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Wednesday – AB

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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Wednesday – AB

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  1. 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 • 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

  2. Wednesday – AB/BC • 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

  3. Wednesday Assignment - AB • 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

  4. Wednesday Assignment – 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

  5. Wednesday Files • 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

  6. 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

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

  8. 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

  9. 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 .

  10. Introducing Integration through the Area Model

  11. 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 .

  12. 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 .

  13. Investigating How to Find Area using Riemann Sums and Trapezoids Using the NUMINT program or LMRRAM and TRAPEZOID program on a TI83/84

  14. 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.

  15. Things You Should have Observed • which sums are always greater than the actual area • Which sums are always less than the actual area

  16. 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

  17. 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.

  18. 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.

  19. 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

  20. Using Technology to Approximate the Definite Integral

  21. Developing an Understanding for the Definite Integral Smartboard File

  22. Fun Finding Volume

  23. Solids with Known Cross Sectional Area

  24. Create a table and a sketch for scale for the grid is 0.5 cm on the x and y axes

  25. Re-sketch the graph of f (x). The scale for this grid is 0.25 cm on both the x and y axes.

  26. 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.

  27. Complete the Finding the Volume of the Solid Activity Sheet with your group members.

  28. Volume of Solids of Revolution

  29. Rotating about a Line Other than the x- or y-axis Pages 2 to 5

  30. Rotating about a Line Above the Region Pages 5 to 7

  31. Rotating about a Line to the Left of the Region Pages 10 and 11

  32. Rotating about a Line to the Right of the Region Pages 8 and 9

  33. Building Understanding for the Average Value of a Function An Activity for

  34. 2013 AB5

  35. What Activity Would You Like to Share

  36. Dan Meyer All Examples Taco Stand

  37. Discussion of Tuesday Homework Problems

  38. 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

  39. Tuesday Assignment – AB/BC • 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

  40. 2014 AB2

  41. Scoring Rubric 2014 AB2

  42. 2014 AB3/BC3

  43. Scoring Rubric 2014 AB3/BC3

  44. 2014 BC2

  45. Scoring Rubric 2014 BC2

  46. 2013 AB3/BC3

  47. Scoring Rubric 2013 AB3/BC3

  48. 2013 AB5

  49. Scoring Rubric AB5

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