Problem 9 - Calculus an Applied Approach

Question

Use a graphing utility to graph the function and find its average rate of change on the interval. Compare this rate with the instantaneous rates of change at the endpoints of the interval. $$ f(x)=\frac{1}{x} ;[1,4] $$

Step-by-Step Solution

Answer

The average rate of change of the function on the interval [1,4] is -0.75, while the instantaneous rates of change at the endpoints are -1 at x=1 and -0.0625 at x=4, respectively. These disparities result from the function's non-linear nature.

Step 1: Graph the function

To graph the function, input the equation \( f(x) = \frac{1}{x} \) into a graphing utility. The resulting graph should be smooth curve diminishing as x increases. The graph will aid in visualizing the average and instantaneous rates of change.

Step 2: Calculate the average rate of change on the interval

The average rate of change of a function on a given interval \([a, b]\) can be found using the formula \( \frac{f(b)-f(a)}{b-a} \). To find the average rate of change of \( f(x) = \frac{1}{x} \) over the interval [1,4], substitute \( b=4 \) and \( a=1 \) into the equation to get \( \frac{f(4)-f(1)}{4-1} = \frac{\frac{1}{4}-1}{3} = -\frac{3}{4} \)

Step 3: Find the instantaneous rates of change at the endpoints

The instantaneous rate of change is the derivative of a function at a given point. For the function \(f(x) = \frac{1}{x}\), the derivative is \(f'(x) = -\frac{1}{x^2}\). At the endpoints x=1 and x=4, the instantaneous rates of change are \(f'(1) = -1\) and \(f'(4) = -\frac{1}{16}\), respectively.

Step 4: Compare the rates of change

The average rate of change was calculated as -0.75, while the instantaneous rates at the endpoints were -1 and -0.0625. This discrepancy is due to the non-linear shape of the graph. The instantaneous rate of change reflects the steepness of the graph at only one point, while the average rate of change reflects the overall slope between two points on the graph.

Key Concepts

Instantaneous Rate of Change

The instantaneous rate of change of a function at a specific point is the slope of the tangent line at that point. It tells us how quickly or slowly the function value is changing at that exact point. To find this rate, we need to calculate the derivative of the function and evaluate it at the point of interest.

For instance, with the function \( f(x) = \frac{1}{x} \), the derivative \( f'(x) = -\frac{1}{x^2} \) helps us determine the instantaneous rate of change at any given \( x \). At \( x = 1 \), this rate is \( f'(1) = -1 \), and at \( x = 4 \), it is \( f'(4) = -\frac{1}{16} \). These values signify how steeply the curve is rising or falling at those points.

To visualize it, imagine zooming in on the curve at a point until it becomes almost a straight line. The slope of this line gives you the instantaneous rate of change. It's a powerful concept often used in physics to understand velocity.

Average Rate of Change

The average rate of change between two points on a function gives us an overall picture of how the function behaves over an interval. It doesn't focus on the intricacies at every single point but rather looks at the net change over a stretch.

To calculate this, we use the formula \( \frac{f(b) - f(a)}{b - a} \) where \( a \) and \( b \) are the endpoints of the interval. In our exercise, we wanted the average rate of change for the function \( f(x) = \frac{1}{x} \) from \( x = 1 \) to \( x = 4 \). By using the formula, we calculated it as \( -\frac{3}{4} \). This represents the slope of the secant line, which passes through the points \( (1, f(1)) \) and \( (4, f(4)) \).

The average rate provides a valuable overview by smoothing out the finer details. This can be particularly useful when trying to understand large-scale trends.

Derivative of a Function

The derivative of a function provides insight into how a function changes as its input changes. It's a fundamental concept in calculus because it allows us to understand the dynamics of function behavior at every point.

Mathematically, the derivative of a function \( f \) is another function, often denoted as \( f' \). For example, if \( f(x) = \frac{1}{x} \), then the derivative \( f'(x) = -\frac{1}{x^2} \) tells us that for every small change in \( x \), there is a corresponding change in \( f(x) \).

Calculating the derivative involves applying rules of differentiation, such as the power rule or the quotient rule. Understanding how to derive these allows you to find the slope of the tangent line to any point on the curve, and also forms the basis for more complex calculus operations. Derivatives can answer questions related to rates of change, like speed or acceleration, making them invaluable tools in both theoretical and applied sciences.