3.8
Función decreciente
Una función decreciente describe una relación en la que la variable dependiente disminuye constantemente a medida que la variable independiente aument…
Una función está disminuyendo cuando su salida disminuye a medida que aumenta la entrada.
Este comportamiento se identifica observando si el gráfico se inclina hacia abajo de izquierda a derecha.
Considere a un hombre corriendo en una pista. El tiempo necesario y la distancia recorrida en cada vuelta se registran para determinar los cambios de velocidad en diferentes intervalos.
La velocidad promedio (o tasa de cambio) entre intervalos se determina calculando el cambio en la distancia y dividiéndolo por el cambio en el tiempo entre dos puntos registrados.
A continuación, para identificar si la velocidad aumenta o disminuye, la velocidad de cada vuelta se calcula dividiendo la distancia recorrida por el tiempo necesario para esa vuelta. Esto ayuda a analizar cómo cambia el ritmo del corredor de una vuelta a la siguiente.
Cuando se trazan como un gráfico de velocidad frente a tiempo, los datos muestran una disminución constante de la velocidad. Esto representa una función decreciente, lo que confirma que el corredor reduce la velocidad con cada vuelta sucesiva.
El concepto de funciones decrecientes modela varias situaciones en las que las salidas disminuyen con el aumento de la entrada, como la duración de la batería o la temperatura de refrigeración.
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.