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Increasing and Decreasing Functions AP Calculus – Section 3.3
Increasing and Decreasing Functions On an interval in which a function f is continuous and differentiable, a function is… increasing if f (x) is positive on that interval, decreasing if f (x) is negative on that interval, and constant if f (x) = 0 on that interval.
Visual Example f (x) < 0 on (-5,-2) f(x) is decreasing on (-5,-2) f (x) = 0 on (-2,1) f(x) is constant on (-2,1) f (x) > 0 on (1,3) f(x) is increasing on (1,3)
Finding Increasing/Decreasing Intervals for a Function To find the intervals on which a function is increasing/decreasing: 1. Find critical numbers. 2. Pick an x-value in each closed interval between critical numbers; find derivative value at each. 3. Test derivative value tells you whether the function is increasing/decreasing on the interval.
Example Find the intervals on which the function is increasing and decreasing. Critical numbers:
Example Test an x-value in each interval. f(x) is increasing on and. f(x) is decreasing on. Interval Test Value f (x)
Assignment p.181: 1-5, 7, 9
The First Derivative Test AP Calculus – Section 3.3
The First Derivative Test If c is a critical number of a function f, then: If f (c) changes from negative to positive at c, then f(c) is a relative minimum. If f (c) changes from positive to negative at c, then f(c) is a relative maximum. If f (c) does not change sign at c, then f(c) is neither a relative minimum or maximum. GREAT picture on page 176!
Find all intervals of increase/decrease and all relative extrema. Test: f is decreasing before -4 and increasing after -4; so f(-4) is a MINIMUM.