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# Wednesday – AB - PowerPoint PPT Presentation

Wednesday – AB. Morning (Part 1) Developing the Concept of a Definite Integral Area Model Riemann Sums and Trapezoidal Method Numerical Integration Break Morning (Part 2) Applications of Integration Solids with Known Cross Sectional Area Discovering the Average Value of a Function

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• Morning (Part 1)

• Developing the Concept of a Definite Integral

• Area Model

• Riemann Sums and Trapezoidal Method

• Numerical Integration

• Break

• Morning (Part 2)

• Applications of Integration

• Solids with Known Cross Sectional Area

• Discovering the Average Value of a Function

• Free-Response Problem (2013 AB 5)

• Lunch

• Afternoon (Part 1)

• Share an Activity

• Dan Meyer

• Discussion of Homework Problems

• Break

• Afternoon (Part 2)

• Curriculum Module: Motion (w/Smartboard)

• Mean Value Theorem

• Morning (Part 1)

• Developing the Concept of a Definite Integral

• Area Model

• Riemann Sums and Trapezoidal Method

• Numerical Integration

• Break

• Morning (Part 2)

• AB:

• Applications of Integration

• Solids with Known Cross Sectional Area

• Discovering the Average Value of a Function

• Free-Response Problem (2013 AB 5)

• BC:

• Polar

• Lunch

• Afternoon (Part 1)

• Share an Activity

• Discussion of Homework Problems

• Break

• Afternoon (Part 2)

• AB:

• Curriculum Module: Motion (w/Smartboard)

• Mean Value Theorem

• BC:

• Parametric & Vectors

• Multiple Choice Questions on the 2013 test: 25, 26, 27, 77, 79, 80, 81, 83, 85, 86, 87, 88, 89, 90, 91, 92

• Free Response:

• 2014: AB4/BC4, AB5

• 2013: AB4

• Multiple Choice Questions on the 2013 test: 25, 26, 27, 77, 79, 80, 81, 83, 85, 86, 87, 88, 89, 90, 91, 92

• Free Response for AB Track:

• 2014: AB4/BC4, AB5

• 2013: AB4

• Free Response for BC Track:

• 2014: AB4/BC4, BC5

• 2013: BC4

• Introducing the Definite Integral Through the Area Model

• Investigation How to Find Area Using Riemann Sums and Trapezoids

• Developing Understanding for a Definite Integral

• Fun Finding Volume

• Numerical Integration

• Solids with Known Cross-Sections

• Building Understanding for the Average Value of a Function

• Share an Activity

• Dan Meyer

• Tuesday Assignment

• Motion

• Mean Value Theorem

• Parametric & Vectors

• A definite integral is the limit of a Riemann sum

• The definite integral is the net accumulation of a rate of change

or

All the important concepts related to definite integrals can be taught and understood without knowing antiderivatives.

• 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

### Introducing Integration through the Area Model understand

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 the

definite integral .

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 .

Investigating at 0.5 second intervals. Estimate How to Find Area using Riemann Sums and Trapezoids

Using the NUMINT program or

LMRRAM and TRAPEZOID program on a TI83/84

Things You Should have Observed at 0.5 second intervals. Estimate

• As the number of rectangles increases on monotonically increasing functions, the left hand sums increase, but remain less then the area.

Things You Should have Observed at 0.5 second intervals. Estimate

• which sums are always greater than the actual area

• Which sums are always less than the actual area

Things You Should have Observed at 0.5 second intervals. Estimate

• 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

Students should be able to at 0.5 second intervals. Estimate

• 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.

• Determine Units of Measure: at 0.5 second intervals. Estimate

• 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.

• Verbal Explanation at 0.5 second intervals. Estimate

• 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

Developing an Understanding for the Definite Integral at 0.5 second intervals. Estimate

Smartboard File

Fun Finding at 0.5 second intervals. Estimate Volume

Solids with Known Cross Sectional at 0.5 second intervals. Estimate Area

Create a table and a sketch for at 0.5 second intervals. Estimate

scale for the grid is 0.5 cm

on the x and y axes

Re-sketch at 0.5 second intervals. Estimate the graph of f (x).

The scale for this grid is 0.25 cm on

both the x and y axes.

Select one of the figures. Cut out the 9 shapes, keeping the tabs on the shape. Fold the trapezoidal 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.

Volume of Solids of Revolution with your group members.

Rotating about a Line Other than the x- or y-axis with your group members.

Pages 2 to 5

Rotating about a Line Above the Region with your group members.

Pages 5 to 7

Rotating about a Line to the Left of the Region with your group members.

Pages 10 and 11

Rotating about a Line to the Right of the Region with your group members.

Pages 8 and 9

Building Understanding for the Average Value of a Function with your group members.

An Activity for

2013 AB5 with your group members.

All Examples

Taco Stand

### Discussion of Tuesday Homework Problems with your group members.

Multiple Choice Questions on the 2013 test: 3, 6, 8, 10, 11, 13, 17, 20, 21, 23, 28, 76, 78, 82, 84

Free Response:

2014: AB2, AB3/BC3

2013: AB3

Tuesday Assignment - AB

Tuesday Assignment – AB/BC 13, 17, 20, 21, 23, 28, 76, 78, 82, 84

• Multiple Choice Questions on the 2013 test: 3, 6, 8, 10, 11, 13, 17, 20, 21, 23, 28, 76, 78, 82, 84

• Free Response for AB Track

• 2014: AB2, AB3/BC3

• 2013: AB3

• Free Response for BC Track

• 2014: AB3/BC3, BC2

• 2013: BC3

2014 AB2 13, 17, 20, 21, 23, 28, 76, 78, 82, 84

Scoring Rubric 2014 AB2 13, 17, 20, 21, 23, 28, 76, 78, 82, 84

2014 AB3/BC3 13, 17, 20, 21, 23, 28, 76, 78, 82, 84

Scoring Rubric 2014 AB3/BC3 13, 17, 20, 21, 23, 28, 76, 78, 82, 84

2014 BC2 13, 17, 20, 21, 23, 28, 76, 78, 82, 84

Scoring Rubric 2014 BC2 13, 17, 20, 21, 23, 28, 76, 78, 82, 84

2013 AB3/BC3 13, 17, 20, 21, 23, 28, 76, 78, 82, 84

Scoring Rubric 2013 AB3/BC3 13, 17, 20, 21, 23, 28, 76, 78, 82, 84

2013 AB5 13, 17, 20, 21, 23, 28, 76, 78, 82, 84

Scoring Rubric AB5 13, 17, 20, 21, 23, 28, 76, 78, 82, 84

2013 BC5 13, 17, 20, 21, 23, 28, 76, 78, 82, 84

Scoring Rubric BC5 13, 17, 20, 21, 23, 28, 76, 78, 82, 84

Scoring Rubric BC5 13, 17, 20, 21, 23, 28, 76, 78, 82, 84

College Board has developed a Curriculum Module to assist you in teaching how to use Calculus to study motion.

Dixie Ross

Pflugerville High School

Pflugerville, TX

Motion Smartboard File

What You Need to Know about Motion you in teaching how to use Calculus to study motion.

• Worksheet 1: page 5

Sample Practice Problems you in teaching how to use Calculus to study motion. Numerical, Graphical, Analytical

• Worksheet 2: pages 7-9

• Worksheet 3: page 13-15

• Worksheet 4: page 21

Sample Practice Problems Acceleration

• Worksheet 5: page 23-26

The Mean Value Theorem - AccelerationSmartboard

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?

Wednesday Assignment - AccelerationAB

• Multiple Choice Questions on the 2013 test: 25, 26, 27, 77, 79, 80, 81, 83, 85, 86, 87, 88, 89, 90, 91, 92

• Free Response:

• 2014: AB4/BC4, AB5

• 2013: AB4

Wednesday Assignment Acceleration–AB/BC

• Multiple Choice Questions on the 2013 test: 25, 26, 27, 77, 79, 80, 81, 83, 85, 86, 87, 88, 89, 90, 91, 92

• Free Response for AB Track:

• 2014: AB4/BC4, AB5

• 2013: AB4

• Free Response for BC Track:

• 2014: AB4/BC4, BC5

• 2013: BC4