Problem 41 - Precalculus

Question

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

Step-by-Step Solution

Answer

Positive integers for which \(f^{(n)}(x) = \sin x\) are \(n = 4k + 3\), where \(k\) is a non-negative integer (e.g., \(n = 3, 7, 11, \ldots \)).

Step 1: Understanding the Problem

We need to determine for which integers of the successive derivatives of the function \(f(x) = \cos x\) equal \(\sin x\).

Step 2: Recognizing the Pattern

The derivatives of \(f(x) = \cos x\) follow the sequence: \(\cos x, -\sin x, -\cos x, \sin x, \cos x, \ldots\). This sequence repeats every four steps.

Step 3: Identifying the Matching Condition

The derivative \(f^{(n)}(x) = \sin x\) appears as the fourth derivative in our repeated sequence: e.g., \(f'(x) = -\sin x\), \(f''(x) = -\cos x\), \(f'''(x) = \sin x\).

Step 4: Finding the General Formula

Since \(\sin x\) corresponds to the third derivative, and every cycle is four derivatives long, the general position where this occurs is at \(n = 4k + 3\) for \(k\) an integer (including zero).

Step 5: Confirming Positive Integers

Since \(n\) has to be a positive integer, appropriate values correspond to choosing \(k = 0, 1, 2, ...\), forming the sequence \(n = 3, 7, 11, ...\). These are all positive integers.

Key Concepts

Trigonometric Functions

Trigonometric functions, including the cosine and sine functions, are mathematical functions with many applications, such as calculating angles in triangles or modeling periodic phenomena like sound waves.
They are typically defined on the unit circle, and their values represent the coordinates of points as you move around the circle.

• Cosine: Represented as \( \cos x \, \), where \( x \) is an angle in radians.
• Sine: Represented as \( \sin x \, \), with the same angle \( x \. \)

These functions have a characteristic wave-like behavior that repeats itself over intervals, which is particularly useful when dealing with cyclic or repetitive processes. In calculus, trigonometric functions often appear in the investigation of waves and oscillations.
By understanding these basic definitions, you can get deeper insights into the role these functions play in mathematics and sciences. For this exercise, knowing the basic function \( f(x) = \cos x \, \) helps us explore its derivatives and the cyclic nature they follow.

Cyclic Patterns in Derivatives

When differentiating trigonometric functions, certain patterns appear. This is crucial for understanding problems like our exercise, where successive derivatives follow a predictable cycle.
In the case of the function \( f(x) = \cos x \, \), its derivatives evolve cyclically:

• \( \cos x \) becomes \(-\sin x \).
• \(-\sin x \) becomes \(-\cos x \).
• \(-\cos x \) becomes \(\sin x \).
• \(\sin x \) becomes back to \(\cos x \).

This cycle repeats every four derivatives, implying that every fourth derivative returns the function to its original or rearranged form. Identifying such cycles assists in solving problems involving trigonometric derivatives effectively.For example, in our problem, rebuilding the function \( f^{(n)}(x) = \sin x \, \) involves recognizing it appears as the third term in every cycle. Each cycle consists of four derivations, so the instances where \( \sin x \) appears align with the formula \( n = 4k + 3 \, \) with \( k \) as any integer.The predictability of these cyclic patterns simplifies many calculus tasks because it eliminates the need for tediously calculating successive terms beyond a determined point.

Integer Solutions in Calculus

Integer solutions in calculus often crop up in exercises involving patterns or sequences. Understanding where these solutions come from aids in finding a meaningful answer. In our problem, we seek positive integers \( n \) such that the nth derivative of \( f(x) = \cos x \, \) is \( \sin x \. \)
In the cyclic derivative pattern of trigonometric functions, we derived the condition \( n = 4k + 3 \), where \( k \) is any integer. The understanding here lies in recognizing this formula's capacity to generate integer solutions
:

• Start with \( k = 0 \, \), yielding \( n = 3 \).
• Move to \( k = 1 \, \), getting \( n = 7 \).
• Continue with \( k = 2, 3, \ldots \, \), yielding values like \( n = 11, 15, \ldots \).

Naturally, only positive integer solutions are of interest as they relate to the number of derivatives taken.Leveraging the concept of integer sequences helps shorten the process of deriving solutions in many calculus problems. Recognizing this sequence and algorithmically calculating the values makes the task simple and efficient. This approach is practical and extends beyond this specific problem to a wide array of mathematical scenarios.