Problem 35 - 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. $$f(t)=2-|t|, \quad-1 \leq t \leq 3$$

Step-by-Step Solution

Answer

Absolute maximum is 2 at \(t=0\); absolute minimum is -1 at \(t=3\).

Step 1: Identify Critical Points and Endpoints

First, we need to identify any critical points in the interval \(-1 \leq t \leq 3\). The function \(f(t) = 2 - |t|\) can have critical points where its derivative is zero or undefined. However, the function changes its form at \(t = 0\), where it is non-differentiable. Additionally, evaluate the function at the endpoints and critical points.

Step 2: Evaluate Function at Endpoints

Calculate \(f(t)\) at the endpoints of the interval: - At \(t = -1\), \(f(-1) = 2 - |-1| = 1\).- At \(t = 3\), \(f(3) = 2 - |3| = -1\).

Step 3: Evaluate Function at Critical Point

Evaluate the function at the critical point \(t = 0\):- \(f(0) = 2 - |0| = 2\).

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

Graph the function \(f(t) = 2 - |t|\) over the interval \(-1 \leq t \leq 3\). Mark the points \((-1, 1)\), \((0, 2)\), and \((3, -1)\) on the graph where the extrema occur. The graph is a V-shaped line with the vertex at \(t = 0\).

Key Concepts

Critical Points

In order to find the absolute extrema, it's crucial to identify the critical points of the function. Critical points occur where a function's derivative is either zero or undefined. In this case, we're dealing with the function \( f(t) = 2 - |t| \). This function is non-differentiable where the absolute value expression within it changes, specifically at \( t = 0 \).
When a function includes an absolute value like this one does, it can change its "direction" at the point where the inside of the absolute value (\( t \)) becomes zero.
It's important to recognize these as potential critical points because the slope of the graph will change, indicating a possible maximum or minimum.

• At \( t = 0 \), the slope changes direction and it's essential to evaluate the function at this point for extrema.

Understanding these critical points allows you to determine where the function might achieve its maximum or minimum value within the given interval.

Endpoints

Endpoints are the boundaries of the interval you are examining. In this exercise, our interval is \( -1 \leq t \leq 3 \).
To determine the absolute extrema of \( f(t) = 2 - |t| \) on this interval, you also need to evaluate the function at these endpoints.

• Calculate the function’s value at \( t = -1 \): \( f(-1) = 2 - |-1| = 1 \).
• Calculate the function’s value at \( t = 3 \): \( f(3) = 2 - |3| = -1 \).

This evaluation at the endpoints is essential because, in some cases, absolute maximum or minimum values can occur right at these boundaries. They ensure that every potential point for an extrema is covered, including those at the very edges of the interval.

Graphing Functions

Once you've identified your critical points and evaluated your endpoints, it's time to visually interpret the data through graphing.
For the function \( f(t) = 2 - |t| \), the graph over the interval \( -1 \leq t \leq 3 \) takes the form of a V-shaped line. Here's a quick step-by-step of how this can be done:

• The vertex of this V occurs at the critical point \( t = 0 \), and here \( f(0) = 2 \).
• Mark the point \( (0, 2) \), indicating the graph reaches its absolute maximum at this location.
• Plot points at the calculated endpoints: \( (-1, 1) \) and \( (3, -1) \), which are the absolute minimum points as well.

This graphing provides a clear visual aid to see where the functions trend and confirms the locations of the extrema. Observing the function this way helps reinforce your analytical findings with a visual perspective, making it easier to locate key features and understand the behavior of the function.