Problem 36 - University Calculus: Early Transcendentals

Question

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

The absolute minimum is 0 at \(t = 5\); the absolute maximum is 2 at \(t = 7\).

Step 1: Understanding Absolute Value Function

Consider the function \( f(t) = |t - 5| \). An absolute value function, like this one, reflects all negative input values to positive, resulting in a V-shaped graph. The critical point is where the expression inside the absolute value equals zero: \( t - 5 = 0 \). Solve this to find the critical point.

Step 2: Solve for Critical Point

Set the expression inside the absolute value equal to zero: \( t - 5 = 0 \). Solving this gives \( t = 5 \). This is the critical point of the function \( f(t) = |t - 5| \) within the interval \( 4 \leq t \leq 7 \).

Step 3: Evaluate Function at Endpoints and Critical Point

Calculate the function values at the endpoints and the critical point within the interval:\[ \begin{align*} f(4) &= |4 - 5| = 1, \ f(5) &= |5 - 5| = 0, \ f(7) &= |7 - 5| = 2. \end{align*} \]

Step 4: Identify Absolute Extremes

From the calculated values: \( f(5) = 0 \) is the absolute minimum, and \( f(7) = 2 \) and \( f(4) = 1 \) are higher values. Hence, the absolute maximum value is 2 (at \( t = 7 \)), and the absolute minimum value is 0 (at \( t = 5 \)).

Step 5: Graph the Function

Plot the function \( f(t) = |t - 5| \) on the interval \( 4 \leq t \leq 7 \). The graph has a V-shape with the lowest point at \( (5, 0) \), reflecting the critical point. Mark the points \((4, 1)\), \((5, 0)\), and \((7, 2)\) to identify where the absolute extrema occur on the graph.

Key Concepts

Understanding Absolute Value Functions

The absolute value function is quite unique and intuitive once you get to know it. This function takes an input and measures how far that input is from zero, always returning a non-negative value.
For example, if you have a function like \( f(t) = |t - 5| \), what this means is you're examining how far the value \( t \) is from 5. If \( t \) is larger than 5, it's simply \( t - 5 \). If \( t \) is less than 5, it flips to \( 5 - t \) to ensure the result is positive.
This often results in a V-shaped graph, characteristic of absolute value functions.

Defining Critical Points

Critical points in calculus are vital for determining where functions change their behavior drastically, such as reaching peaks or troughs.
For the function \( f(t) = |t - 5| \), to find the critical point, you set the inside of the absolute value to zero: \( t - 5 = 0 \). Solving this gives \( t = 5 \), where the function transitions from decreasing to increasing.
On a graph, this point marks where the sharp turn or vertex of the V-shape occurs. Understanding this helps identify potential points where the function might have extreme values.

Locating Absolute Maximum and Minimum

Finding the absolute maximum and minimum on a given interval helps you understand the range of values the function can take.
For \( f(t) = |t - 5| \) on the interval \( 4 \leq t \leq 7 \), examine endpoint values and the critical point.

• When \( t = 4 \), \( f(4) = 1 \).
• At \( t = 5 \), \( f(5) = 0 \), the lowest it gets.
• For \( t = 7 \), \( f(7) = 2 \), the highest value in this range.

Thus, the absolute minimum value is 0 at \( t = 5 \), and the absolute maximum is 2 at \( t = 7 \). This understanding helps when sketching and interpreting the function's graphical representation.

Function Graphing Essentials

Graphing a function provides a visual understanding which can be much clearer than numbers. For the absolute value function \( f(t) = |t - 5| \), the graph on \( 4 \leq t \leq 7 \) has a distinct V-shape.
Key points to plot include:

• (4, 1) where the function starts rising towards the vertex.
• (5, 0), the vertex which is the absolute minimum.
• (7, 2), where it reaches the maximum in this interval.

Marking these coordinates creates a clear picture of how the function behaves in the specified range, highlighting where it dips and peaks.