Lecture Notes: Solving the Equation of a Cubic Graph

Overview

• Learn how to determine the equation of a cubic graph using given information.
• The process involves plugging given values into the cubic equation and its derivatives.

Steps to Solve the Equation

Step 1: Initial Conditions

• Given: When (x = 0), (f(x) = 8).
  • Set all x terms to zero and solve for (d).
  • Result: (d = 8).

Step 2: First Derivative

• Purpose: To find further coefficients using given x values.
• First Derivative: (f'(x) = 3ax^2 + 2bx + c).
• Given Condition: (f'(1) = -5).
  • Substitute x = 1 into the first derivative equation.
  • Simplify to form another equation.
  • No further solutions available at this stage.

Step 3: Second Derivative

• Purpose: Use another x value to form simultaneous equations.
• Second Derivative: (f''(x) = 6ax + 2b).
• Given Condition: (f''(2) = 8).
  • Substitute x = 2 into the second derivative equation.
  • Simplify to create an equation: (8 = 12a + 2b).
  • Divide by 2 to simplify: (4 = 6a + b).

Step 4: Additional Information

• Further Condition: (f''(4) = 20).
  • Substitute x = 4 into the second derivative.
  • Simplify to another equation: (20 = 24a + 2b).
  • Divide by 2 to simplify: (10 = 12a + b).

Solving Simultaneous Equations

• Equations:
  • 1: (4 = 6a + b)
  • 2: (10 = 12a + b)
• Method: Use substitution or elimination to solve for (a) and (b).
  • Rearrange Equation 1: (b = 4 - 6a).
  • Substitute into Equation 2: (10 = 12a + (4 - 6a)).
  • Solve for (a): (a = 1).
  • Substitute (a) back to find (b): (b = -2).

Determining Constant (c)

• Return to original conditions and substitute (a) and (b) into the first derivative.
• Solve for (c) using the derived equations: (c = -4).

Final Equation

• Cubic Equation: [f(x) = x^3 - 2x^2 - 4x + 8]

Key Takeaway

• Plug values into correct derivatives, form equations, and solve using algebraic methods (e.g., simultaneous equations) to derive the cubic equation.