calc unit 3

• Critical numbers


  c is a critical number if f'(c) = 0 or f'(c) DNE
• Not all critical numbers are extrema
• Rolle's Theorem


  If f is continuous on [a, b] and differentiable on (a, b) and f(a) = f(b), then there is a c in (a, b) such that f'(c) = 0
• Mean Value Theorem (MVT)


  If f is continuous on [a, b] and differentiable on (a, b), then there is a c in (a, b) such that f'(c) = (f(b) - f(a)) / (b - a)
• Extreme Value Theorem (EVT)

  If f(x) is continuous on [a,b], then f(x) takes on an (absolute) max AND an (absolute) min value on [a, b].
• Closed Interval Method:

  
  

  Find absolute max and mins

  1. Find and evaluate the critical #’s
  2. Check the endpoints
• Fermat’s Theorem

  If f has a local extrema at x = c and c € (a,b), then f’(c) = 0 or f’(c) DNE.
• First Derivative Test – Defined. The First Derivative Test states that if we are given a continuous and differentiable function f, and c is a critical number of function f, then f(c) can be classified as follows: If f' (x) changes from negative to positive at c, then f(c) is a relative minimum.
• Second derivative Test

  States that if f is a function with continuous second derivative, then: if c is a critical point an f'(c) > 0, then c is a local minimum of f
• no mins or max at vertical asymptotes (not in domain)
• Summary of Curve Sketching

  A)Domain

  B)Intercepts

  C)Symmetry

  D)Asymptotes

  E)Increase/Decrease

  F)Mins & maxs

  G)Concavity & I.P.