Problem 61 - Thomas Calculus

Question

Suppose that \(f^{\prime}(x) \leq 1\) for \(1 \leq x \leq 4 .\) Show that \(f(4)-\) \(f(1) \leq 3 .\)

Step-by-Step Solution

Answer

The inequality is shown by applying the Mean Value Theorem: \(f(4)-f(1) \leq 3\).

Step 1: Understand the Given Information

We are given that the derivative of the function, \(f'(x)\), is less than or equal to 1 for all \(x\) in the interval \([1, 4]\). This implies that the rate of change of the function \(f(x)\) does not exceed 1 at any point within this interval.

Step 2: Interpret the Derivative Information

The derivative \(f'(x)\) being \(\leq 1\) means that the function \(f(x)\) is non-decreasing at a rate not faster than 1 unit change in \(f\) for 1 unit change in \(x\). This forms the fundamental basis for the Mean Value Theorem application.

Step 3: Apply the Mean Value Theorem

The Mean Value Theorem states that there exists at least one \(c\) in \((1, 4)\) such that:\[f'(c) = \frac{f(4) - f(1)}{4 - 1}\]Given \(f'(c) \leq 1\), substitute to get:\[\frac{f(4) - f(1)}{3} \leq 1\]

Step 4: Solve the Inequality

Multiply both sides of the inequality \(\frac{f(4) - f(1)}{3} \leq 1\) by 3 to isolate the expression for \(f(4) - f(1)\):\[f(4) - f(1) \leq 3\]This proves that the change in the function values over the interval \([1, 4]\) is at most 3.

Key Concepts

Derivative

The concept of a derivative is fundamental in calculus and represents the idea of instantaneous rate of change. In the context of the function provided, the derivative is expressed as \( f'(x) \), which tells us how the function \( f(x) \) changes at any point \( x \). When we say \( f'(x) \leq 1 \) within an interval, it implies that for every unit increase in \( x \), the change in \( f(x) \) is at most 1 unit. This is crucial for understanding the function's behavior over any segment.

• The derivative essentially measures the slope of the tangent line to the function at any given point.
• It provides insights into whether the function is increasing or decreasing within a particular range.
• If the derivative remains constant over an interval, the graph of the function is a straight line within that interval.

Rate of Change

The rate of change is closely related to the derivative and is a measure of how one quantity changes with respect to another. In this exercise, the rate of change is defined by the derivative \( f'(x) \), which is not greater than 1.This means:

• For every 1 unit increase in \( x \), \( f(x) \) will increase by no more than 1 unit.
• The function changes smoothly without sudden spikes or drops.
• It allows us to predict and understand the behavior of the function over an interval by knowing that the change is bounded.

This understanding fits into the larger framework of calculus and helps in solving inequalities.

Inequality

Inequalities are mathematical expressions used to compare two values. In the context of this problem, we worked with the inequality \( \frac{f(4) - f(1)}{3} \leq 1 \).Inequalities help in setting up constraints:

• They tell us the range within which a particular expression or function's value lies.
• For \( f(x) \), the inequality ensures that \( f(4) - f(1) \leq 3 \).
• This gives us a ceiling for the function's growth over the specified interval, allowing bounded estimation of the function values from start to end of the interval.

Understanding how to manipulate inequalities is essential for finding limits or bounds within calculus and various applications.

Non-decreasing Function

A non-decreasing function is a function where, as \( x \) increases, \( f(x) \) does not decrease. In the provided interval \([1, 4]\), knowing the derivative \( f'(x) \leq 1 \) doesn't just imply it is non-decreasing but also gives the rate of non-decrease.This characteristic of the function is significant because:

• It assures us that as \( x \) moves from 1 to 4, \( f(x) \) will not fall, ensuring at least a constant or increasing trend.
• Combined with the derivative information, it informs us about the maximum possible increase over the interval.
• Non-decreasing functions are typically used in real-world applications where a consistent or growing rate is expected, such as in population growth, cumulative data patterns, etc.

The concept of a non-decreasing function, especially with a bound like \( f'(x) \), helps set expectations and predictions about the function's behavior over a chosen interval.