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Calculus AB APSI 2015
Day 3
Professional Development
Workshop Handbook
Curriculum Framework
Calculus AB and BC
Professional Development
Integration, Problem Solving, and Multiple
RepresentationsCurriculum Module
2
Wednesday
Morning (Part 1)Introducing the Definite Integral Through the Area Model Investigating How to Find Area Using Riemann Sums and TrapezoidsDeveloping an Understanding for a Definite Integrals Numerical Integration
Break Morning (Part 2)Fun Finding VolumeSolids with Known Cross Sectional AreaFundamental Theorem of CalculusActivity Sheet for Understanding the 2nd Fundamental Theorem of Calculus
Lunch
Afternoon (Part 1)Discussion of Homework ProblemsFree Response Problem 2014 AB5Share an ActivityDan Meyer
Break Afternoon (Part 2)Calculus GamesBuilding Understanding for the Average Value of a FunctionMean Value TheoremCurriculum Module: Motion (w/Smartboard)Big Idea 3 – The Integral and the Fundamental Theorem of Calculus
3
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
4
Key Ideas to Cover on Integration
A definite integral is the limit of a Riemann sum The definite integral is the net
accumulation of a rate of change
or
1
( ) lim ( )b n
inia
f x dx f x x
'( ) ( ) ( )
b
a
f x dx f b f a
( ) ( ) '( )b
a
f b f a f x dx
5
Concept of a Definite Integral
All the important concepts related to definite integrals can be taught and understood without knowing
antiderivatives.
6
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 areaHow functions can be by integralsTechniques for finding indefinite integralsApplications of integrals
7
Deal with graphical and tabular sets of data to find area
That can be represented by a bounded region.
That can be approximated using several
methods.
Relationships between approximations.
How more accurate approximation be found
The units of measure for .
( )b
a
f x dx
( )b
a
f x dx
( )b
a
f x dx
Introducing Integration through the Area
Model
9
Figure 1 shows the velocity of an object, v(t), over a 3-minute interval. Determine the distance traveled over the interval . The area bounded by the graph of v(t) and the t-axis for represents the distance traveled by this object. The distance can be represented by thedefinite integral .
0 3t 0 3t
3
0( )v t dt
Introducing the Definite Integral Through the Area Model
Activity
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( )v t dt
Activity
11
1
( ) lim ( )b n
inia
f x dx f x x
Using the NUMINT program orLMRRAM and TRAPEZOID program on a TI84
Investigating How to Find Area using Riemann Sums and Trapezoids
Activity
12
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.
13
Things You Should have Observed
Which sums are always
greater than the actual
area?
Which sums are always less
than the actual area?
14
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.
area of the region or 1
lim ( )n
ini
f x x
( )b
a
f x dx
15
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.
0
3
√9− 𝑥2𝑑𝑥
16
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
af x dx
17
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
af x dx
18
Using Technology to Approximate the Definite Integral
19
Smartboard File
Developing an Understanding for the Definite Integral
20
Fun Finding Volume
21
Solids with Known Cross Sectional Area
Activity
22
Create a table and a sketch for 𝑓 (𝑥)=√𝑥
scale for the grid is 0.5 cm on the x and y axes
Volume of a Solid with Known Cross Sectional Area
23
Sketch the graph of f (x). The scale for this grid is 0.25 cm on
both the x and y axes.
24
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.
25
Complete the Finding the Volume of the Solid Activity Sheet with your
group members.
Group Work
26
Volume of Solids of Revolution
Section 9 in Notebook
Smartboard File
27
Rotating about a Line Other than the x- or y-axis
Pages 2 to 5
28
Rotating about a Line Above the Region
Pages 5 to 7
29
Rotating about a Line to the Left of the Region
Pages 10 and 11
30
Rotating about a Line to the Right of the Region
Pages 8 and 9
31
Fundamental Theorem of Calculus
Smartboard File
32
Activity Sheet for Understanding the 2nd Fundamental Theorem of Calculus
Activity
Section 5 in Notebook
33
Section 8 in Notebook
34
Areas, Derivatives, and the Fundamental Theorem of Calculus
Worksheet 1: pages 8-11
35
Applying the Second Fundamental Theorem of Calculus: Finding Derivatives of Functions Defined by Integrals
Worksheet 2: page 13
36
Applying the First Fundamental Theorem of Calculus: Definite Integrals as Total Change
Worksheet 3: page 15-16
37
Examples of Multiple Choice Questions
Worksheet 4: page 15-16
38
Applying the Fundamental Theorem of Calculus Exercises
Worksheet 5: page 15-16
39
►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
Tuesday Assignment - AB
40
Scoring Rubric 2014 AB3
41
2014 AB3
42
2014 AB6
43
Scoring Rubric-2014 AB6
44
2015 AB2
45
Scoring Rubric 2015 AB2
46
2015 AB3
47
Scoring Rubric 2015 AB3
48
Share An Activity from Your Classroom
49
Dan Meyer
Taco Stand All Examples
50
Calculus Games
51
Building Understanding for the Average Value of a Function
Activity
52
The Mean Value Theorem
53
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?
54
Dixie RossPflugerville High SchoolPflugerville, TX
Motion Smartboard File
College Board has developed a
Curriculum Module to assist you in
teaching how to use Calculus to study
motion.
55
What You Need to Know about Motion
Worksheet 1: page 5
56
Sample Practice ProblemsNumerical, Graphical, Analytical
Worksheet 2: pages 7-9
57
Understanding the Relationship Among Velocity, Speed and Acceleration
Worksheet 3: page 13-15
58
What You Need to Know about Motion Along the x-axis
Worksheet 4: page 21
59
Sample Practice Problems
Worksheet 5: page 23-26
Big Idea 3: Integrals and the Fundamental Theorem
of Calculus
Page 364-367
68
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