Problem 61 - Thomas Calculus in SI Units

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

By integrating \( f'(x) \leq 1 \) from 1 to 4, we show that \( f(4)-f(1) \leq 3 \).

Step 1: Understand the Given Information

We are given that the derivative of a function, denoted by \( f'(x) \), is less than or equal to 1 for every \( x \) in the interval \([1, 4]\). We need to use this information to prove that \( f(4) - f(1) \leq 3 \).

Step 2: Integrate the Inequality

Since \( f'(x) \leq 1 \) for \( x \in [1, 4] \), integrate both sides from 1 to 4: \[ \int_{1}^{4} f'(x) \, dx \leq \int_{1}^{4} 1 \, dx. \]

Step 3: Apply the Fundamental Theorem of Calculus

By the Fundamental Theorem of Calculus, \( \int_{1}^{4} f'(x) \, dx = f(4) - f(1) \). Therefore, substituting this into the inequality from Step 2 gives us \( f(4) - f(1) \leq \int_{1}^{4} 1 \, dx \).

Step 4: Evaluate the Right Part of Inequality

Calculate \( \int_{1}^{4} 1 \, dx \). This represents the area under the graph of \( y=1 \) from \( x=1 \) to \( x=4 \), which is the width of the interval (4 - 1) times the height 1, therefore \( 4 - 1 = 3 \).

Step 5: Conclude the Solution

Substituting the value calculated in Step 4 back into the inequality derived from Step 3, we have \( f(4) - f(1) \leq 3 \), which is what we needed to prove.

Key Concepts

Differentiation

Differentiation is a core concept in calculus that deals with finding the rate at which a function changes at any given point. It involves determining the derivative of a function, which represents the slope of the tangent line to the function's graph at a particular point. In this exercise, the function's derivative is given as \( f'(x) \leq 1 \) over a specified interval \([1, 4]\). This tells us that the rate of change of the function \( f(x) \) does not exceed 1 anywhere in this interval.

• The derivative, \( f'(x) \), tells us how steep the function is. If \( f'(x) = 1 \), the function increases linearly.
• If \( f'(x) < 1 \), the function increases, but more slowly than a linear function with slope 1.
• If \( f'(x) = 0 \), the function is constant at that point within the interval.

Understanding these aspects of differentiation helps us apply them when proving properties like the one in the exercise, ensuring the function change over the interval does not exceed 3.

Integration

Integration is another fundamental concept in calculus, which essentially performs the reverse operation of differentiation. It involves finding the integral of a function, which can be understood as the accumulation of quantities, such as area under a curve. In this problem, integration is applied to both sides of the inequality \( f'(x) \leq 1 \).
To solve this exercise, we used indefinite integral properties:

• The left side, \( \int_{1}^{4} f'(x) \, dx \), results in \( f(4) - f(1) \), thanks to the Fundamental Theorem of Calculus.
• The right side, \( \int_{1}^{4} 1 \, dx \), evaluates to 3, as it represents the area of a rectangle with height 1 and width \(4 - 1\).

Through integration, we verify that the total change in the function’s value over the interval \([1, 4]\) is indeed less than or equal to 3, as required.

Inequalities in Calculus

Inequalities are an essential aspect of calculus, providing constraints and relationships between functions and their rates of change. The exercise involves understanding the inequality \( f'(x) \leq 1 \) and deriving a connection with another inequality, \( f(4) - f(1) \leq 3 \). By applying calculus concepts, we built an argument based on these inequalities.

• The inequality \( f'(x) \leq 1 \) tells us the steepest the function can increase is by a slope of 1.
• By integrating both sides, we equate this rate to quantify the total change between two endpoints.
• This process transforms a pointwise constraint into an overall constraint for the interval.

Understanding inequalities through calculus allows us to predict and control the behavior of functions, such as ensuring rates or changes do not exceed given limits. This is crucial for real-world applications where calculus models dynamic systems.