Calculus: Curve Sketching and Optimization Problems

Unit 6: Curve Sketching and Optimization

Example 1: Analyze and Sketch f(x) = 2x³ - 3x² - 3x + 2

• Domain: D = {x ∈ ℝ}
• Intercepts:
  • x-intercepts: Set f(x) = 0. Solving (x + 1)(x - 2)(2x - 1) = 0 gives x = -1, 2, 0.5.
  • y-intercept: f(0) = 2. Point: (0, 2).
• Increasing/Decreasing Intervals:
  • f'(x) = 6x² - 6x - 3
  • Setting f'(x) = 0, we find critical points at x ≈ 1.4 and x ≈ -0.4.
  • Local minimum at (1.4, -2.6); local maximum at (-0.4, 2.6).
• Concavity:
  • f''(x) = 12x - 6
  • Setting f''(x) = 0 gives x = 0.5. Inflection point at (0.5, 0).

Example 2: Optimization of an Open-Topped Box

Problem: A box has a base twice as long as its width and a volume of 2400 cm³. Find the dimensions that minimize the surface area (SA).

• Volume Constraint: 2x(x)(h) = 2400 ⇒ 2x²h = 2400 ⇒ h = 1200/x²
• Surface Area Formula: SA = 2x² + 6xh
• Substitution: SA = 2x² + 6x(1200/x²) = 2x² + 7200x^-1
• Optimization: SA' = 4x - 7200x^-2
• Solve for x: 0 = 4x - 7200/x² ⇒ 4x³ = 7200 ⇒ x = ∛1800 ≈ 12.16 cm
• Final Dimensions:
  • Width (w) ≈ 12.16 cm
  • Length (l) ≈ 24.32 cm
  • Height (h) ≈ 8.1 cm