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AP_2014_2a Mathematical Example

AP_2014_2a Mathematical Example
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2017-06-30 03:12
In collections Volumes in AP Calculus
This example demonstrates the geometric and probabilistic relationships involving special points, means, binomial distribution, limits, and isosceles triangles. Key math concepts include special point calculation, mean theorem, binomial distribution model, limit solving, and isosceles triangle properties. GeoGebra commands used are Sequence, Polygon, Rotate, Integral, Surface, and RectangleSum. Suitable for high school and above mathematics learning, illustrating the integration of advanced mathematics and geometry.

GeoGebra Commands & Parameters

  1. Sequence(Polygon[((k * h), f(((k + p) * h))), (((k + 1) * h), f(((k + p) * h))), (((k + 1) * h), g(((k + p) * h))), ((k * h), g(((k + p) * h)))], k, 0, m - 1) → Rectangles
  2. Polygon(((DK * h), f(((DK + p) * h))), (((DK + 1) * h), f(((DK + p) * h))), (((DK + 1) * h), g(((DK + p) * h))), ((DK * h), g(((DK + p) * h)))) → Rectangle1, b, c, d, e
  3. Rotate(Rectangle1, (-α), a) → Rectangle1'
  4. Integral((π * ((2 + g(x))^(2) - (2 + f(x))^(2))), 0, 2.3) → VolInteg
  5. Rotate(Rectangles, (-α), a) → Rectangles'
  6. Rotate(g_1, (-α), a) → i
  7. Rotate(f_1, (-α), a) → j
  8. Surface(u, -2 + ((2 + f(((DK + p) * h))) * cos(β)), ((2 + f(((DK + p) * h))) * sin(β)), u, (DK * h), ((DK + 1) * h), β, 0, α) → Inner1
  9. Surface(u, -2 + ((2 + g(((DK + p) * h))) * cos(β)), ((2 + g(((DK + p) * h))) * sin(β)), u, (DK * h), ((DK + 1) * h), β, 0, α) → Outer1
  10. Surface((DK * h), -2 + ((u + 2) * cos(β)), ((u + 2) * sin(β)), u, f(((DK + p) * h)), g(((DK + p) * h)), β, 0, α) → SideLeft1
  11. Surface(((DK + 1) * h), -2 + ((u + 2) * cos(β)), ((u + 2) * sin(β)), u, f(((DK + p) * h)), g(((DK + p) * h)), β, 0, α) → SideRight1
  12. Surface(u, -2 + ((2 + f(u)) * cos(β)), ((2 + f(u)) * sin(β)), u, 0, 2.3, β, 0, (2 * π)) → InnerSurf
  13. Surface(u, -2 + ((2 + g(u)) * cos(β)), ((2 + g(u)) * sin(β)), u, 0, 2.3, β, 0, (2 * π)) → OuterSurf
  14. Sequence(Surface[u, -2 + ((2 + f(((k + p) * h))) * cos(β)), ((2 + f(((k + p) * h))) * sin(β)), u, (k * h), ((k + 1) * h), β, 0, α], k, 0, m - 1) → Inners
  15. Sequence(Surface[u, -2 + ((2 + g(((k + p) * h))) * cos(β)), ((2 + g(((k + p) * h))) * sin(β)), u, (k * h), ((k + 1) * h), β, 0, α], k, 0, m - 1) → Outers
  16. Sequence(Surface[(k * h), -2 + ((u + 2) * cos(β)), ((u + 2) * sin(β)), u, f(((k + p) * h)), g(((k + p) * h)), β, 0, α], k, 0, m - 1) → SidesLeft
  17. Sequence(Surface[((k + 1) * h), -2 + ((u + 2) * cos(β)), ((u + 2) * sin(β)), u, f(((k + 1 + p) * h)), g(((k + 1 + p) * h)), β, 0, α], k, 0, m - 1) → SidesRight
  18. RectangleSum(A, 0, (m * h), m, p) → WS