Increasing and Decreasing Functions

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Presentation transcript:

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) > 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)

Finding 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.

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. Test:

Assignment p.181: 11-21 odd