:
Campus: Princeton High School
Author(s): Thomas Banschbach
Date Created / Revised: July 2015
Six Weeks Period: 1st
Grade Level & Course: AP Calculus AB
Timeline: 18 days
Unit Title: Unit 2 Rates of Change and Limits
Stated
Objectives:
TEK # and SE
Lesson
#1
(C1) – The teacher has read the most recent AP Calculus Course Description, available as a free
download on the AP Calculus AB Home Page.
(C2) – The course teaches all topics associated with Functions, Graphs, and Limits; Derivatives;
and Integrals as delineated in the Calculus AB Topic Outline in AP Calculus Course Description.
(C3) – The course provides students with the opportunity to work with functions represented in a
variety of ways – graphically, numerically, analytically, and verbally – and emphasizes the
connections among these representations.
(C4) – The course teaches students how to communicate mathematics and explain solutions to
problems both verbally and in written sentences.
(C5) – The course teaches students how to use graphing calculators to help solve problems,
experiment, interpret results, and support conclusions.
See Instructional Focus Document (IFD) for TEK Specificity
Key
Understanding
s
The real life attributes of a limit. The concept and parameters of a limit. The existence or
non-existence of a limit. Limits vs One-Sided limits. Limits as x is allowed to increase
without bound. The formal definition of Continuity
Misconception
s
Some students may think that limits always exist no matter the conditions. Students will
have difficulty with the concept of a gap, jump, hole in regards to functions and their
graphs. Some students will have difficulty with limits as x approaches from one side
versus x approaching from either side.
Key
Vocabulary
Limit, jump, gap, hole, one-sided, removable, continuity, discontinuity. limits and infinity,
horizontal and vertical asymptotes, Sandwich Theorem , Intermediate Value Theorem
Suggested Day
5E Model
Instructional Procedures
(Engage, Explore, Explain, Extend/Elaborate, Evaluate)
Materials,
Resources,
Notes
Tuesday
09/08/15
Topic: Concept
of a limit
OBJECTIVE: We will introduce the concept of a limit.

Engage:INSTRUCTIONAL PROCEDURE:
1. Teacher initiates a brief discussion on what is a
limit and where are they found in real life.

Pencils,
paper,
spirals,
projector
Alternative
Assignmen
t Read Pg.
59-69
Explore:INSTRUCTIONAL PROCEDURE:
1. Teacher will engage the class in discussion and demonstration
of mathematical limit and how a limit is a foundational concept of
calculus.
2. Teacher will stress the idea that we are not interested in the
solution to a function at a value of x (it may not exist), we are
interested in the value of the function as we get close to x. This
is a new concept for most students and is essential to the
understanding of calculus.
 𝑓(𝑥) =
2𝑥, 𝑤ℎ𝑎𝑡 𝑖𝑠 𝑓(. 9), 𝑓(. 99) 𝑓(. 999), 𝑓(1.1), 𝑓(1.01, 𝑓(1.001)

CLOSING TASK: I will pick a real life limit (like a speed limit) and use my
example in mathematical terms.
Wednesday
09/09/15
Topic:
Introduction to
Limits
OBJECTIVE: We will use tables and graphs to determine limits at particular
values of x.
Engage: INSTRUCTIONAL PROCEDURE
1.Teacher posts an expression to be evaluated at a particular
value of the variable, for example:
2
𝑓(𝑥) = 2𝑥 + 1.
𝑔(𝑥) =
𝑥−1
 What is f(.9), f(.99) f (1) , f(1.1) , f(1.01)
 What is g(.9), g(.99) g (1) , g(1.1) , g(1.01)
Explore: INSTRUCTIONAL PROCEDURE:
1. Using an example of a function that does not exist at a particular
value of x, whether table form or from a graph, the teacher will
explain what a limit is and what it does as x approaches that
value from the example.








Pencils,
paper,
spirals,
projector
Lesson T-9
Textbook
2.1
Assignmen
t A-9
Alternative
Assignmen
t Pg.65/1-8
QR
What does it mean for a function not to exist?
When a problem does not exist for a particular value, what
mathematical rule is probably being broken?
What is a piecewise function?
CLOSING TASK: I will the table method to find the limit of a function even
though the function does not exist at that particular value.
Thursday
09/10/15
Topic:
Evaluating
Limits
OBJECTIVE: We will evaluate limits using the evaluation flowchart.

Engage: INSTRUCTIONAL PROCEDURE
1. Prior to students’ arrival, the teacher will post logic problem
and asks students to explain how they arrived at the answer...
a short discussion will following dealing with
analysis of a problem.

Explain: INSTRUCTIONAL PROCEDURE
1. Teacher will demonstrate the flowchart used to solve limits
plug in


Pencils,
paper,
spirals,
projector
Lesson T10,11
Textbook
2.1
Assignmen
t complete
A10 & A11

Alternative
Assignmen
t Pg.66/717 odd
OBJECTIVE: We will evaluate limits from one side only.

Engage: INSTRUCTIONAL PROCEDURE
1. Teacher poses an example where the students must park
their car in a slot where there is a curb involved.Students
will ponder the situation for a while then there will be a short
discussion.

Pencils,
paper,
spirals,
projector
Lesson T10,11
Textbook
2.1
Assignmen
t complete
A10.5
Alternative
Assignmen
t Pg.66/717 odd
Factor

(If indeterminate)
or
Multiply by Conjugate
Plug in
What is indeterminate form? (when a value is plugged into
0
the function the result is )
0
An example of the factoring method of finding the limit.
An example of the conjugate method of finding the limit.




What is polynomial factoring?
𝑥 2 + 5𝑥 + 6 = (𝑥 + 3)(𝑥 + 2)
What is conjugate? (changing the sign in the middle of a
binomial) 𝑓(𝑥) = 2𝑥 + 3, 𝑡ℎ𝑒 𝑐𝑜𝑛𝑗𝑢𝑔𝑎𝑡𝑒 𝑖𝑠 2𝑥 − 3
What does factoring / multiplying by conjugate do?
(eliminates the zero on the bottom of a rational function)
What happens after we remove the error part of the problem?
(now plug in finds the limit)
CLOSING TASK: I will create and solve a limit problem requiring factorization
to solve.
Friday 09/11/15
Topic:
Evaluating
Limits





Can you approach the slot from any direction?
Is there a direction where it is unreasonable to try to park
from there?
Explain: INSTRUCTIONAL PROCEDURE
1. Teacher will demonstrate this idea from graphs and piecewise
functions. The teacher will also explain that one-sided limits
always exist and are plug in problems.

Where does the graph go as x gets close to -1 from the left
side? (1)
 Where does the graph go as x gets close to -1 from the right
side? (-2)
 In the piecewise function, which part is used to find f(-1)?
(𝑥 2 )
 In the piecewise function what is f(5)? (-1)
 How did you arrive at that answer ? (plugged in 4-x
 In the piecewise function what is f(3)? (2)
2. Teacher will introduce the One Sided Limit notation.
lim+ 𝑓(𝑥) 𝑠𝑎𝑦𝑠 𝑤𝑒 𝑎𝑟𝑒 𝑎𝑝𝑝𝑟𝑜𝑎𝑐ℎ𝑖𝑛𝑔 𝑜𝑛𝑒 𝑓𝑟𝑜𝑚 𝑡ℎ𝑒 𝑟𝑖𝑔ℎ𝑡 ℎ𝑎𝑛𝑑 𝑠𝑖𝑑𝑒.
𝑥→1
lim 𝑓(𝑥) 𝑠𝑎𝑦𝑠 𝑤𝑒 𝑎𝑟𝑒 𝑎𝑝𝑝𝑟𝑜𝑎𝑐ℎ𝑖𝑛𝑔 𝑜𝑛𝑒 𝑓𝑟𝑜𝑚 𝑡ℎ𝑒 𝑙𝑒𝑓𝑡 ℎ𝑎𝑛𝑑 𝑠𝑖𝑑𝑒
𝑥→1−
lim 𝑓(𝑥) 𝑠𝑎𝑦𝑠 𝑤𝑒 𝑎𝑟𝑒 𝑎𝑝𝑝𝑟𝑜𝑎𝑐ℎ𝑖𝑛𝑔 𝑜𝑛𝑒 𝑓𝑟𝑜𝑚 𝑠𝑖𝑑𝑒𝑠
𝑥→1
CLOSING TASK: I will create and solve a limit problem requiring factorization
to solve.
Monday
09/14/15
Topic:
Properties of
Limits
OBJECTIVE: We will show that the operations of mathematics holds true in
limit problems.
Engage: INSTRUCTIONAL PROCEDURE
1. Teacher posts examples of addition of numbers functions etc
and ask student to solve them. (REVIEW)
𝑓(2) = 5 , 𝑔(2) = −2 𝑡ℎ𝑒𝑛 𝑓 + 𝑔(2) = 3
Explore: INSTRUCTIONAL PROCEDURE
1. Teacher will facilitate a discussion of the operations as they
pertain to limits.
Operations are +,-,*,/ Give students some limit values and
have them discern the value when added, subtracted etc.
Have the students separate the following problem into parts
and find the limit of each part. Then using the graphic
calculator, determine the limit of the function as a whole.
Compare results.





Pencils,
paper,
spirals,
projector
Lesson T11.5
Textbook
2.1 pg.61
Assignmen
t complete
A-11.5
Alternative
Assignmen
t Pg.67/4950
Explain: INSTRUCTIONAL PROCEDURE
1. Teacher will explain and demonstrate that the operations,
although not used too often, do indeed hold true for limits
(So Called Limit Properties).
2. Teacher will explain that somethings must be memorized. (Get
over it!)
CLOSING TASK: I will create two limits and show that one of the properties
indeed holds true.
Tuesday
09/15/15
OBJECTIVE: We will find limits of trig functions including some that must be
memorized.

Pencils,
paper,
Topic: Limits
of Trig
Functions
Engage: INSTRUCTIONAL PROCEDURE
1. Teacher posts collection difference games where rule require
Memorization. Have students pick a game and list the
appropriate rules.
 What game did you pick?
 What are the rules to the game…how do you know?
Explore: INSTRUCTIONAL PROCEDURE
sin 𝑥
1. Teacher will present limit problem lim
that is indeterminate
𝑥→0



𝑥
spirals,
projector
Lesson T12
Textbook
2.1
Assignmen
t complete
A-12
but the flowchart doesn’t work. Teacher will graph the problem
and seek answers from the students.

Even though the limit is in an indeterminate form and
cannot be solved by traditional methods, the limit as x
approaches 0 appears to be ? (1)

Explain INSTRUCTIONAL PROCEDURE
1. Teacher will explain that the problem is solved by the Squeeze
(Sandwich) theorem. The teacher will also present other
Special Trig Limits that must be memorized.
For a given interval containing point a, where f, g, and h are three functions
that are differentiable and
over the interval
(except, perhaps, at a), then if the limit L of g(x) is equal to the limit L of h(x)
as x approaches a, then that must also be the limit off(x):
if
, then
CLOSING TASK: I will make two flip cards of Special Trig Limits.
Wednesday
09/16/15
College and
Career
OBJECTIVE: Class does not meet.

None
Thursday
09/17/15
Topic: Limits
and Infinity
OBJECTIVE: We will determine limit values as x increases or decreases
without bound.

Pencils,
paper,
spirals,
projector
Lesson T15
Textbook
2.2
Engage: INSTRUCTIONAL PROCEDURE
1.
Teacher initiates a discussion of why people, for example
continue to grow but on reach a particular height. “Mary’s
height increase by 2” inches each of the last 3 years. She is 7
years old and a height of 39’’.





What is her predicted height when she reaches 75 years old?
Is this reasonable? Explain your solution.
2. Just like Mary’s growth rate, math problems seem to ‘level’ out
sometimes

Assignmen
t complete
A-15
Alternative
Assignmen
t Read Pg.
70 - 77
Explore:INSTRUCTIONAL PROCEDURE
1. Teacher will facilitate a discussion of rational function where x
is allowed in increase (or decrease) without bound (So Called
Limit and Infinity) Students are paired up and graphic
calculators are provided.
2𝑥
𝑓(𝑥) =
(𝑥+1)




What is f(1) ? (1)
What is f(10) (20/11)
What is f(1000) (2000/1001)
As x gets larger and larger, where does f(x) appear to go?
(2)
Explain:INSTRUCTIONAL PROCEDURE
1. Teacher will explain to the students that this is end behavior
and involves horizontal asymptotes. (see below)
2.
Teacher will demonstrate how to find the horizontal
2𝑥
asymptote by finding the limit of the function. lim
is found
𝑥→∞ 𝑥+1
by the leading coefficients of each part of the rational function.
1
Note: (student need to memorize lim = 0
𝑥→∞ 𝑥
CLOSING TASK: I will find the horizontal asymptote of a rational function of
my own creation.
Friday 09/18/15
Topic: Limits
and Infinity
OBJECTIVE: We will determine the students’ ability to find limits up to this
point in time.
Engage:INSTRUCTIONAL PROCEDURE
1. Teacher initiates a brief Q & A session on limits and their
properties.
 What is a limit? (value approach but not necessarily
reached)
 What is a one-sided limit? (value approached from only
one side)
 Can you add limits? (yes)
 If indeterminate form, how do we find the limit? (factoring
or conjugate)
Evaluate:INSTRUCTIONAL PROCEDURE
1. Teacher will prepare an assessment to use to determine
students’ knowledge and skills of limits and their properties.
Monday
09/21/15
Topic: Limits to
this point
OBJECTIVE: We will begin preparation for the limit test
Engage:INSTRUCTIONAL PROCEDURE





Pencils,
paper,
spirals,
projector
Lesson T15.5
Quiz on
Limits and
Limit
properties
Textbook
2.1-2.2
Study for
limit Test
1.
Prior to the arrival of students, teacher posts a warm up
dealing with a limit review.
Explore:INSTRUCTIONAL PROCEDURE
1. Teacher will facilitate a discussion nature and format used in
class for unit tests. Unit tests will have two distinct sections; the
first will be multiple choice, the second free response. On many
occasions the multiple choice section will be machine graded.
Grading of the free response section will be done by hand.
Explain:INSTRUCTIONAL PROCEDURE
1. Teacher will prepare a list of general test type questions to use
as review for the test. Using various question and answer
techniques, students will be prepare for the Limit test.

What is a limit? (value approach but not necessarily
reached)
 What is a one-sided limit? (value approached from only
one side)
 Can you add limits? (yes)
 If indeterminate form, how do we find the limit? (factoring
or conjugate)
CLOSING TASK: I will create 2 problems with correct answers from the limit
information to prep my partner.
Tuesday
09/22/15
Topic: Limit
Test


limit test
Read
Section 2.3
OBJECTIVE: We will understand when a function is considered continuous.

Engage:INSTRUCTIONAL PROCEDURE
1. Teacher poses the question ‘has there ever been a time when
your height was exactly 3 feet?
 Discuss with your partner your answer and how you
arrived at that conclusion.

Pencils,
paper,
spirals,
projector
Lesson T13
Textbook
2.3
Assignmen
t Read Pg.
78-82 ,
pg.84/1-10
QR
OBJECTIVE: We will be successful on the limit test
Engage:INSTRUCTIONAL PROCEDURE:
1. Teacher posts test taking aids including, reading the question
first, eliminate bad answer choices, justifying all results in the
free response section etc.
Explore: Teacher will facilitate a brief discussion of what is on the limit test.
Evaluate: Limit Test – Assess student understanding of related concepts and
processes by using the objectives covered in our study of limits
.
CLOSING TASK: I will prepare for the next section by reading textbook
pages on Continuity
Wednesday
09/23/15
Topic:
Introduction to
Continuity


Explore:INSTRUCTIONAL PROCEDURE
1. Teacher will facilitate a discussion of continuity by drawing
various graphs and poses questions..
 How do we determine if a graph is a function? (VLT)
 What do you notice? (no jumps, skips)
 How would you explain continuous? (drawn without lifting
your pencil)
Continuous
Discontinuous
Discontinuous
Explain:INSTRUCTIONAL PROCEDURE
1. Teacher will engage the class in discussion, in layman’s terms
what is continuity..
CLOSING TASK: I will draw some graphs and ask my partner on which are
continuous.
Thursday
09/24/15
Topic:
Continuity
OBJECTIVE: We will determine the requirements necessary for a function to
be considered continuous.
Engage:INSTRUCTIONAL PROCEDURE
1. Prior to students’ arrival, teacher prepares a matching game
of some basic limit ideas.
2. Upon arrival students, with their partners, will match the
game pieces
Explore:INSTRUCTIONAL PROCEDURE
1. Teacher will use the layman’s definition, to determine if a
given function is continuous.
 What was yesterday’s ‘definition’ of a continuous
function? (drawn without lifting pencil)
Explain:INSTRUCTIONAL PROCEDURE
1 Teacher will each of the ‘three part requirements’ by using
graphs to demonstrate when a function is continuous and
when it is discontinuous.





Pencils,
paper,
spirals,
projector
Lesson T13
Textbook
2.3
Assignmen
t complete
A-13
Alternative
Assignmen
t Pg.84/115 odd
Definition of Continuity. Let a be a point in the domain of the
function f(x). Then f is continuous at x=c if and only if
lim 𝑓(𝑥) 𝑒𝑥𝑖𝑠𝑡𝑠
𝑋→𝐶
𝑓(𝑐) 𝑒𝑥𝑖𝑠𝑡𝑠
lim 𝑓(𝑥) = 𝑓(𝑐)
𝑋→𝐶
Short version below
Let c be a point in the domain of the function f(x). Then f
is continuous at x=c if and only if
lim f(x) = f(c)
x --> c
CLOSING TASK: I will explain continuity to someone at home
(documentation required).
Friday 09/25/15
Topic:
Intermediate
Value Theorem
OBJECTIVE: We will be able state the Intermediate Value Theorem and find
applications of the IVT
Engage: Teacher poses the “travel” question of same place at the same
exact time.
Explore: Teacher will facilitate a discussion of the Intermediate Value
Theorem the history, relevance and application..


Pencils,
paper,
spirals,
projector
Lesson T14
Explain: Teacher will engage the class in discussion and demonstration the
uses of the IVT.



Textbook
2.3 pg 83
Assignmen
t complete
A-14
Alternative
Assignmen
t Read
Pg.83
CLOSING TASK: I will recite the IVT to my partner. .
Monday
09/28/15
Topic:
Intermediate
Value Theorem
OBJECTIVE: We will discover the relevance uses of the IVT

Engage: Teacher posts scenarios some of which reflect the IVT Students will
discern which.
Elaborate: Teacher will on the finer points and uses of the IVT Explain:
Teacher will engage the class in discussion and demonstrations of various
examples of the IVT.

CLOSING TASK: I will demonstrate an understanding of the Intermediate
Value Theorem (IVT) by illustration (a graph)



Tuesday
09/29/15
Topic: Review
of Limits
OBJECTIVE: We will review limit facts, rules and uses.
Engage: Teacher posts review of various mathematical principles and
student determine which apply to limits.
Explore: Teacher will facilitate a discussion of limits, demonstrate the rules
and how limits vs one-sided limits are found.





.
Explain: Teacher will engage the class in discussion and demonstration of
how to find limits of functions.
Pencils,
paper,
spirals,
projector
Lesson T14.5
Textbook
2.3 pg 83
Assignmen
t complete
A-14.5
Alternative
Assignmen
t Pg.84/1929 odd
Pencils,
paper,
spirals,
projector
Lesson T16
Textbook
2.1-2.3
Assignmen
t complete
A-16 evens
Alternative
Assignmen
t Re-read
Pg 59-84
CLOSING TASK: I will contrast limits vs one-sided limits in a sentence
format.
Wednesday
09/30/15
Topic: Prep for
Unit 2
Assessment
OBJECTIVE: We will begin preparation for the Unit 2 test
Engage: Teacher posts a warm up review limits, continuity and IVT .
Explore: Teacher will facilitate a discussion nature and format used in class
for unit tests..
Explain: Review and prepare for the Unit Test – Using various question and
answer techniques, students will be prepare for the Unit 2 test.

Study for
Unit 2 Test
CLOSING TASK: I will create 2 problems with correct answers from Unit 2
information to prep my partner.
Thursday
10/01/15
Topic: Unit 2
Assessment
OBJECTIVE: We will be successful on the Unit 2 test

Unit test
Engage: Teacher posts test taking aids.
Explore: Teacher will facilitate a brief discussion of what is on the Unit 1 test.
Evaluate: Unit – 2 Test – Assess student understanding of related concepts
and processes by using the objectives covered in this unit.
CLOSING TASK: I will prepare for the next unit by reading textbook pages on
Rates of Change and Tangent Lines
2
Accommodations
for Special
Populations
Accommodations for instruction will be provided as stated on each student’s (IEP)
Individual Education Plan for special education, 504, at risk, and ESL/Bilingual.