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Chapter 4: Problem 13

Determine whether Rolle's Theorem applies to the following functions on the given interval. If so, find the point(s) guaranteed to exist by Rolle's Theorem. $$f(x)=\cos 4 x ;[\pi / 8,\(x=\frac{\pi}{4}\)3 \pi / 8]$$

Short Answer

Expert verified
What is the value of the point? Answer: Rolle's Theorem guarantees the existence of a point in the interval $[\frac{\pi}{8}, \frac{3\pi}{8}]$ where the derivative of the function $f(x) = \cos(4x)$ is zero if the function is continuous, differentiable, and has equal values at the endpoints of the interval. The value of the point is $x = c = \frac{\pi}{4}$.

Step by step solution

01

Check the given function and interval

We are given the function: $$f(x) = \cos(4x)$$ and the interval: $$[\frac{\pi}{8}, \frac{3\pi}{8}]$$
02

Check if the function is continuous

Since the cosine function is continuous on its entire domain, and the given function is just a composition of the cosine function, it follows that \(f(x) = \cos(4x)\) is continuous on its entire domain.
03

Check if the function is differentiable

The derivative of the cosine function is the negative sine function and the given function is a composition of the cosine function and a linear function. By the chain rule, we can find the derivative of \(f(x)\) with respect to \(x\) as follows: $$f'(x) = -\sin(4x)(4) = -4\sin(4x)$$ Since the derivative exists for all values of \(x\), the function \(f(x)=\cos(4x)\) is differentiable.
04

Check if the function has equal values at the endpoints

To apply Rolle's Theorem, the function needs to have equal values at the endpoints of the interval. So, we check if \(f(\frac{\pi}{8}) = f(\frac{3\pi}{8})\): $$f(\frac{\pi}{8}) = \cos\left(4\left(\frac{\pi}{8}\right)\right) = \cos\left(\frac{4\pi}{2}\right) = \cos(\pi) = -1$$ $$f(\frac{3\pi}{8}) = \cos\left(4\left(\frac{3\pi}{8}\right)\right) = \cos\left(\frac{12\pi}{2}\right) = \cos(3\pi) = -1$$ Since \(f(\frac{\pi}{8}) = f(\frac{3\pi}{8})\), the condition is met. Given that all three conditions are satisfied, we can now apply Rolle's Theorem by finding the point(s) in the interval where \(f'(x) = 0\).
05

Find the point(s) where the derivative is zero

According to Rolle's Theorem, there exists at least one point \(x=c\) in the interval where the derivative is zero: $$-4\sin(4c) = 0$$ This is true when \(\sin(4c) = 0\), which occurs when \(4c = n\pi\) with \(n\) being an integer. Dividing by 4, we have: $$c = \frac{n\pi}{4}$$ Since \(c\) belongs to the interval \(\left[\frac{\pi}{8}, \frac{3\pi}{8}\right]\), we can find the possible values of \(n\). For \(n=1\), we have \(c=\frac{\pi}{4}\), which is inside the interval. For \(n=2\), we have \(c=\frac{\pi}{2}\), which lies outside the interval. Therefore, the point guaranteed to exist by Rolle's Theorem is: $$x = c = \frac{\pi}{4}.$$

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Key Concepts

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

Continuous Function
When we talk about a continuous function, we mean a function that you can graph without ever lifting your pencil from the paper. This characteristic makes continuous functions crucial for many mathematical theorems like Rolle's Theorem.

For a function to be continuous on a certain interval, it must not have any jumps, holes, or breaks within that interval. It's like a smooth road with no interruptions!
  • The cosine function, denoted as \( \cos(x) \), is a classic example of a continuous function. It flows seamlessly along its entire domain.
  • Thus, any stretching, compressing, or translating of the cosine function, such as \( \cos(4x) \), still remains continuous.
  • In our case, \( f(x) = \cos(4x) \) is continuous on the interval \([\frac{\pi}{8}, \frac{3\pi}{8}]\).
Ensuring continuity is the first step in utilizing Rolle鈥檚 Theorem effectively.
Differentiable Function
Differentiability is another key aspect of applying Rolle's Theorem. A function is differentiable at a point if it has a derivative there, meaning its slope can be defined.

Differentiability implies smoothness鈥攖here are no sharp corners or cusps in the graph of the function.
  • The derivative of the cosine function is the sine function, specifically \( -\sin(x) \) for \( \cos(x) \).
  • Using the chain rule, we determine that for \( f(x) = \cos(4x) \), its derivative is \( f'(x) = -4\sin(4x) \).
  • This derivative exists for all values of \( x \), confirming that \( f(x) \) is differentiable on the given interval.
Again, verifying differentiability is necessary to proceed with Rolle鈥檚 Theorem, since the theorem demands both differentiability and continuity.
Trigonometric Function
Trigonometric functions, like sine and cosine, often appear in problems involving calculus due to their periodic and smooth nature.

The function \( f(x) = \cos(4x) \) is a trigonometric function, and understanding its properties can guide us in evaluating functions on specific intervals.
  • Trigonometric functions are particularly predictable; they repeat their values periodically, which helps in tackling problems related to periodicity and symmetry.
  • For example, \( \cos(x) \) completes a full cycle over the interval \([0, 2\pi]\), and modifications such as \( \cos(4x) \) adjust this period but retain the continuous nature.
  • This periodicity assists in determining equal endpoint values crucial to Rolle's Theorem.
Mastery of trigonometric functions is invaluable in evaluating both function behavior and application in calculus.
Endpoint Values
The endpoint values of a function on a specified interval are pivotal in assessing the application of Rolle鈥檚 Theorem. For the theorem to hold, the function must have equal values at the endpoints of the interval.

Let鈥檚 break it down:
  • Given \( f(x) = \cos(4x) \), we need to check \( f(\frac{\pi}{8}) \) and \( f(\frac{3\pi}{8}) \).
  • If these values are the same, we can further explore Rolle's Theorem to find points where the derivative is zero within the interval.
For \( f(x) = \cos(4x) \):
  • \( f(\frac{\pi}{8}) = \cos(\pi) = -1 \)
  • \( f(\frac{3\pi}{8}) = \cos(3\pi) = -1 \)
Since these values are equal, we can confidently say that the conditions for Rolle's Theorem are satisfied, allowing us to find the particular points within the interval.

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