Problem 7 - Calculus for the Life Sciences: A Modeling Approach Volume I

Question

Draw the graph of a continuous function \(F\) with domain [1,5] that has a non- negative derivative at every point between 1 and \(5,\) and for which 5 is a local minimum. The next three exercises demonstrate interesting function equivalents of the Axiom of Completeness 5.2 .1 from Section 5.2 .

Step-by-Step Solution

Answer

Drawn as a constant from [1, 5) and minimum at x = 5.

Step 1: Understand the Function Characteristics

The function \( F \) needs to be continuous over the interval \([1, 5]\). Additionally, \( F \) should have a non-negative derivative, meaning \( F'(x) \geq 0 \) for all \( x \) in the domain. This implies the function is non-decreasing over the interval.

Step 2: Draw the Function

Since 5 is a local minimum, the function should be decreasing up to \( x = 5 \). A simple function that meets these criteria is a constant function from \( x = 1 \) to \( x < 5 \), and then it can slightly dip or remain constant at \( x = 5 \). For example, have \( F(x) = 2 \) for \( x < 5 \) and \( F(x) = 1.5 \) at \( x = 5 \).

Key Concepts

Non-negative Derivative

The concept of a non-negative derivative is crucial when exploring the properties of a function on a given interval. A derivative, in simple terms, represents the rate at which a function is changing at any given point. If the derivative of a function is non-negative across an interval, it indicates that the function is either constant or increasing within that range. This is because a non-negative derivative, signified mathematically as \( F'(x) \geq 0 \), implies there is no downward slope on the function graph.

This idea can be visualized as having a graph that either stays flat or ascends as you move from left to right. If you imagine walking along this curve, you would either stay on the same level or climb upwards, but would never descend.

Key points to remember about non-negative derivatives:

• A non-negative derivative does not automatically mean an increasing function overall—constant sections might exist.
• A positive derivative indicates strictly increasing sections.
• A zero derivative indicates flat sections of the function, where it remains constant.

Understanding the behavior of derivatives provides insight into the overall shape and behavior of the function on its domain.

Local Minimum

A local minimum refers to a point where a function value is lower than adjacent points, creating a dip in the function. It's as if you are standing at a valley where your surroundings (nearby points) are at higher elevations than where you stand. This concept is central in analyzing and understanding functions, particularly in the context of differential calculus.

In the exercise, point 5 is marked as a local minimum. Hence, around this point, any nearby values of \( x \) show \( F(x) \) being higher. For a smooth continuous function, this often means the derivatives surrounding the local minimum exhibit particular behaviors.

Characteristics of a local minimum include:

• The derivative might be zero if this is a smooth curve (indicating a horizontal tangent at that point).
• The function values increase as you move away from the local minimum in either direction.

This understanding aids in drawing and identifying function behavior, ensuring graphs are correctly plotted according to stated conditions.

Axiom of Completeness

The Axiom of Completeness is a fundamental property in real analysis and calculus. It asserts that if any non-empty set of real numbers has an upper bound, then it has a least upper bound or supremum. In terms of functions, this principle means that every bounded, continuous function will have an extreme value—either a maximum or minimum—on a closed interval.

This is vital when considering functions like in the exercise, where the function is constrained to a domain (from 1 to 5). Due to continuity and boundedness of the function within this range, we can confidently assert the existence of minimum (and possibly maximum) values. This aligns with the conditions given in the exercise that identify 5 as a local minimum.

Why the Axiom of Completeness matters:

• Ensures every bounded sequence has limits, securing behavior prediction over intervals.
• Supports the Intermediate Value Theorem—important for discovering function roots within intervals.

This conceptual underpinning forms the basis of much theoretical and practical function investigation, ensuring we can effectively work with real, continuous functions.