Problem 36 - Thomas Calculus

Question

In Exercises \(21-36,\) find the absolute maximum and minimum values of each function on the given interval. Then graph the function. Identify the points on the graph where the absolute extrema occur, and include their coordinates. $$ f(t)=|t-5|, \quad 4 \leq t \leq 7 $$

Step-by-Step Solution

Answer

Minimum at \((5, 0)\); Maximum at \((7, 2)\).

Step 1: Understand the Function

We are given the function \(f(t) = |t - 5|\). This describes the absolute distance of \(t\) from 5. The graph will be V-shaped with a vertex at \(t = 5\).

Step 2: Evaluate at Endpoints and Critical Points

Check the function at the critical point and endpoints of the interval. For \(f(t) = |t - 5|\), critical point is \(t = 5\), and endpoints are \(t = 4\) and \(t = 7\). Calculate: 1. \(f(4) = |4 - 5| = 1\)2. \(f(5) = |5 - 5| = 0\)3. \(f(7) = |7 - 5| = 2\)

Step 3: Determine Absolute Extrema

From the calculations, the minimum value is 0 at \(t = 5\) and the maximum value is 2 at \(t = 7\). These are the absolute minimum and maximum values on the interval \([4, 7]\).

Step 4: Identify Coordinates

The coordinates for the absolute minimum are \((5, 0)\), and for the absolute maximum are \((7, 2)\). These points represent where the extremes occur on the graph of the piecewise function.

Step 5: Sketch the Graph

Plot the points \((5, 0)\), \((4, 1)\), and \((7, 2)\) on a coordinate plane. Connect them to form a V-shaped graph with a vertex at \((5, 0)\). This visually confirms the extrema found.

Key Concepts

Understanding Critical Points

In mathematics, critical points are fundamental in understanding the behavior of functions, especially when identifying local and absolute extrema. A critical point is where the derivative of a function is zero or undefined, indicating a potential minimum, maximum, or a point of inflection.

For a function like \( f(t) = |t-5| \), understanding critical points involves recognizing that the function is non-differentiable at \( t = 5 \). This occurs because the graph has a sharp vertex at this point, which is common in absolute value functions. Despite the lack of a derivative at \( t = 5 \), the function exhibits an absolute minimum here, as it's the lowest value within the defined interval.

The critical point evaluation can be extended by calculating the function's values at the interval's endpoints. This step ensures all potential candidates for extrema are considered, including these boundary points.

Graphing Functions Efficiently

Graphing functions is a crucial step in visually interpreting mathematical results, allowing us to confirm algebraic conclusions. When graphing \( f(t) = |t - 5| \), we anticipate a V-shaped graph, as absolute value functions are known for their linear segments that meet at a vertex.

Breaking down the function into its linear components, it translates to two lines: one with a slope of -1 and the other with a slope of +1, meeting at the vertex \( t = 5 \). Through graphing:

• Plot crucial points, such as calculations at endpoints \( (4, 1) \) and \( (7, 2) \), and identify the critical point \( (5, 0) \).
• Connect these points in a piecewise linear manner to illustrate the V-shape.

This visual representation confirms the exact position of both the absolute minimum and maximum, helping comprehend the function's domain range.

Exploring Piecewise Functions

Piecewise functions are those defined by different expressions on different parts of their domain. They allow for more complex modeling of real-world situations where a single formula cannot suffice for the whole domain.

The function \( f(t) = |t-5| \) effectively behaves as a piecewise function, consisting of two distinct linear equations which define its behavior depending on the value of \( t \):

• For \( t < 5 \), the expression substitutes as \( 5 - t \) due to the nature of absolute value.
• For \( t \geq 5 \), it behaves as \( t - 5 \).

The clear definition of pieces delivers immediate visualization once plotted. Each section on the graph reflects these linear behaviors, and when combined, they form the typical V-shape of an absolute value equation, with their meeting point (critical point) at \( t = 5 \). This insight is pivotal in solving any mathematics problem involving extreme or boundary values for piecewise functions.