Problem 29 - Thomas Calculus

Question

In Exercises \(15-30\) , 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. $$ f(t)=2-|t|, \quad-1 \leq t \leq 3 $$

Step-by-Step Solution

Answer

The absolute maximum value is 2 at \( t = 0 \), and the absolute minimum value is -1 at \( t = 3 \).

Step 1: Understanding the Function

The function given is \( f(t) = 2 - |t| \). This is a piecewise function, where \( |t| \) affects the output based on whether \( t \) is positive or negative. For \( t \geq 0 \), \( |t| = t \), and for \( t < 0 \), \( |t| = -t \). In our interval \([-1, 3]\), both cases occur.

Step 2: Calculate the Function Values at Critical Points

Find any critical points by checking where the piecewise definition changes or slopes might be zero. The critical point for \(|t|\) change is at \(t = 0\). We need to evaluate \(f(t)\) at \(-1\), \(0\), and \(3\).

Step 3: Evaluate at Critical Points

Calculate \( f(-1) = 2 - |-1| = 2 - 1 = 1 \). Then, \( f(0) = 2 - |0| = 2 \). Similarly, \( f(3) = 2 - |3| = 2 - 3 = -1 \).

Step 4: Identify Absolute Maxima and Minima

From the evaluations, observe that \( f(0) = 2 \) is the largest value and is thus the absolute maximum, and \( f(3) = -1 \) is the smallest value and is the absolute minimum in the interval \([-1, 3]\).

Step 5: Graph the Function and Mark Extrema

Plot the piecewise function on the graph. Linearly connect points from \(-1\) to \(0\) and from \(0\) to \(3\). Mark the points where the maximum \((0, 2)\) and minimum \((3, -1)\) values occur. These are the absolute extrema on the graph.

Key Concepts

Absolute Extrema

In calculus, finding the absolute extrema of a function involves identifying the highest and lowest values that the function attains over a given interval. These points are called absolute maximum and minimum, respectively. To find these extrema, evaluate the function at critical points and endpoints within the specified interval. Critical points occur where the derivative is zero or undefined, indicating potential changes in the slope of the graph. Yet, for piecewise functions, critical points might also appear where the function's definition changes, like in the case of the absolute value function given in the problem. Evaluating the function at all critical points and endpoints reveals the extrema, ensuring no potential point is omitted.For the function evaluated in the problem, the maximum value, or the absolute maximum, occurred at the point \( (0, 2) \), while the minimum value, or the absolute minimum, appeared at \( (3, -1) \). These values represent the extrema within the designated interval \([-1, 3]\). Thus, understanding and pinpointing the absolute extrema helps us grasp the function's behavior entirely over a range.

Piecewise Functions

Piecewise functions define separate expressions for different parts of their domain, making them versatile for modeling real-world problems with different conditions. Such is the case with the given function \( f(t) = 2 - |t| \). It's structured to behave differently based on whether \( t \) is positive or negative, serving as segments that together form the whole.For non-negative \( t \), the absolute value \( |t| \) simplifies to \( t \) itself. Consequently, the function expresses as \( f(t) = 2 - t \). Meanwhile, for negative \( t \), \( |t| \) becomes \(-t\), making the function \( f(t) = 2 + t \). Understanding how these pieces connect enables us to analyze function behavior seamlessly over varying intervals.Breaking the function into this piecewise form helps identify points where changes occur, such as \( t = 0 \), and allows for proper assessment of critical points.

Graphing Functions

Graphing a function, especially a piecewise function, involves plotting each segment based on their respective definitions and intervals. Then, connect these points in a manner that reflects the function's actual behavior. In this problem, we graph the function \( f(t) = 2 - |t| \) over the interval \([-1, 3]\).To achieve this, consider the nature of the function in both regions:

• Between \(-1 \) and \( 0\), it is defined as \( f(t) = 2 + t \), a linear equation sloping upwards.
• From \( 0 \) to \( 3\), it switches to \( f(t) = 2 - t \), now sloping downwards.

Start by plotting the known coordinates from evaluating the critical points, such as \( (-1, 1) \), \( (0, 2) \), and \( (3, -1) \). Join these points, ensuring each line segment aligns appropriately to establish the correct shape of the graph. By marking points where absolute extrema are observed, such as \( (0, 2) \) and \( (3, -1) \), the graph achieves more clarity and serves as visual confirmation of the function's properties.