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