AP Calculus AB Course Study Guide

Analytical Applications of Differentiation

Mean Value Theorem

• Conditions: Function f(x) must be continuous on closed interval [a, b] and differentiable on open interval (a, b).
• Example: For f(x) = x³ on [0,3], find c in (0,3).
  • f'(x) = 3x²
  • Solve: 3x² = (3)(0) → 3x² = 9

Increasing or Decreasing Functions

• Criteria:
  • f(x) is increasing when f'(x) > 0
  • f(x) is decreasing when f'(x) < 0
• Example: Where is f(x) increasing/decreasing on [-4,6]?
  • f(x) increasing: (-∞, -3) ∪ (0,4)
  • f(x) decreasing: (-3,0) ∪ (4,5)

Extreme Value Theorem

• Statement: If f(x) is continuous on [a,b], it has a maximum and minimum on that interval.
• Example: Find extrema for f(x)=3-12x on [0,4].
  • f(0) = 0, f(2) = -16, f(4) = 16
  • Absolute min at x=2, max at x=4

First Derivative Test

• Steps:
  • Derive f(x), set derivative to 0, and solve for critical points.
  • Test intervals around critical points.
  • If f'(x) changes from positive to negative at c, f(x) has a relative maximum at c.
  • If f'(x) changes from negative to positive at c, f(x) has a relative minimum at c.
• Example: f(x)=x²+6x+10
  • f'(x)=2x+6 → x=-3

Concavity

• Criteria:
  • Concave up: f''(x) > 0
  • Concave down: f''(x) < 0
• Points of inflection: Change in concavity

Second Derivative Test

• Example: For y=-3x²+6x
  • y'' = -6x + 6
  • Zeros at x=1
  • Interval test: Positive at x=-1, negative at x=2

Critical Numbers

• Definition: Points where the derivative is zero or undefined, indicating potential maxima, minima, or points of inflection.
• Example: f(x)=2secx+tanx
  • Find critical numbers: x = 3/2, π/2, 7/6, 11/6