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CIS.PSMPDI Scalar Multiple Property 'Proof'

CIS.PSMPDI Scalar Multiple Property 'Proof'
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2018-07-10 01:09
VectorFunctionIntegralSlider AnimationGraphing
① This GeoGebra example dynamically demonstrates a geometric 'proof' of the Scalar Multiple Property, typically illustrating the linear relationship under scalar multiplication for vectors or functions. ② Core mathematical concepts include vector operations, scalar multiplication, function graphs, and the relationship between integrals and areas, possibly also involving arithmetic mean and quadratic equations. ③ Construction highlights: from the summary, commands such as Integral and RectangleSum are used to build area approximations; basic objects like Point, Line, Vector, Segment construct the geometry; Sequence, Zip, and TableText generate data tables; Textfield and input boxes enable interactive parameter adjustment. Objects include multiple points, axes, functions, and text labels. ④ Suitable for high school to early college students, this helps understand the geometric impact of scalar multiplication on functions or vectors, and serves as an intuitive teaching tool for the linearity property of integrals in calculus.

GeoGebra Commands & Parameters

  1. Integral(P, a, c) → IPac
  2. Integral(P, c, b) → IPcb
  3. Integral(P, a, b) → IPab
  4. Textfield(P) → inputBox1
  5. Point(yAxis) → D
  6. Point(yAxis) → E
  7. Line(D, E) → Xaxis
  8. Line(Ypt, Xaxis) → Yaxis
  9. Vector(G, J) → u
  10. Vector(I, F) → v
  11. Vector(H, K) → w
  12. Segment(F, K) → f
  13. Segment(F, J) → h
  14. Segment(I, H) → i
  15. Segment(G, I) → j
  16. Integral(αP, a, b) → IαPab
  17. Line(M, Yaxis) → Y2axis
  18. Vector(J, αs_a) → e
  19. Vector(K, αs_b) → d
  20. Segment(αs_a, αs_b) → g
  21. Segment(J, K) → m
  22. Segment(G, H) → p
  23. Textfield(Q) → InputBox1
  24. Integral(Q, a, b) → IQab
  25. Integral(P(x) + Q(x), a, b) → IPQab
  26. Line(O, Xaxis) → q
  27. Vector(G, SQa) → r
  28. Vector(H, SQb) → s
  29. Segment(SQb, SQa) → t
  30. Vector(K, SPQb) → u_1
  31. Vector(SQb, SPQb) → v_1
  32. Vector(SQa, SPQa) → w_1
  33. Segment(SPQb, SPQa) → f_1
  34. Vector(SQb, SQb1) → a_1
  35. Segment(SPQb, SQb1) → g_1
  36. Segment(SQb1, SPQa) → h_1
  37. Vector(J, SQb1) → b_1
  38. Point(f_1) → R
  39. Sequence(1, n) → N
  40. Zip((0, a + (k * dx)), k, N) → x_1
  41. Zip(a - dx / 2 + (k * dx), k, N) → c_*
  42. Zip(Sum[Pcdx_*, k], k, N) → SPcdx_*
  43. Zip(Sum[αPxdx_*, k], k, N) → SαPcdx_*
  44. Zip((x(Ypt), sp), sp, SPcdx_*) → SptPcdx_*
  45. Zip(Vector[A, B], A, x_1, B, SptPcdx_*) → vecSP_*
  46. Zip((x(M), sp), sp, SαPcdx_*) → SptαPcdx_*
  47. Zip(Vector[A, B], A, SptPcdx_*, B, SptαPcdx_*) → vecSαP_*
  48. TableText(Transpose[{Join[{{"c_k"}, c_*}], Join[{{"P(c_k)Δx"}, Pcdx_*}], Join[{{"αP(c_k)Δx"}, αPxdx_*}]}], "|_cc") → TableI
  49. RectangleSum(P, y(G), y(H), n, 1 / 2) → GrMidEstIntP
  50. RectangleSum((α * P(x)), y(G), y(H), n, 1 / 2) → GrMidestIntαP

AI Drawing Prompt Reference

Draw a dynamic geometric proof of the scalar multiple property, showing vector operations, function graphs, and the relationship between integrals and areas. Include basic geometric objects like points, lines, vectors, and segments, with integral and rectangle sum constructions for area approximations. Use sliders for interactive parameter adjustment, generate data tables with labels, and illustrate the geometric impact of scalar multiplication on functions or vectors, as well as the linearity property of integrals in calculus.
The prompt above was compiled from the original drawing commands and is for reference only.