
Teach=
er: CORE
AP CALCULUS AB 
Updat=
ed 2014 
=
; 
<=
/td>
 <=
/td>
 =
; 
=
; 

Cours=
e: AP
CALCULUS AB 
Month=
:
All Months 













S 
Intro=
duction to
Derivatives 






e 
Essen=
tial
Questions 
Conte=
nt 
Skill=
s 
Assessm=
ents 
Lessons=

Learn=
ing
Benchmarks 
Stand=
ards 
p 
Am I =
able to
work with functions represented in a variety of ways (graphical, numerica=
l,
analytical, or verbal) and understand the connections among these
representations? 
Avera=
ge rate of
change 
Find =
the
average rate of change between two points. 


Stude=
nts will
know and be able to find the average rate of change of a function. 
1.2F=
unctions,
Graphs, and Limits ~ Limits of functions (including onesided limits) â€“=
An
intuitive understanding of the limiting process; Calculating limits using
algebra; Estimating limits from graphs or tables of data 
t 
Do I =
understand
the meaning of the derivative in terms of rate of change and local linear
approximation and am I able to use derivatives to solve a variety of
problems? 
Defin=
ition of
Derivative 
Find =
the slope
of the tangent line to a curve at a point. 
=
; 
2.1D=
erivatives
~ Concept of the derivative ~ Derivative presented graphically, numerical=
ly,
and analytically; Derivative interpreted as an instantaneous rate of chan=
ge;
Derivative defined as the limit of the difference quotient; Relationship
between differentiability and continuity 
e 
Am I =
able to
communicate mathematics both orally and in wellwritten sentences and am I
able to explain solutions to problems? 
Diffe=
rentiability 
Find =
the
derivative of a function using the limit definition. 
Stude=
nts will
know and be able to calculate derivatives using the definition of the
derivative. 
2.2D=
erivatives
~ Derivative at a point ~ Slope of a curve at a point. Examples are
emphasized, including points at which there are vertical tangents and poi=
nts
at which there are no tangents; Tangent line approximation to a curve at a
point and local linear approximation; Instantaneous rate of change as the
limit of average rate of change; Approximate rate of change from graphs a=
nd
tables of values 
m 
Am I =
able to
use technology to help solve problems, experiment, interpret results, and
verify conclusions? 
=
; 
Under=
stand the
relationship between differentiability and continuity. 
=
; 
2.6D=
erivatives
~ Computation of Derivatives ~ Knowledge of derivatives of basic function=
s,
including power, exponential, logarithmic, trigonometric, and inverse
trigonometric functions; Basic rules for the derivative of sums, products,
and quotients of functions; Chain rule and implicit differentiation =

b 
Am I =
able to
determine the reasonableness of solutions, including sign, size, relative
accuracy, and units of measurement? 
Power=
Rule 
Find =
the
equations of tangent and normal lines. 
=
; 
=
; 
e 
Have I
developed an appreciation of calculus as a coherent body of knowledge and=
as
a human accomplishment? 
=
; 
Find =
the
derivative of a function using the Constant Rule. 
Stude=
nts will
know and be able to find the equations of the tangent line and normal lin=
e to
a curve at a given point. 
=
; 
r 
=
; 
=
; 
Find =
the
derivative of a function using the Power Rule. 
Stude=
nts will
know and be able to find where a function is not differentiable. 
=
; 

=
; 
Equat=
ions of
tangent and normal lines 
Find =
the
derivative of a function using the Constant Multiple Rule. 
Stude=
nts will
know and be able to approximate derivatives numerically and graphically.<=
/td>
 =
; 

=
; 
Rules=
for
differentiation 
Find =
the
derivative of a function using the Sum and Difference Rules. 
=
; 
=
; 

=
; 
Highe=
rOrder
Derivatives 
Find =
the
derivative of a function using the Product Rule. 
Stude=
nts will
know and be able to use the rules of differentiation to calculate
derivatives, including second and higherorder derivatives. 
=
; 

=
; 
Deriv=
atives of
Trigonometric Functions 
Find =
the
derivative of a function using the Quotient Rule. 
=
; 
=
; 

=
; 
=
; 
Find a
higherorder derivative of a function. 
=
; 
=
; 

=
; 
Motio=
n along a
line 
Use d=
erivatives
to find rates of change. 
Stude=
nts will
know and be able to calculate the instantaneous rate of change. 
=
; 

=
; 
=
; 
=
; 
=
; 
=
; 

=
; 
=
; 
Deriv=
e two
important trigonometry limits. 
=
; 
=
; 

=
; 
Chain=
Rule 
=
; 
Stude=
nts will
know and be able to use derivatives to analyze motion along a straight li=
ne. 
=
; 

=
; 
Impli=
cit
Differentiation 
=
; 
=
; 
=
; 

=
; 
Deriv=
atives of
Exponential and Logarithmic Functions 
Find =
the
derivatives of the sine and cosine functions. 
Stude=
nts will
know and be able to use the rules for differentiating the six trigonometr=
ic
functions. 
=
; 

=
; 
Deriv=
atives of
Inverse Functions (in particular, inverse trigonometric functions) 
Find =
the
derivative of all trigonometric functions. 
=
; 
=
; 

=
; 
Local
Linearity 
Find =
the
derivative of a composite function using the Chain Rule. 
=
; 
=
; 

=
; 
=
; 
=
; 
Stude=
nts will
know and be able to differentiate composite functions using the Chain Rul=
e. 
=
; 

=
; 
=
; 
Disti=
nguish
between fucntions written in implicit form and explicit form. 
Stude=
nts will
know and be able to find derivatives using implicit differentiation. 
=
; 

=
; 
=
; 
=
; 
Stude=
nts will
know and be able to find the derivatives of exponential and logarithmic
functions. 
=
; 

=
; 
=
; 
=
; 
Stude=
nts will
know and be able to calculate derivatives of the inverse trigonometric
functions. 
=
; 

=
; 
=
; 
Use i=
mplicit
differentiation to find the derivative of a function. 
Stude=
nts will
know and be able to find linearizations. 
=
; 

=
; 
=
; 
Find =
the
derivative of the natural logarithm function. 
=
; 
=
; 

=
; 
=
; 
Diffe=
rentiate
the natural exponential function and exponential functions that have bases
other than e. 
=
; 
=
; 

=
; 
=
; 
Find =
the
derivative of an inverse function. 
=
; 
=
; 

=
; 
=
; 
Diffe=
rentiate
an inverse trigonometric function. 
=
; 
=
; 

=
; 
=
; 
Under=
stand the
concept of a tangent line approximation. 
=
; 
=
; 
O 
Appli=
cations of
Derivatives 






c 
Essen=
tial
Questions 
Conte=
nt 
Skill=
s 
Assessm=
ents 
Lessons=

Learn=
ing
Benchmarks 
Stand=
ards 
t 
Am I =
able to
work with functions represented in a variety of ways (graphical, numerica=
l,
analytical, or verbal) and understand the connections among these
representations? 
Mean =
Value
Theorem 
Under=
stand and
use the Mean Value Theorem. 


Stude=
nts will
know and be able to apply the Mean Value Theorem. 
1.1F=
unctions,
Graphs, and Limits ~ Analysis of graphs ~ With the aid of technology, gra=
phs
of functions are often easy to produce. The emphasis is on the interplay
between the geometric and analytic information and on the use of calculus
both to predict and to explain the observed local and global behavior of a
function. 
o 
Do I =
understand
the meaning of the derivative in terms of rate of change and local linear
approximation and am I able to use derivatives to solve a variety of
problems? 
Extre=
ma
(Max/Min) 
Under=
stand the
definitions of absolute and relative extrema on open and closed intervals=
. 
Stude=
nts will
know and be able to determine the relative (local) or absolute (global)
maxima and minima of a function. 
1.2F=
unctions,
Graphs, and Limits ~ Limits of functions (including onesided limits) â€“=
An
intuitive understanding of the limiting process; Calculating limits using
algebra; Estimating limits from graphs or tables of data 
b 
Am I =
able to
communicate mathematics both orally and in wellwritten sentences and am I
able to explain solutions to problems? 
Incre=
asing and
Decreasing using the 1st Derivative 
Deter=
mine
intervals on which a function is increasing or decreasing. 
Stude=
nts will
know and be able to use the First and Second Derivative Tests to determine
local max/min of a function. 
1.3F=
unctions,
Graphs, and Limits ~ Asymptotic and unbounded behavior â€“ Understanding
asymptotes in terms of graphical behavior; Describing asymptotic behavior=
in
terms of limits involving infinity; Comparing relative magnitudes of
functions and their rates of change (for example, contrasting exponential
growth, polynomial growth, and logarithmic growth) 
e 
Am I =
able to
model a written description of a physical situation with a function, a
differential equation, or an integral? 
The 1=
st
Derivative Test 
Apply=
the First
Derivative Test to find relative extrema of a function. 
Stude=
nts will
know and be able to determine the concavity of a function and locate the
points of inflection by analyzing the Second Derivative. 
1.4F=
unctions,
Graphs, and Limits ~ Functions, Graphs, and Limits ~ Continuity as a prop=
erty
of functions â€“ An intuitive understanding of continuity; Understanding
continuity in terms of limits; Geometric understanding of graphs of
continuous functions (Intermediate Value Theorem and Extreme Value Theore=
m) 
r 
Am I =
able to
use technology to help solve problems, experiment, interpret results, and
verify conclusions? 
Conca=
vity and
inflection points using the 2nd derivative 
Deter=
mine
intervals on which a function is concave up or concave down. 
Stude=
nts will
know and be able to graph a function using information about the first
derivative. 
2.1D=
erivatives
~ Concept of the derivative ~ Derivative presented graphically, numerical=
ly,
and analytically; Derivative interpreted as an instantaneous rate of chan=
ge;
Derivative defined as the limit of the difference quotient; Relationship
between differentiability and continuity 

Am I =
able to
determine the reasonableness of solutions, including sign, size, relative
accuracy, and units of measurement? 
The 2=
nd
Derivative Test 
Find =
any points
of inflection of the graph of a function. 
Stude=
nts will
know and be able to solve application problems involving finding maximum =
or
minimum values of functions. 
2.2D=
erivatives
~ Derivative at a point ~ Slope of a curve at a point. Examples are
emphasized, including points at which there are vertical tangents and poi=
nts
at which there are no tangents; Tangent line approximation to a curve at a
point and local linear approximation; Instantaneous rate of change as the
limit of average rate of change; Approximate rate of change from graphs a=
nd
tables of values 

Have I
developed an appreciation of calculus as a coherent body of knowledge and=
as
a human accomplishment? 
Given=
graph of
first derivative 
Apply=
the 2nd
Derivative Test to find relative max/min of a function. 
Stude=
nts will
know and be able to solve related rate problems. 
2.3D=
erivatives
~ Derivative as a function ~ Corresponding characteristics of graphs of f=
and
f'; Relationship between the increasing and decreasing behavior of f and =
the
sign of f'; The Mean Value Theorem and its geometric consequences; Equati=
ons
involving derivatives. Verbal descriptions are translated into equations
involving derivatives and vice versa 

=
; 
Maxim=
um and
Minimum Applied Problems 
Find =
max/min on
a closed interval. 
=
; 
2.4D=
erivatives
~ Second Derivatives ~ Corresponding characteristics of the graphs of f, =
f',
and f''; Relationship between the concavity of f and the sign of f''; Poi=
nts
of inflection as places where concavity changes 

=
; 
Relat=
ed
Rates 
Analy=
ze and
sketch the graph of a function. 
=
; 
2.5D=
erivatives
~ Applications of Derivatives ~ Analysis of curves, including the notions=
of
monotonicity and concavity; Optimization, both absolute (global) and rela=
tive
(local) extrema; Modeling rate of change, including related rates problem=
s;
Use of implicit differentiation to find the derivative of an inverse
function; Interpretation of the derivative as a rate of change in varied
applied contexts, including velocity, speed, and acceleration; Geometric
interpretation of differential equations via slope fields and the
relationship between slope fields and solution curves for differential
equations 

=
; 
=
; 
Solve=
applied
minimum and maximum problems. 
=
; 
2.6D=
erivatives
~ Computation of Derivatives ~ Knowledge of derivatives of basic function=
s,
including power, exponential, logarithmic, trigonometric, and inverse
trigonometric functions; Basic rules for the derivative of sums, products,
and quotients of functions; Chain rule and implicit differentiation =


=
; 
=
; 
Find =
a related
rate. 
=
; 
=
; 

=
; 
=
; 
Use r=
elated
rates to solve problems. 
=
; 
=
; 
N 
Intro=
duction to
Integration 






o 
Essen=
tial
Questions 
Conte=
nt 
Skill=
s 
Assessm=
ents 
Lessons=

Learn=
ing
Benchmarks 
Stand=
ards 
v 
Am I =
able to
work with functions represented in a variety of ways (graphical, numerica=
l,
analytical, or verbal) and understand the connections among these
representations? 
Recta=
ngular
Approximation Methods 
Appro=
ximate a
definite integral using Rectangular Approximation Methods. 


Stude=
nts will
know and be able to approximate the area under a graph using rectangle
approximation methods. 
1.4F=
unctions,
Graphs, and Limits ~ Functions, Graphs, and Limits ~ Continuity as a prop=
erty
of functions â€“ An intuitive understanding of continuity; Understanding
continuity in terms of limits; Geometric understanding of graphs of
continuous functions (Intermediate Value Theorem and Extreme Value Theore=
m) 
e 
Do I =
understand
the meaning of the definite integral both as a limit of Riemann sums and =
as
the net accumulation of change and am I able to use integrals to solve a
variety of problems? 
Trape=
zoidal
Rule 
Appro=
ximate a
definite integral using the Trapezoidal Rule. 
Stude=
nts will
know and be able to interpret the area under a graph as a net accumulatio=
n of
a rate of change. 
3.1I=
ntegrals ~
Interpretations and properties of definite integrals ~ Definite integral =
as a
limit of Riemann sums; Definite integral of the rate of change of a quant=
ity
over an interval interpreted as the change of the quantity over the inter=
val;
Basic properties of definite integrals (examples include additivity and
linearity) 
m 
Am I =
able to
communicate mathematics both orally and in wellwritten sentences and am I
able to explain solutions to problems? 
Defin=
ite
Integrals 
Using=
sigma
notation to write a sum. 
Stude=
nts will
know and be able to approximate a definite integral by using the Trapezoi=
dal
Rule. 
3.2I=
ntegrals ~
Applications of integrals ~ Appropriate integrals are used in a variety of
applications to model physical, biological, or economic situations. Altho=
ugh
only a sample of applications can be included in any specific course,
students should be able to adapt their knowledge and techniques to solve
other similar application problems. Whatever applications are chosen, the
emphasis is on using the method of setting up an approximating Riemann sum
and representing its limit as a definite integral. To provide a common
foundation, specific applications should include using the integral of a =
rate
of change to give accumulated change, finding the area of a region, the
volume of solid with known cross sections, the average value of a functio=
n,
and the distance traveled by a particle along a line 
b 
Am I =
able to
model a written description of a physical situation with a function, a
differential equation, or an integral? 
Rules=
of
Integration 
Under=
stand the
definition of a Riemann sum. 
Stude=
nts will
know and be able to express the area under a curve as a definite integral=
and
as a limit of a Riemann sum. 
3.3I=
ntegrals ~
Fundamental Theorem of Calculus ~ Use of the Fundamental Theorem to evalu=
ate
definite integrals; Use of the Fundamental Theorem to represent a particu=
lar
antiderivative, and the analytical and graphical analysis of functions so
defined 
e 
Am I =
able to
use technology to help solve problems, experiment, interpret results, and
verify conclusions? 
Displ=
acement
and Distance from velocity function 
Evalu=
ate a
definite integral using properties of definite integrals. 
Stude=
nts will
know and be able to compute the area under a curve using a calculator.
 3.6I=
ntegrals ~
Numerical approximations to definite integrals ~ Use of Riemann sums (usi=
ng
left, right, and midpoint evaluation points) and trapezoidal sums to
approximate definite integrals of functions represented algebraically,
graphically, and by tables of values 
r 
Am I =
able to
determine the reasonableness of solutions, including sign, size, relative
accuracy, and units of measurement? 
Funda=
mental
Theorem of Calculus (Part I) 
=
; 
Stude=
nts will
know and be able to apply the rules for definite integrals. 
=
; 

Have I
developed an appreciation of calculus as a coherent body of knowledge and=
as
a human accomplishment? 
=
; 
=
; 
Stude=
nts will
know and be able to solve problems in which a rate is integrated to find =
the
net change over time in a variety of applications. 
=
; 
D 
Mecha=
nics of
Integration 






e 
Essen=
tial
Questions 
Conte=
nt 
Skill=
s 
Assessm=
ents 
Lessons=

Learn=
ing
Benchmarks 
Stand=
ards 
c 
Do I =
understand
the meaning of the definite integral both as a limit of Riemann sums and =
as
the net accumulation of change and am I able to use integrals to solve a
variety of problems? 
Indef=
inite
Integrals 
Use b=
asic
integration rules to find antiderivatives. 


Stude=
nts will
know and be able to apply the Fundamental Theorem of Calculus. 
1.4F=
unctions,
Graphs, and Limits ~ Functions, Graphs, and Limits ~ Continuity as a prop=
erty
of functions â€“ An intuitive understanding of continuity; Understanding
continuity in terms of limits; Geometric understanding of graphs of
continuous functions (Intermediate Value Theorem and Extreme Value Theore=
m) 
e 
Do I =
understand
the relationship between the derivative and the definite integral as
expressed in both parts of the Fundamental Theorem of Calculus? 
Defin=
ite
Integrals 
Evalu=
ate a
definite integral using properties of definite integrals. 
=
; 
3.1I=
ntegrals ~
Interpretations and properties of definite integrals ~ Definite integral =
as a
limit of Riemann sums; Definite integral of the rate of change of a quant=
ity
over an interval interpreted as the change of the quantity over the inter=
val;
Basic properties of definite integrals (examples include additivity and
linearity) 
m 
Am I =
able to
communicate mathematics both orally and in wellwritten sentences and am I
able to explain solutions to problems? 
Basic
Differential Equations 
Evalu=
ate a
definite integral using the Fundamental Theorem of Calculus. 
Stude=
nts will
know and be able to construct antiderivatives using the Fundamental Theor=
em
of Calculus. 
3.2I=
ntegrals ~
Applications of integrals ~ Appropriate integrals are used in a variety of
applications to model physical, biological, or economic situations. Altho=
ugh
only a sample of applications can be included in any specific course,
students should be able to adapt their knowledge and techniques to solve
other similar application problems. Whatever applications are chosen, the
emphasis is on using the method of setting up an approximating Riemann sum
and representing its limit as a definite integral. To provide a common
foundation, specific applications should include using the integral of a =
rate
of change to give accumulated change, finding the area of a region, the
volume of solid with known cross sections, the average value of a functio=
n,
and the distance traveled by a particle along a line 
b 
Am I =
able to
use technology to help solve problems, experiment, interpret results, and
verify conclusions? 
Integ=
ration
through usubstitution 
Write=
the
general solution of a differential equation. 
=
; 
3.3I=
ntegrals ~
Fundamental Theorem of Calculus ~ Use of the Fundamental Theorem to evalu=
ate
definite integrals; Use of the Fundamental Theorem to represent a particu=
lar
antiderivative, and the analytical and graphical analysis of functions so
defined 
e 
Am I =
able to
determine the reasonableness of solutions, including sign, size, relative
accuracy, and units of measurement? 
=
; 
Find a
particular solution of a differential equation. 
=
; 
3.4I=
ntegrals ~
Techniques of antidifferentiation ~ Antiderivatives following directly fr=
om
derivatives of basic functions; Antiderivatives by substitution of variab=
les
(including change of limits for definite integrals) 
r 
Have I
developed an appreciation of calculus as a coherent body of knowledge and=
as
a human accomplishment? 
=
; 
Use a=
change of
variables to find an indefinite integral. 
Stude=
nts will
know and be able to solve initial value problems for basic differential
equations. 
3.5I=
ntegrals ~
Applications of antidifferentiation ~ Finding specific antiderivatives us=
ing
initial conditions, including applications to motion along a line; Solving
separable differential equations and using them in modeling 

=
; 
=
; 
Use a=
change of
variables to evaluate a definite integral. 
Stude=
nts will
know and be able to compute indefinite and definite integrals by the meth=
od
of substitution. 
=
; 
J 
Limit=
Review 






a 
Essen=
tial
Questions 
Conte=
nt 
Skill=
s 
Assessm=
ents 
Lessons=

Learn=
ing
Benchmarks 
Stand=
ards 
n 
Am I =
able to
work with functions represented in a variety of ways (graphical, numerica=
l,
analytical, or verbal) and understand the connections among these
representations? 
Graph=
ical
Interpretation 
Estim=
ate a
limit using a numerical or graphical approach. 


Stude=
nts will
know and be able to define and calculate limits for function values and a=
pply
the properties of limits. 
1.1F=
unctions,
Graphs, and Limits ~ Analysis of graphs ~ With the aid of technology, gra=
phs
of functions are often easy to produce. The emphasis is on the interplay
between the geometric and analytic information and on the use of calculus
both to predict and to explain the observed local and global behavior of a
function. 
u 
Am I =
able to
communicate mathematics both orally and in wellwritten sentences and am I
able to explain solutions to problems? 
Algeb=
raic
Techniques for finding limits 
Learn=
different
ways that a limit can fail to exist. 
Stude=
nts will
know and be able to find and verifiy end behavior models for various
functions. 
1.2F=
unctions,
Graphs, and Limits ~ Limits of functions (including onesided limits) â€“=
An
intuitive understanding of the limiting process; Calculating limits using
algebra; Estimating limits from graphs or tables of data 
a 
Am I =
able to
use technology to help solve problems, experiment, interpret results, and
verify conclusions? 
Appro=
ximating
Limits 
Evalu=
ate a
limit using properties of limits. 
Stude=
nts will
know and be able to calculate limits as x approaches infinity and to iden=
tify
vertical and horizontal asymptotes. 
1.3F=
unctions,
Graphs, and Limits ~ Asymptotic and unbounded behavior â€“ Understanding
asymptotes in terms of graphical behavior; Describing asymptotic behavior=
in
terms of limits involving infinity; Comparing relative magnitudes of
functions and their rates of change (for example, contrasting exponential
growth, polynomial growth, and logarithmic growth) 
r 
Am I =
able to
determine the reasonableness of solutions, including sign, size, relative
accuracy, and units of measurement? 
Infin=
ite Limits 
Devel=
op and use
a strategy for finding limits. 
Stude=
nts will
know and be able to identify the intervals upon which a given function is
continuous and understand the meaning of continuous function. 
1.4F=
unctions,
Graphs, and Limits ~ Functions, Graphs, and Limits ~ Continuity as a prop=
erty
of functions â€“ An intuitive understanding of continuity; Understanding
continuity in terms of limits; Geometric understanding of graphs of
continuous functions (Intermediate Value Theorem and Extreme Value Theore=
m) 
y 
Have I
developed an appreciation of calculus as a coherent body of knowledge and=
as
a human accomplishment? 
Conti=
nuity 
=
; 
Stude=
nts will
know and be able to remove removable discontinuities by extending or
modifying a function. 
2.1D=
erivatives
~ Concept of the derivative ~ Derivative presented graphically, numerical=
ly,
and analytically; Derivative interpreted as an instantaneous rate of chan=
ge;
Derivative defined as the limit of the difference quotient; Relationship
between differentiability and continuity 

=
; 
=
; 
Evalu=
ate a
limit using dividing out and rationalizing techniques. 
Stude=
nts will
know and be able to apply the Intermediate Value Theorem. 
=
; 

=
; 
=
; 
=
; 
=
; 
=
; 

=
; 
=
; 
=
; 
=
; 
=
; 

=
; 
=
; 
Deter=
mine
infinite limits from the left and from the right. 
=
; 
=
; 

=
; 
=
; 
Find =
and sketch
the vertical asymptotes of the graph of a function. 
=
; 
=
; 

=
; 
=
; 
Deter=
mine
finite limits at infinity. 
=
; 
=
; 

=
; 
=
; 
Deter=
mine the
horizontal asymptote, if any, of the graph of a function. 
=
; 
=
; 

=
; 
=
; 
Deter=
mine
infinite limits at infinity. 
=
; 
=
; 

=
; 
=
; 
Deter=
mine
continuity at a point and continuity on an open interval. 
=
; 
=
; 

=
; 
=
; 
Under=
stand and
use the Intermediate Value Theorem. 
=
; 
=
; 
F 
Diffe=
rential
Equations 






e 
Essen=
tial
Questions 
Conte=
nt 
Skill=
s 
Assessm=
ents 
Lessons=

Learn=
ing
Benchmarks 
Stand=
ards 
b 
Am I =
able to
work with functions represented in a variety of ways (graphical, numerica=
l,
analytical, or verbal) and understand the connections among these
representations? 
Separ=
ation of
Variables 
Use s=
eparation
of variables to solve a simple differential equation. 


Stude=
nts will
know and be able to solve differential equations using the method of
separation of variables. 
1.1F=
unctions,
Graphs, and Limits ~ Analysis of graphs ~ With the aid of technology, gra=
phs
of functions are often easy to produce. The emphasis is on the interplay
between the geometric and analytic information and on the use of calculus
both to predict and to explain the observed local and global behavior of a
function. 
r 
Do I =
understand
the meaning of the derivative in terms of rate of change and local linear
approximation and am I able to use derivatives to solve a variety of
problems? 
Slope=
Fields 
Use s=
lope
fields to approximate solutions of differential equations. 
Stude=
nts will
know and be able to construct slope field and interpret slope fields as a
visualization of differential equations. 
1.4F=
unctions,
Graphs, and Limits ~ Functions, Graphs, and Limits ~ Continuity as a prop=
erty
of functions â€“ An intuitive understanding of continuity; Understanding
continuity in terms of limits; Geometric understanding of graphs of
continuous functions (Intermediate Value Theorem and Extreme Value Theore=
m) 
u 
Am I =
able to
communicate mathematics both orally and in wellwritten sentences and am I
able to explain solutions to problems? 
Expon=
ential
Growth and Decay 
Use e=
xponential
functions to model growth and decay in applied problems. 
Stude=
nts will
know and be able to solve problems involving exponential growth and decay=
in
a variety of applications. 
2.1D=
erivatives
~ Concept of the derivative ~ Derivative presented graphically, numerical=
ly,
and analytically; Derivative interpreted as an instantaneous rate of chan=
ge;
Derivative defined as the limit of the difference quotient; Relationship
between differentiability and continuity 
a 
Am I =
able to
model a written description of a physical situation with a function, a
differential equation, or an integral? 
=
; 
Use
differential equations to model and solve applied problems. 
=
; 
2.2D=
erivatives
~ Derivative at a point ~ Slope of a curve at a point. Examples are
emphasized, including points at which there are vertical tangents and poi=
nts
at which there are no tangents; Tangent line approximation to a curve at a
point and local linear approximation; Instantaneous rate of change as the
limit of average rate of change; Approximate rate of change from graphs a=
nd
tables of values 
r 
Am I =
able to
use technology to help solve problems, experiment, interpret results, and
verify conclusions? 
=
; 
=
; 
=
; 
2.3D=
erivatives
~ Derivative as a function ~ Corresponding characteristics of graphs of f=
and
f'; Relationship between the increasing and decreasing behavior of f and =
the
sign of f'; The Mean Value Theorem and its geometric consequences; Equati=
ons
involving derivatives. Verbal descriptions are translated into equations
involving derivatives and vice versa 
y 
Am I =
able to
determine the reasonableness of solutions, including sign, size, relative
accuracy, and units of measurement? 
=
; 
=
; 
=
; 
3.1I=
ntegrals ~
Interpretations and properties of definite integrals ~ Definite integral =
as a
limit of Riemann sums; Definite integral of the rate of change of a quant=
ity
over an interval interpreted as the change of the quantity over the inter=
val;
Basic properties of definite integrals (examples include additivity and
linearity) 

Have I
developed an appreciation of calculus as a coherent body of knowledge and=
as
a human accomplishment? 
=
; 
=
; 
=
; 
3.3I=
ntegrals ~
Fundamental Theorem of Calculus ~ Use of the Fundamental Theorem to evalu=
ate
definite integrals; Use of the Fundamental Theorem to represent a particu=
lar
antiderivative, and the analytical and graphical analysis of functions so
defined 

=
; 
=
; 
=
; 
=
; 
3.4I=
ntegrals ~
Techniques of antidifferentiation ~ Antiderivatives following directly fr=
om
derivatives of basic functions; Antiderivatives by substitution of variab=
les
(including change of limits for definite integrals) 

=
; 
=
; 
=
; 
=
; 
3.5I=
ntegrals ~
Applications of antidifferentiation ~ Finding specific antiderivatives us=
ing
initial conditions, including applications to motion along a line; Solving
separable differential equations and using them in modeling 
M 
Appli=
cations of
Integration 






a 
Essen=
tial
Questions 
Conte=
nt 
Skill=
s 
Assessm=
ents 
Lessons=

Learn=
ing
Benchmarks 
Stand=
ards 
r 
Am I =
able to
work with functions represented in a variety of ways (graphical, numerica=
l,
analytical, or verbal) and understand the connections among these
representations? 
Area =
between
two curves 
Find =
the area
of a region between two curves using integration. 


Stude=
nts will
know and be able to use integration to calculate areas of regions in a pl=
ane. 
1.1F=
unctions,
Graphs, and Limits ~ Analysis of graphs ~ With the aid of technology, gra=
phs
of functions are often easy to produce. The emphasis is on the interplay
between the geometric and analytic information and on the use of calculus
both to predict and to explain the observed local and global behavior of a
function. 
c 
Do I =
understand
the meaning of the derivative in terms of rate of change and local linear
approximation and am I able to use derivatives to solve a variety of
problems? 
Volum=
e of
Solids with Known Cross Sections 
Find =
the area
of a region between intersecting curves using integration. 
Stude=
nts will
know and be able to use integration to calculate volume of solids. 
3.1I=
ntegrals ~
Interpretations and properties of definite integrals ~ Definite integral =
as a
limit of Riemann sums; Definite integral of the rate of change of a quant=
ity
over an interval interpreted as the change of the quantity over the inter=
val;
Basic properties of definite integrals (examples include additivity and
linearity) 
h 
Do I =
understand
the meaning of the definite integral both as a limit of Riemann sums and =
as
the net accumulation of change and am I able to use integrals to solve a
variety of problems? 
Volum=
e of
Solids of Revolution 
Find =
the volume
of a solid with known cross sections. 
Stude=
nts will
know and be able to find the average value of a function over a closed
interval. 
3.2I=
ntegrals ~
Applications of integrals ~ Appropriate integrals are used in a variety of
applications to model physical, biological, or economic situations. Altho=
ugh
only a sample of applications can be included in any specific course,
students should be able to adapt their knowledge and techniques to solve
other similar application problems. Whatever applications are chosen, the
emphasis is on using the method of setting up an approximating Riemann sum
and representing its limit as a definite integral. To provide a common
foundation, specific applications should include using the integral of a =
rate
of change to give accumulated change, finding the area of a region, the
volume of solid with known cross sections, the average value of a functio=
n,
and the distance traveled by a particle along a line 

Do I =
understand
the relationship between the derivative and the definite integral as
expressed in both parts of the Fundamental Theorem of Calculus? 
Avera=
ge Value 
Find =
the volume
of a solid of revolution using the disk/washer method. 
Stude=
nts will
know and be able to solve problems in which a rate is integrated to find =
the
net change over time in a variety of applications. 
3.3I=
ntegrals ~
Fundamental Theorem of Calculus ~ Use of the Fundamental Theorem to evalu=
ate
definite integrals; Use of the Fundamental Theorem to represent a particu=
lar
antiderivative, and the analytical and graphical analysis of functions so
defined 

Am I =
able to
communicate mathematics both orally and in wellwritten sentences and am I
able to explain solutions to problems? 
=
; 
Find =
the
average value of a function over a closed interval. 
=
; 
3.4I=
ntegrals ~
Techniques of antidifferentiation ~ Antiderivatives following directly fr=
om
derivatives of basic functions; Antiderivatives by substitution of variab=
les
(including change of limits for definite integrals) 

Am I =
able to
model a written description of a physical situation with a function, a
differential equation, or an integral? 
Misce=
llaneous
Applications 
Descr=
ibe
integration as an accumulation process. 
Stude=
nts will
know and be able to apply the Fundamental Theorem of Calculus (Part II).<=
/td>
 3.5I=
ntegrals ~
Applications of antidifferentiation ~ Finding specific antiderivatives us=
ing
initial conditions, including applications to motion along a line; Solving
separable differential equations and using them in modeling 

Am I =
able to
use technology to help solve problems, experiment, interpret results, and
verify conclusions? 
Funda=
mental
Theorem of Calculus (Part II) 
Under=
stand and
use the Fundamental Theorem of Calculus (Part II). 
Stude=
nts will
know and be able to understand the relationship between the derivative and
definite integral as expressed in both parts of the Fundamental Theorem of
Calculus. 
3.6I=
ntegrals ~
Numerical approximations to definite integrals ~ Use of Riemann sums (usi=
ng
left, right, and midpoint evaluation points) and trapezoidal sums to
approximate definite integrals of functions represented algebraically,
graphically, and by tables of values 

Am I =
able to
determine the reasonableness of solutions, including sign, size, relative
accuracy, and units of measurement? 
=
; 
=
; 
=
; 
=
; 

Have I
developed an appreciation of calculus as a coherent body of knowledge and=
as
a human accomplishment? 
=
; 
=
; 
=
; 
=
; 
A 
AP Ex=
am
Preparation 






p 
Essen=
tial
Questions 
Conte=
nt 
Skill=
s 
Assessm=
ents 
Lessons=

Learn=
ing
Benchmarks 
Stand=
ards 
r 
Am I =
able to
work with functions represented in a variety of ways (graphical, numerica=
l,
analytical, or verbal) and understand the connections among these
representations? 
Multiple
Choice and Free Response Problems 




1.1F=
unctions,
Graphs, and Limits ~ Analysis of graphs ~ With the aid of technology, gra=
phs
of functions are often easy to produce. The emphasis is on the interplay
between the geometric and analytic information and on the use of calculus
both to predict and to explain the observed local and global behavior of a
function. 
i 
Do I =
understand
the meaning of the derivative in terms of rate of change and local linear
approximation and am I able to use derivatives to solve a variety of
problems? 
1.2F=
unctions,
Graphs, and Limits ~ Limits of functions (including onesided limits) â€“=
An
intuitive understanding of the limiting process; Calculating limits using
algebra; Estimating limits from graphs or tables of data 
l 
Do I =
understand
the meaning of the definite integral both as a limit of Riemann sums and =
as
the net accumulation of change and am I able to use integrals to solve a
variety of problems? 
1.3F=
unctions,
Graphs, and Limits ~ Asymptotic and unbounded behavior â€“ Understanding
asymptotes in terms of graphical behavior; Describing asymptotic behavior=
in
terms of limits involving infinity; Comparing relative magnitudes of
functions and their rates of change (for example, contrasting exponential
growth, polynomial growth, and logarithmic growth) 

Do I =
understand
the relationship between the derivative and the definite integral as
expressed in both parts of the Fundamental Theorem of Calculus? 
1.4F=
unctions,
Graphs, and Limits ~ Functions, Graphs, and Limits ~ Continuity as a prop=
erty
of functions â€“ An intuitive understanding of continuity; Understanding
continuity in terms of limits; Geometric understanding of graphs of
continuous functions (Intermediate Value Theorem and Extreme Value Theore=
m) 

Am I =
able to
communicate mathematics both orally and in wellwritten sentences and am I
able to explain solutions to problems? 
2.1D=
erivatives
~ Concept of the derivative ~ Derivative presented graphically, numerical=
ly,
and analytically; Derivative interpreted as an instantaneous rate of chan=
ge;
Derivative defined as the limit of the difference quotient; Relationship
between differentiability and continuity 

Am I =
able to
model a written description of a physical situation with a function, a
differential equation, or an integral? 
2.2D=
erivatives
~ Derivative at a point ~ Slope of a curve at a point. Examples are
emphasized, including points at which there are vertical tangents and poi=
nts
at which there are no tangents; Tangent line approximation to a curve at a
point and local linear approximation; Instantaneous rate of change as the
limit of average rate of change; Approximate rate of change from graphs a=
nd
tables of values 

Am I =
able to
use technology to help solve problems, experiment, interpret results, and
verify conclusions? 
2.3D=
erivatives
~ Derivative as a function ~ Corresponding characteristics of graphs of f=
and
f'; Relationship between the increasing and decreasing behavior of f and =
the
sign of f'; The Mean Value Theorem and its geometric consequences; Equati=
ons
involving derivatives. Verbal descriptions are translated into equations
involving derivatives and vice versa 

Am I =
able to
determine the reasonableness of solutions, including sign, size, relative
accuracy, and units of measurement? 
2.4D=
erivatives
~ Second Derivatives ~ Corresponding characteristics of the graphs of f, =
f',
and f''; Relationship between the concavity of f and the sign of f''; Poi=
nts
of inflection as places where concavity changes 

Have I
developed an appreciation of calculus as a coherent body of knowledge and=
as
a human accomplishment? 
2.5D=
erivatives
~ Applications of Derivatives ~ Analysis of curves, including the notions=
of
monotonicity and concavity; Optimization, both absolute (global) and rela=
tive
(local) extrema; Modeling rate of change, including related rates problem=
s;
Use of implicit differentiation to find the derivative of an inverse
function; Interpretation of the derivative as a rate of change in varied
applied contexts, including velocity, speed, and acceleration; Geometric
interpretation of differential equations via slope fields and the
relationship between slope fields and solution curves for differential
equations 

=
; 
2.6D=
erivatives
~ Computation of Derivatives ~ Knowledge of derivatives of basic function=
s,
including power, exponential, logarithmic, trigonometric, and inverse
trigonometric functions; Basic rules for the derivative of sums, products,
and quotients of functions; Chain rule and implicit differentiation 

=
; 
3.1I=
ntegrals ~
Interpretations and properties of definite integrals ~ Definite integral =
as a
limit of Riemann sums; Definite integral of the rate of change of a quant=
ity
over an interval interpreted as the change of the quantity over the inter=
val;
Basic properties of definite integrals (examples include additivity and
linearity) 

=
; 
3.2I=
ntegrals ~
Applications of integrals ~ Appropriate integrals are used in a variety of
applications to model physical, biological, or economic situations. Altho=
ugh
only a sample of applications can be included in any specific course,
students should be able to adapt their knowledge and techniques to solve
other similar application problems. Whatever applications are chosen, the
emphasis is on using the method of setting up an approximating Riemann sum
and representing its limit as a definite integral. To provide a common
foundation, specific applications should include using the integral of a =
rate
of change to give accumulated change, finding the area of a region, the
volume of solid with known cross sections, the average value of a functio=
n,
and the distance traveled by a particle along a line 

=
; 
3.3I=
ntegrals ~
Fundamental Theorem of Calculus ~ Use of the Fundamental Theorem to evalu=
ate
definite integrals; Use of the Fundamental Theorem to represent a particu=
lar
antiderivative, and the analytical and graphical analysis of functions so
defined 

=
; 
3.4I=
ntegrals ~
Techniques of antidifferentiation ~ Antiderivatives following directly fr=
om
derivatives of basic functions; Antiderivatives by substitution of variab=
les
(including change of limits for definite integrals) 

=
; 
3.5I=
ntegrals ~
Applications of antidifferentiation ~ Finding specific antiderivatives us=
ing
initial conditions, including applications to motion along a line; Solving
separable differential equations and using them in modeling 

=
; 
3.6I=
ntegrals ~
Numerical approximations to definite integrals ~ Use of Riemann sums (usi=
ng
left, right, and midpoint evaluation points) and trapezoidal sums to
approximate definite integrals of functions represented algebraically,
graphically, and by tables of values 
M 
AP Ex=
am 






a 
Essen=
tial
Questions 
Conte=
nt 
Skill=
s 
Assessm=
ents 
Lessons=

Learn=
ing
Benchmarks 
Stand=
ards 
y 
=
; 
AP Ex=
am 
=
; 
<=
/td>
 <=
/td>
 =
; 
=
; 







