Problem 42 - Calculus - AP Edition

Question

Let \(f(x)=\sin x .\) Find all positive integers \(n\) for which \(f^{(n)}(x)=\sin x\)

Step-by-Step Solution

Answer

The positive integers are all multiples of 4: 4, 8, 12, 16, ...

Step 1: Understand the Derivative Pattern

The function given is \(f(x) = \sin x\). First, let's find the pattern of derivatives for the sine function. Compute the first few derivatives: 1. \(f'(x) = \cos x\) 2. \(f''(x) = -\sin x\) 3. \(f'''(x) = -\cos x\) 4. \(f^{(4)}(x) = \sin x\). Notice that \(f^{(4)}(x) = \sin x\), which means the derivatives repeat every four terms.

Step 2: Determine the Pattern's Cycle

We've determined that the cycle length is 4 because after every four derivatives, the function returns to the original \(\sin x\). This gives us the sequence: 1. \(f^{(0)}(x) = \sin x\) 2. \(f^{(4)}(x) = \sin x\)... This pattern repeats indefinitely.

Step 3: Identify the Positive Integers

Since the derivatives repeat every four steps, we need to find all positive integers \(n\) such that \(n\equiv0\pmod{4}\). Thus, the positive integers \(n\) are multiples of 4.

Step 4: List the Values of n

The positive integers that satisfy the condition \(f^{(n)}(x) = \sin x\) are all positive multiples of 4, so they are: 4, 8, 12, 16, and so on.

Key Concepts

Derivatives of Trigonometric Functions

When dealing with trigonometric functions like the sine function, their derivatives follow predictable patterns. To fully grasp how these derivatives work, let's explore the derivatives of the sine function step-by-step.
Given the function:

• \(f(x) = \sin x\)
• The first derivative becomes \(f'(x) = \cos x\)
• The second derivative is \(f''(x) = -\sin x\)
• The third derivative is \(f'''(x) = -\cos x\)
• Finally, the fourth derivative returns to \(f^{(4)}(x) = \sin x\)

Here, we see that taking the derivative of a sine function successively leads to a repeating sequence every four derivatives. Each derivative shifts between the sine and cosine functions and alternates their signs.
Understanding these patterns is essential in calculus as it allows us to determine higher-order derivatives without repeatedly differentiating. It also sets up foundational knowledge for solving complex differentiation problems in calculus.

Differentiation Patterns

Differentiation patterns occur when a function's derivatives display a cyclic behavior. In our example, the derivative of \(f(x)=\sin x\) returns to the original after every fourth derivative.
This can be described as a periodic sequence that features important traits:

• Cycle Length: The length of the cycle for \(f(x) = \sin x\) is 4 since it repeats every four derivatives.
• Predictability: Knowing the cycle's length helps easily predict the function's higher-order derivatives.

These patterns make complex calculus problems more manageable by providing a shortcut to find higher-order derivatives efficiently. By recognizing that the pattern repeats, one does not need to compute each derivative manually for large \(n\), saving both time and effort.

Positive Integer Solutions

In the context of the problem involving the sine function, identifying positive integer solutions means finding all positive integers \(n\) that satisfy \(f^{(n)}(x)=\sin x\).
Based on the cycle established from the differentiation pattern, we see that every four derivatives, the function returns to its original form. Thus, the integers \(n\) must be such that \(n\equiv 0 \pmod{4}\).
This translates to:

• \(n = 4k\), where \(k = 1, 2, 3, \ldots\)
• Examples include 4, 8, 12, 16, and so on

Identifying these integer solutions allows us to predict at which specific derivatives the function will behave like the original sine function again. Incorporating modular arithmetic in this process simplifies finding these integers, providing an efficient method applicable to other periodic differentiation sequences as well.