Problem 63 - Thomas Calculus in SI Units

Question

Show that \(|\cos x-1| \leq|x|\) for all \(x\) -values.

Step-by-Step Solution

Answer

The inequality \(|\cos x-1| \leq|x|\) holds for all \(x\).

Step 1: Understand the Function Behavior

We start by understanding the behavior of the function \(|\cos x - 1|\). The cosine function, \( \cos x \), is always between \(-1\) and \(1\) for all real \(x\). So \( \cos x - 1 \) ranges from \(-2\) to \(0\). Thus, \(|\cos x - 1|\) ranges from \(0\) to \(2\).

Step 2: Use the Taylor Series Approximation

Consider the Taylor series expansion of \( \cos x \) around \(0\). The Taylor series is \( \cos x = 1 - \frac{x^2}{2} + \frac{x^4}{24} - \cdots \). Thus, \( \cos x - 1 \approx -\frac{x^2}{2} \) for small values of \(x\). This shows that \(|\cos x - 1| \approx \frac{x^2}{2} \leq |x|\) for small \(x\).

Step 3: Evaluate at Special Points

Check some specific values, such as \(x = 0\), where \( \cos(0) = 1 \) and hence \(|\cos(0)-1| = 0 \leq 0 = |0|\). For \(x = \pi\), \(\cos(\pi) = -1\), so \(|\cos(\pi) - 1| = 2\), but \(|\pi| > 2\). Thus inequality holds.

Step 4: Verify the Inequality using Derivative

The function \( f(x) = \cos x - 1 + x \) and \( g(x) = \cos x - 1 - x \) should both be non-negative for the inequality to hold. Compute the derivatives: \( f'(x) = -\sin x + 1 \) and \( g'(x) = -\sin x - 1 \). Analyze where these derivatives are zero or non-positive to confirm these functions do not change sign improperly, showing \(f(x)\) does not fall below zero and \(g(x)\) does not rise above zero.

Key Concepts

Trigonometric Functions

Trigonometric functions, including cosine (\( \cos x \)), play a critical role in analyzing periodic phenomena. The cosine function oscillates between \(-1\) and \(1\) for any real number \(x\). This range is what makes the cosine function predictable and highly useful in various applications like engineering and physics.

• When dealing with the expression \(\cos x - 1\), we see that it shifts the cosine graph down by 1 unit, resulting in a range from \(-2\) to \(0\).
• Thus, the absolute value expression \(|\cos x - 1|\) takes on values from 0 to 2, reflecting how far the result is from zero.

Understanding the motion and oscillation of trigonometric functions helps provide intuition into their behavior in inequalities and other mathematical problems.

Taylor Series

The Taylor series is a fundamental concept that assists in approximating complex functions using polynomials. It's like expressing the function as an infinite sum of terms made from its derivatives at a specific point.

• In the case of \(\cos x\) around zero, the Taylor series is \(\cos x = 1 - \frac{x^2}{2} + \frac{x^4}{24} - \cdots\)
• When considering \(\cos x - 1\), we approximate it using the first few terms: \(-\frac{x^2}{2}\) for small values of \(x\)

This approximation \(|\cos x - 1| \approx \frac{x^2}{2} \leq |x|\) is useful for small \(x\), providing a means to compare magnitudes easily.

• The series provides the insight that for sufficiently small \(x\), \(|\cos x - 1|\) is dominated by \(\frac{x^2}{2}\)

This makes it easier to establish inequalities involving trigonometric functions near specific points.

Derivative Analysis

Derivatives are powerful tools in calculus that help understand how functions change, indicating where they increase, decrease, or remain constant.

• In the exercise, we analyze the functions \( f(x) = \cos x - 1 + x \) and \( g(x) = \cos x - 1 - x \) to verify the inequality \(|\cos x - 1| \leq |x|\).
• The derivative \( f'(x) = -\sin x + 1 \) helps determine where \( f(x) \) increases or decreases.
• Similarly, \( g'(x) = -\sin x - 1 \) lets us analyze how \( g(x) \) behaves across its domain.

By exploring the sign and critical points of these derivatives, we can confirm that \( f(x) \) does not decrease past zero and \( g(x) \) does not increase beyond zero, ensuring that the inequality holds for all \(x\).

• In essence, derivative analysis provides a way to verify function behavior without having to rely entirely on function evaluations.