Presentation on theme: "Increasing and Decreasing Functions"— Presentation transcript:
1Increasing and Decreasing Functions AP Calculus – Section 3.3
2Increasing 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, andconstant if f ‘(x) = 0 on that interval.
3Visual Example f ‘(x) < 0 on (-5,-2) f ‘(x) > 0 on (1,3) f(x) is decreasing on (-5,-2)f ‘(x) > 0 on (1,3)f(x) is increasing on (1,3)f ‘(x) = 0 on (-2,1)f(x) is constant on (-2,1)
4Finding Increasing/Decreasing Intervals for a Function To find the intervals on which a function is increasing/decreasing:Find critical numbers.Pick an x-value in each closed interval between critical numbers; find derivative value at each.Test derivative value tells you whether the function is increasing/decreasing on the interval.
5ExampleFind the intervals on which the function is increasing and decreasing. Critical numbers:
6ExampleTest an x-value in each interval. f(x) is increasing on and . f(x) is decreasing on .IntervalTest Valuef ‘(x)
8The First Derivative Test AP Calculus – Section 3.3
9The 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!
10Find 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.Test: