Problem 52 - Precalculus : Building Concepts and Connections

Question

Find the average rate of change of each ficnetion on the given interval. $$f(x)=2|x|+4 ; \text { interval: }[3,5]$$

Step-by-Step Solution

Answer

The average rate of change of the function \(f(x) = 2|x|+4\) on the interval [3,5] is 2.

Step 1: Evaluate the function at the endpoints of the interval

Firstly, substitute the values of the end points of the interval (3 and 5) into the function \(f(x) = 2|x|+4\). This gives us \(f(3) = 2*3 + 4 = 10\) and \(f(5) = 2*5 + 4 = 14\).

Step 2: Apply the average rate of change formula

Now, put the values obtained from step 1 into the formula for the average rate of change, which is \((f(b) - f(a)) / (b - a)\). So, the average rate of change is \((f(5) - f(3)) / (5 - 3) = (14 - 10) / 2 = 2\).

Step 3: Interpret the result

The average rate of change is 2. This means that for each unit change in x, the value of the function increases by 2 on average over the interval [3,5].

Key Concepts

Absolute Value Function

An absolute value function is a mathematical function that takes any real number as input and outputs its absolute value. The absolute value of a number is the distance between the number and zero on the number line. This function is always non-negative. For example, the absolute value of both 3 and -3 is 3.

In the exercise presented, the function is defined as \( f(x) = 2|x| + 4 \). The expression inside the absolute value, \(|x|\), ensures that the input \(x\) remains positive or zero before it is multiplied by 2 and then increased by 4. This transformation effectively shifts the graph vertically by 4 units due to the addition of 4. This kind of transformation is typical for absolute value functions where each transformation modifies how the graph looks on a coordinate plane.

The structure of this function allows it to accommodate a wide range of input values while maintaining its piecewise nature, where the overall direction of the "V" shape is dictated by the multiplicative constant outside the absolute value, in this case, the 2, which indicates the steepness or rate of change in the output values.

Function Evaluation

Function evaluation is the process of determining the output of a function for particular input values. It's essential when solving problems involving functions, as it helps to understand how the function behaves over specific intervals.

Consider the problem at hand, where we evaluated the function \( f(x) = 2|x| + 4 \) at specific points \(x = 3\) and \(x = 5\). Here's a step-by-step guide on function evaluation for these inputs:

• First, substitute \(x = 3\) into the function: \( f(3) = 2|3| + 4 = 6 + 4 = 10 \).
• Next, substitute \(x = 5\) into the function: \( f(5) = 2|5| + 4 = 10 + 4 = 14 \).

This evaluation allows us to determine the function's output at these specific points, which is crucial for further analysis, such as calculating the average rate of change. By evaluating the endpoints of the given interval, you can understand how the function behaves over that range, aiding in deriving real insights.

Interpreting Results

Interpreting the results of mathematical calculations is key to understanding the implications of those calculations in real-world terms. For this exercise, the interpretation revolves around the average rate of change.

The average rate of change is a concept used to determine how fast a function's values are changing,on average, over a specific interval. Here, the formula is:
\[ \text{Average Rate of Change} = \frac{f(b) - f(a)}{b - a} \]
In our exercise, substituting the values we've evaluated gives: \((f(5) - f(3)) / (5 - 3) = (14 - 10) / 2 = 2 \).

Understanding this result is essential. It signifies that, between \(x=3\) and \(x=5\), the function's value increases by 2 units for every one unit increase in \(x\). This insight can be extended to any practical application where one needs to understand how quickly or slowly a value changes over a period, providing a clearer picture of performance or behavior. Such concepts are pivotal in fields like physics, economics, and engineering, where rates of change are often analyzed.