Finding Average Rate Of Change Of F(x) = 8x^2 - 9 On [4, T]
Understanding Average Rate of Change
In mathematics, the average rate of change of a function over an interval represents the average amount that the function changes per unit change in the input variable. Geometrically, it corresponds to the slope of the secant line connecting the points on the function's graph at the endpoints of the interval. Calculating the average rate of change is a fundamental concept in calculus and has applications in various fields, including physics, economics, and engineering. In this article, we will delve into the process of finding the average rate of change of the quadratic function f(x) = 8x^2 - 9 over the interval [4, t]. This exploration will involve applying the definition of average rate of change and simplifying the resulting expression to obtain a clear understanding of how the function's rate of change varies with the value of t.
The average rate of change of a function f(x) over an interval [a, b] is defined as the change in the function's value divided by the change in the input variable, which can be expressed as the formula:
(Average Rate of Change) = (f(b) - f(a)) / (b - a)
This formula calculates the slope of the line that connects the two points (a, f(a)) and (b, f(b)) on the graph of the function. The average rate of change provides valuable information about how the function's output changes in relation to its input over the specified interval. It's a crucial concept in calculus and helps in understanding the behavior of functions, especially when dealing with non-linear functions where the rate of change isn't constant.
In our specific problem, we are tasked with finding the average rate of change of the function f(x) = 8x^2 - 9 over the interval [4, t]. This means that a = 4 and b = t. To proceed, we need to calculate f(4) and f(t), which are the values of the function at the endpoints of the interval. Then, we can substitute these values into the average rate of change formula and simplify the expression to obtain the desired result. This process will give us an expression in terms of t, which represents how the average rate of change varies as the upper endpoint of the interval changes. Understanding this variation is essential for analyzing the function's behavior over different intervals.
Applying the Formula
To find the average rate of change of the function f(x) = 8x^2 - 9 on the interval [4, t], we will use the formula we previously discussed:
(Average Rate of Change) = (f(t) - f(4)) / (t - 4)
First, we need to evaluate f(t) and f(4). Let's start with f(t). To find f(t), we substitute t for x in the function's expression:
f(t) = 8t^2 - 9
This expression represents the value of the function at the point t. Next, we need to find f(4). We substitute 4 for x in the function's expression:
f(4) = 8(4)^2 - 9
Now, we can simplify this expression:
f(4) = 8(16) - 9
f(4) = 128 - 9
f(4) = 119
So, f(4) = 119. Now that we have both f(t) and f(4), we can substitute these values into the average rate of change formula:
(Average Rate of Change) = ((8t^2 - 9) - 119) / (t - 4)
This expression represents the average rate of change of the function f(x) over the interval [4, t]. To simplify this expression further, we need to perform some algebraic manipulations. This will involve combining like terms in the numerator and looking for opportunities to factor and cancel common factors. The goal is to obtain a simplified expression that clearly shows how the average rate of change depends on the value of t. This simplified form will make it easier to analyze the function's behavior and understand its rate of change over different intervals.
Simplifying the Expression
Now that we have the expression for the average rate of change:
(Average Rate of Change) = ((8t^2 - 9) - 119) / (t - 4)
Let's simplify the numerator by combining the constant terms:
(Average Rate of Change) = (8t^2 - 128) / (t - 4)
We can see that there is a common factor of 8 in the numerator. Let's factor it out:
(Average Rate of Change) = 8(t^2 - 16) / (t - 4)
The expression t^2 - 16 is a difference of squares, which can be factored as (t - 4)(t + 4). So, we can rewrite the expression as:
(Average Rate of Change) = 8(t - 4)(t + 4) / (t - 4)
Now, we can cancel the common factor of (t - 4) in the numerator and denominator, provided that t ≠4:
(Average Rate of Change) = 8(t + 4)
This is the simplified expression for the average rate of change of the function f(x) = 8x^2 - 9 on the interval [4, t]. The expression 8(t + 4) clearly shows how the average rate of change depends on the value of t. It's a linear function of t, which means that the average rate of change increases linearly as t increases. This simplified form is much easier to work with and provides a clearer understanding of the function's behavior over the interval [4, t]. It also highlights the importance of algebraic simplification in calculus problems, as it can often lead to more manageable and insightful expressions.
Final Answer
The average rate of change of f(x) = 8x^2 - 9 on the interval [4, t] is:
8(t + 4)
This expression represents the slope of the secant line connecting the points (4, f(4)) and (t, f(t)) on the graph of the function. It tells us how much the function's value changes, on average, for each unit change in x over the interval [4, t]. The result is a linear function of t, indicating that the average rate of change increases linearly as t increases. This is a significant finding, as it reveals the behavior of the quadratic function over different intervals. Understanding the average rate of change is crucial in calculus for analyzing the function's overall behavior, including its increasing and decreasing intervals, and for approximating the instantaneous rate of change at a specific point.
In conclusion, by applying the definition of average rate of change, evaluating the function at the endpoints of the interval, and simplifying the resulting expression, we have successfully found the average rate of change of the given function. This process demonstrates the power of algebraic manipulation in calculus and provides a valuable tool for understanding the behavior of functions.
