Problem 23 - Calculus Early Transcendentals

Question

(a) Use the Constant Difference Theorem \((4.8 .3)\) to show that if \(f^{\prime}(x)=g^{\prime}(x)\) for all \(x\) in the interval \((-\infty,+\infty)\) and if \(f\) and \(g\) have the same value at some point \(x_{0}\) then \(f(x)=g(x)\) for all \(x\) in \((-\infty,+\infty)\) (b) Use the result in part (a) to confirm the trigonometric identity \(\sin ^{2} x+\cos ^{2} x=1\)

Step-by-Step Solution

Answer

Due to the same derivatives, identical point values lead to identical functions over the interval. Therefore, \(\sin^2 x + \cos^2 x = 1\) is confirmed for all \(x\).

Step 1: Understand the Constant Difference Theorem

The Constant Difference Theorem states that if two functions have the same derivative over an interval, they differ by at most a constant on that interval. If there is a point where these functions also have the same value, then they are exactly the same function across the whole interval.

Step 2: Application to functions f and g

Given that for all \(x\), \(f^{\prime}(x) = g^{\prime}(x)\), this means that \(f\) and \(g\) differ only by a constant, according to the Constant Difference Theorem. If \(f(x_0) = g(x_0)\) for some \(x_0\), then the constant difference is zero. Therefore, \(f(x) = g(x)\) for all \(x\) in \((-fty, +fty)\).

Step 3: Confirm with given identity

According to part (a), if two functions have the same derivative and they are equal at some point, they are equal everywhere. Consider \(f(x) = \sin^2(x) + \cos^2(x)\) and \(g(x) = 1\). Then, \(f^{\prime}(x) = 2\sin(x)\cos(x) - 2\cos(x)\sin(x) = 0\) and \(g^{\prime}(x) = 0\). Since both derivatives are equal and both functions equal 1 when \(x=0\), by the logic of the Constant Difference Theorem, \(f(x) = g(x) = 1\) for all \(x\).

Key Concepts

Trigonometric Identities

Trigonometric identities are equations that involve trigonometric functions and are true for every value of the variables involved. These identities are crucial in simplifying expressions and solving trigonometric equations. One of the most well-known trigonometric identities is \[\sin^2(x) + \cos^2(x) = 1\]This identity states that the sum of the squares of sine and cosine of an angle is always equal to 1. It is derived from the Pythagorean theorem and is an essential part of trigonometry. Understanding and being able to apply these identities is important in various fields such as physics, engineering, and mathematics.

• They help simplify complex trigonometric expressions.
• They are useful in calculus, especially when working with derivatives and integrals.
• They form the foundation for more advanced topics, such as Fourier analysis.

Given the identity \(\sin^2(x) + \cos^2(x) = 1\), using the Constant Difference Theorem helps confirm that this relationship holds true for every possible value of \(x\).

Derivatives

Derivatives are a fundamental concept in calculus. They represent the rate of change or the slope of a function at a given point. When dealing with functions, the derivative tells us how the function is changing - whether it is increasing or decreasing and by how much at any particular point. For a function \(f(x)\), its derivative is denoted as \(f'(x)\). If two functions have the same derivative across an interval, they differ by at most a constant. This is precisely what the Constant Difference Theorem tells us. For example, if \(f'(x) = g'(x)\), then there is some constant \(C\) such that \(f(x) = g(x) + C\). If additionally \(f(x_0) = g(x_0)\) for some point \(x_0\), then the constant \(C\) is zero, signifying that \(f(x) = g(x)\) across the whole interval.

• Helps measure the rate at which changes occur in functions.
• Essential for finding optimal solutions such as maximums or minimums.
• Used in many scientific and engineering applications to model behaviors of real-world systems.

Using derivatives within the framework of trigonometric identities, as in the given example, showcases how they can be applied to prove equality between different functional expressions.

Function Equality

Function equality is a concept where two functions are said to be equal if they produce the same output for every input within a specific domain. This is crucial when we are trying to confirm that two seemingly different expressions truly represent the same mathematical phenomenon.Using the Constant Difference Theorem provides a powerful way to establish function equality when it states that two functions are equal if:

• They have identical derivatives over an entire interval.
• There is at least one point within that interval where they share the same value.

Applying this to determine that \(f(x) = \sin^2(x) + \cos^2(x)\) is equal to \(g(x) = 1\) effectively leverages this theorem. We derive that both functions' derivatives are zero, and they equal each other at \(x = 0\). These criteria affirm their equality across all values of \(x\). Recognizing function equality is important not only in verifying trigonometric identities like those commonly encountered in calculus but also in ensuring that mathematical models accurately describe persistent and uniform behaviors throughout their defined domains.