AP Calculus

Section 4.3 How Derivatives Affect the Shape of a Graph

• What the first derivative tells you:
  • Increasing/Decreasing
    • If f '(x) > 0 for all x in an interval, the function is increasing on the interval. (See Section 1.1, p. 21 for the definition of "increasing")
    • If f '(x) < 0 for all x in an interval, the function is decreasing on the interval.
• The First Derivative Test
  • If x = c is a critical number of f, and f ' changes from positive to negative at c, then x = c is a local (relative) maximum of f.
  • If x = c is a critical number of f and f ' changes from negative to positive at c, then x = c is a local (relative) minimum of f.
  • Step-by-step:
    • Find the critical numbers of f by: (See Section 4.1 p. 227) (see Critical Numbers on the TI-89)
      • finding f '(x)
    • setting f '(x) = 0
    • solve for x
    • Plot the critical numbers on a number line.
    • Pick a value of x in each interval and determine if f ' is positive or negative for that value.
      (See Increasing/Decreasing Functions on the TI-89)
    • Use your "sign chart" to locate relative maxima and minima for the function.
  • Note: The sign chart is not sufficient justification by itself. You must state "x = c is a local maximum because f '(c) = 0 and f ' changes from positive to negative at x = c." Statements such as "x = c is a local maximum because f changes from increasing to decreasing at x = c" will not be accepted as full justification!
• What the second derivative tells you:
• Concavity
• Concave up
  • smiley-face shaped or bowl shaped)
  • Definition: A function is concave up on an interval if the graph lies above all of its tangents in the interval.
• Concave down
  • frowny-face shaped or umbrella shaped
  • Definition: A function is concave down on an interval if the graph lies below all of its tangents in the interval.
• Concavity test:
  • If f "(x) > 0 for all x in an interval, the graph of f is concave up in the interval.
  • If f "(x) < 0 for all x in an interval, the graph of f is concave down in the interval.
• Points of Inflection
• A point P is an inflection point of f if f is continuous at P and the curve changes from concave up to concave down (or vice-versa) at P.

last update November 18, 2005 by JL Stanbrough