3.8
递减函数
递减函数指当输入值增加时，输出值持续减小的函数关系。换言之，对于任意两个输入值，若其中一个输入值较大，则其对应的输出值较小。从数学角度看，若在区间 I 上对于所有满足 x_1 < x_2 的任意实数，都有 f(x_1) > f(x_2)，则称函数 f 在该区间上为递减函数。这种特性在图像上表现为从左…
当函数的输出值随着输入值的增加而减小，即沿 x 轴从左向右移动时，图像上的 f(x) 值变小，则称该函数为减函数。
通过观察图形是否从左到右呈下降趋势，可以确定这种行为。
考虑一名男子在跑道上跑步。记录每一圈所用的时间和跑过的距离，以确定不同时间段内速度的变化。
区间之间的平均速度（或变化率）通过计算距离的变化量除以两个记录时间点之间的时间变化量来确定。
接下来，为了确定速度是增加还是减小，通过将每圈所覆盖的距离除以该圈所用的时间，来计算每圈的速度。这有助于分析跑步者的配速如何从一圈到下一圈发生变化。
以速度-时间图像绘制数据时，图像显示速度持续下降。这表明函数呈递减趋势，证实了跑者在每一圈中速度逐渐减慢。
递减函数的概念可用于模拟多种输出随输入增加而减少的情况，例如电池电量或冷却温度的变化。
Q1: What does it mean for a function to be decreasing?
A decreasing function has outputs that get smaller as inputs increase. Mathematically, for any two input values where x1 < x2, the corresponding outputs satisfy f(x1) > f(x2). On a graph, this appears as a downward slope from left to right, visually confirming the declining relationship between input and output.
Q2: How do you identify a decreasing function on a graph?
A decreasing function is identified by observing whether the graph slopes downward from left to right. As you move along the x-axis from left to right, the f(x)-values consistently get smaller. This visual pattern on a graphs of functions clearly indicates the function is decreasing across that interval.
Q3: What does a negative rate of change indicate about a function?
A negative rate of change indicates a decreasing function. The average rate of change is calculated by dividing the change in output by the change in input. When this ratio is negative across intervals, the function is decreasing, meaning outputs decline as inputs increase.
Q4: How does the derivative help determine if a function is decreasing?
For continuous functions, the derivative f′(x) serves as an indicator of whether a function is decreasing. If f′(x) < 0 for all x in an interval, the function is decreasing on that interval. A negative derivative confirms that outputs are declining as inputs increase.
Q5: What real-world situations can be modeled using decreasing functions?
Decreasing functions model many natural and technological phenomena where outputs decline with increasing input. Examples include the temperature of a cooling object, the voltage of a discharging battery, and the height of a falling object after its peak. These scenarios involve quantities that reduce as time or another input progresses.
Q6: How can you calculate speed changes using the rate of change concept?
Speed changes are calculated by determining the average rate of change between intervals. Divide the change in distance by the change in time between two recorded points to find average speed. Comparing speeds across successive intervals reveals whether the rate is increasing or decreasing over time.
Q7: How does a decreasing function differ from an increasing function?
A decreasing function has outputs that get smaller as inputs increase, while an increasing function has outputs that get larger. On a graph, a decreasing function slopes downward from left to right, whereas an increasing function slopes upward. Understanding both types helps analyze how quantities change across different intervals.