3.3a Increasing and Decreasing Functions and the 1 st Derivative Test <ul><li>Determine intervals on which a function is increasing or decreasing </li></ul><ul><li>Apply the First Derivative Test to find relative extrema of functions . </li></ul>
Understanding that the first derivative produces a function that will calculate slope at any x, this next theorem should be no surprise! Critical numbers divide graph into intervals to test.
Ex 1 p. 180 Intervals on which f is increasing or decreasing Find the open intervals on which is increasing or decreasing. Solution : This function if differentiable everywhere. To find critical numbers, set first derivative = 0
Interval - ∞ < x < 0 0 < x < 1 1 < x < ∞ Test value x = -1 x = ½ x = 2 Sign of f’(x) f’(-1) = 6, >0 f’(1/2) = -3/4, <0 f’(2) = 6, >0 Increasing Decreasing Increasing
Once you know where graph is increasing, it is not too difficult to determine which of critical numbers is at relative min and which is at relative max.
Strictly monotonic: a function is strictly monotonic on an interval if it is increasing on the entire interval, or decreasing on the entire interval. IS IS NOT
Ex 2 p. 183 Applying the first derivative test Find the relative extrema of the function Decision? Interval 0 < x < π /3 π /3 < x < 5 π /3 5 π /3 < x < 2 π Test Value x = π /4 x = π x = 7 π /4 Sign of f’(x) Conclusion