Problem 47 - Advanced Engineering Mathematics

Question

Find a function whose square plus the square of its derivative is \(1 .\)

Step-by-Step Solution

Answer

The function is \( f(x) = \sin(x) \).

Step 1: Understand the Problem

We are asked to find a function \( f(x) \) such that the sum of the square of the function and the square of its derivative equals 1. Mathematically, this means we need to satisfy the equation: \( f(x)^2 + (f'(x))^2 = 1 \).

Step 2: Recall Pythagorean Identity

Recall a fundamental identity from trigonometry: \( \sin^2(x) + \cos^2(x) = 1 \). This identity might be useful because it has a similar structure to our problem.

Step 3: Test Possible Functions

Let us test \( f(x) = \sin(x) \) as a solution. The derivative of \( \sin(x) \) is \( \cos(x) \), so we substitute these into the equation: \( \sin^2(x) + \cos^2(x) = 1 \).

Step 4: Verify the Solution

The expression \( \sin^2(x) + \cos^2(x) = 1 \) is true by the Pythagorean identity. This means that our test function \( f(x) = \sin(x) \) satisfies the original equation.

Key Concepts

Trigonometric Identities

Trigonometric identities are mathematical equations involving the trigonometric functions. They're fundamental in understanding the properties of angles and circles. One common pair includes the sine and cosine functions, which are periodic and oscillate between -1 and 1. These functions are defined as:

• Sine of an angle in a right triangle: Opposite side divided by the hypotenuse.
• Cosine of an angle: Adjacent side divided by the hypotenuse.

Trigonometric identities are used to simplify expressions and solve equations involving trigonometric functions. The most notable is the Pythagorean identity, but there are others like double-angle and sum-to-product identities.
These identities are handy because they allow the transformation and simplification of complex trigonometric equations. By understanding these identities, solving problems such as the one given in the exercise becomes a feasible task.

Pythagorean Identity

The Pythagorean identity is a core concept in trigonometry. It states that for any angle, the following holds true: \[ \sin^2(x) + \cos^2(x) = 1 \] This identity is derived from the Pythagorean theorem, which relates the sides of a right triangle. Here, the hypotenuse is the diameter of the unit circle (radius equals 1), which frames the identity.
This identity is pivotal when solving trigonometric equations, like those found in calculus problems. Since the identity mirrors the structure of our problem, it offers a solid lead to finding the solution.
Understanding and recognizing such identities allow us to replace more complex expressions with simpler ones, ensuring that calculations remain straightforward. It's these sorts of shortcuts that make manipulating trigonometric functions more intuitive.

Function Derivatives

Function derivatives are a cornerstone of calculus, representing the rate at which a function changes. For trigonometric functions, derivatives have unique properties that align with the periodic nature of these functions.
The derivative of \( \sin(x) \) is \( \cos(x) \), while the derivative of \( \cos(x) \) is \( -\sin(x) \). These relationships form a cycle, showing how the trigonometric functions are interconnected.
Derivatives help us understand the behavior of functions graphically. They tell us about the slope of the function at any given point and provide insights into maximum and minimum values and points of inflection.

• For example, if \( f(x) = \sin(x) \), then its derivative \( f'(x) = \cos(x) \).
• When combined as \( f(x)^2 + (f'(x))^2 = 1 \), they satisfy the problem's requirement. This is due to the trigonometric identity.

Mastering derivatives, especially those of trigonometric functions, is essential for problem solving, as they often intersect with identities like the Pythagorean identity in various mathematical problems.