Lake County Schools
Investing In Excellence!
College and Career Readiness
Academic Services
April 2013
AP Seminar
For
Advanced
Placement
Calculus AB
_______
WELCOME
Community Builder:
Just like me…..
1.
2.
3.
4.
One person begins introducing
themselves to the group
When another person in the
audience hears something the
previous person shared that is
“just like them” they chime in and
say “ that’s just like me”
The person that chimed in saying
“ that’s just like me” begins
introducing themselves starting
with the connection from the
previous person.
Repeat step 2 and 3 until all
members in the audience have
introduced themselves.
Bellwork: Community Builder
•Instructor will review the 21st
Century skills and AP success
stats with students and make a
connection to the AP Seminar
•Instructor will review specific
content for AP course of study
Learning Goal: Learners will
understand and implement effective test
taking strategies for passing AP exams.
I DO
Benchmarks:
Strategic Plan Goal # 1
Increased Student Achievement
Objective
April 6 & April 27 2013
AP Seminar
CBC
•Learners will: utilize content
knowledge learned in AP courses coupled
with effective test taking strategies to
increase pass rate by completing practice
AP test questions
WE DO
•Students will utilize materials
YOU DO from the AP Seminar to study for
AP exam
Essential Question:
How do we revolutionize the way we teach, lead,
and learn for 21st century success?
Common Language:
•Advanced Placement
•Effective Strategies
•Instructor and students will
utilize test taking strategies to
answer multiple choice and free
response answers
Exit Activity
Students will share with the class one
strategy or tip they will use on exam day
NEXT STEPS:
1. Utilize new learning and implement on AP exam
2. Continue to study for AP exam
21st Century Skills
Tony Wagner, The Global Achievement Gap
1. Critical Thinking and Problem Solving
2. Collaboration and Leadership
3. Agility and Adaptability
4. Initiative and Entrepreneurialism
5. Effective Oral and Written Communication
6. Accessing and Analyzing Information
7. Curiosity and Imagination
Academic Services
Positive Statistics
Hurray Lake County Schools
Lake County Schools….
– Named to the College Board District
Honor Roll
What to Bring
•Several #2 pencils and a good eraser - NO
MECHANICAL PENCILS!
•Black or blue colored pens are allowed for the freeresponse section
•One or two graphing calculators with fresh batteries
•A watch so you can monitor your time
•A simple snack if the test site permits it
•A light jacket in case the room is cold
•Tissues
•ABSOLUTELY NO CELL PHONES or any
electronic devices.
How the test is graded:
• Multiple Choice: 50% of your grade
– 1.2 points for each correct
– 0 points for incorrect/blank
• Free Response: 50% of your grade
– Each question is graded on a scale of 0-9
– No “bald” answers are given credit –
show work
Parts of Test:
Multiple Choice:
Part A 28 problems in 55 min (NO calculator)
Part B 17 problems in 50 min (calculator allowed)
Free Response:
Part A
Part B
2 problems in 30 mins (calculator allowed)
4 problems in 1 hour (NO calculator) *
*You may go back and work on first two problems
but you may NOT use calculator.
Multiple Choice
Strategies
•
•
•
•
•
•
•
Do easy problems first!
Do not linger-let the hard ones go.
Read the question carefully.
Graph or chart- is it f (x) or f '(x)?
Do not round until the final answer
Perhaps you can work backwards
If all else fails – guess!
Content Overview
•
•
•
•
•
Limits and their Properties
Derivative Concepts
Applications of Derivatives
Integration
Applications of Integration
Choose a partner.
Have your packet and calculator
out. Write yourself notes about
what to study before the exam.
•Let’s review
Limits.
•Ready?
sin x
lim
x 0
x
1
1  cos x
lim
x 0
x
0
0
f ( x) 
If lim
x c
0
what can you do?
•
•
•
•
•
•
Factor
Rationalize numerator (conjugate)
Simplify
Graph
Make a table
THINK!
3 ways a limit fails to exist
• Limit from the left is not equal to the
limit from the right (break)
• Unbounded Behavior (vertical
asymptote)
• Oscillating Behavior
1
lim
x2 x  2

lim f ( x)
x 
means
Find horizontal asymptotes
rational functions
non-rational functions
Partner Time
Ready to review derivatives?
A derivative is
the slope of the
tangent line
Definition of
Derivative
(formula)
f  x  x   f ( x)
lim
x 0
x
3 ways a derivative
fails to exist
1.
2.
3.
Discontinuities
Sharp turn
Vertical tangent
Instantaneous rate
of change
derivative
Average
rate of change
slope of secant
line
Acceleration
a(t) = v’(t)
a(t) = s’’(t)
Derivative Rules
Derivatives to
Remember!
d
sin x
dx
d
cos x
dx
d
tan x
dx
d
sin x
dx
cos x
d
cos x
dx
d
tan x
dx
 sin x
2
sec x
More Derivatives to
Remember
d
sec x
dx
d
csc x
dx
d
cot x
dx
d
sec x
dx
d
csc x
dx
d
cot x
dx
d
cot x
dx
secxtanx
-cscxcotx
 csc x
2
And more derivatives
d
ln x
dx
d x
e
dx
And more derivatives
d
ln x
dx
d x
e
dx
1
x
e
x
Don’t forget derivatives of logs, inverse trig functions…
Find equation of
tangent line
Find f’(c) m
Find ordered pair
Plug into y  y1  m( x  x1 )
Related Rates
Know:
Want:
Eq:
Deriv:
Don’t forget:
when:
d
dt
Partner Time
f (b)  f (a ) f (3)  f (0) 66  0


 22
ba
30
3
Partner Time
f '(5)  4  m
f (5)  3  (5,3)
y  3  4( x  5)
y  3  4( x  5)
f (4.8)  3  4(4.8  5)  2.2
Let’s look at applications
of derivatives.
Find
Absolute Extrema
on [a,b]
(also known as global)
Check endpoints and
critical numbers to find
extreme y values.
Relative Extrema
(also called local)
May occur where f’=0
or f’ undefined
Find inflection points:
Find f’’(x) to see where
concavity changes
signs.
Partner Time
dy
 0  decreasing
dx
d2y
 0  concave down
2
dx
Point B
Partner Time
dy
dy
2
2
y (1)  x 3 y
 y 2x  x 2 y
0
dx
dx
2 dy
2 dy
3xy
 2x y
  y 3  2 xy 2
dx
dx
dy
(3 xy 2  2 x 2 y )   y 3  2 xy 2
dx
dy  y 3  2 xy 2 1  4 5



2
2
dx 3xy  2 x y 6  8 14
3
2
Here come the integrals!
Trig Integrals
sin
xdx


cos
xdx


tan
dx


Trig Integrals
sin
xdx


 cos x  C
cos
xdx


sin x  C
tan
dx


 ln cos x  C
More Trig Integrals
sec
xdx


csc
xdx


cot
xdx


sec
xdx


ln secx  tan x  C
csc
xdx


 ln cscx  cot x  C
cot
xdx


ln sin x  C
Don’t forget the easy
ones!
sec
xdx


2
csc
xdx


2
 sec x tan xdx 
 csc x cot xdx 
sec
xdx


tan x  C
csc
xdx


 cot x  C
sec
x
tan
xdx


sec x  C
csc
x
cot
xdx


 csc x  C
2
2
And more Integrals
1
dx

x
e
dx


x
1
dx

x
ln x  C
e
dx


e C
x
x
Applications of
Integration
Solve a differential equation
Separate
and
Integrate
Find a particular solution to a
differential equation:
• Separate and Integrate + C
• Use initial condition to solve for C.
• Plug C into answer.
Area under a curve
b

a
f  x dx
Area between two curves
b
A    f  x   g  x  dx
a
b
A   (top )  (bottom)dx
a
Partner Time
Intersection Point:
3
A   [(5 x  x )  2 x ]dx
2
0
9

2
5x  x  2 x
3x  x 2  0
x(3  x)  0
x  0,3
2
Disk Method
b
2
V     R  x   dx
a
Washer Method
b
V     R  x     r  x   dx
a
2
2
Solids with known
cross sections
b
V   A  x dx
a
or
d
V   A  y dy
c
Accumulation
Function
x

a
f (t )dt
Partner Time
1
g (1) 

0
1

f (t )dt    f (t )dt    1 2   1
2

1
0
Particle Motion
b
Total distance =
how far did you actually walk?

v  t  dt
a
b
Displacement =
how far are you from where you started?
 v(t )dt
a
A particle is at rest when
A particle is moving right when
Particle changes direction when
Speed =
A particle is at rest when velocity = 0
A particle is moving right when velocity >0
Particle changes direction when velocity = 0
Speed = |velocity|
Speed increases when
velocity and acceleration have
same signs
Speed decreases when
velocity and acceleration have
opposite signs
Partner Time
v(t )  6t 2  48t  90  6(t 2  8t  15)  6(t  3)(t  5)  0
a(t )  12t  48  12(t  4)  0
t  3, t  5
t 4
v(t )    3      5   
a (t ) <      4       
v(t ) and a(t ) have same signs on 3  t  4 and t  5
Strategies and Practice:
Free Response
•
•
•
•
•
Show all work
Do not round partial answers
Answers correct to nearest thousandth
Attempt all parts of each question
Write the equation before using
calculator to solve
• Write the integral before using
calculator to solve
Strategies and Practice:
Free Response
• Don’t erase, just cross out work you
don’t want graded.
• Calculators are only for graphs,
derivatives, integrals, and equations
• Be sure you have answered the question
• “Justify your answer” means to explain
using calculus concepts (sign charts are
not justification)
Be sure you know the theorems!
Intermediate Value Theorem
Mean Value Theorem
Rolle’s Theorem
First Derivative Test
2nd Derivative Test
Mean Value Theorem for Integrals
Fundamental Theorem of Calculus
2005 AB Free Response
Scores
Time for Group Work
You need a group of 4.
Each person will be responsible for
working out one part of the problem and
explaining it to the others.
Group Work
Free Response 2005 AB1
calculator allowed
Free Response 2005 AB1
Group Work
Free Response 2005 AB2
calculator allowed
Free Response 2005 AB2
Group Work
Free Response 2005 AB4
no calculator allowed
Free Response 2005 AB4
Group Work
Free Response 2005 AB6
no calculator allowed
Free Response 2005 AB6
Exit Activity
• With a shoulder partner, turn and talk
about one strategy you will utilize on
your AP exam and why.
Lake County Schools
Investing In Excellence!
College and Career Readiness
Academic Services
April 2013