Problem 3 - Calculus - AP Edition

Question

Verify that the hypotheses of Rolle's Theorem are satisfied on the given interval, and find all values of \(c\) in that interval that satisfy the conclusion of the theorem. $$f(x)=\cos x ;[\pi / 2,3 \pi / 2]$$

Step-by-Step Solution

Answer

The value of \(c\) is \(\pi\).

Step 1: Verify Continuity

Rolle's Theorem requires that the function be continuous on the closed interval \([rac{\pi}{2}, \frac{3\pi}{2}]\). The cosine function \(f(x) = \cos x\) is continuous for all real numbers, including this interval.

Step 2: Verify Differentiability

The function \(f(x) = \cos x\) must be differentiable on the open interval \((\frac{\pi}{2}, \frac{3\pi}{2})\). Since \(f(x)\) is differentiable everywhere on the real number line (differentiable implies continuity), this condition is also satisfied.

Step 3: Check Endpoints Equality

Check if \(f(x)\) has equal values at the endpoints of the interval. Calculate \(f(\frac{\pi}{2}) = \cos(\frac{\pi}{2}) = 0\) and \(f(\frac{3\pi}{2}) = \cos(\frac{3\pi}{2}) = 0\). Since both are equal, the hypothesis of Rolle's Theorem is fulfilled.

Step 4: Find f'(x)

Calculate the derivative of the function: \(f'(x) = -\sin x\).

Step 5: Solve f'(c) = 0

Set \(f'(c) = -\sin c = 0\) to find values of \(c\) in \((\frac{\pi}{2}, \frac{3\pi}{2})\). This implies \(\sin c = 0\).

Step 6: Identify Valid c Values

Solve \(\sin c = 0\) and identify solutions within the interval \((\frac{\pi}{2}, \frac{3\pi}{2})\). The solutions are \(c = \pi\), as \(\sin \pi = 0\).

Step 7: Conclusion

The hypotheses of Rolle's Theorem are satisfied and \(c = \pi\) is the value that fulfills the theorem’s conclusion in the interval \((\frac{\pi}{2}, \frac{3\pi}{2})\).

Key Concepts

Continuity

To understand Rolle's Theorem, we first need to discuss the concept of continuity. A function is continuous on an interval if there are no sudden jumps or breaks in its graph within that interval. For the function \(f(x) = \cos x\), which represents a wave-like curve, continuity is inherent. The cosine function is continuous for all real numbers, which means there are no interruptions or gaps in its values.

• Continuous functions adhere to the intuitive idea that you can draw their graph without lifting your pencil off the paper.
• On the closed interval \([\pi/2, 3\pi/2]\), \(\cos x\) does not exhibit any discontinuity.

This property of the cosine function satisfies the first hypothesis of Rolle's Theorem, ensuring that \(\cos x\) is continuous over the entire specified interval.

Differentiability

Differentiability of a function is closely tied to continuity, but with an added requirement: the function's graph should be smooth without any sharp corners or cusps. Simply put, a function is differentiable at a point if it has a well-defined tangent line at that point.

• The function \(\cos x\) is differentiable everywhere on the real number line.
• This means on the open interval \((\pi/2, 3\pi/2)\), \(\cos x\) has a continuous derivative.
• For differentiability, we compute the derivative, which for \(\cos x\) is \(f'(x) = -\sin x\).

The ability to take a derivative everywhere on \((\pi/2, 3\pi/2)\) confirms the second condition of Rolle's Theorem.

Trigonometric Functions

Trigonometric functions like \(\cos x\) and \(\sin x\) are fundamental to understanding periodic phenomena. They repeat at regular intervals, known as their period. The cosine function, in particular, repeats every \(2\pi\), meaning every \(2\pi\) radians the function's values begin anew.

• An important property of trigonometric functions is their specific values at key points, such as \(\cos(\pi/2) = 0\) and \(\cos(3\pi/2) = 0\), which are used in this problem to check endpoint equality.
• Another relevant property is the zeroes of \(\sin x\), occurring at integer multiples of \(\pi\), such as \(\sin(\pi) = 0\).
• These properties make trigonometric functions essential tools in calculus and calculus-based physics.

Understanding the behavior of sine and cosine through their repetitions and zeroes is integral in applying the conclusion of Rolle's Theorem, which in this problem finds \(c = \pi\) as where the derivative \(f'(c) = 0\).