Question

In Exercises 93–96, find the average rate of change of the function over the given interval. Compare this average rate of change with the instantaneous rates of change at the endpoints of the interval. $$ f(x)=\frac{-1}{x}, \quad[1,2] $$

Short answer

The average rate of change over the interval [1,2] is 0.5. The instantaneous rates of change at the interval's endpoints are 1 and 0.25 respectively.

Step-by-step solution

Calculating the average rate of change

The formula for calculating the average rate of change over an interval \([a,b]\) for a function \(f(x)\) is \(\frac{f(b) - f(a)}{b - a}\). For this exercise, \(a = 1\) and \(b = 2\). Hence, \[f(b) - f(a) = f(2) - f(1) = \frac{-1}{2} - \left(-\frac {1}{1}\right) = \frac{-1}{2} + 1 = \frac{1}{2}\] and \(b - a = 2 - 1 = 1\). Therefore, the average rate of change on the interval \([1,2]\) is \(\frac{1}{2}/1 = 0.5\).

Finding the instantaneous rate of change at the endpoints

The instantaneous rate of change of function \(f(x)\) at a point \(x = a\) is represented by the derivative of the function at that point. The derivative of this function \(f(x)\), which is \(\frac{-1}{x}\), is \(f'(x) = \frac{1}{x^2}\). Hence, the instantaneous rate of change at the endpoints \(a = 1\) and \(b = 2\) are \(f'(1) = 1\) and \(f'(2) = \frac{1}{4}\) respectively.

Comparing the average rate of change with the instantaneous rates of change at the endpoints

The average rate of change is 0.5, while the instantaneous rates of change at the endpoints are 1 and 0.25 respectively. This clearly illustrates how the instantaneous rate of change can vary across different points in the interval, whereas the average rate of change provides an overall rate of change for the entire interval. Regardless, their comparison might not be directly helpful to infer a clear observation as they simply reflect different types of change.

Key concepts

Instantaneous Rate of Change

Understanding the instantaneous rate of change in calculus is akin to measuring the speed of a car at a precise moment in time. It's about pinpointing how fast a function is changing at a single point. For a mathematical function, this rate is given by the derivative at a specific point.

For our function, which is defined as
\( f(x) = \frac{-1}{x} \),
the instantaneous rate of change at any point x is simply the derivative evaluated at that point,
\( f'(x) = \frac{1}{x^2} \).
By finding the derivative, we determine a formula that tells us how the function behaves instantaneously throughout its domain. When evaluating
\( f'(x) \) at the endpoints of the interval [1,2], we get different values—illustrating the fact that the instantaneous change varies at every point.

Derivative of a Function

The derivative of a function is a cornerstone concept in calculus that gives us the instantaneous rate of change of the function with respect to its variable. It's the equivalent of finding the slope or steepness of the function at any given point.

In our case, the function
\( f(x) = \frac{-1}{x} \)
expresses a relationship where the derivative,
\( f'(x) = \frac{1}{x^2} \),
can be interpreted as the function's rate of change at any point along its curve. Differentiating functions allows us to analyze motion and change in a wide array of fields.

Rate of Change Comparison

Comparing different types of rates of change—like average and instantaneous—can deepen our understanding of a function's behavior over an interval. While the average rate of change consolidates the overall variation of a function from one point to another, the instantaneous rate describes the function's behavior at each specific point within the interval.

In the given exercise, the average rate of change was found to be 0.5 over the interval [1,2], while the instantaneous rates of change were 1 at x = 1 and 0.25 at x = 2. This contrast reflects the detailed and nuanced nature of instantaneous rates when compared to the broader, averaged out change.

Calculus Interval Analysis

Interval analysis in calculus involves studying the behavior of a function over a specific range of values. It's not merely about evaluating the function at certain points, but rather understanding how and why it changes across intervals.

Revisiting our function
\( f(x) = \frac{-1}{x} \),
we glance at the interval [1,2] and notice that the function decreases as x increases. Average and instantaneous rates of change provide different insights; the former averages out the behavior across the entire interval, while the latter gives us the 'moment-to-moment' changes a function undergoes. Such analysis is essential for thoroughly understanding the complexities of function behaviors in calculus.