Problem 35 - Thomas Calculus in SI Units

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

Step-by-Step Solution

Answer

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

Step 1: Understanding the Function

The function given is \( f(t) = 2 - |t| \). The absolute value function \(|t|\) is piecewise: \(|t| = t\) if \(t \geq 0\) and \(|t| = -t\) if \(t < 0\). Therefore, the function can be rewritten as a piecewise function: - \( f(t) = 2 - (-t) = 2 + t \) for \(-1 \leq t < 0\) - \( f(t) = 2 - t \) for \(0 \leq t \leq 3\).

Step 2: Finding Critical Points and Evaluating Endpoints

To find the extrema, evaluate the function at critical points and the interval's endpoints. For this piecewise linear function, check the endpoints first: - At \(t = -1\), \(f(-1) = 2 - |-1| = 2 - 1 = 1\) - At \(t = 3\), \(f(3) = 2 - |3| = 2 - 3 = -1\).

Step 3: Evaluating at the Piecewise Junction

Evaluate the function at the point where the piecewise definition changes, which is \(t = 0\). - At \(t = 0\), \(f(0) = 2 - |0| = 2\).

Step 4: Determine Absolute Extrema

Compare the function values obtained: - \(f(-1) = 1\) - \(f(0) = 2\) - \(f(3) = -1\) The absolute maximum value is 2 at \(t = 0\), and the absolute minimum value is -1 at \(t = 3\).

Step 5: Graph the Function and Identify Extrema

Graph \(f(t) = 2 - |t|\) over \([-1, 3]\). - For \(-1 \leq t < 0\), the graph is a line with equation \(f(t) = 2 + t\), sloping upwards. - For \(0 \leq t \leq 3\), the graph is a line with equation \(f(t) = 2 - t\), sloping downwards. The point (0, 2) is the absolute maximum, and the point (3, -1) is the absolute minimum.

Key Concepts

Piecewise Function

A piecewise function is a function that has different expressions for different intervals of the domain. In the given problem, the function is expressed as \( f(t) = 2 - |t| \), an absolute value function. An absolute value function itself can be considered piecewise because it changes its expression based on the sign of the variable inside. For the interval \(-1 \leq t < 0\), since \(|t| = -t\), the function becomes \( f(t) = 2 + t \). On the other hand, for the interval \(0 \leq t \leq 3\), it changes to \( f(t) = 2 - t\) because \( |t| = t\). Switching between these segments, depending on the interval, gives us a piecewise form. This helps us address each section of the domain separately when analyzing the behavior of the function.

Critical Points

Critical points are values of the variable in the domain where the function's derivative changes its sign, or where the derivative does not exist. They indicate potential locations of maxima or minima. For the function \(f(t) = 2 - |t|\), since it is piecewise linear, its derivative is constant on each segment, which simplifies the process. Thus, instead of solely depending on derivatives, critical points are usually found at endpoints or points where the expression changes. In this example, evaluating function behavior at \(t = -1\), \(t = 0\), and \(t = 3\) is essential. Here, \(t = 0\) is a special point that emerges as a critical point due to its nature of being at the junction of piecewise definitions.

Graphing Functions

Graphing a function helps you visually interpret its behavior across its domain. For the function \(f(t) = 2 - |t|\), two linear pieces are graphically represented. These are:

• From \(-1 \leq t < 0\), the piece \(f(t) = 2 + t\) forms an upward-sloping line.
• From \(0 \leq t \leq 3\), the piece \(f(t) = 2 - t\) forms a downward-sloping line.

By plotting these segments, you can see how they meet at \(t = 0\) and how the slopes differ. The graph will also clearly highlight the absolute maximum and minimum values. The key points, or turning points, are emphasized, marking where the absolute extrema occur.

Absolute Value Function

The absolute value function, denoted \(|x|\), outputs the non-negative value of \(x\). This means \(|x| = x\) when \(x \geq 0\) and \(|x| = -x\) when \(x < 0\). Because of this two-part nature, the absolute value function is inherently piecewise. In the problem, \(|t|\) is a central component of the function \(f(t) = 2 - |t|\). Understanding the absolute value is crucial, as it transitions the function from one linear expression to another at the critical point \(t = 0\). This insight into how absolute value functions behave aids in predicting points of change such as bends or cusps on the graph.