HARRISON SCHOOL
DISTRICT
CURRICULUM OFFICE
Mathematics Department
AP Calculus
Page 1 of 86
May 16, 2013
HARRISON SCHOOL DISTRICT
Board of Education
Commissioner James A. Fife, President
Commissioner Maria Villa, Vice President
Commissioner Anthony R. Comprelli
Commissioner Brigite Goncalves
Commissioner Lily Wang
Commissioner Vincent Franco
Commissioner Arthur Pettigrew
Commissioner Kimberly Woods
James P. Doran, Ed.D. Superintendent
Curriculum Writing Committee
Director of Curriculum
Deborah Ronan
Dr. Cynthia Baumgartner
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May 16, 2013
HARRISON SCHOOL DISTRICT
PHILOSOPHY FOR CURRICULUM
The goal of the Harrison School District is to prepare students for the 21st century demands of the world at large. In this regard, the
curricula has been designed to make it relevant and accessible to our students by focusing each unit of study around enduring
understandings and essential questions. Based upon the New Jersey Common Core Curriculum Standards, the enduring understandings
are the starting points to the backward curriculum model utilized for this guide and espoused by Grant Wiggins and Jay McTighe. The
curriculum is predicated by essential questions which are developed to challenge our students to think on a higher level and use the inquiry
method to gain a deeper understanding of the content material.
The Advanced Placement Calculus course will allow students to encompass and extend topics of intermediate algebra, apply algebraic
techniques to geometric problems, gain experience with methods and applications of calculus, integrate technological applications, and
improve critical thinking. It will prepare students to achieve desired results on the AP Calculus AB exam.
COURSE DESCRIPTION
Calculus is the mathematics of change and motion. There are two types, differential calculus, finding the rate of change of a function and,
integral calculus, finding the function when its rate of change is given. Science, engineering, and business use these branches for expressing
physical laws in exact mathematical terms and as a tool for studying ramifications of those laws.
The Advanced Placement Calculus (AP Calculus) course is intended for those students who have successfully completed precalculus and
desire to further their studies in mathematics. It includes such topics as limits, derivatives, applications of derivatives, integrals,
applications of integrals, and slope fields.
Students will use technology regularly to assist in solving calculus problems. Graphing calculators* will be utilized to represent functions,
check written work, execute experimentation, and interpret results. Students will recognize and appreciate how technological tools can be
used to solve real world mathematical problems.
*As part of the College Board requirement during the entire course students will use graphing calculators to assist in solving problems.
For computational purposes and/or
graphing functions, students will demonstrate the ability to find a numerical derivative, evaluate a definite integral, find the zeros of functions, find the maximums, minimums, and
intersections of curves, alter a viewing window, zoom a function, use trace and table options along with other operations basic to calculator applications.
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May 16, 2013
Grade Level:
12th
Department:
Mathematics
Course Code:
0347
Credits:
5
GRADING
Student achievement will be evaluated using multiple assessment tools as described in the individual units of study. Marking period grades
will be determined according to the following:
Criteria
Tests
Quizzes/ Projects
Homework/ Classwork
Total:
Percentage
50%
40%
10%
100%
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May 16, 2013
HARRISON SCHOOL DISTRICT
Curriculum Based on NJCCCS
A P Calculus
Domain
F-IF
Interpreting Functions
Cluster
F-IF
Interpreting Functions
Understand the concept of a function and
use function notation.
2. Use function notation, evaluate
functions for inputs in their domains, and
interpret statements that use function
notation in terms of a context.
F-IF
Interpreting Functions
Understand the concept of a function and
use function notation.
3. Recognize that sequences are
functions, sometimes defined recursively,
whose domain is a subset of the integers.
Interpret functions that arise in
applications in terms of the context.
4. For a function that models a relationship
between two quantities, interpret key
features of graphs and tables in terms of the
quantities, and sketch graphs showing key
features given a verbal description of the
relationship. Key features include:
F-IF
Interpreting Functions
Understand the concept of a function and
use function notation.
Standard
1. Understand that a function from one set
(called the domain) to another set (called
the range) assigns to each element of the
domain exactly one element of the range. If
f is a function and x is an element of its
domain, then f(x) denotes the output of f
corresponding to the input x. The graph of
f is the graph of y = f(x).
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intercepts; intervals where the function is
increasing, decreasing, positive, or negative;
relative maximums and minimums;
symmetries; end behavior; and periodicity.
F-IF
Interpreting Functions
Interpret functions that arise in
applications in terms of the context.
5. Relate the domain of a function to its
graph and, where applicable, to the
quantitative relationship it describes.
F-IF
Interpreting Functions
Interpret functions that arise in
applications in terms of the context.
6. Calculate and interpret the average rate of
change of a function (presented symbolically
or as a table) over a specified interval.
Estimate the rate of change from a graph.
F-IF
Interpreting Functions
Analyze functions using different
representations.
7. Graph functions expressed symbolically
and show key features of the graph, by
hand in simple cases and using technology
for more complicated cases.
a. Graph linear and quadratic functions
and show intercepts, maxima, and
minima.
b. Graph square root, cube root, and
piecewise-defined functions and
absolute value functions.
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c.
Graph polynomial functions
identifying zeros when suitable
factorizations are available, and
showing end behavior.
d. (+) Graph rational functions, identifying
zeros and asymptotes when suitable
factorizations are available, and
showing end behavior.
e. Graph exponential and logarithmic
functions, showing intercepts and end
behavior, and trigonometric functions,
showing period, midline, and
amplitude.
F-IF
Interpreting Functions
Analyze functions using different
representations.
8. Write a function defined by an expression
in different but equivalent forms to reveal
and explain different properties of the
function.
a. Use the process of factoring and
completing the square in a quadratic
function to show zeros, extreme values,
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and symmetry of the graph, and
interpret these in terms of a context.
b. Use the properties of exponents to
interpret expressions for exponential
functions.
F-IF
Interpreting Functions
Analyze functions using different
representations.
9. Compare properties of two functions
each represented in a different way
(Algebraically, graphically, numerically in
tables, or by verbal descriptions.)
F-BF
Building Functions
Build a function that models a relationship
between two quantities
1. Write a function that describes a
relationship between two quantities.
a. Determine an explicit expression, a
recursive process, or steps for
calculation from a context.
b. Combine standard function types using
arithmetic operations.
c. (+) Compose functions.
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F-BF
Building Functions
F-BF
Building Functions
Build new functions from existing functions.
3. Identify the effect on the graph of
replacing f(x) by f(x) + k, kf(x), f(kx), and
f(x + k) for specific values of k (both
positive and negative); find the value of k
given the graphs. Experiment with cases
and illustrate an explanation of the effects
on the graph using technology. Include
recognizing even and odd functions from
their graphs and algebraic expressions for
them.
Build new functions from existing functions.
4. Find inverse functions.
a. Solve an equation of the form f(x) = c
for a simple function f that has an
inverse and write an expression for the
inverse.
b. (+) Verify by composition that one
function is the inverse of another.
c. (+) Read values of an inverse function
from a graph or a table, given that the
function has an inverse.
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d. (+) Produce an invertible function from
a non-invertible function by restricting
the domain.
F-BF
Building Functions
Build new functions from existing functions.
F-LE
Linear, Quadratic, and Exponential
Models
Construct and compare linear, quadratic, and
exponential models and solve problems.
5. (+) Understand the inverse relationship
between exponents and logarithms and use
this relationship to solve problems
involving logarithms and exponents.
1. Distinguish between situations that can
be modeled with linear functions and with
exponential functions.
a. Prove that linear functions grow by
equal differences over equal intervals,
and that exponential functions grow by
equal factors over equal intervals.
b. Recognize situations in which one
quantity changes at a constant rate per
unit interval relative to another.
c. Recognize situations in which a quantity
grows or decays by a constant percent
rate per unit interval relative to another.
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F-LE
Linear, Quadratic, and Exponential
Models
Construct and compare linear, quadratic, and
exponential models and solve problems.
2. Construct linear and exponential
functions, including arithmetic and
geometric sequences, given a graph, a
description of a relationship, or two inputoutput pairs (include reading these from a
table).
F-LE
Linear, Quadratic, and Exponential
Models
Construct and compare linear, quadratic, and
exponential models and solve problems.
3. Observe using graphs and tables that a
quantity increasing exponentially eventually
exceeds a quantity increasing linearly,
quadratically, or (more generally) as a
polynomial function.
F-LE
Linear, Quadratic, and Exponential
Models
Construct and compare linear, quadratic, and
exponential models and solve problems.
4. For exponential models, express as a
logarithm the solution to abct = d where a,
c, and d are numbers and the base b is 2, 10
or e; evaluate the logarithm using
technology.
Interpret expressions for functions in terms of
the situation they model.
5. Interpret the parameters in a linear or
exponential function in terms of a context.
F-LE
Linear, Quadratic, and Exponential
Models
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F-TF
Trigonometric Functions
Extend the domain of trigonometric functions
using the unit circle.
1. Understand radian measure of an angle as
the length of the arc on the unit circle
subtended by the angle.
F-TF
Trigonometric Functions
Extend the domain of trigonometric functions
using the unit circle.
2. Explain how the unit circle in the
coordinate plane enables the extension of
trigonometric functions to all real numbers,
interpreted as radian measures of angles
traversed counterclockwise around the unit
circle.
F-TF
Trigonometric Functions
Extend the domain of trigonometric functions
using the unit circle.
3. (+) Use special triangles to determine
geometrically the values of sine, cosine
tangle for π/3, π/4 and π/6, and use the
unit circle to express the values of sine,
cosines, and tangent for x, π + x, and 2 π - x
in terms of their values for x, where x is any
real number.
F-TF
Trigonometric Functions
Extend the domain of trigonometric functions
using the unit circle.
4. (+) Use the unit circle to explain
symmetry (odd or even) and periodicity of
trigonometric functions.
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F-TF
Trigonometric Functions
Model periodic phenomena with trigonometric
functions.
5. Choose trigonometric functions to model
periodic phenomena with specified
amplitude, frequency, and midline.
F-TF
Trigonometric Functions
Model periodic phenomena with trigonometric
functions.
6. (+) Understand that restricting a
trigonometric function to a domain on
which it is always increasing or always
decreasing allows its inverse to be
constructed.
F-TF
Trigonometric Functions
Model periodic phenomena with trigonometric
functions.
7. (+) Use inverse functions to solve
trigonometric equations that arise in
modeling contexts; evaluate the solutions
using technology, and interpret them in
terms of the context.
F-TF
Trigonometric Functions
Prove and apply trigonometric identities.
F-TF
Trigonometric Functions
Prove and apply trigonometric identities.
8. Prove the Pythagorean identity sin2(θ) +
cos2(θ) = 1 and use it to calculate
trigonometric ratios.
9. (+) Prove the addition and subtraction
formulas for sine, cosine, and tangent and
use them to solve problems.
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HARRISON SCHOOL DISTRICT
Curriculum Based on NJCCCS
AP Calculus
ASSESSMENTS
Enduring
Understandings
NJCCCS
Unit 1
F.IF.1.2.5.6.7a.7b.7c.7d.8a.9
F.BF.1a.1b.1c(+).3



Parentheses indicate
open interval notation
and brackets indicate
closed interval
notation.
A relation is a
function when each
member of the
domain is assigned
exactly one member
of the range.
Examples of
functions with
domain restrictions
are square root
functions because the
radicand cannot equal
a negative number
and rational functions
Essential
Questions
 What symbols
are used to
indicate open or
closed interval
notation?
Diagnostic
(before)
Anticipatory
set exercise
Formative
(during)
Summative
(after)
Lesson
closure
assessment
Cumulative
test
Unit test
 What is a
function?
Guided
practice
exercises
during lesson
 What are
examples of
functions with
domain
restrictions?
Why?
Independent
class
assignments
and
homework
 What makes a
function even or
an odd?
Sample
multiplechoice and
free-response
Quarterly
exam
Released
College
Board AP
Calculus AB
exam
Project
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May 16, 2013






because the
denominator cannot
equal zero.
Addition, subtraction,
multiplication,
division, and
composition of two
functions produce a
third function.
Functions are even if
f(x) = f(-x) and odd if
f( -x) = -f(x).
Some functions have
limits; others have no
boundaries.
The limit as x
approaches c of f(x)
exists if the limit as x
approaches c from the
left of f(x) equals the
limit as x approaches
c from the right of
f(x).
Limits that approach a
constant can be
evaluated with a table,
a graph, or
algebraically.
Limits may exist on
one side of c because
a function may be
honing in on a
different value on the
 How do you
determine if a
limit exists?
questions
(with and
without
calculator
applications)
 How do you
evaluate limits?
Quizzes
 What is a onesided limit?
 How do you
evaluate the limit
approaching
infinity of a
rational
function?
 How is a
function proven
continuous?
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May 16, 2013




other side of c or be
nonexistent on the
other side of c.
The limit of 1/x as x
approaches 0 does not
exist.
The limit of 1/x as x
approaches positive
or negative infinity is
0.
To evaluate the limit
approaching infinity
of a rational function,
divide every term by
the variable to the
highest power,
simplify each rational
expression within the
newly created
complex fraction, and
apply the limit
properties.
When the substitution
approach for finding a
limit of a rational
function results as the
indeterminate form
0/0, the process of
factoring, reducing,
then substituting can
be used to provide an
answer.
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Unit 2
F.IF.2.3.5.6.7a.7b.7c.7d.8a.8b.9
F.BF.1a.1b.1c(+)
F.TF.8

A function, f(x), is
continuous if f(x) is
defined for all values
c, if the limit as x
approaches c from the
left equals the limit as
x approaches c from
the right of f(x), and if
f(c) = f(x).

The slope of a secant
line is an average rate
of change and the
slope of a tangent
line, the derivative, is
an instantaneous rate
of change.
The difference
quotient is used to
find a general
derivative and the
derivative at a
particular point.
Derivatives of
trigonometric
functions, sine and
cosine, are proven by
the difference
quotient, angle
addition formulas,
and the limiting
process.



What is the
derivative?

What are the
derivative short
cuts?

What is
L’Hopital’s Rule?
How is it used?

Are Δx and dx
equal?

What is the
differential?

How do Δy and
dy differ?

How is linear
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




The remaining
trigonometric
derivatives are proven
by renaming using
trigonometric
relationships and
applying the product
and quotient rules.
The short cut rules
for differentiating
algebraic explicit,
algebraic implicit, and
transcendental
functions are power
rule, product rule,
quotient rule, and
chain rule.
Implicit
differentiation is used
when it’s difficult or
impossible to isolate
the y variable.
The chain rule is
applied to
differentiate a
composition of
functions.
L’Hopital’s Rule is a
derivative algorithm
used to evaluate the
limits of rational
functions with
indeterminate forms.
approximation
used?
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

Unit 3
F.IF.2.4.5.6.7a.7b.7c.7d. 8b.9
F.BF.1a.1b.1c(+).3


After using
substitution if the
limit is the
indeterminate form,
0/0, the limit can be
obtained by applying
L’Hopital’s Rule; that
is differentiating the
parts of the quotient
separately, and
substituting.
Delta x and dx
represent a horizontal
change and are equal.
Delta y is the vertical
change to a point on
the curve and dy, the
differential, is the
vertical change to the
tangent line.
Linear approximation
is used to evaluate
functions without a
calculator.
Derivatives are used 
to curve sketch
functions as well as

find related rates,
optimization, velocity,
and acceleration.
When the first
How are
derivatives used?
How do you
analyze
derivatives to
curve sketch
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



derivative equals zero
a relative maximum or
minimum point may
occur in a graph; if a
max or min occurs
the line tangent is
horizontal.
If the first derivative
is positive the
function is increasing,
if the first derivative is
negative the function
is decreasing.
When the second
derivative equals zero
the function has a
point of inflection,
which means the
graph is changing
concavity.
If the second
derivative is positive
the function is
concave up, if the
second derivative is
negative the function
is concave down.
To find vertical
asymptotes of a
rational function, set
the denominator
equal to zero and
solve for x, to find the
horizontal asymptotes
functions?

Where do cusps
or vertical
tangents occur?

What is an
absolute max and
absolute min?

Where do
absolute extrema
occur? What
method is used to
locate them?

What is the Mean
Value Theorem?

What is Rolle’s
Theorem?

What is Newton’s
Approximation
Method?

How are
optimization
problems solved?

How are
derivatives used
to describe
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May 16, 2013




take the limit as x
approaches infinity of
f(x)/g(x).
Cusps and vertical
tangents occur at
singular points, where
f '(x) is undefined.
The absolute max and
absolute min, also
known as absolute
extrema, are the
highest and lowest
points on a
continuous graph.
The absolute max and
absolute min will
occur at the critical
points on a
continuous closed
interval. These
include:
1. Points where f '(x)
= 0, called stationary
points.
2. Points where f '(x)
is undefined, called
singular points.
3. Endpoints of the
interval of definition.
To determine an
absolute extreme on
the continuous closed
interval, find the
critical points,
velocity and
acceleration?
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May 16, 2013




substitute them and
the end points in the
original function, the
highest is where the
absolute maximum
occurs and the lowest
is where the absolute
minimum occurs.
The Mean Value
Theorem states that if
f(x) is defined and
continuous on a
closed interval, [a,b]
and differentiable on
the open interval,
(a,b), there exists at
least one value c in
(a,b) where the slope
of the tangent line
equals the slope of the
secant line.
Rolle’s Theorem is a
special case of the
Mean Value Theorem,
when the slope of the
secant is zero.
Newton’s Method is
an iterative process
used to find the zeros
of a function.
A particle is stopped
(changing direction)
when the 1st
derivative, v(t), equals
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May 16, 2013

Unit 4
F.IF.2.8b.9
F.BF.1a.1b.1c(+)



zero, moving to the
right when it is
positive and moving
to the left when it is
negative.
A change in motion
occurs when the 2nd
derivative, a(t), equals
zero, speeding up
when v(t) and a(t)
have the same signs,
and slowing down
when v(t) and a(t)
have opposite signs.

Antidifferentiation is
the reverse process of
differentiation; when
given a derivative
taking its integral to

find the original
function.
Sigma notation is used 
to represent a series
which is the sum of a
sequence.
Area of an enclosed
figure can be

approximated with
inscribed rectangles,
circumscribed
rectangles, and
What is the
process of antidifferentiation?
How is sigma
notation used?
What methods
are used to
approximate
areas?
What happens
when an infinite
number of
rectangles are
used for finding
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May 16, 2013

trapezoids.
The First

Fundamental
Theorem of Calculus
states that, if f is
continuous on the
closed interval [a,b]
and F is the indefinite
integral of f on [a,b], 
then
= F (b) – F(a);
the definite integral
can be evaluated by
taking the
antiderivative of the
integrand then finding
the difference of the
substituted upper limit
value in the
antiderivative and the
lower limit value in
the antiderivative.
 A definite integral
involving a
composition of
functions can be
evaluated using usubstitution with the
same limits or by
changing the limits of
the definite integral.
 The Second
Fundamental
an irregular area?
What is the First
Fundamental
Theorem of
Calculus and
how is it used?
What is the
Second
Fundamental
Theorem of
Calculus?
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Theorem of Calculus
holds for f, a
continuous function
on an open interval,
and a any point in the
interval, and states
that d/dx (
= f(x); the derivative
of the antiderivative is
the original function.
Unit 5
F.IF.2.4.5.6.7a.7b.7c.7d.9
F.BF.1a.1b.1c(+)
F.LE.2.5




The Mean Value
Theorem for Integrals
states that if f(x) is
continuous on the
closed interval [a,b]
and differentiable on 
the open interval,
(a,b), there exists a
value c such that
F(c) =
)dx.
The definite integral is
used to find the area 
under a curve and the
area between curves.
Volume of a solid of
revolution around the
x- and y- axes is
obtained by using the
disk, washer, or shell
methods.

What is the Mean
Value Theorem
for Integral
Calculus?
How can the
First
Fundamental
Theorem of
Calculus be used
to find area?
What methods
can be used to
find volume of a
solid of
revolution
around the xand y- axes?
How do you
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


Unit 6
F.IF.2.4.5.6.7e.8b.9
F.BF.1a.1b.1c(+).4a.4c(+).4d(+).5(+)
F.LE.1a,1b,1c.2.3.4.5


Volume of the solid
with known cross
sections is obtained
by first determining
the area from the
shape of one cross

section perpendicular
to an axis, then using
the definite integral to
find the combined
areas of all the cross
sections.
Position of a particle 
and distance traveled
are found by using a
definite integral.
To find displacement
take the definite
integral of v(t)d(t);
distance traveled is
obtained by taking the
definite integral of the
absolute value of
v(t)d(t).
find volume of a
solid with
known cross
sections?
Euler’s, e, is the limit 
as n approaches
infinity of (1+ 1/n)n. 
Logarithms are
exponents; properties
of logarithms are used
What is e?
How do you find
the position of a
particle when the
velocity or
acceleration is
given?
What is the
difference
between
displacement and
distance
traveled? How
do you calculate
distance
traveled?
What are
logarithms?
How are they
used?
Page 26 of 86
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






to perform
computations more
easily.
The log ex is the same
as the natural log of x,
ln x.
The inverse of f(x) =
ex is f (x) = ln x.
Properties of
common and natural
logarithms are used to
expand or contract
logarithmic
expressions to
equivalent forms.
The change of base
formula enables
renaming any base to
a calculator friendly
base.
The derivative of ln u
is (1/u)(du/dx); the
integral of du/u is
ln│ u│+ c.
Logarithmic
differentiation is the
application of the
properties of
logarithms to simplify
or enable the process
of taking the
derivative.
The derivative of eu is

What is the
inverse of f(x) =
ex? What do
their graphs look
like?

How is the
change of base
formula useful?

What is the
derivative of the
ln u? What is the
integral of du/u?

Why use
logarithmic
differentiation?

What is the
derivative and
antiderivative of
eu? The
exponential
function, bu ?

What method is
used to find the
derivative of y =
f(x)g(x)?

How do you
solve a first
Page 27 of 86
May 16, 2013




eu(du/dx); the integral
of eu du is eu +c.
The derivative of bu is
bu ln b (du/dx); the
integral of bu du is
( bu/ln b)+ c.

To find the derivative
of y = f(x)g(x), (use
logarithmic
differentiation) take

the natural log of both
sides of the equation,
use the properties of
logs to enable the
differentiation
process, differentiate,
then isolate dy/dx,
and rename the
derivative explicitly.
To solve an initial
value problem, set up
the differential
equation, cross
multiply to isolate dy,
take the antiderivative
of both sides, then
substitute the initial
value.
A slope field, also
called direction field,
is a graphical
representation of the
solutions of a first
order differential
order differential
equation initial
value problem?
What is a slope
field? Direction
curve?
What is Euler’s
method?
Page 28 of 86
May 16, 2013
equation. A direction
curve is traced
through an initial
value in a slope field.
 Euler’s Rule is the
simplest iterative
numerical method for
solving a first order
differential equation; it
gives exact solutions
of slopes in a slope
field.
Unit 7
F.IF.2.4.5.6.8b.9
F.BF.1a.1b.1c(+).4a.4b(+).4c(+).4d(+)
F.TF.1.2.3(+).4(+).5.6(+).7(+).8.9(+)
 If restrictions are put 
on the domain and
range, trigonometric
functions will have
inverses and the
inverses will be

functions.
 Inverse trigonometric
functions have
derivatives and the
derivatives have
integrals.
 By parts integration is 
used on a product of
functions that cannot
be integrated in its
present form.
Do the
trigonometric
functions have
inverses?
Do inverse
trigonometric
functions have
derivatives? Do
the derivatives
have integrals?
When would you
use by parts
integration?
Page 29 of 86
May 16, 2013
HARRISON SCHOOL DISTRICT
Pacing Guide
AP Calculus
UNIT
Unit 1
Functions
and
Limits
UNIT OBJECTIVES & ENDURING UNDERSTANDINGS
Unit Objectives:
Students will know and do:
 Use properties and apply the algebra to solve equations and
inequalities.
 Use set and interval notation.
 Define function, domain, and range.
 Find domain, range, zeros, inverse, and composition of functions.
 Define even- and odd-functions.
 Graph functions that include linear, quadratic, polynomial, rational,
radical, trigonometric, absolute value, and piecewise functions.
 Find limits of functions reading graphs, using tables or algebra.
 Find right- and left-hand limits.
 Identify indeterminate forms, limits 0/0, and find limits using
algebraic techniques.
 Define and show continuity or discontinuity.
 Prove lim
sin Ө = 1
Ө→0
Ө
and


TIME FRAME
Block Schedule Pacing
8 days
lim
cos Ө – 1 = 0
Ө→0
Ө
.
Find vertical and horizontal asymptotes of rational functions.
Graph rational functions, y=1/f(x) and y=f(x)/g(x) using limits.
Page 30 of 86
May 16, 2013

Practice multiple-choice and free-response questions (with and
without calculator applications) pertaining to the unit.
Enduring Understandings:
 Parentheses indicate open interval notation and brackets indicate
closed interval notation.
 A relation is a function when each member of the domain is assigned
exactly one member of the range.
 Examples of functions with domain restrictions are square root
functions because the radicand cannot equal a negative number and
rational functions because the denominator cannot equal zero.
 Addition, subtraction, multiplication, division, and composition of
two functions produce a third function.
 Functions are even if f(x) = f(-x) and odd if f( -x) = -f(x).
 Some functions have limits; others have no boundaries.
 The limit as x approaches c of f(x) exists if the limit as x approaches
c from the left of f(x) equals the limit as x approaches c from the
right of f(x).
 Limits that approach a constant can be evaluated with a table, a
graph, or algebraically.
 Limits may exist on one side of c because a function may be honing
in on a different value on the other side of c or be nonexistent on the
other side of c.
 The limit of 1/x as x approaches 0 does not exist.
 The limit of 1/x as x approaches positive or negative infinity is 0.
 To evaluate the limit approaching infinity of a rational function,
divide every term by the variable to the highest power, simplify each
rational expression within the newly created complex fraction, and
apply the limit properties.
 When the substitution approach for finding a limit of a rational
function results as the indeterminate form 0/0, the process of
factoring, reducing, then substituting can be used to provide an
Page 31 of 86
May 16, 2013

Unit 2
Differentiation
answer.
A function, f(x), is continuous if f(x) is defined for all values c, if the
limit as x approaches c from the left equals the limit as x approaches
c from the right of f(x), and if f(c) = f(x).
Unit Objectives:
Students will know and do:
 Find tangent lines as instantaneous rates of change to a graph.
 Define the derivative.
 Find derivatives of functions using the difference quotient,
f ´(x)= lim f(x+h) – f(x)
h→0
h
,
f ´(a)= lim
x→a
f ´(a)=








lim
h→0
f(x) – f(a)
x–a
Block Schedule Pacing
12 days
and
f(a+h) – f(a)
h
.
Calculate the slope of a curve at a point.
Find an equation of a tangent line to a curve at a point.
Determine points of non-differentiability, such as corners, vertical
tangents, and cusps.
Use differentiation rules, power, product, quotient, and chain rule on
algebraic explicit and implicit functions.
Find nth derivatives.
Find derivatives of trigonometric functions including proofs.
Use L’Hopital’s Rule on indeterminate forms. (optional to College
Board AB exam)
Practice multiple-choice and free-response questions (with and
without calculator applications) pertaining to the unit and prior unit.
Page 32 of 86
May 16, 2013
Enduring Understandings:
 The slope of a secant line is an average rate of change and the slope
of a tangent line, the derivative, is an instantaneous rate of change.
 The difference quotient is used to find a general derivative and the
derivative at a particular point.
 Derivatives of trigonometric functions, sine and cosine, are proven
by the difference quotient, angle addition formulas, and the limiting
process.
 The remaining trigonometric derivatives are proven by renaming
using trigonometric relationships and applying the product and
quotient rules.
 The short cut rules for differentiating algebraic explicit, algebraic
implicit, and transcendental functions are power rule, product rule,
quotient rule, and chain rule.
 Implicit differentiation is used when it’s difficult or impossible to
isolate the y variable.
 The chain rule is applied to differentiate a composition of functions.
 L’Hopital’s Rule is a derivative algorithm used to evaluate the limits
of rational functions with indeterminate forms. After using
substitution if the limit is the indeterminate form, 0/0, the limit can
be obtained by applying L’Hopital’s Rule; that is differentiating the
parts of the quotient separately, and substituting.
 Delta x and dx represent a horizontal change and are equal. Delta y
is the vertical change to a point on the curve and dy, the differential,
is the vertical change to the tangent line.
 Linear approximation is used to evaluate functions without a
calculator.
Page 33 of 86
May 16, 2013
Unit 3
Applications
of
the Derivative
Unit Objectives:
Students will know and do:
 Find differentials and compare ∆y to dy.
 Use linear approximation to estimate functions.
 Identify critical points, stationary and singular points, and use
number line analysis to justify and describe the increase/decrease of
functions.
 Find points of inflection and use number line analysis to justify and
describe concavity.
 Find relative and absolute maximum and minimum values.
 Curve sketch quadratic, polynomial, rational, and radical functions
using first and second derivatives.
 Tell information about a function from the graph of its 1st or 2nd
derivative.
 Find related rates, velocity, speed, acceleration, and instantaneous
rates of change of functions relevant to real life problems.
 Solve optimization problems relevant to real life situations.
 Use Rolle’s Theorem and the Mean Value Theorem.
 Apply Newton’s Method to find roots of equations. (optional to
College Board AB exam)
 Practice multiple-choice and free-response questions (with and
without calculator applications) pertaining to the unit and prior units.
Block Schedule Pacing
9 days
Enduring Understandings:
 Derivatives are used to curve sketch functions as well as find related
rates, optimization, velocity, and acceleration.
 When the first derivative equals zero a relative maximum or
minimum point may occur in a graph; if a max or min occurs the line
tangent is horizontal.
 If the first derivative is positive the function is increasing, if the first
derivative is negative the function is decreasing.
 When the second derivative equals zero the function has a point of
Page 34 of 86
May 16, 2013











inflection, which means the graph is changing concavity.
If the second derivative is positive the function is concave up, if the
second derivative is negative the function is concave down.
To find vertical asymptotes of a rational function, set the
denominator equal to zero and solve for x, to find the horizontal
asymptotes take the limit as x approaches infinity of f(x)/g(x).
Cusps and vertical tangents occur at singular points, where f '(x) is
undefined.
The absolute max and absolute min, also known as absolute extrema,
are the highest and lowest points on a continuous graph.
The absolute max and absolute min will occur at the critical points
on a continuous closed interval. These include:
1. Points where f '(x) = 0, called stationary points.
2. Points where f '(x) is undefined, called singular points.
3. Endpoints of the interval of definition.
To determine an absolute extrema on the continuous closed interval,
find the critical points, substitute them and the end points in the
original function, the highest is where the absolute maximum occurs
and the lowest is where the absolute minimum occurs.
The Mean Value Theorem states that if f(x) is defined and
continuous on a closed interval, [a,b] and differentiable on the open
interval, (a,b), there exists at least one value c in (a,b) where the
slope of the tangent line equals the slope of the secant line.
Rolle’s Theorem is a special case of the Mean Value Theorem, when
the slope of the secant is zero.
Newton’s Method is an iterative process used to find the zeros of a
function.
A particle is stopped (changing direction) when the 1st derivative,
v(t), equals zero, moving to the right when it is positive and moving
to the left when it is negative.
A change in motion occurs when the 2nd derivative, a(t), equals zero,
speeding up when v(t) and a(t) have the same signs, and slowing
down when v(t) and a(t) have opposite signs.
Page 35 of 86
May 16, 2013
Unit 4
Integration
Unit Objectives:
Block Schedule Pacing
Students will know and do:
11 days
 Find antiderivatives using the power algorithm.
 Find antiderivatives of trigonometric functions.
 Evaluate definite integrals.
 Approximate the area under a curve using Riemann Sums, inscribed/
circumscribed rectangles, and the trapezoidal rule.
 Use the First Fundamental Theorem of Calculus on definite
integrals.
 Antidifferentiate an indefinite integral using u-substitution.
 Evaluate a definite integral using substitution with the same limits or
by changing the limits of the definite integral.
 Use the Second Fundamental Theorem of Calculus.
 Practice multiple-choice and free-response questions (with and
without calculator applications) pertaining to the unit and prior units.
Enduring Understandings:
 Antidifferentiation is the reverse process of differentiation; when
given a derivative taking its integral to find the original function.
 Sigma notation is used to represent a series which is the sum of a
sequence.
 Area of an enclosed figure can be approximated with inscribed
rectangles, circumscribed rectangles, or trapezoids.
 The First Fundamental Theorem of Calculus states that, if f is
continuous on the closed interval [a,b] and F is the indefinite integral
of f on [a,b], then
= F (b) – F(a) ); the definite integral can
be evaluated by taking the antiderivative of the integrand then
finding the difference of the substituted upper limit value in the
antiderivative and the lower limit value in the antiderivative.
 A definite integral involving a composition of functions can be
evaluated using u-substitution with the same limits or by changing
the limits of the definite integral.
Page 36 of 86
May 16, 2013

Unit 5
Applications
of the
Definite Integral
The Second Fundamental Theorem of Calculus holds for f, a
continuous function on an open interval, and a any point in the
interval, and states that d/dx (
= f(x); the derivative of the
antiderivative is the original function.
Unit Objectives:
Students will know and do:
 Use the Mean-Value Theorem for Integrals to find the average value
of a function.
 Find area under a curve and between curves using the definite
integral.
 Find volume of a solid of revolution around x- or y-axes using disk,
washer, and shell methods.
 Find volume of a solid with known cross sections.
 Apply integral calculus to find position of a particle and distance
traveled relevant to real life problems.
 Practice multiple-choice and free-response questions (with and
without calculator applications) pertaining to the unit and prior units.
Block Schedule Pacing
8 days
Enduring Understandings:
 The Mean Value Theorem for Integrals states that if f(x) is
continuous on the closed interval [a,b] and differentiable on the open
interval, (a,b), there exists a value c such that
F(c) =



)dx.
The definite integral is used to find the area under a curve and the
area between curves.
Volume of a solid of revolution around the x- and y- axes is obtained
by using the disk, washer, or shell methods.
Area is determined from the shape of one cross section
perpendicular to the x- or y-axis. Using the definite integral to find
the combined area of all the cross sections, volume of the solid is
obtained.
Page 37 of 86
May 16, 2013


Unit 6
Logarithmic
and
Exponential
Functions
Position of a particle and distance traveled are found by using a
definite integral.
To find displacement take the definite integral of v(t)d(t); distance
traveled is obtained by taking the definite integral of the absolute
value of v(t)d(t).
Unit Objectives:
Students will know and do:
 Define e as a limit.
 Define f(x) = 1n x.
 Prove the derivative of the ln x and ex.
 Find derivatives of the functions eu, bu, and ln u and integrals of eu,
bu and du/u.
 Find derivatives of y = f(x)g(x).
 Use logarithmic differentiation.
 Apply all calculus studied previously using exponential and
logarithmic functions.
 Apply first order separable differential equations to solve initial-value
problems.
 Find exponential growth and decay relevant to real life situations.
 Graph slope-fields and find direction curves as a graphical approach
to solve differential equations.
 Use Euler’s Method as a numerical approach to solve differential
equations. (optional to College Board AB exam)
 Practice multiple-choice and free-response questions (with and
without calculator applications) pertaining to the unit and prior units.
Block Schedule Pacing
12 days
Enduring Understandings:
 Euler’s, e, is the limit as n approaches infinity of (1+ 1/n)n.
 Logarithms are exponents; properties of logarithms are used to
perform computations more easily.
 The log ex is the same as the natural log of x, ln x.
Page 38 of 86
May 16, 2013


The inverse of f(x) = ex is f (x) = ln x.
Properties of common and natural logarithms are used to expand or
contract logarithmic expressions to equivalent forms.
 The change of base formula enables renaming any base to a
calculator friendly base.
 The derivative of ln u is (1/u)(du/dx); the integral of du/u is
ln│ u│+ c.
 Logarithmic differentiation is the application of the properties of
logarithms to simplify the process of taking the derivative.
 The derivative of eu is eu(du/dx); the integral of eu du is eu +c.
 The derivative of bu is bu ln b (du/dx); the integral of bu du is
( bu/ln b)+ c.
 To find the derivative of y = f(x)g(x), take the natural log of both sides
of the equation, use the properties of logs to enable the
differentiation process, differentiate, then isolate dy/dx and rename
the derivative explicitly.
 To solve an initial value problem, set up the differential equation,
cross multiply to isolate dy, take the antiderivative of both sides, then
substitute the initial value.
 A slope field, also called direction field, is a graphical representation
of the solutions of a first order differential equation. A direction
curve is traced through an initial value in a slope field.
 Euler’s Rule is the simplest iterative numerical method for solving a
first order differential equation; it gives exact solutions of slopes in a
slope field.
Block Schedule Pacing
6 days
Page 39 of 86
May 16, 2013
Unit 7
Inverse
Trigonometric
Functions
Unit Objectives:
Students will know and do:
 Prove derivatives of inverse trigonometric functions.
 Find derivatives of inverse trigonometric functions.
 Find anti-derivatives of inverse trigonometric functions.
 Apply all of calculus studied previously using inverse trigonometric
functions.
 Integrate functions by parts. (optional to College Board AB exam)
 Practice multiple-choice and free-response questions (with and
without calculator applications) pertaining to the unit and prior units.
Enduring Understandings:
 If restrictions are put on the domain and range, trigonometric
functions will have inverses and the inverses will be functions.
 Inverse trigonometric functions have derivatives and the derivatives
have integrals.
 By parts integration is used on a product of functions that cannot be
integrated in its present form.
Page 40 of 86
May 16, 2013
HARRISON SCHOOL DISTRICT
AP Calculus AB
Unit 1-Functions and Limits
UNIT GOAL(S): Students will know and do:
DURATION:

Use properties and apply the algebra to solve equations and inequalities.
Block Schedule Pacing

Use set and interval notation.
8 days

Define function, domain, and range.

Find domain, range, zeros, inverse, and composition of functions.

Define even- and odd-functions.

Graph functions that include linear, quadratic, polynomial, radical, trigonometric, absolute value,
and piecewise functions.

Find limits of functions reading graphs, using tables or algebra.

Find right- and left-hand limits.

Identify indeterminate forms, limits 0/0, and find limits using algebraic techniques.

Define and show continuity or discontinuity.
Page 41 of 86
May 16, 2013

Prove lim
Ө→0
sin Ө = 1
Ө
and
lim
Ө→0
cos Ө – 1 = 0.
Ө

Find vertical and horizontal asymptotes of rational functions.

Graph rational functions, y=1/f(x) and y=f(x)/g(x) using limits.

Practice multiple-choice and free-response questions (with and without calculator applications)
pertaining to the unit.
NJCCCS:
F.IF.1.2.5.6.7a.7b.7c.7d.8a.9, F.BF.1a.1b. 1c(+).3
ENDURING UNDERSTANDINGS:



Parentheses indicate open interval notation
and brackets indicate closed interval
notation.
A relation is a function when each member
of the domain is assigned exactly one
member of the range.
Examples of functions with domain
restrictions are square root functions
because the radicand cannot equal a
negative number and rational functions
ESSENTIAL QUESTIONS:
 What symbols are used to indicate open or closed interval notation?
 What is a function?
 What are examples of functions with domain restrictions? Why?
 What makes a function even or an odd?
 How do you determine if a limit exists?
Page 42 of 86
May 16, 2013








because the denominator cannot equal zero.
Addition, subtraction, multiplication,

division, and composition of two functions
produce a third function.

Functions are even if f(x) = f(-x) and odd if

f( -x) = -f(x).
Some functions have limits; others have no

boundaries.
The limit as x approaches c of f(x) exists if
the limit as x approaches c from the left of
f(x) equals the limit as x approaches c from
the right of f(x).
Limits that approach a constant can be
evaluated with a table, a graph, or
algebraically.
Limits may exist on one side of c because a
function may be honing in on a different
value on the other side of c or be
nonexistent on the other side of c.
The limit of 1/x as x approaches 0 does not
exist.
The limit of 1/x as x approaches positive or
negative infinity is 0.
How do you evaluate limits?
What is a one-sided limit?
How do you evaluate the limit approaching infinity of a rational function?
How is a function proven continuous?
Page 43 of 86
May 16, 2013



To evaluate the limit approaching infinity of
a rational function, divide every term by the
variable to the highest power, simplify each
rational expression within the newly created
complex fraction, and apply the limit
properties.
When the substitution approach for finding
a limit of a rational function results as the
indeterminate form 0/0, the process of
factoring, reducing, then substituting can be
used to provide an answer.
A function, f(x), is continuous if f(x) is
defined for all values c, if the limit as x
approaches c from the left equals the limit
as x approaches c from the right of f(x), and
if f(c) = f(x).
Page 44 of 86
May 16, 2013
Guiding Question
Content, Concepts and
Skill
Instructional Materials
and Resources
Instructional Strategies

What methods
are used to solve
absolute value
rational and
quadratic
equations and
inequalities?
Students will know and
do:
Textbooks
Demonstration of sample
problems
 Use properties and
apply the algebra to
solve equations and
inequalities.
Worksheets involving
multiple choice, fill in,
and open-ended
questions.
What is set and
interval notation?
 Use set and interval
Whiteboard
What are the
definitions of
function, domain,
and range?
 Define function,
domain, and range.





How do you find
domain, range,
zeros, inverse,
and composition
of functions?
What are evenand oddfunctions?
What techniques
are used for
notation.
 Find domain, range,
zeros, inverse, and
composition of
functions.
 Define even- and oddfunctions.
 Graph functions that
include linear,
quadratic, polynomial,
radical, trigonometric,
absolute value, and
piecewise functions.
Workbooks
Overhead projector and
screen
Overhead transparencies
Vis-Tablet
Television
CDs/DVDs/Video tapes
Scientific and graphic
calculators
Computers and software
Manipulative devises
Communicator kit
Guided practice
Discovery approach
Cooperative
learning/group work
Assessments
Guided practice with
overt or covert
responses
Check and grade
homework /class
assignments
Notebook examination
Individualized attention
Quizzes
Question and answer
Projects
CDs/DVDs/Video
presentations
Tests
Vis-Tablet/TV
presentations
Exams
Power point
presentations
Computer laboratory
activities
Internet activities
Homework and class
assignments
Page 45 of 86
May 16, 2013
graphing
functions that
include linear,
quadratic,
polynomial,
radical,
trigonometric,
absolute value,
and piecewise
functions?





How do you find
limits of
functions ?
Do one sided
limits exist?
 Find limits of
functions reading
graphs, using tables or
algebra.
How do you
Guest speakers
 Find right- and lefthand limits.
 Identify indeterminate
forms, limits 0/0, and
find limits using
algebraic techniques.
 Define and show
continuity or
discontinuity.
 Prove
How can you find
lim
limits of
functions with
Ө→0
indeterminate
and
form, 0/0?
What is
continuity or
discontinuity and
how do you
prove it?
Field trips
lim
Ө→0
sin Ө = 1
Ө
cos Ө – 1 = 0.
Ө
 Find vertical and
horizontal asymptotes
of rational functions.
Page 46 of 86
May 16, 2013
prove
lim
Ө→0
sin Ө = 1
Ө
and
lim
Ө→0
cos Ө – 1 = 0?
Ө

How do you find
vertical and
horizontal
asymptotes of
rational
functions?

Graph y=1/f(x)
and y=f(x)/g(x)
using limits.
 Graph rational
functions, y=1/f(x)
and y=f(x)/g(x) using
limits.
 Practice multiplechoice and freeresponse questions
(with and without
calculator applications)
pertaining to the unit.
Page 47 of 86
May 16, 2013
HARRISON SCHOOL DISTRICT
AP Calculus
Unit 2-Differentiation
UNIT GOAL(S): Students will know and do:
DURATION:

Find tangent lines as instantaneous rates of change to a graph.
Block Schedule Pacing

Define the derivative.
12 days

Find derivatives of functions using the difference quotient,
f ´(x)= lim
h→0
f ´(a)= lim
x→a
f ´(a)=
lim
h→0

f(x+h) – f(x)
h
,
f(x) – f(a)
x–a
and
f(a+h) – f(a)
h
.
Calculate the slope of a curve at a point.
Page 48 of 86
May 16, 2013

Find an equation of a tangent line to a curve at a point.

Determine points of non-differentiability, such as corners, vertical tangents, and cusps.

Use differentiation rules, power, product, quotient, and chain rule on algebraic explicit and
implicit functions.

Find nth derivatives.

Find derivatives of trigonometric functions including proofs.

Use L’Hopital’s Rule on indeterminate forms. (optional to College
Board AB exam)

Practice multiple-choice and free-response questions (with and without calculator applications)
pertaining to the unit and prior unit.
NJCCCS:
F.IF.2.3.5.6.7a.7b.7c.7d.8a.8b.9, F.BF.1a.1b.1c(+), F.TF.8
ENDURING UNDERSTANDINGS:




The slope of a secant line is an average rate of
change and the slope of a tangent line, the
derivative, is an instantaneous rate of change.
The difference quotient is used to find a general
derivative and the derivative at a particular point.
Derivatives of trigonometric functions, sine and
cosine, are proven by the difference quotient,
angle addition formulas, and the limiting process.
The remaining trigonometric derivatives are
ESSENTIAL QUESTIONS:
 What is the derivative?

What are the derivative short cuts?

What is L’Hopital’s Rule? How is it used?

Are Δx and dx equal?
Page 49 of 86
May 16, 2013






proven by renaming using trigonometric

relationships and applying the product and
quotient rules.

The short cut rules for differentiating algebraic
explicit, algebraic implicit, and transcendental

functions are power rule, product rule, quotient
rule, and chain rule.
Implicit differentiation is used when it’s difficult
or impossible to isolate the y variable.
The chain rule is applied to differentiate a
composition of functions.
L’Hopital’s Rule is a derivative algorithm used to
evaluate the limits of rational functions with
indeterminate forms. After using substitution if
the limit is the indeterminate form, 0/0, the limit
can be obtained by applying L’Hopital’s Rule;
that is differentiating the parts of the quotient
separately, and substituting.
Delta x and dx represent a horizontal change and
are equal. Delta y is the vertical change to a
point on the curve and dy, the differential, is the
vertical change to the tangent line.
Linear approximation is used to evaluate
functions without a calculator.
What is the differential?
How do Δy and dy differ?
How is linear approximation used?
Page 50 of 86
May 16, 2013
Guiding Question
Content, Concepts and
Skill
Instructional Materials
and Resources
Instructional Strategies

Students will know and
do:
Textbooks
Demonstration of sample
problems





How do you find
tangent lines as
instantaneous
rates of change to
a graph?
What is the
derivative?
What is the
difference
quotient?
How do you
calculate the slope
of a curve at a
point?
How do you find
an equation of a
tangent line to a
curve at a point?
How are points
of nondifferentiability,
such as corners
determined?
Workbooks
 Calculate the numerical
Worksheets involving
slope and an angle of
multiple choice, fill in,
inclination.
and open-ended
questions.
 Determine parallel and
perpendicular lines.
Whiteboard
 Find tangent lines as
instantaneous rates of
change to a graph.
Overhead projector and
screen
Overhead transparencies
 Define the derivative.
Vis-Tablet
 Find derivatives of
functions using the
difference quotient,
Television
CDs/DVDs/Video tapes
f ´(x)= lim f(x+h) – f(x)
h→0
h
,
Scientific and graphic
calculators
Computers and software
f ´(a)= lim
x→a
f(x) – f(a)
x–a
Manipulative devises
Communicator kit
Guided practice
Discovery approach
Cooperative
learning/group work
Assessments
Guided practice with
overt or covert
responses
Check and grade
homework /class
assignments
Notebook examination
Individualized attention
Quizzes
Question and answer
Projects
CDs/DVDs/Video
presentations
Tests
Vis-Tablet/TV
presentations
Exams
Power point
presentations
Computer laboratory
activities
Internet activities
Homework and class
assignments
Page 51 of 86
May 16, 2013




What are the
short cut rules for
differentiation?
How do you find
nth derivatives?
What are the
derivatives of
trigonometric
functions?
and
Field trips
f ´(a)= lim f(a+h) – f(a)
Guest speakers
h→0
h
.
 Calculate the slope of a
curve at a point.
 Use slope-intercept
form or point-slope
formula to write a
linear equation.
What is
L’Hopital’s Rule
 Find an equation of a
and how is it used?
tangent line to a curve
(optional to
at a point.
College Board
 Determine points of
AB exam)
non-differentiability,
such as corners,
vertical tangents, and
cusps.
 Use differentiation
rules, power, product,
quotient, and chain
rule on algebraic
explicit and implicit
functions.
 Find nth derivatives.
Page 52 of 86
May 16, 2013
 Use trigonometric
ratios and identities to
manipulate, simplify
and evaluate
expressions.
 Apply trigonometric
angle addition and
double angle formulas.
 Find derivatives of
trigonometric
functions including
proofs.
 Use L’Hopital’s Rule on
indeterminate forms.
(optional to College
Board AB exam)
 Practice multiplechoice and freeresponse questions
(with and without
calculator applications)
pertaining to the unit
and prior unit.
Page 53 of 86
May 16, 2013
HARRISON SCHOOL DISTRICT
AP Calculus
Unit 3-Applications of the Derivative
UNIT GOAL(S): Students will know and do:
DURATION:

Find differentials and compare ∆y to dy.
Block Schedule Pacing

Use linear approximation to estimate functions.
9 days

Identify critical points, stationary and singular points, and use number line analysis to justify and
describe the increase/decrease of functions.

Find points of inflection and use number line analysis to justify and describe concavity.

Find relative and absolute maximum and minimum values.

Curve sketch quadratic, polynomial, rational, and radical functions using first and second
derivatives.

Tell information about a function from the graph of its 1st or 2nd derivative.

Find related rates, velocity, speed, acceleration, and instantaneous rates of change of functions
relevant to real life problems.

Solve optimization problems relevant to real life situations.

Use Rolle’s Theorem and the Mean Value Theorem.
Page 54 of 86
May 16, 2013

Apply Newton’s Method to find roots of equations. (optional to College Board AB exam)

Practice multiple-choice and free-response questions (with and without calculator applications)
pertaining to the unit and prior units.
NJCCCS:
F.IF.2.4.5.6.7a.7b.7c.7d. 8b.9
F.BF.1a.1b. 1c(+).3
ENDURING UNDERSTANDINGS:





Derivatives are used to curve sketch
functions as well as find related rates,
optimization, velocity, and acceleration.
When the first derivative equals zero a
relative maximum or minimum point may
occur in a graph; if a max or min occurs the
line tangent is horizontal.
If the first derivative is positive the function
is increasing, if the first derivative is
negative the function is decreasing.
When the second derivative equals zero the
function has a point of inflection, which
means the graph is changing concavity.
If the second derivative is positive the
function is concave up, if the second
derivative is negative the function is
concave down.
ESSENTIAL QUESTIONS:

How are derivatives used?

How do you analyze derivatives to curve sketch functions?

Where do cusps or vertical tangents occur?

What is an absolute max and absolute min?

Where do absolute extrema occur? What method is used to locate them?

What is the Mean Value Theorem?

What is Rolle’s Theorem?

What is Newton’s Approximation Method?
Page 55 of 86
May 16, 2013






To find vertical asymptotes of a rational

function, set the denominator equal to zero
and solve for x, to find the horizontal

asymptotes take the limit as x approaches
infinity of f(x)/g(x).
Cusps and vertical tangents occur at
singular points, where f '(x) is undefined.
The absolute max and absolute min, also
known as absolute extrema, are the highest
and lowest points on a continuous graph.
The absolute max and absolute min will
occur at the critical points on a continuous
closed interval. These include:
1. Points where f '(x) = 0, called stationary
points.
2. Points where f '(x) is undefined, called
singular points.
3. Endpoints of the interval of definition.
To determine an absolute extrema on the
continuous closed interval, find the critical
points, substitute them and the end points
in the original function, the highest is where
the absolute maximum occurs and the
lowest is where the absolute minimum
occurs.
The Mean Value Theorem states that if f(x)
is defined and continuous on a closed
interval, [a,b] and differentiable on the open
interval, (a,b), there exists at least one value
How are optimization problems solved?
How are derivatives used to describe velocity and acceleration?
Page 56 of 86
May 16, 2013




c in (a,b) where the slope of the tangent
line equals the slope of the secant line.
Rolle’s Theorem is a special case of the
Mean Value Theorem, when the slope of
the secant is zero.
Newton’s Method is an iterative process
used to find the zeros of a function.
A particle is stopped (changing direction)
when the 1st derivative, v(t), equals zero,
moving to the right when it is positive and
moving to the left when it is negative.
A change in motion occurs when the 2nd
derivative, a(t), equals zero, speeding up
when v(t) and a(t) have the same signs, and
slowing down when v(t) and a(t) have
opposite signs.
Page 57 of 86
May 16, 2013
Guiding Question
Content, Concepts and
Skill
Instructional Materials
and Resources
Instructional Strategies

Students will know and
do:
Textbooks
Demonstration of sample
problems



What is the
differential and
how does ∆y and
dy compare?
How is linear
approximation
used to estimate
functions?
How do you
identify critical
points, stationary
and singular
points, and use
number line
analysis to justify
and describe the
increase/decrease
of functions?
How do you
identify points of
inflection and use
number line
analysis to justify
and describe
 Find differentials and
compare ∆y to dy.
 Convert angles
between degrees and
radians for
computational
purposes.
 Use linear
approximation to
estimate functions,
such as trigonometric
and square root
functions.
 Identify critical points,
stationary and singular
points, and use
number line analysis to
justify and describe the
increase/decrease of
functions.
 Find points of
Workbooks
Worksheets involving
multiple choice, fill in,
and open-ended
questions.
Whiteboard
Overhead projector and
screen
Overhead transparencies
Vis-Tablet
Television
CDs/DVDs/Video tapes
Scientific and graphic
calculators
Computers and software
Manipulative devises
Communicator kit
Guided practice
Discovery approach
Cooperative
learning/group work
Assessments
Guided practice with
overt or covert
responses
Check and grade
homework /class
assignments
Notebook examination
Individualized attention
Quizzes
Question and answer
Projects
CDs/DVDs/Video
presentations
Tests
Vis-Tablet/TV
presentations
Exams
Power point
presentations
Computer laboratory
activities
Internet activities
Homework and class
assignments
Page 58 of 86
May 16, 2013
concavity?




How do you
identify relative
and absolute
maximum and
minimum values?
What methods
are used to curve
sketch quadratic,
polynomial,
rational, and
radical functions
using first and
second
derivatives?
What information
is described from
the graph of a
function’s 1st or
2nd derivative?
How do you find
related rates,
velocity, speed,
acceleration, and
instantaneous
rates of change of
inflection and use
number line analysis to
justify and describe
concavity.
Field trips
Guest speakers
 Find relative and
absolute maximum and
minimum values.
 Curve sketch
quadratic, polynomial,
rational, and radical
functions using first
and second derivatives.
 Tell information about
a function from the
graph of its 1st or 2nd
derivative.
 Apply the Pythagorean
Theorem to find a
missing part of a right
triangle.
 Find related rates,
velocity, speed,
acceleration, and
instantaneous rates of
change of functions
relevant to real life
Page 59 of 86
May 16, 2013
functions?
problems.

How are
optimization
problems solved?
 Solve optimization
problems relevant to
real life situations.

What is Rolle’s
Theorem and the
Mean Value
Theorem?
 Use Rolle’s Theorem
and the Mean Value
Theorem.

 Apply Newton’s
Method to find roots of
equations.
(optional to College
Board AB exam)
How is Newton’s
Method used to
find roots of
equations?
(optional to College
Board AB exam)  Practice multiplechoice and freeresponse questions
(with and without
calculator applications)
pertaining to the unit
and prior units.
Page 60 of 86
May 16, 2013
HARRISON SCHOOL DISTRICT
AP Calculus
Unit 4-Integration
UNIT GOAL(S): Students will know and do:
DURATION:

Find antiderivatives using the power algorithm.
Block Schedule Pacing

Find antiderivatives of trigonometric functions.
11 days

Evaluate definite integrals.

Approximate the area under a curve using Riemann Sums, inscribed/ circumscribed rectangles,
and the trapezoidal rule.

Use the First Fundamental Theorem of Calculus on definite integrals.

Antidifferentiate an indefinite integral using u-substitution.

Evaluate a definite integral using substitution with the same limits or by changing the limits of
the definite integral.

Use the Second Fundamental Theorem of Calculus.

Practice multiple-choice and free-response questions (with and without calculator applications)
pertaining to the unit and prior units.
Page 61 of 86
May 16, 2013
NJCCCS:
F.IF.2.8b.9
F.BF.1a.1b.1c(+)
ENDURING UNDERSTANDINGS:






Antidifferentiation is the reverse process of
differentiation; when given a derivative taking its
integral to find the original function.
Sigma notation is used to represent a series which is the
sum of a sequence.
Area of an enclosed figure can be approximated with
inscribed rectangles, circumscribed rectangles, or
trapezoids.
The First Fundamental Theorem of Calculus states
that, if f is continuous on the closed interval [a,b] and F
ESSENTIAL QUESTIONS:
 What is the process of antidifferentiation?
 How is sigma notation used?
 What methods are used to approximate areas?
 What happens when an infinite number of rectangles are used for
finding an irregular area?
 What is the First Fundamental Theorem of Calculus and how is it
is the indefinite integral of f on [a,b], then
=
used?
F (b) – F(a) ); the definite integral can be evaluated by
taking the antiderivative of the integrand then finding
the difference of the substituted upper limit value in the  What is the Second Fundamental Theorem of Calculus?
antiderivative and the lower limit value in the
antiderivative.
A definite integral involving a composition of functions
can be evaluated using u-substitution with the same
limits or by changing the limits of the definite integral.
The Second Fundamental Theorem of Calculus holds
for f, a continuous function on an open interval, and a
any point in the interval, and states that d/dx
(
= f(x); the derivative of the antiderivative is
the original function.
Page 62 of 86
May 16, 2013
Guiding Question
Content, Concepts and
Skill
Instructional Materials
and Resources
Instructional Strategies

Students will know and
do:
Textbooks
Demonstration of sample
problems




How do you find
antiderivatives
using the power
algorithm?

What are the
antiderivatives of
the trigonometric 
functions?
How do you
evaluate definite
integrals?
How do you
approximate the
area under a
curve using
Riemann Sums,
inscribed/
circumscribed
rectangles, and
the trapezoidal
rule?
What is the First
Fundamental
Theorem of



Find antiderivatives
using the power
algorithm.
Find antiderivatives of
trigonometric
functions.
Evaluate definite
integrals.
Use sigma notation
and find values of
sums.
Prove:
n
∑ k = n(n+1)
k=1
2
n
∑ k2 = n(n+1)(2n+1)
k=1
6
Workbooks
Worksheets involving
multiple choice, fill in,
and open-ended
questions.
Whiteboard
Overhead projector and
screen
Overhead transparencies
Vis-Tablet
Television
CDs/DVDs/Video tapes
Scientific and graphic
calculators
Computers and software
Manipulative devises
Communicator kit
Guided practice
Discovery approach
Cooperative
learning/group work
Assessments
Guided practice with
overt or covert
responses
Check and grade
homework /class
assignments
Notebook examination
Individualized attention
Quizzes
Question and answer
Projects
CDs/DVDs/Video
presentations
Tests
Vis-Tablet/TV
presentations
Exams
Power point
presentations
Computer laboratory
activities
Internet activities
Homework and class
assignments
n
Page 63 of 86
May 16, 2013
∑ k3 = n(n+1) 2
k=1
2
Calculus and how
is it applied?



How do you

integrate an
indefinite integral
using usubstitution?
How is a definite
integral evaluated
using substitution

or by changing
the limits of the
definite integral?
What is the

Second
Fundamental
Theorem of
Calculus and how

is it applied?

Field trips
Guest speakers
Approximate the area
under a curve using
Riemann Sums,
inscribed/
circumscribed
rectangles, and the
trapezoidal rule.
Use the First
Fundamental
Theorem of Calculus
on definite integrals.
Antidifferentiate an
indefinite integral
using u-substitution.
Evaluate a definite
integral using
substitution with the
same limits or by
changing the limits of
the definite integral.
Use the Second
Fundamental
Page 64 of 86
May 16, 2013
Theorem of Calculus.

Practice multiplechoice and freeresponse questions
(with and without
calculator
applications)
pertaining to the unit
and prior units.
Page 65 of 86
May 16, 2013
HARRISON SCHOOL DISTRICT
AP Calculus
Unit 5-Applications of the Definite Integral
UNIT GOAL(S): Students will know and do:
DURATION:

Use the Mean-Value Theorem for Integrals to find the average value of a function.
Block Schedule Pacing

Find area under a curve and between curves using the definite integral.
8 days

Find volume of a solid of revolution around x- or y-axes using disk, washer, and shell methods.

Find volume of a solid with known cross sections.

Apply integral calculus to find position of a particle and distance traveled relevant to real life
problems.

Practice multiple-choice and free-response questions (with and without calculator applications)
pertaining to the unit and prior units.
NJCCCS:
F.IF.2.4.5.6.7a.7b.7c.7d.9
F.BF.1a.1b.1c(+)
F.LE.2.5
Page 66 of 86
May 16, 2013
ENDURING UNDERSTANDINGS:
ESSENTIAL QUESTIONS:

The Mean Value Theorem for Integrals
 What is the Mean Value Theorem for Integral Calculus?
states that if f(x) is continuous on the closed
interval [a,b] and differentiable on the open  How can the First Fundamental Theorem of Calculus be used to find area?
interval, (a,b), there exists a value c such
that
 What methods can be used to find volume of a solid of revolution around the xand y- axes?
F(c) =
)dx.

The definite integral is used to find the area
under a curve and the area between curves.

How do you find volume of a solid with known cross sections?

Volume of a solid of revolution around the
x- and y- axes is obtained by using the disk,
washer, or shell methods.

How do you find the position of a particle when velocity or acceleration is given?

How do you find displacement and distance traveled?

Area is determined from the shape of one
cross section perpendicular to the x- or yaxis. Using the definite integral to find the
combined area of all the cross sections,
volume of the solid is obtained.

Position of a particle and distance traveled
are found by using a definite integral.

To find displacement take the definite
integral of v(t)d(t); distance traveled is
obtained by taking the definite integral of
the absolute value of v(t)d(t).
Page 67 of 86
May 16, 2013
Guiding Question





Content, Concepts and
Skill
What is the
 Use the Mean-Value
Mean-Value
Theorem for Integrals
Theorem for
to find the average
Integrals and how
value of a function.
is it used to find
the average value  Graph conic sections.
of a function?
 Graph linear,
polynomial, rational,
How do you find
radical, absolute value,
area under a
trigonometric and
curve and
piecewise relations
between curves
using algebraic
using the definite
techniques.
integral?.
How do you find
volume of a solid
of revolution
around x- or yaxes?
How do you find
volume of a solid
with known cross
sections?
How do you find
position of a
Instructional Materials
and Resources
Instructional Strategies
Textbooks
Demonstration of sample
problems
Workbooks
Worksheets involving
multiple choice, fill in,
and open-ended
questions.
Whiteboard
Overhead projector and
screen
Overhead transparencies
 Solve one and two
variable equations.
Vis-Tablet
 Solve systems of
equations using
substitution.
CDs/DVDs/Video tapes
 Find area under a
curve and between
curves using the
definite integral.
 Find volume of a solid
Television
Scientific and graphic
calculators
Computers and software
Manipulative devises
Communicator kit
Guided practice
Discovery approach
Cooperative
learning/group work
Assessments
Guided practice with
overt or covert
responses
Check and grade
homework /class
assignments
Notebook examination
Individualized attention
Quizzes
Question and answer
Projects
CDs/DVDs/Video
presentations
Tests
Vis-Tablet/TV
presentations
Exams
Power point
presentations
Computer laboratory
activities
Internet activities
Homework and class
assignments
Page 68 of 86
May 16, 2013
particle and
distance traveled?
of revolution around
x- or y-axes using disk,
washer, and shell
methods.
Field trips
Guest speakers
 Find volume of a solid
with known cross
sections.
 Apply integral calculus
to find position of a
particle and distance
traveled relevant to
real life problems.
 Practice multiplechoice and freeresponse questions
(with and without
calculator applications)
pertaining to the unit
and prior units.
Page 69 of 86
May 16, 2013
HARRISON SCHOOL DISTRICT
AP Calculus
Unit 6-Logarithmic and Exponential Functions
UNIT GOAL(S): Students will know and do:
DURATION:

Define e as a limit.
Block Schedule Pacing

Define f(x) = 1n x.
12 days

Prove the derivative of the ln x and ex.

Find derivatives of the functions eu, bu, and ln u and integrals of eu, bu and du/u.

Find derivatives of y = f(x)g(x).

Use logarithmic differentiation.

Apply all calculus studied previously using exponential and logarithmic functions.

Apply first order separable differential equations to solve initial-value problems.

Find exponential growth and decay relevant to real life situations.

Graph slope-fields and find direction curves as a graphical approach to solve differential
equations.

Use Euler’s Method as a numerical approach to solve differential equations. (optional to College
Board AB exam)
Page 70 of 86
May 16, 2013

Practice multiple-choice and free-response questions (with and without calculator applications)
pertaining to the unit and prior units.
NJCCCS:
F.IF.2.4.5.6.7e.8b.9, F.BF.1a.1b.1c(+).4a.4c(+).4d(+).5(+),F.LE.1a,1b,1c.2.3.4.5
ENDURING UNDERSTANDINGS:

Euler’s, e, is the limit as n approaches infinity of
(1+ 1/n)n.

Logarithms are exponents; properties of logarithms
are used to perform computations more easily.



The log ex is the same as the natural log of x, ln x.
ESSENTIAL QUESTIONS:

What is e?

What are logarithms? How are they used?

What is the inverse of f(x) = ex? What do their graphs look like?

How is the change of base formula useful?
x
The inverse of f(x) = e is f (x) = ln x.

Properties of common and natural logarithms are
used to expand or contract logarithmic expressions

to equivalent forms.


The change of base formula enables renaming any
base to a calculator friendly base.

The derivative of ln u is (1/u)(du/dx); the integral 
of du/u is
ln│ u│+ c.


Logarithmic differentiation is the application of the
properties of logarithms to simplify the process of 
What is the derivative of the ln u? What is the integral of du/u?
Why use logarithmic differentiation?
What is the derivative and antiderivative of eu? The exponential function,
bu ?
What method is used to find the derivative of y = f(x)g(x)?
How do you solve a first order differential equation initial value problem?
What is a slope field? Direction curve?
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taking the derivative.

The derivative of eu is eu(du/dx); the integral of eu
du is eu +c.

The derivative of bu is bu ln b (du/dx); the integral
of bu du is

What is Euler’s method?
( bu/ln b)+ c.

To find the derivative of y = f(x)g(x), take the natural
log of both sides of the equation, use the properties
of logs to enable the differentiation process,
differentiate, then isolate dy/dx and rename the
derivative explicitly.

To solve an initial value problem, set up the
differential equation, cross multiply to isolate dy,
take the antiderivative of both sides, then substitute
the initial value.

A slope field, also called direction field, is a
graphical representation of the solutions of a first
order differential equation. A direction curve is
traced through an initial value in a slope field.
 Euler’s Rule is the simplest iterative numerical
method for solving a first order differential
equation; it gives exact solutions of slopes in a slope
field.
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Guiding Question
Students will know and
do:






What is e?
What is f(x) = 1n
x?
What is the
relationship
between f(x) = ex
and f (x) = ln x?
How can the
derivatives of the
ln x and ex be
proven?
What are
derivatives of the
functions eu, bu,
and ln u and
integrals of eu, bu
and du/u?
Content, Concepts and
Skill
Instructional Materials
and Resources
Instructional Strategies
Students will know and
do:
Textbooks
Demonstration of sample
problems
 Define e as a limit.
 Define f(x) = 1n x.
 Graph logarithmic and
exponential functions.
 Use properties of
common/natural
logarithms and
exponential functions.
 Evaluate logarithms
and solve exponential
and logarithmic
equations.
 Prove the derivative of
the ln x and ex.
 Find derivatives of the
functions eu, bu, and ln
u and integrals of eu, bu
How do you take
and du/u.
the derivative of y
= f(x)g(x)?
 Find derivatives of y =
Workbooks
Worksheets involving
multiple choice, fill in,
and open-ended
questions.
Whiteboard
Overhead projector and
screen
Overhead transparencies
Vis-Tablet
Television
CDs/DVDs/Video tapes
Scientific and graphic
calculators
Computers and software
Manipulative devises
Communicator kit
Guided practice
Discovery approach
Cooperative
learning/group work
Assessments
Guided practice with
overt or covert
responses
Check and grade
homework /class
assignments
Notebook examination
Individualized attention
Quizzes
Question and answer
Projects
CDs/DVDs/Video
presentations
Tests
Vis-Tablet/TV
presentations
Exams
Power point
presentations
Computer laboratory
activities
Internet activities
Homework and class
assignments
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May 16, 2013





Why use
logarithmic
differentiation,
and how is it
applied?
f(x)g(x).
 Use logarithmic
differentiation.
Field trips
Guest speakers
 Apply all calculus
studied previously
How is all
using exponential and
calculus studied
logarithmic functions.
previously applied
to exponential
 Apply first order
and logarithmic
separable differential
functions?
equations to solve
initial-value problems.
How do you
apply first order
 Find exponential
separable
growth and decay
differential
relevant to real life
equations to solve
situations.
initial-value
problems?
 Graph slope-fields and
find direction curves as
Find exponential
a graphical approach to
growth and decay
solve differential
relevant to real
equations.
life situations.
 Use Euler’s Method as a
Graph slopenumerical approach to solve
fields and find
differential equations.
direction curves
(optional to College
Board AB exam)
as a graphical
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May 16, 2013
approach to solve
differential
equations.

How do you
use Euler’s
Method as a
numerical
approach to
solve differential
equations?
(optional to
College Board
AB exam)
 Practice multiplechoice and freeresponse questions
(with and without
calculator applications)
pertaining to the unit
and prior units.
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May 16, 2013
HARRISON SCHOOL DISTRICT
AP Calculus
Unit 7-Inverse Trigonometric Functions
UNIT GOAL(S): Students will know and do:
DURATION:

Prove derivatives of inverse trigonometric functions.
Block Schedule Pacing

Find derivatives of inverse trigonometric functions.
6 days

Find anti-derivatives of inverse trigonometric functions.

Apply all of calculus studied previously using inverse trigonometric functions.

Integrate functions by parts. (optional to College Board AB exam)

Practice multiple-choice and free-response questions (with and without calculator applications)
pertaining to the unit and prior units.
NJCCCS: F.IF.2.4.5.6.8b.9, F.BF.1a.1b.1c(+).4a.4b(+).4c(+).4d(+),
F.TF.1.2.3(+).4(+).5.6(+).7(+).8.9(+)
ENDURING UNDERSTANDINGS:
 If restrictions are put on the domain and
range, trigonometric functions will have
inverses and the inverses will be functions.
 Inverse trigonometric functions have
derivatives and the derivatives have
ESSENTIAL QUESTIONS:

Do the trigonometric functions have inverses?

Do inverse trigonometric functions have derivatives? Do the derivatives have
integrals?
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May 16, 2013
integrals.

When would you use by parts integration?
 By parts integration is used on a product of
functions that cannot be integrated in its
present form.
Guiding Question




How can the
derivative of the
inverse of the sin
x be proven?
What are the
derivatives of
inverse
trigonometric
functions?
Content, Concepts and
Skill


What are antiderivatives of
inverse
trigonometric
functions?
How is all
calculus studied
previously applied
to inverse

Find the missing
part of a right
triangle using
right triangle or
circle
trigonometry.
Instructional Materials
and Resources
Instructional Strategies
Textbooks
Demonstration of sample
problems
Workbooks
Worksheets involving
multiple choice, fill in,
and open-ended
Use trigonometric questions.
reciprocal
identities, tangent Whiteboard
and cotangent
Overhead projector and
identities,
screen
Pythagorean
identities,
Overhead transparencies
cofunction
identities, and
Vis-Tablet
negative angle
identities.
Television
Prove derivatives
of inverse
trigonometric
functions.
CDs/DVDs/Video tapes
Scientific and graphic
Guided practice
Discovery approach
Assessments
Guided practice with
overt or covert
responses
Check and grade
homework /class
assignments
Cooperative
learning/group work
Notebook examination
Individualized attention
Quizzes
Question and answer
Projects
CDs/DVDs/Video
presentations
Tests
Exams
Vis-Tablet/TV
presentations
Power point
presentations
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May 16, 2013
trigonometric
functions?

How do you
integrate
functions by
parts? (optional to
College Board AB
exam)



Graph inverse
trigonometric
functions.
calculators
Solve inverse
trigonometric
equations.
Manipulative devises
Find derivatives
of inverse
trigonometric
functions.

Find antiderivatives of
inverse
trigonometric
functions.

Apply all of
calculus studied
previously using
inverse
trigonometric
functions.

Integrate
functions by
parts. (optional to
College Board AB
Computers and software
Communicator kit
Computer laboratory
activities
Internet activities
Homework and class
assignments
Field trips
Guest speakers
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May 16, 2013
exam)

Practice multiplechoice and freeresponse
questions (with
and without
calculator
applications)
pertaining to the
unit and prior
units.
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May 16, 2013
HARRISON SCHOOL DISTRICT
Special Education Modifications /Accommodations
AP Calculus
In addition to the specific modifications noted in a student’s IEP, the following accommodations may be implemented as
needed to enable special education students to meet with success.
Instructional Accommodations/Modifications:
1.
Use an alternative instructional method to address individual learning styles.
2.
Use directed reading activity – provide Study Guides.
3.
Use guided reading lesson.
4.
Teach strategies for using context clues.
5.
Provide additional time to complete assignment.
6.
Organize an exercise to reinforce and review lesson content.
7.
Break task down and provide guidance through steps needed for task.
8.
Allow additional processing time.
9.
Check for understanding of direction and/or concepts.
10.
Check organization of notebook/planner.
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May 16, 2013
11.
Check content of notes for accuracy.
12.
Photocopy notes and/or provide extended time to write notes.
13.
Modify length of assignment.
14.
Provide advance notice for upcoming test/projects.
15.
Encourage participation and provide positive feedback.
16.
Prompt student to stay on task.
17.
Modify seating arrangement.
18.
Provide written directions to reinforce oral directions.
19.
Have student verbalize steps in a mathematical process.
20.
Provide a calculator to assist in computing math problems.
21.
Allow use of a laptop or computer/assistive technology.
22.
Encourage student to highlight/ color code notes.
23.
Have students discuss/write brief summaries of chapters.
24.
Provide manipulative materials to reinforce concepts presented.
25.
Provide computer access for assignments.
26.
Avoid oral reading in group situations.
27.
Encourage, but do not force, oral reading.
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May 16, 2013
28.
Provide written cues during lecture/discussion.
29.
Provide student with a second set of textbooks for home use.
30.
Provide one-on-one instruction.
Testing & Grading Accommodations
31.
Do not penalize for spelling errors.
32.
Administer tests orally.
33.
Allow student to re-take tests as appropriate.
34.
Provide extra time for test.
35.
Provide test with modifications.
36.
Allow open book or open note tests
37.
Repeat, clarify, or reword directions.
38.
Provide a word bank for test.
39.
Utilize help period for testing.
40.
Allow use of mnemonic techniques during testing.
41.
Provide student with an alternate setting for test administration.
42.
Utilize an alternative assessment.
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May 16, 2013
43.
Utilize an individual grading system, providing the accommodation is a requirement of the student's IEP.
Social & Emotional Accommodations
44.
Use a private visual cue to stop an inappropriate behavior.
45.
Refrain from reprimanding student in front of others.
46.
Maintain communication with case manager and parent.
47.
Allow student to see counselor as needed.
48.
Refer to individual behavior plan.
49.
Provide student with choices.
50.
Provide self-checking materials to student.
51.
Clearly define limits and expectations.
52.
Redirect student when off task.
53.
Provide verbal praise and reinforcement.
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May 16, 2013
HARRISON SCHOOL DISTRICT
DIFFERENTIATING INSTRUCTION
AP Calculus
UNIT
Suggestions for Differentiation
List the main objective on the assignment sheet or board, but offer two or three different ways that students can learn or
master the objective.
Provide formative assessments that are graded, but not recorded, in order to provide students the opportunity to monitor
their progress and understanding.
Provide opportunities for small group discussion of homework to go over what they have learned and/or what they may
still be struggling with, and to receive feedback from their peers who may have a better understanding of the concepts
studied.
Use peer buddies in which pairs of students check each other’s readiness to begin the next task.
Offer a variety of ways students can present what they have learned: formal writing, group presentations, group debate,
video presentations, journalistic publications, animated tales, etc.
Develop a list of alternative teaching activities/assessments including, but not limited to:
Demonstrate learning using a pamphlet, brochure of newsletter.
Provide art supplies for creative student projects.
Establish an area of the classroom to display student designed work.
Have students make posters displaying key points of a lesson.
Assign students the task of creating class bulletin boards that teach one or more concepts to their classmates.
Involve the class in a unit newspaper pertaining to one unit of study in which students are responsible for articles and
sections such as news, classified ads, feature articles, lay-out, editing, art-work, etc.
Hold a mock trial.
Design advertisements for specific concepts taught.
Put a lecture on audiotape or video and offer it as an option.
Have students teach a concept or chapter to a small group of peers.
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May 16, 2013
Have students make murals, timelines, or other large-scale visuals.
Make a magazine on the topic, either alone or in small groups, ensuring that all students have a specific responsibility.
Have students search online and develop an annotated bibliography of the Websites they discover that pertain to the
topic.
Create and play board games that teach concepts.
Create and perform musical or dramatic works that explain a concept.
Offer choices for homework assignments.
Provide two or three choices for how students practice or apply what they have learned from a class lecture or
demonstration.
Have students supplement the information gleaned from their textbook with either supporting or conflicting information
from a different text or online source.
Use popular adolescent art forms (music, films, video games) to help students apply and understand concepts.
REFERENCES
Anton H., Bivens I., Davis S. (2005). Calculus Eighth Edition. Hoboken, NJ: John Wiley & Sons, Inc.
Thomas, George B. (1972). Calculus and Analytic Geometry. Reading, MA: Addison-Wesley Publishing Company, Inc.
Nunley, K. (2006). Differentiating the high school classroom. Thousand Oaks, CA: Corwin Press.
Bender, W. (2008). Differentiating Instruction for students with learning disabilities: Best teaching practices for general and special educators.
Thousand Oaks, CA: Corwin Press.
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