Advanced Placement Calculus AB/BC Outline 2012-13
The following will serve as the outline guide for the AB/BC course that
succeeds/enhances the International Baccalaureate SL class which has already
included approximately 10 weeks of investigative calculus. The student in the spring
semester will choose either the AB or BC Exam to take in May 2013.
TOPIC
TEXT
TIME
ASSESSMENT HW/due
REFERENCE/READING
date
Chapter 1, including
Ongoing
Chapter 1
I.
Review
Precalculus/Function emphasis on Parametric
for 1st
Equations & Inverse Trig; couple of
Exercises
Study
Key terms on p. 51 includes
p.52-53) even
weeks
even vs. odd function,
problems only
symmetry about y-axis vs.
origin
II. Limit Theory
Chapter 2
4-5 days
TOPIC
TEXT
REFERENCE/READING
TIME
ASSESSMENT
HW/due
date
i. limit at a point
1
ii. limit at infinity
iii. Squeeze Theorem
2 days
III. Continuity
i. definition
ii. types of
discontinuity
iii. Intermediate Value
Th.
IV. Existence of
f ‘ (x)
* average rate of
change vs.
instantaneous rate
i. Continuity of f (x)
w/ 2 sided limit @ a
point
TOPIC
EXAM on
Limits &
Continuity
Chapter 2.4 plus
Chapter 3 – 4
10-12 days
Diagrams to
complete
TEXT
REFERENCE/READING
TIME
ASSESSMENT
HW/due
date
ii. given f (x) sketch
f ‘ (x)
2
iii. Rules of
differentiation
iv. 2nd derivative
v. Average vs.
Instantaneous
Rate of Change
vi. Chain Rule
vii. Implicit
Differentiation
viii. Inverse Trig
Derivatives
V. Extreme Value
Th. – Abs. Extreme
Values
EXAM on IV.
Items i. – v.
EXAM on IV.
Items vi. – viii.
Chapter 4.1
2 days
VI. Mean Value Th.
VII. 1st & 2nd
Derivative Testing
TOPIC
VIII. Modeling &
Optimization
4-5 days
p.187 - 206
4-5 days
EXAM on VII.
TEXT
REFERENCE/READING
Section 4.4
TIME
ASSESSMENT
HW/due
date
8-10 days
3
i. Max/Min w/both
restricted domains and
unrestricted domains
ii. Linear
Section 4.5
Approximation
Model w/
Differentials
iii. Newton’s Method
Section 4.5
iv. Related Rates
Section 4.6 w/Supplements
Chapter 5
IX. Intro to
Integration
i. Riemann Sums
ii. LRAM, MRAM,
RRAM
approximations
iii. Definite Integrals
&
Antiderivatives
TOPIC
TEXT
REFERENCE/READING
10 days
TIME
ASSESSMENT
HW/due
date
iv. Average (Mean)
Value
4
v. Fundamental
Theorem
vi. Trapezoidal
Approx.
X. Differential
Equations
Chapter 6.1
5 days
Chapter 6.2 & 6.3
4 days
i. Initial Value
Problems
ii. Slope Fields
XI. Integration
Techniques
i. by Substitution
ii. by Parts
5
TIME
TOPIC
TEXT
REFERENCE
XI. Integration
Techniques
iii. dy  ky <----->
dt
y  Ae
Chapter 6.4
1 day
Chapter 6.5
2 day
Chapter 6.6
1 day
kt
iv. Logistics Curves
with
y
M
1  Ae
 kt
v. Numerical Methods
of Integration
XIII. Applications of
Definite Integrals
i. Area between two
curves
ii. Volumes of
Revolution
Chapter 7
Chapter 7.2
2 days
Chapter 7.3
3 days
a. Disk technique
6
b. Washer technique
iii. Volumes by Cross
Sections
XIV. Additional Integration Applications Chapter 7.4, back
to 7.3
i.
Length of Curve
ii. Surface Area
iii. Math modeling with various solids of
revolution
Chapter 8
XV. L’Hopital’s Rule & Improper
Integrals
Prelude: Investigation of Gabriel’s Horn
i. Indeterminate form of 0/0
Chapter 8.1
3 days
2-3 days w/ extensive GDC
investigations
ii. Indeterminate forms of
Independent
iii. Indeterminate forms of
Independent
  ,   0, &   
00 and  0
iv. Improper Integrals with infinite
integration
2/3 days
7
2/3 days
v. Improper Integrals with Infinite
Discontinuities @ an interior point
vi. The p-Integral of the form
1 dx
0 x p
&

1
2 days
dx
xp
vii. Direct Comparison Test
viii. Limit Comparison Test
1/2 days
1/2 days
Chapter 9
XVI. Infinite Series
Prelude: the p-series
2
6
 1
1
2
2

i. Infinite geometric Series
ii. Infinite Series in General
1
3
2

1
4
2
 ...
p.459
½ day review of finite and
infinite geometric series
½ day

 ak  a1  a2  a3  a4  ...  an  ...
n 1
8
iii. Power Series centered at
x = 0 & centered at x = a
2/3 days

1
 x n  1  x  x 2  x3  ...  x n  1  x
a)
n 0
converges on (-1, 1)
b) finding power series for other
functions
iv. Finding a Power Series by differentiation
v. Finding a Power Series by integration
vi. Taylor Series centered at x = 0
(Maclaurin Series) OR centered at
x = a:
f '' ( a )
x  a 2  ...
f (a )  f (a )( x  a ) 
2!
2 days
2 days
Chapter 9.2
3 days
'
n

k
f (a)
x  a n  ...   f (a) x  a k
n!
k 0 k!
OR
** all Maclaurin Series for:
1
1
,
, e x , sin x, cos x, ln(1  x), tan 1 x
1 x 1 x
9
vii. Taylor’s Theorem
Including the remainder of order n or the
error term leading to Lagrange Error Bounds
Chapter 9.3
4 days
viii. Radius of Convergence & Interval of
Convergence – Tests for Convergence
a) Convergence Theorem for a Power
Series – 3 possibilities exist for
Chapter 9.4
5 days

 cn x  a n
n 0
1. R exists such that x  a  R and
the series diverges &
x  a  R and the series converges.
Convergence may or may not occur at the
endpoints x  a  R
2. The series converges for all x
3. The series converges at x = a
and diverges elsewhere.
10
b) The nth term Test for Divergence
c) The Direct Comparison Test with no
negative terms
** Absolute vs. Conditional Convergence
d) The Ratio Test
** Testing Convergence @ Endpoints as the
series tests are investigated
e) The Integral Test
** The p-series Test for
1 dx
0 x p
&

1
Chapter 9.5
4 days
dx
xp
f) Limit Comparison Test
g. The Alternating Series Test
Full Summary of Infinite Series, radius of
convergence, and interval of convergence
p.505 and
Chapter Review
exercises
XVII. Parametric, Polar, & Vector
Functions
i. Derivative at a point for a parametrized
curve x = f(t) , y = g(t)
Chapter 10
1 day
1 day
11
ii. 2nd Derivative at a point for a
d2y
dy ' / dt
parametrized curve: 2 
dx / dt
dx
iii. Arc Length of a smooth parametrized
curve: L  
b
a
2
1 day
2
 dx   dy 
     dt
 dt   dt 
iv. Surface Area from a smooth Parametrized
Curve w/ revolution about x-axis:
2
1 day
2
 dx   dy 
S   2y      dt
a
 dt   dt 
b
v. Vectors in the Plane
vi. Derivative at a Point for a vector
function r(t) = f(t) i + g(t) j =
df
dt
Chapter 10.3A
general review of
the algebra &
geometry of
vectors
1 day
1 day
i  dg j
dt
12
vii. related discussion on velocity, speed,
acceleration, distance traveled and direction
of motion for a position vector r
viii. Polar Coordinates and Recognition of
Polar Graphs
a) symmetry tests
b) equations relating Polar and
Cartesian Coordinates
1) circles
2) rose curves
ix. Calculus of Polar Curves
a) slope of the tangent to a polar curve
r  f ( )
b) Area in Polar Coordinates
A


1 day
Chapter 10.5
Chapter 10.6
3 days
1 2
r d
2
c) Area between Polar Curves
d) Length of a Polar Curve
e) Area of a surface of Revolution
13
Review for AP Calculus Exam
AB or BC
1 week or more
Multiple Methods of review
including previous AP exams,
Chapter summaries, & class
exams
Textbook:
Calculus by Finney, Demana, Waits, Kennedy
Prentice Hall 2003
Associated Supplemental Guides: Workbooks and Teacher Guidebooks from the same
edition as textbook above.
14