Problem 15 - Calculus for Biology and Medicine

Question

In Problems , use a graphing calculator or spreadsheet to plot the function and determine all local and global extrema. $$ f(x)=x^{2}-2, x \in[-1,1] $$

Step-by-Step Solution

Answer

Global minimum is at \(x = 0\) with \(f(x) = -2\). Global maxima are at \(x = -1\) and \(x = 1\) with \(f(x) = -1\).

Step 1: Identify the function and interval

The given function is \( f(x) = x^2 - 2 \) and the interval is \( x \in [-1, 1] \). We will find the extrema within this interval.

Step 2: Find the derivative

To find the critical points where extrema might occur, first compute the derivative of the function. The derivative is \( f'(x) = \frac{d}{dx}(x^2 - 2) = 2x \).

Step 3: Set the derivative to zero

Find the critical points by setting the derivative equal to zero: \( 2x = 0 \). This simplifies to \( x = 0 \).

Step 4: Evaluate the function at critical points and endpoints

We evaluate \( f(x) \) at the critical point and the interval endpoints: - \( f(-1) = (-1)^2 - 2 = -1 \) - \( f(0) = (0)^2 - 2 = -2 \) - \( f(1) = (1)^2 - 2 = -1 \).

Step 5: Determine local and global extrema

Based on our evaluations, the smallest value is \(-2\) at \(x = 0\), making it the global minimum within the interval. Both \(-1\) at \(x = -1\) and \(x = 1\) are the global maxima. There is no local maximum different from the endpoints.

Key Concepts

Critical Points

Critical points are points on the graph of a function where the derivative is zero or undefined. These points are crucial because they often indicate where the function changes direction, potentially leading to local maxima or minima. In our exercise, we considered the function \( f(x) = x^2 - 2 \). To find its critical points, we computed the derivative \( f'(x) = 2x \). By setting the derivative equal to zero, \( 2x = 0 \), we found \( x = 0 \) as a critical point.

• Critical points can denote the top of a hill or the bottom of a valley on a graph.
• They are important in determining the extremities of a function within a specific interval.
• The interval in our task was \([-1, 1]\), focusing our search to find extrema only in this range.

Understanding critical points helps predict where a function might achieve its highest or lowest values within the given constraints.

Derivative

The derivative of a function provides us with the rate at which the function's value changes. In simple terms, it's like a speedometer for a function, showing how quickly the function is increasing or decreasing at any point. For our function \( f(x) = x^2 - 2 \), the derivative is \( f'(x) = 2x \). This tells us how fast the function is climbing or dropping at any \( x \) value.

• When the derivative is zero, it indicates a halt in change — a potential maximum or minimum.
• For our function, \( f'(x) = 0 \) at \( x = 0 \), marking it as a critical point and possible extrema.
• Derivatives are also used to determine concavity, helping understand the graph's shape and direction.

Thus, the derivative is a powerful tool in analyzing and predicting a function's behavior, crucial for finding extremas.

Function Evaluation

Function evaluation involves substituting different values of \( x \) into the function to get resulting \( f(x) \) values. This step is essential for comparing outputs at critical points and interval endpoints to determine extrema. We evaluated \( f(x) = x^2 - 2 \) at \( x = -1, 0, \) and \( 1 \). The results were:

• \( f(-1) = (-1)^2 - 2 = -1 \)
• \( f(0) = (0)^2 - 2 = -2 \)
• \( f(1) = (1)^2 - 2 = -1 \)

This evaluation shows that \( x = 0 \) provides the smallest, thus minimum, function value within the interval. Understanding how to evaluate a function ensures we can explore its behavior across different points, identify maximum and minimum values, and outline the complete range of outputs.

Graphing Calculator

A graphing calculator is a valuable tool in visualizing functions and their behavior on a graph. Using this technology, students can plot functions to observe maximum and minimum points visually, alongside critical points within a chosen interval. When investigating \( f(x) = x^2 - 2 \), the graphing calculator helps:

• Highlight critical points and observe their graphical significance, such as peaks or troughs.
• Verify analytical solutions by displaying the curve and ensuring calculations match the visual graph.
• Experiment with changes in domains and parameters, offering interactive learning and a deeper understanding.

The use of a graphing calculator bridges algebraic solutions with graphic interpretations making complex concepts more tangible and easier to grasp for students.