Understanding Average Rate of Change in Mathematics: Definition, Formula, and Calculation Steps

Average Rate of Change of f(x) on [a,b]

The average rate of change of a function f(x) on the interval [a,b] is defined by the formula:

Average Rate of Change = (f(b) – f(a)) / (b – a)

To understand this concept better, let’s break it down step by step:

1

The average rate of change of a function f(x) on the interval [a,b] is defined by the formula:

Average Rate of Change = (f(b) – f(a)) / (b – a)

To understand this concept better, let’s break it down step by step:

1. First, determine the values of f(x) at the endpoints of the interval, f(a) and f(b).

2. Next, subtract the value of f(a) from f(b), which gives you the change in the y-values of the function between the points a and b. This represents the vertical change.

3. Then, subtract a from b, which gives you the change in the x-values between the points a and b. This represents the horizontal change.

4. Finally, divide the vertical change (f(b) – f(a)) by the horizontal change (b – a) to get the average rate of change of the function over the interval [a,b].

The average rate of change gives you the average slope of the function on the interval [a,b]. It represents the rate at which the function is changing over that interval.

For example, let’s find the average rate of change of the function f(x) = 2x + 3 on the interval [1, 4]:

1. Evaluate f(1) = 2(1) + 3 = 5 and f(4) = 2(4) + 3 = 11.

2. Calculate the vertical change: f(4) – f(1) = 11 – 5 = 6.

3. Calculate the horizontal change: 4 – 1 = 3.

4. Finally, divide the vertical change by the horizontal change: 6 / 3 = 2.

Therefore, the average rate of change of f(x) = 2x + 3 on the interval [1, 4] is 2. This means that, on average, the function increases by 2 units for every 1 unit increase in x over this interval.

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