Estimating with Derivatives, Part 2

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• Questions?
• Extreme Values
  • Section 4.1
    • Definitions
      • Max = highest point
      • Min = lowest point
      • Local vs absolute
        • Absolute = highest of the high / lowest of low
        • Local = bigger / smaller than nearby values
    • Theorem: if f has maximum/minimum at c and f′(c) defined then f′(c) = 0
    • Critical points?
      • Derivative = 0 or derivative undefined
  • Examples
    • Where is/are maxima/minima of y = x³ - 3x for -1.5 ≤ x ≤ 1.5?
      • Graph on calculator?
        • Accuracy issues
      • dy/dx = 3x² - 3
      • Solve for 0
        • 0 = 3x² - 3
          • 3 = 3x²
          • 1 = x²
          • x = √1 = +1, -1
        • Or factor
          • 3x² - 3
          • = 3( x² - 1 )
          • = 3 (x-1) (x+1)
      • Plug solutions into equation for y
        • (-1)³ - 3(-1) = 2
        • 1³ - 3 = -2
    • How about for -5 ≤ x ≤ 5?
      • Max/min might be at endpoints (which are technically critical)
    • Where is/are maxima/minima of y = x³ + x for -1 ≤ x ≤ 1?
      • dy/dx = 3x² + 1
      • Solve for 0… no real solutions
      • So look only at endpoints
    • Does y = 1 / ( x² - x - 2) have any absolute maxima/minima? Where? What about local maxima/minima?
      • Denominator might be 0
        • x² - x - 2 = 0?
        • (x-2)(x+1) = 0
        • x = -1, x = 2
      • No absolute maximum/minimum—function value is unbounded
        
      • Local maximum at x = 1/2
        
• Mean Value Theorem
  • Section 4.2
  • Differentiable function f between x=a and x=b, then somewhere between a and b f′(c) = slope of secant line from f(a) to f(b)
  • Implications:
    • If f′(x) = 0 then f(x) = c
    • If f′(x) = g′(x), then f(x) = g(x) + C
      • This guarantees that our informal approach to antiderivatives actually works, i.e., finds all anti-derivatives of any function it can handle at all.
        
• Next
  • More Mean Value Theorem
  • Starting to distinguish maxima and minima
    • Read section 4.3

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