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Chapter 5: Problem 39

Using Mean value theorem, show that \(|\cos a-\cos b| \leq|a-b|\).

Short Answer

Expert verified
We apply the Mean Value Theorem to the function \(f(x) = \cos x\) over the interval \([a, b]\), which states that there exists a point \(c \in (a, b)\) such that \(f'(c) = \frac{f(b) - f(a)}{b - a}\). Since \(f'(x) = -\sin x\), we have \(-\sin c = \frac{\cos b - \cos a}{b - a}\). Taking absolute values, we obtain \(|\sin c| \geq \left|\frac{\cos b - \cos a}{b - a}\right|\). Multiplying both sides by \(|b - a|\), we have \(|\sin c||b - a| \geq |\cos b - \cos a|\). Since \(|\sin c| \leq 1\), the final inequality is \(|\cos a - \cos b| \leq |a - b|\).

Step by step solution

01

The Mean Value Theorem states that if a function \(f(x)\) is continuous on the closed interval \([a, b]\) and differentiable on the open interval \((a, b)\), then there exists a point \(c\) in \((a, b)\) such that the derivative of the function at point \(c\) equals the average rate of change of the function over the interval \([a, b]\), i.e., \(f'(c) = \frac{f(b) - f(a)}{b - a}\). #Step 2: Verify conditions for the Mean Value Theorem with cosine function#

In our case, we are given the function \(f(x) = \cos x\). The cosine function is continuous and differentiable on its entire domain, which is \((-\infty, \infty)\). Therefore, the conditions to apply the Mean Value Theorem are satisfied for any interval \([a, b]\) with \(a < b\). #Step 3: Apply the Mean Value Theorem to the cosine function#
02

Since the conditions are met, we can apply the Mean Value Theorem. Let's apply it to the cosine function on the interval \([a, b]\). According to the theorem, there exists a point \(c \in (a, b)\) such that \(f'(c) = \frac{f(b) - f(a)}{b - a}\), where \(f(x) = \cos x\). The derivative of the cosine function is \(-\sin x\). Therefore, we have the following equality: \(-\sin c = \frac{\cos b - \cos a}{b - a}\). #Step 4: Use inequality to show the desired result#

We know that \(-1 \leq \sin c \leq 1\). Taking the absolute value of the equation from Step 3, we get: \(|-\sin c| = \left|\frac{\cos b - \cos a}{b - a}\right|\). Since \(|\sin c| \leq 1\), we can then write: \(|\sin c| \geq \left|\frac{\cos b - \cos a}{b - a}\right|\). Now, multiply both sides by the positive value \(|b - a|\): \(|\sin c||b - a| \geq |\cos b - \cos a|\). Since \(\sin c \leq 1\), we can replace \(|\sin c|\) with 1: \(|b - a| \geq |\cos b - \cos a|\). And this is our desired result: \( |\cos a - \cos b| \leq |a - b| \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Cosine Function
The cosine function, denoted as \(\cos x\), is one of the fundamental trigonometric functions, describing the ratio of the adjacent side to the hypotenuse in a right-angle triangle. It's an even function, meaning that \(\cos(-x) = \cos(x)\), and it has a period of \(2\pi\), which means the function repeats its values every \(2\pi\) radians.

One of the remarkable properties of the cosine function is its behavior on a graph 鈥� it creates a wave-like pattern oscillating between \(-1\) and \(1\). This reflects its range, ensuring that for any real number x, \(\cos x\) will fall within this interval. This characteristic plays a significant role when we examine the inequalities in trigonometry.

Real-world Applications

In real life, cosine functions are used to model periodic phenomena such as sound waves, light, and the movement of waves. Its predictability and the continuity make it a reliable mathematical tool for engineers and scientists.
Derivatives
In calculus, derivatives represent the rate at which a function is changing at any point on its curve. If you think of a function as a path covered over time, the derivative at any point would tell you the speed of an object moving along that path at a specific moment.

For the cosine function, the derivative is particularly interesting. Let's denote function f with \(f(x) = \cos x\). The derivative of this function, notated as \(f'(x)\), is \(f'(x) = -\sin x\). This means that at any point along the cosine curve, the rate of change is equal to the negative sine value at that point.

Practical Import

Understanding derivatives is vital for fields like physics, where it can represent acceleration, which is the rate of change of velocity. In economics, derivatives can indicate the change in cost or revenue in response to the change in some variable, like price or quantity.
Inequality in Trigonometry
Trigonometry is full of inequalities that allow mathematicians to understand and bound the values of trigonometric functions. These inequalities are invaluable in proving statements about trigonometric expressions without knowing the exact values.

As for the cosine function, we know that its values are confined to the range between \(-1\) and \(1\). This restriction leads to important inequalities like the one featured in the given exercise, where using the Mean Value Theorem helps us show that \( |\cos a - \cos b| \leq |a - b| \).

Constructive Constraints

Such constraints are critical when we need to ensure that certain conditions are met within a problem. Inequalities provide a 'safe zone' where we can confidently say that our trigonometric values will fall, aiding in solving many real-world problems, such as optimizing angles for engineering projects or analyzing periodic functions in financial models.

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Most popular questions from this chapter

If \(f(x)\) is differentiable and \(\lim _{x \rightarrow \infty} f(x)\) is finite and \(\lim _{x \rightarrow \infty} f^{\prime}(x)\) is finite, then show that \(\lim _{x \rightarrow \infty} f^{\prime}(x)=0\).

With the aid of Lagrange's theorem prove the inequalities \(\frac{a-b}{a} \leq \ln \frac{a}{b} \leq \frac{a-b}{b}\), for the condition \(0

Show that the equation \(x^{3}-3 x+c=0\) cannot have two different roots in the interval \((0,1)\).

Given \(g(x)=f\left(x^{2}-x-10\right)+f\left(14+x-x^{2}\right)\) and \(f^{\prime \prime}(x)>0\) for all real \(x\), except at finite no. of real values of \(x\) for which \(f^{\prime \prime}(x)=0\). Discuss the monotonicity of the function \(g(x)\). \\{ns. \((\infty,-3)\) decreasing, \(\left(-3, \frac{1}{2}\right)\) increasing, \(\left(\frac{1}{2}, 4\right)\) decreasing, \((4, \infty)\) increasing \(\\}\)

Find the intervals of concavity of the following functions:- i. \(f(x)=x^{4}+x^{3}-18 x^{2}+24 x-12\). ii. \(f(x)=3 x^{5}-5 x^{4}+3 x-2\). iii. \(f(x)=x^{6}-10 x^{4}\). iv. \(f(x)=\ln \left(x^{2}-1\right)\). v. \(f(x)=(x+1)^{4}+e^{x}\). vi. \(f(x)=x^{2} \ln x .\) vii. \(f(x)=x+x^{\frac{4}{3}}\). viii. \(f(x)=x+x^{\frac{5}{3}}+1\). ix. \(f(x)=x+x^{\frac{2}{3}}\). x. \(\quad f(x)=x^{2}, \quad x \leq 0\) \(=x^{3}, \quad x>0\). \(=x^{2}, \quad x>0\). \(=x^{2}, \quad x>1\).
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