using direct substitution and
factoring
lim f ( x)  L
x a
( x  5)( x  5)
 x  5,
( x  5)
( x  5)( x  5)
but lim
 lim( x  5)
x 5
x 5
( x  5)
Explain why
True/False: As x increases to 100, f(x)
=
1
gets closer to 0, so the limit as x
x
goes to 100 of f(x) is zero. Be
prepared to justify your answer.
If a function f is not defined at x = a,
A) lim f ( x ) cannot exist
xa
B) lim f ( x ) could be 0
xa
C) lim f ( x ) must approach 
xa
Introduction to Calculus
August 2011
12
Learning objective
14. Students will
represent and
solve problems
involving
continuity.
Content outline
Concepts
What does it mean to say a function
is continuous at a point, over an
interval and at an endpoint? What
are the types of discontinuities?
What does the Intermediate Value
Theorem say?
Instructional Materials
Calculus: Concepts & Contexts
Section 2.4
Skills
Identify where a function is
continuous given its graph or its
algebraic equation
Sketch functions given
information about intervals of
continuity
Find values for constants so
that a piecewise function will be
continuous at a given point
Assessment – sample questions
Concept check:
A drippy faucet adds one milliliter to the
volume of water in a tub at precisely
one second interval. Let f be the
function that represents the volume of
water in the tub at time t. Which of the
following is true?
(a) f is a continuous function at every
time t
(b) f is continuous for all t other than
the precise instants when the water
drips into the tub
(c) f is not continuous at any time t
(d) there is not enough information to
discern anything about the continuity of f
A drippy faucet adds one milliliter to the
volume of water in a tub at precisely
one second intervals. Let g be the
function that represents the volume of
the water in the tub as a function of the
depth of the water, w, in the tub. Which
of the following is true?
x
a)good idea - y = e is a continuous
function
b) g is continuous at every depth w
c) there are some values of w at which
g is not continuous
d) g is not continuous at any depth w
e) not enough information to discern
anything about the continuity of g
Introduction to Calculus
August 2011
13
Learning objective
15. Students will
describe the
limits involving
infinity with and
without
representation
or context.
Introduction to Calculus
Content outline
Concepts
What are infinite limits and limits at
infinity? What are the definitions of
vertical and horizontal asymptotes?
Instructional Materials
Calculus: Concepts & Contexts
Section 2.5
Assessment – sample questions
Concept Check:
Why is lim sin x undefined?
x 
What would the graph of f(x) look
like if lim f ( x)   ?
x 
Skills
Identify infinite limits given the graph of
a function or the equation of a function
Identify limits at infinity given the graph
of a function or the equation of a
function
Sketch the graph of a function given
information about infinite limits and limits
at infinity
Find a formula for a function given
horizontal and vertical asymptotes
True/False: A graph can cross its
horizontal asymptote.
August 2011
14
Learning objective
16. Students will
represent and
solve problems
involving
tangents,
velocities and
other rates of
change
Content outline
Concepts
What is the tangent line drawn to a
curve at a point? What is the secant
line of a curve? What is the definition of
the slope of a tangent line drawn to a
curve at x=a and what is its relationship
to the instantaneous rate of change of a
function? What is the definition of
velocity?
Instructional Materials
Calculus: Concepts & Contexts
Section 2.6
http://www.calculus-help.com/tutorials
True/False: The function f(x) = x
continuous at x=0
1
3
is
1
True/False: The function f(x) = x 3 is
differentiable at x=0
Skills
1. Find the slope of the tangent line
drawn to a curve at a given point
using the limit definition
2. Find the equation of the tangent line
drawn to a curve at a given point
3. Find the velocity of a moving particle
at a specific point given its position
function
4. Sketch a graph of the position of a
moving particle given information
about its velocity and a starting point
Introduction to Calculus
Assessment – sample questions
Concept Check:
Is the instantaneous rate of change of a
function equal to the average of two
average rates of change? Why or
why not?
The function y = f(x) measures the fish
population in Blue Lake at time x,
where x is measured in years since
January 1, 1950. If f ′ (25) = 500, it
means that
a. there are 500 fish in the lake in
1975
b. there are 500 more fish in 1975
than there were in 1950
c. on the average, the fish population
increased by 500 fish per year over
the first 25 years following 1950
d. on January 1, 1975, the fish
population was growing at a rate of
500 fish per year
e. none of the above
August 2011
15
Learning objective
17. Students will
interpret and
represent the
derivative of a
function.
Content outline
Concepts
What is the derivative of a function at
x=a? What is the geometric
interpretation of the derivative of f at
x=a? What is the interpretation of the
derivative as a rate of change of a
function?
Instructional Materials
Calculus: Concepts & Contexts
Section 2.7
Assessment – sample questions
Concept Check:
Given common business applications
including C(x), the cost of
manufacturing x items, what is the
proper interpretation of C ′ (c)?
http://www.calculus-help.com/tutorials
If f ′ (a) exists then lim f ( x )
xa
Equals f(a)
Equals f ′ (a)
May not exist
Explain the difference between these
Skills
1. Find f ′ (c) given f(x) as a graph or as
an equation
2. Correctly identify the units of f ′ (c)
3. Use technology to find f ′ (c)
two statements:
and
lim
h 0
f ( x  h)  f ( x )
( x  h)  x
f ( x  h)  f ( x )
( x  h)  x
Which of the following statements are
always true?
!) A function that is continuous at x = c
must be differentiable at
x=c
II) A function that is differentiable at x =
c must be continuous at
x=c
III) A function that is not continuous at x
= c must not be differentiable at x = c
IV) A function that is not differentiable at
x = c must not be continuous at x = c
a) None of them b) I and II
c) III and IV
d) I and IVII and III
e) I, II, III and IV
Introduction to Calculus
August 2011
16
Learning objective
18. Students will
recognize and
use the derivative
as a function.
Content outline
Concepts
What is the definition of f ′ (x)? How can
we find f ′ (x) given f as a graph, a table
or as an equation? What are the
alternate notations for f ′ (x)? Are all
functions differentiable everywhere?
What is the relationship between being
differentiable and continuous at a point?
What is a second derivative and how is
it found?
Instructional Materials
Calculus: Concepts & Contexts
Section 2.8
http://www.calculus-help.com/tutorials
xa
1)it must exist, but there is not enough
information to determine it exactly
equals f.
2)(a) equals f ′
3)(a)it may not exist
Skills
1. Given the graph of a function estimate
f ′ (c) for various values of c
2. Given the graph of a function, create
a sketch of its derivative function
3. Given an equation for f(x), find f ′ (x)
algebraically
4. Given a sketch of a function, identify
where it is not differentiable and why
5. Given a sketch of a function and its
first and second derivatives, identify
which is which
Your mother says “If you eat your
dinner, you can have dessert.” You
know this means “If you don’t eat your
dinner, you cannot have dessert.”
Your calculus teacher says, “If f is
differentiable at x, f is continuous at x.”
You know this means
1)if f is not continuous at x, f is not
differentiable at x
2)if f is not differentiable at x, f is not
continuous at x
3)knowing f is not continuous at x,
does not give enough information to
deduce anything about whether the
derivative of f exists at x.
a)
Introduction to Calculus
Assessment – sample questions
Concept check:
What is the derivative of a function
telling us about the behavior of the
function?
True/False: The function f(x) = lxl has
a derivative at x = 0
True/False: The function g(x) = xlxlhas
a derivative at x= 1.
If f ′ (a) exists, lim f ( x ) then…
August 2011
17
Learning objective
19. Through
investigations,
students will
surmise an
antiderivative
(What does f ′
say about f?).
Content outline
Concepts
What is a local maximum and minimum?
What are absolute extrema? What is
concavity and what does it represent?
What is an antiderivative?
Skills
1. Given the graph of a derivative
function, f ′ , identify the places where
f is increasing/decreasing, concave
up/down, local max/mins, points of
inflection
2. Sketch a graph of a function given
information about its derivative
3. Sketch a graph of the antiderivative of
a given function
Instructional Materials
Calculus: Concepts & Contexts
Section 2.9
http://www.calculus-help.com/tutorials
Assessment – sample questions
Concept check:
What do you know about the value of
the derivative function on those
intervals where the original function f is
increasing/decreasing?
Explain why these observations make
sense in terms of the slope of f
When the graph of f ′ crosses the xaxis, what does this tell you about the
graph of f? Explain, in terms of slope,
why this happens.
Based on the appearance of the graph,
which one of these functions looks like
it could be its own derivative?
a. f(x) = sin x
b. f(x) = cos x
c.
x
f(x) = e -3
x5
d. f(x) = e
e. none of the above
Introduction to Calculus
August 2011
18
Unit Three: Differentiation Rules
Learning objective
20. Students will
represent and
solve problems
involving
derivatives of
polynomials and
exponential
functions
Content outline
Concepts
What are the derivatives of a constant
function, a power function, a constant
times a function, a sum/difference of
functions and an exponential function?
Skills
1. Find derivatives of these types of
functions.
2. Write the equation of tangents &
normals.
3. Find higher order derivatives.
Instructional Materials
Calculus: Concepts & Contexts
Section 3.1
http://www.calculus-help.com/tutorials
Assessment – sample questions
Concept check:
Explain how differentiation and e are
related.
Use the derivative to draw conclusions
about increasing/decreasing, concave
up/down
2
True/False: If f(x) = e , then f ′ (x) = 2e
True/False: An equation of the tangent
2
line to the parabola y = x at (-2,4) is
y – 4 = 2x ( x + 2 )
Why is the natural exponential function
x
y = e used more often in calculus
than the other exponential function
x
y=a ?
21. Students will
determine when
to apply and then
use the product
and quotient
rules.
Concepts
What is the derivative of a function that is
the product/quotient of two differentiable
functions?
Calculus: Concepts & Contexts
Section 3.2
Concept check:
Understand when it is appropriate to
use the product or quotient rule
3
2
(i.e. f(x) = x (x +3x-4) and
3x 4  5 x  2
g(x) =
2x
Skill
Find derivatives of this type
Be able to state the product/quotient
rules in English
Introduction to Calculus
August 2011
19
Learning objective
22. Students will
apply rates of
change in the
natural & social
sciences
situations.
Content outline
Concepts
What is the interpretation of the
derivative in physics, chemistry, biology,
economics, psychology, sociology etc?
Instructional Materials
Calculus: Concepts & Contexts
Section 3.3
Skills
1. Find the derivative given an initial
function in one of these disciplines.
2. Correctly state the units in which the
derivatives should be expressed.
3. Understand the information that is
conveyed by the numerical derivative
at a given point.
Concept
OPTIONAL OBJECTIVE What are the derivatives of the trig
23. Students will find
the derivatives of
trigonometric
functions.
Introduction to Calculus
functions?
Calculus: Concepts & Contexts
Section 3.4
Skills
Find the derivatives of trig functions as
well as sum/difference and
product/quotient functions involving trig
functions
August 2011
Assessment – sample questions
Concept check:
If y = f(x) is a profit function
measuring the amount of profit (in
dollars) as a result
of
manufacturing and selling x
basketballs, what is the significance of f
′ (550)? Make sure you use specific
units.
The water level, W(t), (where t is
measured in hours) is falling at 3
inches every hour. Write an equation
involving a derivative to describe this
situation.
Comment on the statement:
“Mathematics compares the most
diverse phenomena and discovers the
secret analogies that unite them”
(Joseph Fourier)
Concept Check:
Graphically demonstrate that the
derivative of the sine function is the
cosine function
Using the derivative of the sine function
determine increasing/decreasing.
Compare these conclusions to the ‘unit
circle’ definition of the sine function.
20
Learning objective
24. Students will
recognize when
to apply the
chain rule, and
then use it.
Content outline
Concepts
What is a composed function? When is
the use of the chain rule appropriate?
Write a complicated function as the
composition of other simpler functions.
What is the derivative of a composed
function?
Skills
Correctly find the derivative of a
composed function given algebraically, in
a table, or graphically.
Concepts
What is the difference between a
function expressed explicitly and
implicitly? When is it appropriate to use
implicit differentiation?
25.Students will
distinguish function
expressed explicitly
and implicitly, and
recognize when to
Skills
use and then use
implicit differentiation.
dy
Correctly find
by implicit differentiation
Instructional Materials
Calculus: Concepts & Contexts
Section 3.5
Assessment – sample questions
Concept Check:
Be able to state the Chain Rule in
English
If f and g are both differentiable
function a h = f g, then h ′ (2) equals
A) f ′ (2)
g ′ (2)
B) f ′ (2) g ′(2)
C) f ′ (g(2)) * g ′ (2)
D) f ′ (g(x)) * g ′ (2)
Calculus: Concepts & Contexts
Section 3.6
NOTE: This objective should be done
only as a precursor to Related Rates
problems.
dx
Concept
OPTIONAL OBJECTIVE What is a logarithmic function? How can
26. Students will
represent and
solve problems
involving the
derivatives of
logarithmic
functions.
Introduction to Calculus
it be expressed exponentially? What is
logarithmic differentiation and when is its
use appropriate? What is the definition
of e as a limit?
Calculus: Concepts & Contexts
Section 3.7
http://www.calculus-help.com/tutorials
Skills
1. Correctly find the derivative of a
logarithmic function
2. Correctly use logarithmic
differentiation
August 2011
Concept check:
Understand the difference between the
derivative of a logarithmic function and
logarithmic differentiation.
Recognize that implicit differentiation is
used to develop the formula for the
derivative of a logarithmic function
Why is the natural logarithmic function
y = ln x used more often in calculus
than the other logarithmic function y =
log a x?
21
Learning objective
27. Students will
approximate the
values along the
tangent line for
given functions.
Content outline
Concepts
What is the meaning of the phrase
“approximating along a tangent line” and
its connection to linear approximation.
Discuss why one would choose to
approximate a function when you can
easily evaluate it on a calculator.
Emphasize that this is the first, of many,
applications of the derivative.
Instructional Materials
Calculus: Concepts & Contexts
Section 3.8
The line tangent to the graph f(x) = sin
x at (0,0) is y = x. This implies that
A) sin (.005)  .005
B) The line y = x touches the graph
of f(x) = sin x at exactly one point
(0,0)
C) y = x is the best straight line
approximation to the graph y = sin
x for all x
Skills
Linearize a function for a given value
and use this to approximate the function
for nearby values.
Introduction to Calculus
Assessment – sample questions
Concept check:
Some linearizations provide
underestimates and some provide
overestimates. What characteristic of
the function will predict when you get
each?
August 2011
22
Unit Four: Applications of Differentiation
Learning objective
28. Students will
represent and
calculate related
rates of change
to solve
problems.
Introduction to Calculus
Content outline
Concepts
How can we calculate rates of change
we cannot measure from rates of change
that we already know?
Instructional Materials
Calculus: Concepts & Contexts
Section 4.1
Skills
1. Develop a mathematical model of a
problem.
2. Write an equation relating the
variable whose rate of change you
seek with the variable whose rate of
change is known.
3. Differentiate both sides of an
equation implicitly with respect to
time.
4. Interpret a solution by translating the
mathematical result into the problem
setting to determine whether the
result makes sense
Assessment – sample questions
Concept check:
Would it be reasonable to say that the
rate at which water enters a drip
coffeemaker is equivalent to the rate at
which it leaves? What kind of coffee
would that produce? What piece of a
coffeemaker is designed so that there
is no overflow?
Gravel is poured into a canonical pile.
The rate at which the gravel is added to
the pile is:
A)
dV
dt
B)
dr
dt
C)
dV
dr
Peeling an orange changes its volume
V. What does V represent?
A) the volume of the rind
B) the surface area of the orange
C) the volume of the ‘edible’ part of
the orange
D) -1 times the volume of the rind
August 2011
23
Learning objective
29. Students will
identify
maximum and
minimum values
of a function
given an
equation or
graph.
Content outline
Concepts
What are absolute extreme values of a
function? What are local extreme values
of a function? What is a critical point of a
function? Under what circumstances
does a function NOT have absolute
extrema? What is the Extreme Value
Theorem?
Instructional Materials
Calculus: Concepts & Contexts
Section 4.2
Skills
1. Given the equation of a function, find
local and absolute extreme values
using transformations and graphs.
2. Sketch functions given information
about extreme values.
3. Find absolute max/min of a function
defined on a closed interval
Introduction to Calculus
Assessment – sample questions
Concept check:
Is it always true that f ′ (c) = 0 if f has a
local extreme value at c?
If a function is defined on a closed
interval, explain why it must have a
global maximum?
True/False: Every global max/min is a
local max/min
True/False: If f(x) is continuous on a
closed interval, then it is enough to look
at the points where f ′ (x) = 0 in order to
find its absolute maximum and
minimum. Be prepared to justify your
answer.
If f is continuous on [a,b], then
A) there must be number m and M
such that m  f ( x)  M for all x
in [a,b]
B) there must be local extreme vales,
but there may or may not be an
absolute maximum or an absolute
minimum value for the function
C) any absolute max or min would be
at either endpoint of the interval,
or at places in the domain where
f ′ (x) = 0
August 2011
24
Learning objective
30. Given the graph
of a function or
derivative, or the
equation of a
function, students
will indicate
intervals at which
a function is
increasing or
decreasing, and
turning up or
down.
Content outline
Concepts
What is the Mean Value Theorem?
What is the relationship between the first
derivative of a function and whether it is
increasing or decreasing? What is the
first derivative test? What is the
definition of concavity? What are the
tests that determine the concavity of a
function?
Instructional Materials
Calculus: Concepts & Contexts
Section 4.3
Skills
1. Given the graph of a function, indicate
intervals of increasing/decreasing and
concave up/down.
2. Given the graph of the derivative of a
function, indicate intervals at which
the function is increasing/decreasing
and concave up/down.
3. Given the equation of a function, find
local/global max/min, intervals of
increasing/decreasing, intervals of
concave up/down and a sketch using
the derivative
Assessment – sample questions
Concept check:
What result tells us that if a biker travels 45
km in 3 hours, then her speedometer must
have read 15 km/hr at least once during the
trip? Can we guarantee that it will read 12
km/hr during the trip? How about 18 km/hr?
Is it true that if f ′ (c) = 0, x = c is a
maximum?
Is it true that if f ′′ (c) = 0, x = c is a point of
inflection.
What can you conclude about x =c if f ′ (c)
does not exist?
True/False: If f ′ (c) = 0 and f(c) is not a
local maximum, it must be a local minimum.
A continuous function f has domain [1, 25]
and range [3, 30]. If f ′ (x) < 0, for all x
between 1 and 25, what is f (25)?
The following is known about the function f:
on the interval [-5, 5], f is continuous and
differentiable and
2
f ′ (x) = (x+1)(2x+1)(x+3) . Briefly explain
each of the following conclusions:
a. There is a local minimum of f at x=-1/2
b. There is a horizontal tangent but no
extrema at x = -3
c. If f(2) = 7, then f(3) > 7
Water is being poured into a “Dixie Cup” (a
standard cup that is smaller on the bottom
than at the top). The height of the water in
the cup is a function of the volume of the
water in the cup.The graph of this function is
a. increasing concave up
b. increasing concave down
c. a straight line with positive slope
Introduction to Calculus
August 2011
25
Learning objective
Content outline
Concepts
OPTIONAL OBJECTIVE Why would anyone use Calculus to
sketch a curve when it can be done
31. Students will
accurately and quicker using a graphing
integrate graphing calculator?
calculator and
analytic methods Skills
of calculus to
1. Combine graphical methods and
solve problems.
analytic methods of calculus.
2. Distinguish between the graphing
calculator as a means of estimating
local extremes and inflections
points, contrasted with the use of
calculus for precise computation of
such points.
Concepts
What does it mean to optimize a function?
32. Students will
What kinds of functions would one most
represent and
solve optimization likely want to minimize/maximize?
problems.
Skills
1. Develop a mathematical model of a
situation.
2. Clearly identify the quantity to be
maximized/minimized.
3. Write a function whose extreme value
gives the information sought
4. Graph this function.
5. Using calculus methods, identify the
critical points and endpoints of this
function.
6. Interpret your solutions into the
problem setting and decide if your
results make sense
Introduction to Calculus
Instructional Materials
Calculus: Concepts & Contexts
Section 4.4
NOTE: Although this is optional
material, it does remind students that
there are instances when a calculator
should not be trusted.
Calculus: Concepts & Contexts
Section 4.6
August 2011
Assessment – sample questions
Concept check:
Is the calculator always reliable?
Of what use can calculus be when you
cannot find a ‘good’ viewing window for
a function?
Concept check:
Correct the numbering for steps of the
optimization process as they should be:
1)Justify that your solution provides a
max or min.
2)Find the derivative of the varying
quantity.
3)If necessary, substitute from the fixed
quantity into the varying quantity.
4)Make sure you have answered the
original question, with appropriate units
5)Define variables to be used.
6)Read the problem, noting especially
what is to be optimized.
7)Find the zeros of the derivative
equation.
8) Write equations for all fixed and
varying quantities
26
Learning objective
Content outline
Concepts
What is a cost function, a marginal cost
33. Given situations
function, an average cost function, a
or graphs,
students will solve demand function, a revenue function, a
marginal revenue function, a profit
problems
involving functions function and a marginal profit function?
What are the geometric and graphical
that are
applications from interpretations of average and marginal
cost?
business and
economics.
Skills
1. Given a cost and revenue sketch,
identify the point at which the profit is
maximized and sketch a profit function.
2. Create a linear demand function given
information about cost and sales.
3. Use calculus to find the level of
production that will maximize profits
Introduction to Calculus
Instructional Materials
Calculus: Concepts & Contexts
Section 4.7
Assessment – sample questions
Concept check:
Here are three different cost functions.
Suggest reasons why they look the
way they do:
NOTE: Although this is optional
material with respect to the AP Exam,
it should not be omitted in this class as Here are three different revenue
many of our students will be pursuing
functions. Suggest reasons why they
business studies.
look the way they do:
Here is a sketch of typical revenue
and cost functions. Find the maximum
profit graphically (where the difference
between the curves is greatest and
revenue is larger than cost). Show
that this maximum profit always occurs
at the point where the tangent lines to
the curves are parallel to each other.
Then sketch the profit function and the
marginal profit function.
August 2011
27
34. Students will
recognize the
value and use
of Newton’s
Method for
finding the
zeros of a
function.
Concepts
How does a graphing calculator find the real
zeros of a function? What is meant by an
iterative process?
Skills
1. Write an equation whose root is the value
you seek.
2. Choose a reasonable initial estimate, either
by hand or by programming a graphing
calculator. Find successive
approximations until the desired level of
accuracy is reached
Calculus: Concepts & Contexts
Section 4.8
Concept check:
Under what circumstances would
Newton’s Method fail?
Suppose your first guess in using
Newton’s Method is lucky in the sense
that x 1 is a root of f(x) = 0. What
happens to x 2 and later
NOTE: Although this material is
optional for the AP Exam, it is
recommended here as it explains
how a calculator finds zeros. This
should be a demonstration lesson
only.
approximations?
Newton’s Method is a cool technique,
because
A) it can help us get decimal
representations of number like
4
3,
8
5 , and 5 13
B) it can be used to find a solution to
x7  3x3  1
C) both (A) and (B)
Concepts
35. Students will What is an antiderivative? Does every
represent and function have a unique antiderivative? What
solve problemsis a slope field and how is it used to find an
antiderivative?
involving
antidervatives
Skills
Find general antiderivatives
Find antiderivatives given a function
and an initial condition
Given a slope field, use it to
determine an antiderivative if you
know an initial condition
Create a slope field given an
equation for f ′ (x)
Introduction to Calculus
Calculus: Concepts & Contexts
Section 4.9
August 2011
Concept check:
If F (x) is an antiderivative of f (x), what
is F ′ (0) ?
T/F: An antiderivative of a product of
functions, fg, is an antiderivative of f
times an
antiderivative of g
T/F: An antiderivative of the sum of two
functions, f + g, is the antiderivative of f
plus the antiderivative of g
If f is an antiderivative of g and g is an
antiderivative of h then, position
function h is
a. an antiderivative of f
b. the second derivative of f
c. is the derivative of f ′′
28
Recommended Unit Sequencing and Pacing Guide
Timeframe
Review of Basic Algebra
Q1
Unit 1: Functions and Models
1.1 Four Ways to Represent A Function
1.2 Mathematical Models: A Catalog of Essential Functions
1.3 New Functions From Old Functions
1.4 Graphing Calculators and Computers
1.5 Exponential Functions
1.6 Inverse Functions and Logarithms
Q2
Unit 2: Limits & Derivatives
2.1 The Tangent & Velocity Problems
2.2 The Limit of a Function
2.4 Continuity
2.5 Limits Involving Infinity
2.6 Tangents, Velocity and Other Rates of Change
2.7 Derivatives
2.8 The Derivative As A Function
2.9 What Does f ′ Say About f?
Midterm
Introduction to Calculus
August 2011
29
Timeframe
Q3
Unit 3:Differentiation Rules
3.1 Derivatives of Polynomial & Exponential Functions
3.2 The Product & Quotient Rules
3.3 Rates of Change in the Natural & Social Sciences
3.4 Derivatives of Trig Functions
3.5 The Chain Rule
3.6 Implicit Differentiation
3.7 Derivatives of Logarithmic Functions
3.8 Linear Approximations
Q4
Unit 4: Applications of Differentiation
4.1 Related Rates
4.2 Maximum & Minimum Values
4.3 Derivatives & The Shape of Curves
4.4 Graphing With Calculus & Calculators
4.6 Optimization Problems
4.7 Applications to Business & Economics
4.8 Newton’s Method
4.9 Antiderivatives
Final
Introduction to Calculus
August 2011
30
Here is a suggestion for the opening week of school. This will give the kids an idea of what calculus can accomplish. It will also highlight the skills
needed to get to a solution.
OPENING WEEK ACTIVITIES:
One of the main problems faced by a manufacturer is that of determining the level of production: that is, how many units x
of the product should be manufactured during a fixed time period, such as a day, week, month and so on. In some
businesses the variable x can take on only integer values, for examples, an automobile manufacturer cannot manufacture
720.2 cars. Often, however, the variable x can take on any real number. For example it is certainly possible to
manufacture 7.87 million barrels of oil or 2.6 tons of steel. During a given time period the manufacturer is concerned with
the following three quantities:
C(x) = total cost of producing x units of the product
C is the cost function
R(x) = total revenue received from selling x units of the product
R is the revenue function
P(x) = total profit derived from selling x units of the product
P is the profit function
The total profit is the difference between the total revenue and the total cost, so that if all the units that are manufactured
are sold, then
P(x) = R(x) – C(x)
Let us examine the cost function C closely. Even if no items were produced (the level of production is kept at zero),
certain costs, such as insurance, rent and the administrative salaries, would be incurred. Costs of this type, which are
incurred when no items are produced, are called fixed costs and are denoted by F(x). The costs that vary with the level of
production are called variable costs and are denoted by V(x). Variable costs include the cost of materials, labor and
transportation of the manufactured product. Hence we may write
C(x) = F(x) + V(x)
Introduction to Calculus
August 2011
31
ACTIVITY #1:
Consider a manufacturer of Red Bull Energy drink whose fixed costs F(x) is $4000 and whose variable costs (in dollars) is
given by V(x) = 80x  x 2 when the level of production is x gallons per day. This manufacturer sells each gallon of Red Bull
Energy drink for $30. How many gallons should be manufactured in order to maximize profit?
ANALYSIS OF THE COST FUNCTION:
A) Name some components of fixed and variable costs in this scenario.
B) If you were to graph the fixed costs, what would be measured along the horizontal axis? Why? …the vertical axis?
Why?
C) If you were to graph the fixed costs, what shape would the graph have?
D) If you were to graph the variable costs, what shape would the graph have? How do you know this?
E) If you wanted to graph the cost function what would you have to do with the graphs you have produced in parts (C) and
(D)? What shape would this graph have? How do you know this?
ANALYSIS OF THE REVENUE FUNCTION
F) If you were to graph the revenue function what would be measured along the horizontal axis? Why? …the vertical
axis? Why?
G) If you were to graph the revenue function, what shape would the graph have? How do you know this?
Introduction to Calculus
August 2011
32
ANALYSIS OF BOTH FUNCTION LOOKED AT TOGETHER
H) If you graphed BOTH the cost and revenue functions on the same Cartesian Plane, would they intersect? How do you
know this?
I) At what point would they intersect?
J) What is the significance of one (cost or revenue) being over the other?
K) What is the significance of the point where they cross? What is the natural name of this point?
ACTIVITY #2
Consider a manufacturer of calculators whose fixed costs F(x) is $4000 and whose variable costs (in dollars) is given by
V(x) = .02 x 2  50 x when the level of production is x calculators per week. This manufacturer sells each calculator for
$79.95. Because this output has not entirely met the demand, the manufacturer is considering expanding production.
The belief is that this expansion will increase profit. The level of production today is 749 calculators per week. Should
production be expanded?
ACTIVITY #3:
A manufacturer of dog treats wants to determine the weekly level of production that will maximize profits. Based on past
sales information and consumer surveys, if x is the number of pounds of dog treats that will be bought per week at a price
of p dollars each, then
x
p = 60 
.
100
Suppose that the fixed cost of production is $8000 per week and the variable costs are $10 per pound of dog treats
produced. What level of production will maximize profits?
Introduction to Calculus
August 2011
33
ACTIVITY #4:
MATH IN THE CORPORATE WORLD…YOU ARE THE CEO OF A SUPERMARKET CHAIN!!!
Would you buy cheerios that cost $40 a box? Would you buy Cheerios that cost $8 per box? Do you think more people
would buy the $40 box or the $8 box?
What you have just discovered is that the demand (how many boxes will be sold) of a product (Cheerios) is a function of
its price. This is often called the demand function.
Stores often conduct market analysis in order to set a price that will produce maximum demand. Here are the results of
such a market analysis.
Winn Dixie, a big supermarket chain in Florida, is considering makings its own store brand doughnut shaped oat breakfast
cereal. The chain sets various prices for its 15 ounce box at its different stores over a period of time. Then, using this
data, Winn Dixie researchers project the weekly sales at the entire chain of stores for each price. The data is displayed
below:
Price Per Box
Boxes Sold
$2.40
35,761
$2.60
32,586
$2.80
32,377
$3.00
29,854
$3.20
26,993
$3.40
25,723
$3.60
25,000
$3.80
20,145
$4.00
18,051
$4.20
16,841
Introduction to Calculus
August 2011
34
Using the data create a LINEAR model for demand (in boxes sold per week) as a function of selling price (in dollars)
Now realistically Winn Dixie could sell a gazillion boxes of its cereal if it charged $.01 per box. Why is this a very bad
idea?
If you were the president of Winn Dixie would you be interested in how many boxes of cereal you sell or in something
else? Consider the data above…if you price your cereal at $2.40 per box, you will sell 35,761 boxes and you will have
$___________. This amount is called the revenue. What is the revenue if you price the cereal at $2.60 per box?
___________. If the only choices were to price the cereal at $2.40 or $2.60 per box, what would you do?
Clearly, revenue is what it is ALL ABOUT!!!
Write an equation for the revenue of the cereal just using the words REVENUE, PRICE PER BOX AND NUMBER OF
BOXES SOLD.
REVENUE =
Now let’s revisit the data above, but with a third column.
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Price Per Box
$2.40
$2.60
$2.80
$3.00
$3.20
$3.40
$3.60
$3.80
$4.00
$4.20
Boxes Sold
35,761
32,586
32,377
29,854
26,993
25,000
25,685
20,145
18,051
16,841
Revenue
Now let’s create model for revenue as a function of price. Using the word equation above, we substitute in x for price per
box and _______________________________for number of boxes sold, to get:
Revenue = _________________________________________
Simplify this equation…that means all like terms are combined and there are no parentheses.
Revenue = __________________________________________
This is called a _________________________function.
If we put this on the graphing calculator, and look at the graph it will tell us an amazingly important piece of information.
Sketch the graph here:
This important piece of information is:
YOU ARE NOW A SUCCESSFUL SUPERMARKET TYCOON!
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