The average rate of change, which is represented by the symbol "A (x)", measures the average rate at which one element changes relative to another changing element. You can also use this concept to determine the rate of change of a mathematical function, the average rates of change of certain physical qualities (for example, velocity is simply the average rate of change of an object's position), and growth rates. average of plants and animals.
Steps
Method 1 of 3: Calculate Average Velocity
Step 1. Establish the formula to obtain the average velocity
If you want to determine average speed without a speedometer, you can calculate it by getting some basic measurements. For example, you can find the average velocity of an object by dividing the change in position by the change in time, which is expressed mathematically as:
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Velocity = ΔxΔt { displaystyle { text {Velocity}} = { frac { Delta x} { Delta t}}}
- Δx{displaystyle \Delta x}
corresponde al cambio en la posición del objeto o a la distancia que se haya desplazado y Δt{displaystyle \Delta t}
corresponde al cambio en el tiempo.
Step 2. Set the starting position of the object
Average speed is calculated by dividing the change in the position of an object over a certain period of time, so the first thing you should do is establish from where you will start to measure the displacement of the object.
- For example, to calculate the average speed of the commute from home to school on foot, you should start by measuring the commute at home.
- It is not necessary to use the position where the journey actually begins. For example, to calculate the average speed of a race car, its starting position can be anywhere on the track.
Step 3. Measure the distance from the initial position to the final position of the object
You can use any distance or period of time to calculate the average speed; you simply have to take into account the precision of your measuring instruments. In other words, if you are calculating the average speed of a runner, the distance must be accurate to within a few centimeters, while if you are calculating the average speed of a race car, the distance must be accurate to within plus or minus a certain amount of meters.
- For example, if you want to calculate the average speed of your trip from home to school, determine the distance traveled using a map or by traveling along that distance in a car equipped with an odometer. Suppose the distance turns out to be 1 km (0.6 miles).
- If you are calculating the average speed of a race car, imagine that each lap of the track measures 2.5 miles (2.5 miles). So regardless of where you start calculating the path of the car, each time it passes the same place it will have traveled 2.5 miles (2.5 miles).
Step 4. Measure the elapsed time
To calculate the average velocity, you must measure the time that elapses as the object travels from its initial position to its final position. Again, the precision of these measurements will depend on the calculation you are going to perform. That is, if you are calculating the average speed of a racer, you must measure time with an accuracy of up to tenths or hundredths of a second using a stopwatch, while, to calculate the average speed of a race car, you will only need to use the second hand of a clock.
- Following the example of calculating the speed of the journey from home to school, a wristwatch will be enough to measure time. So, imagine that the journey takes 15 minutes.
- Following the example of the race car, you can use a stopwatch or a regular watch to measure the time it takes for the car to complete a complete lap around the track. Imagine that a car at a relatively high speed makes one lap in 45 seconds.
Step 5. Find the average velocity
Once you have all the necessary measurements, you must replace them in the formula established above to calculate the average speed. Make sure the units of speed are consistent with those of the individual measurements of distance and time.
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In the example of the commute from your home to school, the distance was set to be 0.6 miles (1 km) and that it took 15 minutes to travel. Replace the values in the formula as follows:
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Velocity = ΔxΔt = 115 = 0.07 { displaystyle { text {Velocity}} = { frac { Delta x} { Delta t}} = { frac {1} {15}} = 0.07}
km/minuto (0, 04 millas/minuto).
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En el ejemplo del auto de carreras, se estableció que este recorría 4 km (2, 5 millas) en 45 segundos. Reemplaza los valores en la fórmula de la siguiente manera:
- Velocidad=ΔxΔt=445=0, 09{displaystyle {text{Velocidad}}={frac {Delta x}{Delta t}}={frac {4}{45}}=0, 09}
km/segundo (0, 0556 millas/segundo).
Step 6. Convert the units accordingly
The final result units may not be the best for you, so you can change them by multiplying them by a conversion factor.
- For example, speed in the context of racing cars is usually expressed in kilometers per hour, not per second. Therefore, you can convert the speed to kilometers per hour by multiplying it by a factor of 3600, since one hour has 3600 seconds.
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0.09 km per second ∗ 3600 seconds per hour = 324km / h { displaystyle 0.09 { text {km per second}} * 3600 { text {seconds per hour}} = 324 { text {km / h }}}
(200, 16 millas por hora).
Método 2 de 3: Calcular la tasa promedio de crecimiento
Step 1. Establish the formula to calculate the growth rate
You can find the change in the height or weight of the plants or animals and divide it by the change in time to determine the growth rate. Mathematically, this formula is expressed as follows:
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Rate = ΔhΔt { displaystyle { text {Rate}} = { frac { Delta h} { Delta t}}}
o ΔpΔt{displaystyle {frac {Delta p}{Delta t}}}
- En este caso, h{displaystyle h}
y p{displaystyle p}
corresponden a la altura (por su inicial en inglés) o al peso, respectivamente, y t{displaystyle t}
corresponde al tiempo transcurrido.
Step 2. Determine the length of time over which you want to measure the growth rate
For example, it will only take a few hours to measure changes in certain fast-growing plants, such as Asian bamboo. On the other hand, if you want to calculate the growth rate of a child, you will need at least several months or even years. Therefore, choose the most appropriate time course for the calculation you are going to perform.
- For example, imagine that an elementary class conducts an activity in which students plant bean seeds. If the starting point for your measurement is as soon as the plant begins to germinate, you can determine a reasonable time lapse of about a month counted in days.
- On the other hand, if a group of scientists wanted to calculate the growth rate of a baby elephant, they would measure it over the course of its first 90 days of age.
Step 3. Determine your starting height or weight
In order to measure your growth rate, you must establish a starting point that is the first action you will take.
- Following the example of bean plants, since the students decided to start the measurement on the day the first shoots appeared, the initial size will be 0 cm.
- Following the example of the baby elephant, veterinarians most likely weighed the elephant when it was born. Suppose you weighed 200 pounds (90 kg).
Step 4. Measure the final height or weight
Allow the set time to elapse and then measure the height or weight of the object of study.
- For the bean plants, imagine that, by the end of the 30th day of the established month, they had an average height of 24 inches (60 cm). Therefore, because the initial height was 0 cm, they grew a total of 60 cm (24 inches).
- In the case of the elephant, imagine that at the end of the 90th day of the established period, the elephant was found to weigh 180 kg (400 pounds).
Step 5. Plug these values into the formula for the growth rate
Substitute the values that you obtained from your measurements in the established formula to find the growth rate.
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Following the example of bean plants, the formula will look like this:
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Rate = 60cm30days = 2cm day { displaystyle { text {Rate}} = { frac {60 { text {cm}}} {30 { text {days}}}} = 2 { frac { text { cm}} { text {day}}}}
(0, 8 pulgadas al día).
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Siguiendo el ejemplo del elefante, primero debes encontrar la diferencia en el peso:
- Tasa=180−90kg90 días{displaystyle {text{Tasa}}={frac {180-90{text{kg}}}{90{text{ días}}}}}
- Tasa=90kg90 días{displaystyle {text{Tasa}}={frac {90{text{kg}}}{90{text{ días}}}}}
- Tasa=1kgdía{displaystyle {text{Tasa}}=1{frac {text{kg}}{text{día}}}}
(2, 22 libras al día)
Método 3 de 3: Calcular la tasa de cambio de una función
Step 1. Establish the function for which you will perform the calculation
Functions are mathematical relationships between numbers. That is, if you replace a number in the function, it will result in another number. You can usually graph the functions, which will give you different shapes, such as straight lines, parabolas, or curves with no apparent definition.
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Here are some examples of functions:
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y (x) = 3x + 4 { displaystyle y (x) = 3x + 4}
(línea recta)
- y(x)=sin(x){displaystyle y(x)=sin(x)}
(onda)
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- y(x)=x2{displaystyle y(x)=x^{2}}
(parábola)
Step 2. Determine the values of x
To determine the rate of change of a function, you must determine the result of the function for two different values of x. Therefore, you must choose an initial value and a final value of x depending on how big you want the difference between the two to be.
This difference will largely depend on the type of calculation you want to perform. For this exercise, the initial value of x will be 0 and the final value will be 3
Step 3. Find the result of the function for each value of x
Actually, the rate of change of the function measures the change in the results (that is, the values of y) over the range of values of x that you have established. Therefore, you must obtain the values of y that correspond to the initial and final values of x.
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For example, x = 0 and x = 3 were the values chosen for the function y (x) = x2 { displaystyle y (x) = x ^ {2}}
. Entonces, los valores de y(x){displaystyle y(x)}
que corresponden a cada uno de ellos son:
- y(0)=02=0{displaystyle y(0)=0^{2}=0}
- y(3)=32=9{displaystyle y(3)=3^{2}=9}
Step 4. Find the average rate of change of the function
You can express it in the following way:
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A (x) = ΔyΔx = f (x + h) −f (x) h { displaystyle A (x) = { frac { Delta y} { Delta x}} = { frac {f (x + h) -f (x)} {h}}}
- f(x){displaystyle f(x)}
- También puedes expresar h{displaystyle h}
- En el caso de la función y(x)=x2{displaystyle y(x)=x^{2}}
- A(x)=ΔyΔx=9−03−0=3{displaystyle A(x)={frac {Delta y}{Delta x}}={frac {9-0}{3-0}}=3}
corresponde al resultado de la función para el primer valor de x, f(x+h){displaystyle f(x+h)}
corresponde al resultado de la función para el segundo valor de x y h{displaystyle h}
corresponde a la diferencia entre ambos valores de x.
como Δx{displaystyle \Delta x}
, ya que, básicamente, corresponde al cambio en los valores de x.
, la tasa de cambio promedio entre los valores de x 0 y 3 puede calcularse de la siguiente forma:
Step 5. Interpret the results
In this particular example, the rate of change measures the vertical change in the function as it moves horizontally along the x-axis. According to the calculation performed, the function y (x) = x2 { displaystyle y (x) = x ^ {2}}
empieza en (0, 0) y termina en (3, 9) a lo largo del rango establecido. en el caso de las funciones que no produzcan una línea recta al graficarlas, la tasa de cambio promedio constituye la pendiente de la línea recta que se puede dibujar entre estos dos puntos, la cual, en esta función, es un incremento de 3 unidades de y por cada unidad de x.
la tasa de cambio promedio puede variar según el lugar en donde la midas. por ejemplo, si bien la tasa de cambio promedio fue de 3 entre x = 0 y x = 3 en la parábola anterior, el mismo rango de valores de 3 unidades entre x = 3 y x = 6 producirá una tasa de cambio promedio de 8, 33
consejos
- siempre debes prestar atención a las unidades del cálculo que vayas a realizar.
- si estudias cálculo, aprenderás a obtener la tasa de cambio instantánea de una función por medio de su derivada. esto te permitirá encontrar la tasa de cambio en cualquier instante en lugar de como un promedio calculado a lo largo de un rango de valores de x o de un determinado periodo de tiempo, lo que significa que, en teoría, el rango de valores de x es 0. puedes leer el artículo cómo calcular derivadas para encontrar más información.