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Week 6 Calculus: Geometry and Function Fitting

Week 6 Calculus: Geometry and Function Fitting
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2020-07-30 05:54
FunctionCalculusDerivativeGeometric ConstructionGraphing
In collections Length, Area, & Volume MAT 6715 by Laura Anderson
This example demonstrates a comprehensive mathematical construction combining geometric figures with calculus concepts, using points, circles, polygons, and polynomial fitting to show dynamic relationships between functions and geometric objects. Core concepts include derivatives (object f' represents the derivative function), function fitting via FitPoly, and geometry with circles, segments, and intersections, also touching on sequences and combinatorics. Construction uses FitPoly, Point, Circle, Intersect, and Segment commands, with objects A through S, functions f and f', auxiliary lines k-t, circles c, d, e, etc., forming a complex multi-object system. Suitable for pre-university or early undergraduate calculus courses as a comprehensive exercise or demonstration, helping students understand connections between functions, derivatives, and geometric figures, and useful for classroom presentations.

GeoGebra Commands & Parameters

  1. FitPoly({A, B, C, D}, 3) → f
  2. Point(g) → E
  3. Point(h) → F
  4. Point(i) → G
  5. Point(j) → H
  6. Circle(A, 1) → c
  7. Circle(B, 1) → d
  8. Circle(C, 1) → e
  9. Circle(D, 1) → p
  10. Intersect(c, k, 1) → I
  11. Intersect(c, k, 2) → J
  12. Intersect(d, l) → K, L
  13. Intersect(e, m) → M, N
  14. Intersect(p, n) → O, P
  15. Segment(I, J) → q
  16. Segment(K, L) → r
  17. Segment(M, N) → s
  18. Segment(O, P) → t
  19. Point(f) → Q
  20. Point(a) → R
  21. Circle(Q, 1) → c_1
  22. Intersect(c_1, b) → S, T
  23. Segment(S, T) → f_1
  24. Point(f) → V
  25. Point(f) → W
  26. Point(f) → Z
  27. Point(f) → A_1
  28. Point(g_1) → B_1
  29. Point(h_1) → C_1
  30. Point(i_1) → D_1
  31. Point(j_1) → E_1
  32. Circle(V, 1) → d_1
  33. Circle(W, 1) → e_1
  34. Circle(Z, 1) → p_1
  35. Circle(A_1, 1) → q_1
  36. Intersect(p_1, m_1) → F_1, G_1
  37. Intersect(q_1, n_1) → H_1, I_1
  38. Intersect(e_1, l_1) → J_1, K_1
  39. Intersect(d_1, k_1) → L_1, M_1
  40. Segment(L_1, M_1) → r_1
  41. Segment(J_1, K_1) → s_1
  42. Segment(F_1, G_1) → t_1
  43. Segment(H_1, I_1) → a_1

AI Drawing Prompt Reference

Draw a graph with a polynomial fitting function f and four unit circles c, d, e, p around points A, B, C, and D. Use the intersection points I, J, K, L, M, N, O, P and auxiliary lines k, l, m, n to construct several segments q, r, s, t. Then, with the derivative function f', points E, F, G, H, I, J, K, L, M, N, O, P, Q, R, S, T, V, W, Z, A1, B1, C1, D1, and E1, further construct new circles and segments to form a complex geometric figure.
The prompt above was compiled from the original drawing commands and is for reference only.