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Sigma Notation

ET 4.2a. Sigma Notation. Upper Bound. Summation. Counting Mechanism. I ndex # = Lower Bound. Begins development of classic area problem. Sigma Notation. Upper Bound. Summation. Counting Mechanism. I ndex # = Lower Bound. EXAMPLE :. EXAMPLE :. EXAMPLE :. 4(5). = 20. + 5. = 5.

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Sigma Notation

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  1. ET 4.2a Sigma Notation Upper Bound Summation Counting Mechanism Index # = Lower Bound

  2. Begins development of classic area problem Sigma Notation Upper Bound Summation Counting Mechanism Index # = Lower Bound EXAMPLE : EXAMPLE :

  3. EXAMPLE : 4(5) = 20 + 5 = 5 + 5 + 5 i=2 i=3 i=4 i=1

  4. EXAMPLE :

  5. EXAMPLE :

  6. More Summation Formulas EXAMPLE :

  7. EXAMPLE : constant = 1/3 N(x) = D(x)

  8. trick

  9. = (1/6)(2) =1/3

  10. Assignments 4.2 • Day 1: 15-21 odd, 37, 39, 45, 49-53 odd • Day 2: 24, 25, 31, 33, 41, 47 • Day 3: 49-53 odd, 61, 73, 75, 85

  11. Explain geometrically why the red triangle has half the area of the blue rectangle. ET 4.2b h h b b Congruent Triangles A = bh A = ½ bh Explain how you could find the area of the green irregular polygon.

  12. Estimate the area of a circle. Actual is somewhere between the under & over. Underestimate (Inscribed Triangles): Overestimate (Circumscribed): If you want a better estimate… increase the # of triangles.

  13. Let’s try this same method to find the area under the curve on the interval [0,2]. Underestimate: Right Hand Rectangles (2/5, 4.84) (4/5, 4.36) (6/5, 3.56) (8/5, 2.44) (10/5, 1) (2/5)*f(2/5)+(2/5)*f(4/5)+(2/5)*f(6/5)+(2/5)*f(8/5)+(2/5)*f(10/5) (2/5)(4.84)+(2/5)(4.36)+(2/5)(3.56)+(2/5)(2.44)+(2/5)(1) = 6.48

  14. Another way to write it. (2/5)*f(2/5)+(2/5)*f(4/5)+(2/5)*f(6/5)+(2/5)*f(8/5)+(2/5)*f(10/5) (2/5)(4.84)+(2/5)(4.36)+(2/5)(3.56)+(2/5)(2.44)+(2/5)(1) = 6.48

  15. Let’s try this same method to find the area under the curve on the interval [0,2]. Overestimate: Left Hand Rectangles (0, 5) (2/5, 4.84) (4/5, 3.36) (6/5, 3.56) (8/5, 2.44)) (2/5)*f(0/5)+(2/5)*f(2/5)+(2/5)*f(4/5)+(2/5)*f(6/5)+(2/5)*f(8/5) (2/5)(5)+(2/5)(4.84)+(2/5)(4.36)+(2/5)(3.56)+(2/5)(2.44) = 8.08

  16. How can you make a better estimate with either right or left hand rectangles? Increase the number of rectangles!

  17. Assignments 4.2 • Day 1: 15-21 odd, 37, 39, 45, 49-53 odd • Day 2: 24, 25, 27, 31, 33, 41, 47 • Day 3: 49-53 odd, 61, 73, 75, 85 When we return from break we will emphasize the use of right hand rectangles, left hand rectangles, and midpoint rectangles to estimate the area under a curve.

  18. Definition of the area of a region in the plane. Curve is above x-axis. Let f be continuous and nonnegative on the interval [a, b]. The area of the region bounded by the graph of f, the x-axis, and the vertical lines x = a and x = b is (ci, f(ci)) a b xi-1 xi ci Width of each rectangle x

  19. Find the area under f(x) = 1 - x2 on [-1, 1] =b a= a1 a2 a3… x

  20. Find the area under f(x) = 1 - x2 on [-1, 1]

  21. 4.2 Assignments • Day 1: 15-21 odd, 37, 39, 45, 49-53 odd • Day 2: 24, 25, 27, 31, 33, 41, 47 • Day 3: 50-54 even, 61, 73, 75, 85 When we return from break we will emphasize the use of right hand rectangles, left hand rectangles, and midpoint rectangles to estimate the area under a curve.

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