Problem 43 - Calculus Early Transcendentals

Question

If \(f\) is increasing on an interval \([0, b),\) then it follows from Definition 4.1 .1 that \(f(0)< f(x)\) for each \(x\) in the interval \((0, b) .\) Use this result in these exercises. Use a graphing utility to make a conjecture about the relative sizes of \(x\) and \(\sin x\) for \(x \geq 0,\) and prove your conjecture.

Step-by-Step Solution

Answer

For all \(x \geq 0\), \(x \geq \sin x\).

Step 1: Understanding the Problem

We need to compare the values of \(x\) and \(\sin x\) for \(x \geq 0\). The problem asks us to make a conjecture using a graphing utility and then to prove the conjecture.

Step 2: Graphing to Make a Conjecture

Using a graphing utility, plot the functions \(y = x\) and \(y = \sin x\) for \(x \geq 0\). Notice that \(\sin x\) starts at 0 and oscillates between -1 and 1, while \(x\) is a straight line passing through the origin. The conjecture is that \(x \geq \sin x\) for \(x \geq 0\).

Step 3: Formalizing the Conjecture

The conjecture based on graphing is \(x \geq \sin x\) for \(x \geq 0\). To explicitly state this, note that for small positive \(x\), \(\sin x\) is approximately \(x\), but as \(x\) grows, \(\sin x\) cannot exceed 1, establishing \(x \geq \sin x\).

Step 4: Proving the Conjecture

To prove \(x \geq \sin x\) for \(x \geq 0\), consider the derivative of \( \sin x - x \). The derivative is \( \cos x - 1 \), which is non-positive since \(\cos x \leq 1\). This derivative indicates \( \sin x - x \) is non-increasing. Evaluating at \(x = 0\), \( \sin 0 - 0 = 0\). So, \( \sin x \leq x\) for all \(x \geq 0\) as \( \sin x - x \leq 0\).

Step 5: Conclusion

Based on the graphing and the derivative test, the conjecture is true: \(x \geq \sin x\) for all \(x \geq 0\). This demonstrates the relative size based on the provided functions and derivative analysis.

Key Concepts

Increasing Functions

An increasing function is a function where the value of the function grows as the input value increases. If a function \( f \) is increasing over an interval \([a, b)\), then for any two points, \( x_1 \) and \( x_2 \) within this interval where \( x_1 < x_2 \), it follows that \( f(x_1) < f(x_2) \). This property is crucial for evaluating how functions behave across specific ranges.

In the context of this exercise, understanding that \( f(0) < f(x) \) for each \( x \) in \( (0, b) \) serves as a foundational rule. It underscores how, as \( x \) moves from zero and beyond, the function's output tends to increase if the function is said to be increasing. This basic principle helps us compare different functions over a specified range, such as \( x \) and \( \sin x \) in this problem.

Graphing Utility

Graphing utilities are powerful tools, whether online or handheld, that allow us to visualize mathematical functions. By plotting function graphs, we can easily observe behavior that might be difficult to discern numerically.

For this exercise, using a graphing utility helps make a visual conjecture about \(x\) and \(\sin x\) for \(x \geq 0\). When both functions \(y = x\) and \(y = \sin x\) are graphed:

• \(y = x\) is a straight line through the origin.
• \(y = \sin x\) starts at 0 and follows an oscillating pattern between -1 and 1.

From the graph, it becomes apparent that \(x\) retains larger or equal values as compared to \(\sin x\), except at certain intersections. This visual insight facilitates forming a conjecture about their relationship for \(x \geq 0\).

Derivative

A derivative represents the rate of change of a function as its input changes. It can show whether a function is increasing or decreasing at any given point. In this problem, the derivative is used to prove the conjecture that \( x \geq \sin x \) for \( x \geq 0 \). We start by considering the function \( f(x) = \sin x - x \). Its derivative, \( f'(x) = \cos x - 1 \), reveals crucial information.

Since \( \cos x \leq 1 \) for all \(x\), the derivative \( \cos x - 1 \leq 0\) implies \( f(x) \) is non-increasing as \(x\) increases. This means the function \( \sin x - x \) does not go upwards as \( x \) increases, indicating that \( \sin x \) is consistently less than or equal to \( x \). Therefore, we conclude that \( x \geq \sin x \) for all \( x \geq 0 \).

Conjecture Proof

In mathematics, a conjecture is an educated guess based on observation, requiring proof to confirm its validity. The conjecture in this exercise states that \( x \geq \sin x \) for all \( x \geq 0 \). Starting from a graphical observation and using calculus to prove it ensures no counterexamples exist. After forming the conjecture by visualizing the graphs, we employ mathematical proof to substantiate it.

The process involves using derivatives, specifically, analyzing the function \( f(x) = \sin x - x \) and proving \( x \geq \sin x \) by showing \( f(x) \) is non-increasing using its derivative. The approach is logical and relies on making connections between graphical patterns and analytical methods. Thus, successfully proving the conjecture secures our understanding of the functions' comparative behaviors over the specified interval.