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Increasing/ Decreasing Functions. Section 4.2a. Definition: Increasing/Decreasing Functions. Let be a function defined on an interval I and let and be any two points in I . (a) f increases on I if . As x gets bigger, y gets bigger…. (b) f de creases on I if .
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Increasing/ Decreasing Functions Section 4.2a
Definition: Increasing/Decreasing Functions Let be a function defined on an interval I and let and be any two points in I. (a) fincreases on I if As x gets bigger, y gets bigger… (b) fdecreaseson I if As x gets bigger, y gets smaller… • A function that is always increasing or decreasing on • a particular interval is monotonic on that interval
The “Do Now” Writing: How can the derivative help in identifying when a function is increasing or decreasing? Let be continuous on and differentiable on . • If at each point of , then • increases on . (b) If at each point of , then decreases on .
“Seeing” this new tool… Consider the function Next, the derivative: First, look at the graph: y on , so the function is decreasing on . on , so the function is increasing on . x
Guided Practice Use analytic methods to find (a) the local extrema, (b) the intervals on which the function is increasing, and (c) the intervals on which the function is decreasing. The derivative: • There are no endpoints, and the only critical point occurs at: Minimum of at
Guided Practice Use analytic methods to find (a) the local extrema, (b) the intervals on which the function is increasing, and (c) the intervals on which the function is decreasing. The derivative: (b) Since on , is increasing on . (c) Since on , is decreasing on .
Guided Practice Use analytic methods to find (a) the local extrema, (b) the intervals on which the function is increasing, and (c) the intervals on which the function is decreasing. The derivative: • Since the derivative is never zero and is undefined only • where is undefined, there are no critical points. Also, the • domain has no endpoints. Therefore has no local extrema.
Guided Practice Use analytic methods to find (a) the local extrema, (b) the intervals on which the function is increasing, and (c) the intervals on which the function is decreasing. The derivative: (b) on . So, the function is increasing on . (c) on . So, the function is decreasing on .
Guided Practice Find (a) the local extrema, (b) the intervals on which the function is increasing, and (c) the intervals on which the function is decreasing. First, graph the function… The derivative: Critical Points: x = –2, x = 0 • The local extrema can occur at the critical points, but the • graph shows that no extrema occurs at x = 0. There is a local • (and absolute) minimum at
Guided Practice Find (a) the local extrema, (b) the intervals on which the function is increasing, and (c) the intervals on which the function is decreasing. (b) Since on the intervals: … And since is continuous at x = 0… is increasing on . (c) Since on the interval: … is decreasing on .
Guided Practice Find (a) the local extrema, (b) the intervals on which the function is increasing, and (c) the intervals on which the function is decreasing. First, graph the function… The derivative:
Guided Practice Find (a) the local extrema, (b) the intervals on which the function is increasing, and (c) the intervals on which the function is decreasing. Since the derivative is never zero and is undefined only where the function is undefined, there are no critical points. Since there are no critical points and the domain includes no endpoints, the function has no local extrema. (b) Since the derivative is never positive, the function is not increasing on any interval. (c) Since the derivative is whenever it is defined, the function is decreasing on:
