Problem 21 - Calculus Concepts

Question

For the linear function $$ f(x)=3 x+4 $$ a. Calculate the average rate of change and the percentage change in \(f\) 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 linear functions presented in Chapter 1 , what generalizations can be made about percentage change and average rate of change for a linear function?

Step-by-Step Solution

Answer

The average rate of change is constant (3) for all intervals, while percentage change decreases as the start value increases.

Step 1: Evaluate the Function at Given Points

For each interval, evaluate the function at the starting and ending points. - For interval (i) from \(x=1\) to \(x=3\): - \(f(1) = 3(1) + 4 = 7\) - \(f(3) = 3(3) + 4 = 13\) - For interval (ii) from \(x=3\) to \(x=5\): - \(f(3) = 13\) - \(f(5) = 3(5) + 4 = 19\) - For interval (iii) from \(x=5\) to \(x=7\): - \(f(5) = 19\) - \(f(7) = 3(7) + 4 = 25\)

Step 2: Calculate the Average Rate of Change

The average rate of change for a function \(f(x)\) over the interval \([a, b]\) is given by the formula: \[ \text{Average Rate of Change} = \frac{f(b)-f(a)}{b-a} \] - For interval (i) from \(x=1\) to \(x=3\): \(\frac{13-7}{3-1} = \frac{6}{2} = 3\)- For interval (ii) from \(x=3\) to \(x=5\): \(\frac{19-13}{5-3} = \frac{6}{2} = 3\)- For interval (iii) from \(x=5\) to \(x=7\): \(\frac{25-19}{7-5} = \frac{6}{2} = 3\)

Step 3: Calculate the Percentage Change

The percentage change in \(f(x)\) over the interval from \(x=a\) to \(x=b\) is calculated as: \[ \text{Percentage Change} = \left( \frac{f(b)-f(a)}{f(a)} \right) \times 100\% \] - For interval (i) from \(x=1\) to \(x=3\): \(\left( \frac{13-7}{7} \right) \times 100\% = \left( \frac{6}{7} \right) \times 100\% \approx 85.71\%\)- For interval (ii) from \(x=3\) to \(x=5\): \(\left( \frac{19-13}{13} \right) \times 100\% = \left( \frac{6}{13} \right) \times 100\% \approx 46.15\%\)- For interval (iii) from \(x=5\) to \(x=7\): \(\left( \frac{25-19}{19} \right) \times 100\% = \left( \frac{6}{19} \right) \times 100\% \approx 31.58\%\)

Step 4: Generalize About Average Rate and Percentage Changes for Linear Functions

The results show that the average rate of change for the linear function \(f(x)=3x+4\) is constant across all intervals, being equal to the slope, which is 3. However, the percentage change decreases as the starting point increases, illustrating that the function's relative change size depends on the starting point, even as the absolute change is consistent in linear functions.

Key Concepts

Linear Functions Explained

A linear function is one of the simplest forms of mathematical expressions you will encounter. It's called 'linear' because its graph is a straight line. The general form of a linear function is

• \( f(x) = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept.

The slope \( m \) tells you how steep the line is. It also indicates the rate of change for the function's output. In the example given, \( f(x) = 3x + 4 \), the slope is 3. This means for every unit you move to the right along the x-axis, the value of \( f(x) \) increases by 3. The y-intercept \( b \) is the point where the line crosses the y-axis. Here, it is 4, which is the value of \( f(x) \) when \( x \) is 0. Linear functions are straightforward, and their constant rate of change makes calculations easier.

Understanding Percentage Change

Percentage change provides a way of expressing a change in terms of a percentage, which can be very helpful for understanding growth or reduction relative to the initial value.
The formula for percentage change is:

• \[ \text{Percentage Change} = \left( \frac{f(b)-f(a)}{f(a)} \right) \times 100\% \]

This formula measures how much a value has increased or decreased from its original amount by transforming this relative change into a percentage.
In the exercise, as you move to a higher starting point on the x-axis, the percentage change decreases even though the actual change in \( f(x) \) is constant at 6 units. This highlights that while linear functions have a consistent rate of change, the percentage representation of that change shrinks as the base value grows.

Key Calculus Concepts: Rate of Change

Calculus concepts revolve significantly around understanding rates of change. The term 'average rate of change' in calculus is similar to the slope of a line in algebra. For linear functions, this concept is crystal clear because the average rate of change over any interval is constant and equal to the slope.

• The average rate of change is calculated as:\[ \text{Average Rate of Change} = \frac{f(b)-f(a)}{b-a} \]

In linear functions, since the slope is constant, the average rate of change will always be the same across any intervals. For \( f(x) = 3x + 4 \), the average rate of change is always 3.
This constant rate signifies how predictable linear functions are. Even in calculus, where functions often curve and change rate, linear functions remain steady, making them a cornerstone for initial learning in calculus and a helpful predictor in constant motion scenarios.