Problem 1 - Calculus for Biology and Medicine

Question

\mathrm{\\{} I n ~ P r o b l e m s ~ \(1-8\), 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.) $$ f(x)=2 x, 0 \leq x \leq 1 $$

Step-by-Step Solution

Answer

Global minimum: \( f(0) = 0 \); Global maximum: \( f(1) = 2 \).

Step 1: Understand the Problem

We are asked to find the global extrema for function \( f(x) = 2x \) on the interval \( 0 \leq x \leq 1 \). The function is continuous and satisfies the conditions of the extreme-value theorem.

Step 2: Applying the Extreme-Value Theorem

We will first check the endpoints of the interval since, by the extreme-value theorem, a continuous function on a closed interval will have both a global maximum and minimum within that interval. For this problem, the endpoints are \( x = 0 \) and \( x = 1 \).

Step 3: Evaluate at the Endpoints

Calculate \( f(x) \) at the endpoints: - For \( x = 0 \), \( f(0) = 2 \times 0 = 0 \). - For \( x = 1 \), \( f(1) = 2 \times 1 = 2 \). These calculations determine the outputs of the function at the boundary of our interval.

Step 4: Determine Global Extrema

Based on our evaluations, we have two points: - \( f(0) = 0 \) which is a candidate for the global minimum. - \( f(1) = 2 \) which is a candidate for the global maximum.

Step 5: Confirm Using a Graph

Graph the function \( f(x) = 2x \) over the interval \( 0 \leq x \leq 1 \). Observing the graph, we see that it is an increasing linear function. Thus, - The point \( (0, 0) \) is indeed the global minimum. - The point \( (1, 2) \) is the global maximum.

Key Concepts

Global Extrema

The term 'global extrema' refers to the highest and lowest values that a function reaches on a given interval. These maxima and minima are called the global maximum and global minimum, respectively. In simpler terms, global extrema are the peaks and valleys a function hits over its entire domain.

To find the global extrema of a function, you need to check the entire interval over which the function is defined. Typically, this involves evaluating the function at the endpoints of a closed interval as well as at any critical points within the interval. Critical points are where the derivative of the function, if it exists, is zero or undefined.

In the context of the problem, since the function \( f(x) = 2x \) on the interval \( 0 \leq x \leq 1 \) is linear, the maximum and minimum occur precisely at the endpoints. Evaluating the function at these endpoints gives the global minimum at \( x = 0 \) and the global maximum at \( x = 1 \).

Continuous Function

A continuous function is one that you can draw without lifting your pen from the paper. This means the function has no breaks, jumps, or holes within its domain. Mathematically, a function \( f(x) \) is continuous at a point \( x = c \) if three conditions are met:

• \( f(c) \) is defined.
• The limit of \( f(x) \) as \( x \) approaches \( c \) exists.
• The limit of \( f(x) \) as \( x \) approaches \( c \) is equal to \( f(c) \).

Continuous functions are especially important in calculus because they follow predictable patterns, making theorems about maxima, minima, and integration more applicable.

The function \( f(x) = 2x \) is a continuous function because it’s a simple linear function with an unbroken straight line. This continuity over the interval \( 0 \leq x \leq 1 \) means we can confidently apply the extreme-value theorem to find the global extrema.

Closed Interval

A closed interval is a set of numbers that includes all the numbers between two endpoints and also includes the endpoints themselves. It's typically denoted as \([a, b]\), meaning every number \( x \) satisfies \( a \leq x \leq b \).

This is an essential concept in calculus because many theorems, such as the extreme-value theorem, specifically require functions to be evaluated over closed intervals. If the endpoint values weren't included, you might miss the actual maximum or minimum values of the function.

In the given exercise for the function \( f(x) = 2x \), the closed interval is \([0, 1]\). This means the function is examined from \( x = 0 \) to \( x = 1 \) inclusive, thus including these boundary values in consideration for determining global maxima and minima.