Problem 60 - Thomas Calculus

Question

Let \(f\) be a function defined on an interval \([a, b] .\) What conditions could you place on \(f\) to guarantee that $$\min f^{\prime} \leq \frac{f(b)-f(a)}{b-a} \leq \max f^{\prime}$$ where \(\min f^{\prime}\) and \(\max f^{\prime}\) refer to the minimum and maximum values of \(f^{\prime}\) on \([a, b] ?\) Give reasons for your answers.

Step-by-Step Solution

Answer

Continuity on \([a, b]\) and differentiability on \((a, b)\) guarantee it.

Step 1: Understanding the Problem

We need to find conditions on the function \( f \) such that the average rate of change, \( \frac{f(b)-f(a)}{b-a} \), lies between the minimum and maximum values of the derivative \( f' \) on the interval \([a, b]\). This relates to the behavior of derivatives within this interval.

Step 2: Recall the Mean Value Theorem

The Mean Value Theorem (MVT) states that if function \( f \) is continuous on \([a, b]\) and differentiable on \((a, b)\), then there exists at least one point \( c \in (a, b) \) where \( f'(c) = \frac{f(b) - f(a)}{b-a} \). This theorem is key to understanding the condition for the average rate of change.

Step 3: Condition of Continuity and Differentiability

For the Mean Value Theorem to apply, function \( f \) must be continuous on the closed interval \([a, b]\) and differentiable on the open interval \((a, b)\). This ensures that there is at least one point \( c \) where the instantaneous rate of change \( f'(c) \) is equal to the average rate of change \( \frac{f(b)-f(a)}{b-a} \).

Step 4: Defining Minimum and Maximum Derivative Values

The minimum and maximum of \( f' \) on \([a, b]\), denoted as \( \min f' \) and \( \max f' \), must encompass the value at \( f'(c) \) for some \( c \in (a, b) \). Therefore, the values of \( f' \) must be bounded and have a defined range within \( [a, b] \). This guarantees that \( \min f' \leq f'(c) \leq \max f' \).

Step 5: Conclusion of Conditions

The conditions required are that \( f \) is continuous on \([a, b]\) and differentiable on \((a, b)\). These allow the application of the Mean Value Theorem, ensuring that the average rate of change is within the range of the derivative values as described.

Key Concepts

Continuity

Continuity is an essential characteristic of a function that ensures it behaves predictably. For a function to be continuous on an interval \([a, b]\), it must not have any breaks, jumps, or holes in its graph. This means you can draw the entire curve of the function from a to b without lifting your pencil off the paper.

Mathematically, continuity at a point \(c\) within an interval means:

• \(\lim_{{x \to c^{-}}} f(x) = \lim_{{x \to c^{+}}} f(x) = f(c)\)
• The left and right-hand limits at \(c\) must exist and be equal to \(f(c)\).

Ensuring continuity on \([a, b]\) is crucial for applying the Mean Value Theorem (MVT). Without continuity, there might be obstacles preventing the average rate of change from existing within the interval. When a function is continuous, it means the inputs are smoothly connected, and the MVT can be effectively used to find a point where the instantaneous rate of change equals the average rate of change.

Differentiability

A function's differentiability on an interval \((a, b)\) indicates that it has a defined derivative at every point within this interval. Differentiability implies that the function has a tangent line at each point, and the graph can be tracked smoothly without sharp corners or cusps.

For a function \(f\) to be differentiable:

• \(f'\) must exist for every point in \(a, b\).
• There should be no sharp turns or vertical tangents in the graph.

Differentiability is a stronger condition than continuity; every differentiable function is continuous, but not every continuous function is differentiable. Differentiability over \((a, b)\) ensures that the Mean Value Theorem can be applied readily by guaranteeing that the function behaves "nicely" across the interval. This ensures that there is at least one point \(c\) where the tangent to the function aligns with the average rate of change across the interval.

Average Rate of Change

The average rate of change of a function \ f \ over an interval \ [a, b] \ is a measure of how much the function's value changes, on average, as you move between two points in that interval. It is calculated as the difference in the function's values at the endpoints divided by the length of the interval:

\[ \frac{f(b) - f(a)}{b-a} \]

This formula gives a numerical value representing the slope of the secant line connecting two points \(a, f(a)\) and \(b, f(b)\) on the function's graph. It provides a broad overview of the function's behavior over a specified range rather than instantaneous value changes.

The average rate of change is crucial in various applications, such as understanding trends, making predictions, or even in finance for measuring performance over time. In the context of calculus, particularly regarding the Mean Value Theorem, it establishes a benchmark against which the instantaneous rate of changes at various points in the interval can be compared.

Instantaneous Rate of Change

The instantaneous rate of change of a function at a specific point measures how fast the function's value is changing at that point. This is equivalent to the derivative of the function at that point, denoted as \(f'(x)\).

The instantaneous rate of change is akin to finding the slope of the tangent line to the function at a given x-value. Unlike the average rate of change, which covers an interval, the instantaneous rate of change is localized to one specific point. This exactness provides detailed insight into the function's momentary behavior.

In physics, this concept is used to calculate things like velocity (the rate of change of position) or acceleration (the rate of change of velocity). In the context of the Mean Value Theorem, finding a point \(c\) where the instantaneous rate of change \(f'(c)\) is equal to the average rate of change on \([a, b]\) validates where the curve "matches" the secant line slope, painting a precise relationship between overall trends and specific moments.