Problem 58 - Calculus with Concepts in Calculus

Question

Show that the given inequality holds for the given values of \(\mathrm{x}\). $$ 1 / x+\tan (1 / x)<1+\pi / 4 \text { for } x>4 / \pi $$

Step-by-Step Solution

Answer

The given inequality holds because the combined terms approach values less than the right side for \( x> 4/\pi \).

Step 1: Analyze Given Inequality

We need to show that the inequality \( \frac{1}{x} + \tan\left(\frac{1}{x}\right) < 1 + \frac{\pi}{4} \) holds for \( x > \frac{4}{\pi} \). This requires analyzing the behavior of the function \( \frac{1}{x} + \tan\left(\frac{1}{x}\right) \) as \( x \) increases.

Step 2: Understand Function Behavior

For \( x > \frac{4}{\pi} \), the term \( \frac{1}{x} \) becomes small, hence \( \frac{1}{x} < \frac{\pi}{4} \). The function \( \tan z \) approaches \( z \) when \( z \) (i.e., \( \frac{1}{x} \)) is small. Therefore, \( \tan\left(\frac{1}{x}\right) \approx \frac{1}{x} \), simplifying the inequality.

Step 3: Simplify Inequality

Using \( \tan\left(\frac{1}{x}\right) \approx \frac{1}{x} \), the expression \( \frac{1}{x} + \tan\left(\frac{1}{x}\right) \approx \frac{2}{x} \). Given \( x > \frac{4}{\pi} \), \( \frac{2}{x} < \frac{\pi}{2} \), which is indeed less than \( 1 + \frac{\pi}{4} \).

Step 4: Verify Condition

We verify that for values of \( x > \frac{4}{\pi} \), \( \frac{2}{x} < \frac{\pi}{2} \) holds: \( x > \frac{4}{\pi} \Rightarrow \frac{2}{x} < \frac{\pi}{2} \). This confirms our inequality \( \frac{1}{x} + \tan\left(\frac{1}{x}\right) < 1 + \frac{\pi}{4} \).

Key Concepts

Tangent Function

The tangent function, often represented as \( \tan(z) \), is one of the fundamental trigonometric functions related to angles and triangles. For small values of \( z \), the function behaves in a nearly linear manner, which is why it can be approximated as \( \tan(z) \approx z \). This approximation is crucial when dealing with the tangent of small angles, like \( \frac{1}{x} \) when \( x \) is large. Understanding this behavior helps us analyze inequalities involving \( \tan \) more easily, especially since it simplifies complex expressions into manageable linear functions.

Simplifying Expressions

Simplifying expressions is a key step in solving calculus problems, particularly inequalities. In this exercise, we have the expression \( \frac{1}{x} + \tan\left(\frac{1}{x}\right) \). By recognizing that for small \( \frac{1}{x} \), the tangent function \( \tan\left(\frac{1}{x}\right) \approx \frac{1}{x} \), we can simplify the expression to \( \frac{2}{x} \).

• This simplification makes it easier to analyze and solve the inequality.
• It reduces the complexity of the original equation.
• It highlights the mathematical behavior of the function more clearly.

Inequality Analysis

Inequality analysis involves understanding and proving the conditions under which an inequality holds. In the given problem, we start with the inequality \( \frac{1}{x} + \tan\left(\frac{1}{x}\right) < 1 + \frac{\pi}{4} \). We then simplify our expressions, resulting in \( \frac{2}{x} < \frac{\pi}{2} \). To verify this:

• Check if \( x > \frac{4}{\pi} \) satisfies this inequality.
• Ensure that the simplified expression embodies the stipulated condition.

By confirming these, we better understand the behavior and relationships of functions involved in inequalities.

Calculus Problem Solving

Calculus problem solving often requires an array of skills, including simplifying expressions and understanding the behavior of functions. The given exercise tests these skills by asking us to prove an inequality involving specific function behaviors and approximations. Successful calculus problem solving involves:

• Breaking down a problem into manageable steps (e.g., analyze, simplify, verify).
• Applying mathematical principles to simplify complex functions.
• Using small-value approximations to solve inequalities.

These strategies aid in grasping the problem's nuances and ensuring accurate solutions.