Understanding the Average Rate of Change of a Function Over an Interval

In calculus, the average rate of change (ARC) of a function over an interval is a fundamental concept that helps to estimate the slope of a curve over that period. This article will explore how to calculate the average rate of change, the significance of this concept, and provide examples to illustrate the process.

Definition and Formula

The average rate of change of a function f(x) over an interval (a, b) is defined as follows:

ARC frac{f(b) - f(a)}{b - a}

Step-by-Step Calculation

To calculate the average rate of change, you need to follow these steps:

Identify the function f(x) Determine the interval [a, b] Evaluate the function at the endpoints of the interval: f(a) and f(b) Apply the formula to find the average rate of change

Example Calculations

Let's apply the formula to a concrete function and interval.

Example 1: Quadratic Function

Consider the function f(x) x^3 - 7x^2 4x - 2, and we want to find the average rate of change over the interval [1, 5].

Identify the function: f(x) x^3 - 7x^2 4x - 2 The interval: [a, b] [1, 5] Evaluate the function at the endpoints: At x 1: f(1) 1^3 - 7(1)^2 4(1) - 2 1 - 7 4 - 2 -4 At x 5: f(5) 5^3 - 7(5)^2 4(5) - 2 125 - 175 20 - 2 -32 Calculate the average rate of change:

ARC frac{f(5) - f(1)}{5 - 1} frac{-32 - (-4)}{4} frac{-32 4}{4} frac{-28}{4} -7

The average rate of change for the function f(x) x^3 - 7x^2 4x - 2 over the interval [1, 5] is -7.

Example 2: Simplified Function

For the function f(x) 17 - x^2, over the interval [1, 5], we proceed as follows.

Identify the function: f(x) 17 - x^2 The interval: [a, b] [1, 5] Evaluate the function at the endpoints: At x 1: f(1) 17 - (1)^2 17 - 1 16 At x 5: f(5) 17 - (5)^2 17 - 25 -8 Calculate the average rate of change:

ARC frac{f(5) - f(1)}{5 - 1} frac{-8 - 16}{4} frac{-24}{4} -6

The average rate of change for the function f(x) 17 - x^2 over the interval [1, 5] is -6.

Geometric Interpretation

The average rate of change over an interval [a, b] can also be interpreted geometrically as the slope of the secant line connecting the points (a, f(a)) and (b, f(b)) on the graph of the function. This slope gives us a linear approximation of how the function changes over the interval.