Use your knowledge of the derivative to describe the graph.

5.1: Increasing/Decreasing Functions Objective: Students will be able to… Relate the derivative to intervals of increase and decrease

Perform the Indicated Operation. Write your answer as a single rational expression.

Let f be a function defined on some interval. Then for any 2 numbers x 1 and x 2 in the interval:  f is increasing on the interval if f(x 1 ) < f(x 2 ) whenever x 1 < x 2  f is decreasing on the interval if f(x 1 ) > f(x 2 ) whenever x 1 < x 2

Find the open intervals where the function is increasing and decreasing.

Testing for Intervals where f(x) is Increasing and Decreasing…Using the derivative!!!! Suppose a function f has a derivative at each point in an interval, then:  f is increasing if f’(x) > 0 for each x in interval  f is decreasing if f’(x) < 0 for each x in interval  f is constant if f’(x) = 0 for each x in interval (must hold true for entire interval)

Values of x that are in the domain of f but where f’(x)=0 or where f’(x) is not defined are called CRITICAL NUMBERS. These are possible points where the graph changes from increasing to decreasing or vice versa. We use critical numbers to test for intervals of increase or decrease.

Test for Intervals of Increase/Decrease 1.Determine the domain of f. 2.Find f’(x) 3.Find critical numbers (where f’(x)=0 or where f’(x) does not exist). 4.Make a number line and put critical numbers on it. Pay attention to domain, and include any vertical asymptotes. 5.Choose x values to the left and right of critical numbers and test these values in f’(x) 6.If f’(x) > 0 on interval →increasing If f’(x) < 0 on interval → decreasing

Find the critical numbers and the open intervals of increase and decrease.