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)
- 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