Problem 15 - Essential Calculus Early Transcendentals

Question

\(15-22=\) Sketch the graph of \(f\) by hand and use your sketch to find the absolute and local maximum and minimum values of \(f .\) (Use the graphs and transformations of Sections \(1.2 . )\) $$f(x)=\frac{1}{2}(3 x-1), \quad x \leqslant 3$$

Step-by-Step Solution

Answer

The absolute maximum value is 4 at \(x=3\); there's no absolute minimum.

Step 1: Understanding the Function

The function given is \( f(x) = \frac{1}{2}(3x - 1) \), which is a linear function because it can be simplified to \( f(x) = \frac{3}{2}x - \frac{1}{2} \). The graph of this function is a straight line. The restriction \( x \leq 3 \) means that the graph is only drawn for \( x \) values less than or equal to 3.

Step 2: Identify the Slope and Y-Intercept

The slope (m) of the line is \( \frac{3}{2} \), and the y-intercept (b) is \( -\frac{1}{2} \). This tells us that the line inclines upwards from left to right and crosses the y-axis at \(-\frac{1}{2}\).

Step 3: Sketch the Graph

To sketch the graph, start by plotting the y-intercept point \((0, -\frac{1}{2})\) on the y-axis. Then, use the slope \( \frac{3}{2} \) to plot another point: from \( (0, -\frac{1}{2}) \), move up 3 units and right 2 units to reach point \( (2, 2.5) \). Considering \( x \leq 3 \), continue the line until \( x = 3 \), getting the point \( (3, 4) \). For \( x > 3 \), the line does not exist.

Step 4: Determine Maximum and Minimum Values

Since this is a linear function on a bounded interval (\( x \leq 3 \)), the maximum value occurs at the boundary. Substitute \( x=3 \) into the function: \( f(3) = \frac{1}{2}(3 \times 3 - 1) = 4 \). When \( x \to -\infty \), \( f(x) \to -\infty \) since the line continues indefinitely downwards. Therefore, the maximum value is 4 at \( x = 3 \), and there is no absolute minimum because the function decreases without bound.

Key Concepts

Slope of a Line

The slope of a line is a measure of its steepness and direction. It tells us how much the line rises or falls as we move along the x-axis. For a linear function like \( f(x) = \frac{3}{2}x - \frac{1}{2} \), the slope is \( \frac{3}{2} \). This means that for every two units you move to the right on the x-axis, the function value increases by three units.
This positive slope indicates an upward incline from left to right.

• A larger positive slope results in a steeper line.
• If the slope were negative, the line would slope downwards.

Understanding the slope is critical in predicting the behavior of a linear function and graphing it accurately.

Y-Intercept

The y-intercept of a line is the point where it crosses the y-axis. For the function \( f(x) = \frac{3}{2}x - \frac{1}{2} \), the y-intercept is \( -\frac{1}{2} \). This means that when the x-value is 0, the function value is \( -\frac{1}{2} \).
The y-intercept provides a starting point for graphing the line.

• It shows where the function's graph intersects the vertical axis.
• This point is always given by \( (0, b) \), where \( b \) is the y-intercept in the function's equation.

When sketching a graph, the y-intercept offers an anchor, ensuring that the linear function touches the y-axis at this specific point.

Graph Sketching

Graph sketching involves plotting a line or curve based on a function. For linear functions like \( f(x) = \frac{3}{2}x - \frac{1}{2} \), we start by identifying key features, such as the slope and y-intercept.
Begin by plotting the y-intercept \( (0, -\frac{1}{2}) \) on the y-axis. Using the slope \( \frac{3}{2} \), plot another point by moving up three units and two units to the right to reach \( (2, 2.5) \).
Once two points are plotted, draw the line, and extend it accordingly. In this example, since \( x \leq 3 \), continue until reaching \( (3, 4) \).

• Graph sketching is a quick method to visualize how a function behaves.
• Make sure to respect any restrictions, such as \( x \leq 3 \).

Good graph sketching allows for immediate visual insights into a function's behavior and helps in identifying key values.

Absolute and Local Extrema

In the context of linear functions with restrictions, understanding extrema is essential. Absolute extrema are the highest or lowest points over the entire domain, while local extrema are relative high or low points within a specific region.
For the function \( f(x) = \frac{1}{2}(3x - 1), x \leq 3 \), the absolute maximum is found by checking the endpoint \( x = 3 \). Substituting gives \( f(3) = 4 \), marking 4 as the highest point.

• Linear functions typically do not have local extrema unless bounded.
• In our case, the function has no absolute minimum as it decreases towards negative infinity.

Recognizing these values provides vital information on the function's overall behavior and is particularly important in finding maxima and minima on bounded intervals.