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  1. Anthony Chun
  2. For Ned: Calculus 3/1/2015
  3.  
  4. f(x) = Ax^3 + Bx^2 + Cx + D
  5.  
  6. The tangent line to it at x = 0 is y = x
  7. The tangent line at x = 2 is y = 2x - 3
  8.  
  9. When we plug x = 0 to y = x, we get y = 0.
  10. This is one point: (0,0)
  11.  
  12. When we plug in x = 2 to y = 2x - 3, we get y = 2(2) - 3 = 1
  13. This is another point, (2,1)
  14.  
  15. Ok so tangent lines involve the derivative.
  16. The derivative becomes the coefficient of x for the tangent line.
  17. Coefficient of the first one is 1, and the second one is 2.
  18.  
  19. So F'(x) = 3Ax^2 + 2Bx + C
  20.  
  21. Now that we have all of this analyzed, we can solve the problem.
  22.  
  23. We know that one point on the graph is (0,0).
  24. We can plug this in to the first equation f(x).
  25.  
  26. 0 = A(0)^3 + B(0)^2 + C(0) + D
  27. 0 = D
  28.  
  29. We also know from y = x being the tangent line, the slope(derivative) at 0 is 1.
  30.  
  31. 1 = 3A(0)^2 + 2B(0) + C
  32. This means C = 1
  33.  
  34. We now have two variables already, C = 1, D = 0.
  35.  
  36. Let's continue.
  37. Now we have specific points that need to be on the graph, lets plug them into f(x) and f'(x).
  38.  
  39. We know that the slope(derivative) at x = 2 is 2.
  40. We plug (2,2) into f'(x).
  41. 2 = 3A(2)^2 + 2B(2) + 1
  42. 1 = 12A + 4B
  43.  
  44. We also have another point on the graph: (2,1). We plug it into f(x).
  45. 1 = A(2)^3 + B(2)^2 + 1(2) + 0
  46. 1 = 8A + 4B + 2
  47. -1 = 8A + 4B
  48.  
  49. With the above two we have
  50.  
  51. 1 = 12A + 4B
  52. -1 = 8A + 4B
  53.  
  54. Subtract the two equations.
  55.  
  56. 2 = 4A
  57.  
  58. A = 1/2
  59.  
  60. Plug it back into one of the equations
  61. -1 = 8(1/2) + 4B
  62. -1 = 4 + 4B
  63. -5 = 4B
  64.  
  65. B = -5/4
  66.  
  67. We have
  68.  
  69. A = 1/2
  70. B = -5/4
  71. C = 1
  72. D = 0
  73.  
  74. Our final equation is
  75. y = (1/2)x^3 - (5/4)x^2 + x
  76.  
  77. Here are the graphs from x = -2 to 4
  78.  
  79. http://aton2.com/cap/8601bcbd1a9e9f751902.png
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