Problem 5 - Calculus for Biology and Medicine

Question

\mathrm{\\{} I n ~ P r o b l e m s ~ , each function is continuous and defined on a closed interval. It therefore satisfies the assumptions of the extreme- value theorem. With the help of a graphing calculator or spreadsheet, graph each function and locate its global extrema. (Note that a function may have more than one global minimum or maximum point.) $$ f(x)=|x|,-1 \leq x \leq 1 $$

Step-by-Step Solution

Answer

The global minimum is 0 at \(x=0\), and the global maxima are 1 at \(x=-1\) and \(x=1\).

Step 1: Understanding the Function

The function given is \( f(x) = |x| \), which is the absolute value function. It's defined on the interval \(-1 \leq x \leq 1\). We know the absolute value function has a 'V' shape with its vertex at the origin (0,0).

Step 2: Determine Critical Points and Endpoints

To find global extrema, we need to consider critical points and endpoints of the interval. The critical point here is where the slope changes, which occurs at \(x = 0\). Additional critical points are at the endpoints \(x = -1\) and \(x = 1\).

Step 3: Evaluate the Function at Critical Points

Evaluate \(f(x)\) at the critical points and endpoints: - \(f(-1) = |-1| = 1\).- \(f(0) = |0| = 0\).- \(f(1) = |1| = 1\).

Step 4: Graphing and Visualization

Graph \(f(x) = |x|\) using a graphing tool. The graph should show a V-shape with the vertex at (0,0) and endpoints at (1,1) and (-1,1). The x-axis runs from -1 to 1.

Step 5: Identify Global Extrema

Identify the highest and lowest points on the graph: - The global minimum is \(f(0) = 0\) at \(x=0\).- The global maxima occur at \(f(-1) = 1\) and \(f(1) = 1\). These maxima happen at the endpoints of the interval.

Key Concepts

Absolute Value Function

The absolute value function, denoted as \( f(x) = |x| \), is among the fundamental building blocks in mathematics. It is especially important in real analysis and calculus. This function maps every real number to its non-negative value.
For any real number \( x \), the absolute value function is defined as:

• \( |x| = x \), if \( x \geq 0 \)
• \( |x| = -x \), if \( x < 0 \)

The graph of the absolute value function has a distinctive 'V' shape, with its point or vertex located at the origin, (0,0). This shape reflects the way the function treats positive and negative numbers symmetrically, always resulting in a non-negative value.
Understanding this function is essential when analyzing trends, data, or solving equations that involve non-negative constraints.

Global Extrema

The concept of global extrema is crucial in calculus and optimization. Global extrema refer to the highest or lowest values that a function can attain over a given interval. For the function \( f(x) = |x| \) on the interval \([-1, 1]\), searching for global extrema involves:

• Identifying critical points where the derivative is zero or undefined.
• Evaluating function values at these critical points as well as at the endpoints of the interval.

In terms of the absolute value function, the critical point is at \( x = 0 \), where the function reaches its lowest value or global minimum, which is 0.
The endpoints \( x = -1 \) and \( x = 1 \) both yield the global maximum value of 1, as both points result in the absolute value being 1.
This symmetric property of the absolute value function illustrates its predictable behavior with regard to extrema, simplifying the analysis of functions over specified intervals.

Graphing Calculator

A graphing calculator is an invaluable tool for visualizing functions and optimizing problem-solving in mathematics. When dealing with the absolute value function \( f(x) = |x| \), a graphing calculator can quickly illustrate its characteristic 'V' shape.
Here's how to make the most of a graphing calculator:

• Enter the function into the calculator, ensuring the correct interval is specified.
• Adjust the viewing window to include the entire interval of interest, here \([-1, 1]\).
• Observe the graph to identify key features such as the vertex and endpoints.
• Use the calculator's trace or analysis tools to pinpoint exact values of critical points and extrema.

These tools allow students to directly "see" the mathematical behavior of functions. They can confirm their algebraic calculations and develop a deeper understanding through visual representation.
Particularly, for the absolute value function, recognizing the global minimum at the vertex and the global maxima at the endpoints can be visually and academically satisfying.