Problem 22 - Calculus Concepts

Question

For the exponential function $$ f(x)=3\left(0.4^{x}\right) $$ a. Calculate the percentage change and average rate of change of \(f\) for each of the following intervals: $$ \begin{array}{l} \text { i. From } x=1 \text { to } x=3 \\ \text { ii. From } x=3 \text { to } x=5 \end{array} $$ iii. From \(x=5\) to \(x=7\) b. On the basis of the results in part \(a\) and the characteristics of exponential functions presented in Chapter 1 , what generalizations can be made about percentage change and average rate of change for an exponential function?

Step-by-Step Solution

Answer

The percentage change is constant at -84% for all intervals, while the average rate of change decreases. Exponential decay has a consistent percentage change.

Step 1: Calculate Function Values at Given Points

First, calculate \(f(x)\) at the specified points for each interval.\(f(1) = 3(0.4^1) = 1.2\)\(f(3) = 3(0.4^3) = 3\times0.064 = 0.192\)\(f(5) = 3(0.4^5) = 3\times0.01024 = 0.03072\)\(f(7) = 3(0.4^7) = 3\times0.0016384 = 0.0049152\)

Step 2: Calculate Percentage Change for Each Interval

Next, find the percentage change for each interval. The formula is:\[\text{Percentage Change} = \frac{f(x_2) - f(x_1)}{f(x_1)} \times 100\%\]i. From \(x=1\) to \(x=3\):\[\frac{0.192 - 1.2}{1.2} \times 100\% = -84\%\]ii. From \(x=3\) to \(x=5\):\[\frac{0.03072 - 0.192}{0.192} \times 100\% = -84\%\]iii. From \(x=5\) to \(x=7\):\[\frac{0.0049152 - 0.03072}{0.03072} \times 100\% = -84\%\]

Step 3: Calculate Average Rate of Change for Each Interval

Find the average rate of change using the formula:\[\text{Average Rate of Change} = \frac{f(x_2) - f(x_1)}{x_2 - x_1}\]i. From \(x=1\) to \(x=3\):\[\frac{0.192 - 1.2}{3 - 1} = \frac{-1.008}{2} = -0.504\]ii. From \(x=3\) to \(x=5\):\[\frac{0.03072 - 0.192}{5 - 3} = \frac{-0.16128}{2} = -0.08064\]iii. From \(x=5\) to \(x=7\):\[\frac{0.0049152 - 0.03072}{7 - 5} = \frac{-0.0258048}{2} = -0.0129024\]

Step 4: Generalizations About Exponential Functions

Based on the calculations, notice that the percentage change for each interval remains constant at \(-84\%\). This illustrates a consistent rate of change for exponential decay, where each interval changes by the same percentage, reflecting a fixed base multiplier less than one. However, the average rate of change decreases, indicating a more rapid decline in function value as \(x\) increases.

Key Concepts

Percentage Change

When analyzing an exponential function, understanding the percentage change is crucial to see how rapidly a function grows or decays over a specific range. Percentage change measures how much a function's value changes in terms of percentage in one interval compared to its starting value at the beginning of that interval.

• The formula for percentage change is given by \[ \text{Percentage Change} = \frac{f(x_2) - f(x_1)}{f(x_1)} \times 100\% \]
• This formula helps us calculate the relative change between two function values, \(f(x_1)\) and \(f(x_2)\).
• In the original exercise, when we computed the percentage change for the interval from \(x=1\) to \(x=3\), \(x=3\) to \(x=5\), and \(x=5\) to \(x=7\), each resulted in a consistent \(-84\%\) change.

This results from the nature of exponential decay—since the function decreases by a fixed percentage each time, the percentage change remains constant across equal intervals of \(x\). Such consistent change illustrates properties of exponential functions, making this concept a vital part of understanding how these functions behave.

Average Rate of Change

Another important concept in analyzing functions is the average rate of change. This gives us a sense of how the function's value changes on average over a specific interval of the variable \(x\). Unlike percentage change, which is relative, the average rate of change gives us an absolute measure.

• The formula for the average rate of change is \[ \text{Average Rate of Change} = \frac{f(x_2) - f(x_1)}{x_2 - x_1} \]
• This depicts how the function's value shifts from \(f(x_1)\) to \(f(x_2)\) over the interval \(x_2 - x_1\).
• In our specific problem, the average rate of change diminished with each interval from \(x=1\) to \(x=7\).

For exponential decay, this means the function's absolute change per unit of \(x\) gets smaller as \(x\) increases. While the percentage change stays constant, the smaller average rate of change reflects how the function becomes less steep, indicating a slower absolute loss of value over each successive interval.

Exponential Decay

Exponential decay is a type of exponential function where the value of the function decreases by a consistent factor for every increase in \(x\).

• In the standard exponential decay function \(f(x) = a(b^x)\), the base \(b\) is a positive number less than 1, indicating decay.
• As \(b\) approaches zero, the function decreases very quickly, which is reflected in the original exercise's constant percentage change of \(-84\%\) across intervals.
• The consistent factor by which the function's value decreases can also describe how sharply or gradually a function decays.

An understanding of exponential decay is useful in many real-world applications such as calculating depreciation in finance, determining half-lives in science, and more. This constant dying down of values means that, over every equal increment of \(x\), the percentage drop remains the same, but the actual drop becomes smaller, showcasing the reliable predictability of exponential decay.