Calculus AB APSI 2015
Day 3
Professional
Development
Workshop
Handbook
Curriculum
Framework
Calculus AB and
BC
Professional
Development
Integration, Problem
Solving, and Multiple
Representations
Curriculum Module
Wednesday
Morning (Part 1)
• Introducing the Definite Integral Through
the Area Model
• Investigating How to Find Area Using
Riemann Sums and Trapezoids
• Discussion of Homework Problems
• Free Response Problem 2014 AB5
• Share an Activity
• Dan Meyer
• Developing an Understanding for a
Definite Integrals
Break
• Numerical Integration
Afternoon (Part 2)
Break
• Calculus Games
Morning (Part 2)
• Building Understanding for the Average
Value of a Function
• Fun Finding Volume
• Solids with Known Cross Sectional Area
2
Afternoon (Part 1)
• Mean Value Theorem
• Fundamental Theorem of Calculus
• Curriculum Module: Motion
(w/Smartboard)
• Activity Sheet for Understanding the 2nd
Fundamental Theorem of Calculus
• Big Idea 3 – The Integral and the
Fundamental Theorem of Calculus
Lunch
Wednesday Assignment - AB
 Multiple Choice Questions on the 2014 test: 1, 4, 8, 12,
14, 18, 26, 76, 77, 70, 80, 81, 83, 84, 85
 Free Response:
2014: AB4, AB6
2015: AB4/BC4, AB5
3
Key Ideas to Cover on Integration
A definite integral is the limit of
a Riemann sum
b
n
 f (x)dx  lim  f (x )x
a
n 
i 1
i
The definite integral is the net
accumulation of a rate of
change
b
 f '(x)dx  f (b)  f (a)
a
or
b
f (b)  f (a)   f '(x )dx
a
4
Concept of a Definite Integral
All the important concepts related to definite integrals
can be taught and understood without knowing
antiderivatives.
5
Calculus AP should include opportunities for students to
understand
• Area under a graph
- Riemann Sum – Definition of a Definite Integral
• Ways to Evaluate a Definite Integral
- Fundamental Theorem
• How integrals accumulate area
• How functions can be by integrals
• Techniques for finding indefinite integrals
• Applications of integrals
6
• Deal with graphical and tabular sets of data to find area
b
• That  f (x )dx can be represented by a bounded region.
a
b
• That  f (x )dx can be approximated using several
a
methods.
• Relationships between approximations.
• How more accurate approximation be found
b
• The units of measure for  f (x )dx .
a
7
Introducing Integration through the Area
Model
Introducing the Definite Integral Through the Area Model
Figure 1 shows the velocity of an object, v(t), over a 3-minute
interval. Determine the distance traveled over the interval
0  t  3 . The area bounded by the graph of v(t) and the t-axis
for 0  t  3 represents the distance traveled by this object.
The distance can be represented by the
3
definite integral  v (t )dt.
0
Activity
9
The following chart gives the velocity of a particle, v(t), at 0.5 second intervals.
Estimate the distance traveled by the particle in the three seconds using three
different methods. Each method is an approximation for

3
0
Activity
v (t )dt .
Investigating How to Find Area using Riemann Sums
and Trapezoids
b
n
 f (x)dx  lim  f (x )x
a
n 
i 1
i
Using the NUMINT program or
LMRRAM and TRAPEZOID program on a TI84
Activity
11
Things You Should have Observed
As the number of rectangles increases on monotonically
increasing functions, the left hand sums increase, but
remain less then the area.
12
Things You Should have Observed
Which sums are always
greater than the actual
area?
Which sums are always less
than the actual area?
13
Things You Should have Observed
The limit of the left hand sum equals the limit of the
right hand sum and equals the area of the region.
n
lim  f ( xi )x 
n 
14
i 1
area of the region or
b
 f (x)dx
a
Students should be able to
Set up and evaluate left,
right and midpoint
Riemann sums from
analytical data, tabular
data, or graphical data.
Set up and evaluate a
trapezoidal sum
approximation from
analytical data, tabular
data, or graphical data.
15
3
9 − 𝑥 2 𝑑𝑥
0
Determine Units of Measure
• The units of the definite integral are the units of the
Riemann Sum
- The units of the function multiplied by the units of
the independent variable.

b
a
16
f (x )dx
Verbal Explanation
• Students need to be able to tell what a definite
integral represents in the context of the problem and
identify the units of measure.
• Very common AP question on Free Response
Questions

b
a
17
f (x )dx
Using Technology to Approximate the Definite Integral
18
Developing an Understanding for the Definite Integral
Smartboard File
19
Fun Finding Volume
20
Solids with Known Cross Sectional Area
Activity
21
Volume of a Solid with Known Cross Sectional Area
Create a table and a sketch for
𝑓(𝑥) = 𝑥
scale for the grid is 0.5 cm
on the x and y axes
22
Sketch the graph of f (x).
The scale for this grid is 0.25 cm on
both the x and y axes.
23
Select one of the figures. Cut out the 9 shapes,
keeping the tabs on the shape. Fold the trapezoidal
tab. Glue the tab on the graph so that the edge of
the shape is the f(x) segment. Face all the colored
faces in the same direction.
24
Group Work
Complete the Finding the Volume of
the Solid Activity Sheet with your
group members.
25
Volume of Solids of Revolution
Section 9 in Notebook
Smartboard File
26
Rotating about a Line Other than the x- or y-axis
Pages 2 to 5
27
Rotating about a Line Above the Region
Pages 5 to 7
28
Rotating about a Line to the Left of the Region
Pages 10 and 11
29
Rotating about a Line to the Right of the Region
Pages 8 and 9
30
Fundamental Theorem of Calculus
Smartboard File
31
Activity Sheet for Understanding the 2nd Fundamental
Theorem of Calculus
Section 5 in Notebook
Activity
32
Section 8 in Notebook
33
Areas, Derivatives, and the Fundamental Theorem of
Calculus
Worksheet 1: pages 8-11
34
Applying the Second Fundamental Theorem of
Calculus: Finding Derivatives of Functions Defined by
Integrals
Worksheet 2: page 13
35
Applying the First Fundamental Theorem
of Calculus: Definite Integrals as Total
Change
Worksheet 3: page 15-16
36
Examples of Multiple Choice Questions
Worksheet 4: page 15-16
37
Applying the Fundamental Theorem of Calculus Exercises
Worksheet 5: page 15-16
38
Tuesday Assignment - AB
► Multiple
Choice Questions on the 2014
test: 9, 11, 15, 19, 21, 22, 23, 27, 28, 82,
88, 89, 90, 91, 92
► Free
Response:
► 2014:
AB3, AB6
► 2015:
AB2, AB3
39
Scoring Rubric 2014 AB3
40
2014 AB3
41
2014 AB6
42
Scoring Rubric-2014 AB6
43
2015 AB2
44
Scoring Rubric 2015 AB2
45
2015 AB3
46
Scoring Rubric 2015 AB3
47
Share An Activity from Your Classroom
48
Dan Meyer
Taco Stand
49
All Examples
Calculus Games
50
Building Understanding for the Average Value of a
Function
Activity
51
The Mean Value Theorem
52
The Mean Value Theorem - Smartboard
What is guaranteed?
What must be true
for the guarantee?
Can parts be true if
the conditions are not
met?
How does it apply to
real data?
53
College Board has
developed a
Curriculum Module
to assist you in
teaching how to use
Calculus to study
motion.
Motion
Smartboard File
54
Dixie Ross
Pflugerville High School
Pflugerville, TX
What You Need to Know about Motion
Worksheet 1: page 5
55
Sample Practice Problems
Numerical, Graphical, Analytical
Worksheet 2: pages 7-9
56
Understanding the Relationship Among Velocity, Speed
and Acceleration
Worksheet 3: page 13-15
57
What You Need to Know about Motion Along the x-axis
Worksheet 4: page 21
58
Sample Practice Problems
Worksheet 5: page 23-26
59
Big Idea 3: Integrals and the
Fundamental Theorem of
Calculus
Page 364-367
Wednesday Assignment - AB
 Multiple Choice Questions on the 2014 test: 1, 4, 8, 12,
14, 18, 26, 76, 77, 70, 80, 81, 83, 84, 85
 Free Response:
2015: AB4, AB5
2014: AB4, AB6
68