Problem 41 - Applied Calculus

Question

Decide which function is an antiderivative of the other. $$f(x)=-\sin x-\cos x ; g(x)=\cos x-\sin x$$

Step-by-Step Solution

Answer

\( g(x) \) is an antiderivative of \( f(x) \).

Step 1: Identify Derivatives

Identify the derivative of each function to determine if one is an antiderivative of the other. The process involves differentiating each given function individually.

Step 2: Differentiate f(x)

Differentiate the function \( f(x) = -\sin x - \cos x \). The derivative is \( f'(x) = -\cos x + \sin x \) using the rules that the derivative of \( \sin x \) is \( \cos x \) and the derivative of \( \cos x \) is \( -\sin x \).

Step 3: Differentiate g(x)

Differentiate the function \( g(x) = \cos x - \sin x \). The derivative is \( g'(x) = -\sin x - \cos x \).

Step 4: Compare Derivatives

Compare the derivatives found in Steps 2 and 3. \( f'(x) = -\cos x + \sin x \) is not equal to \( g(x) \), but \( g'(x) = -\sin x - \cos x \) is equal to \( f(x) \). This indicates that \( g(x) \) is the antiderivative of \( f(x) \).

Key Concepts

Differentiation

Differentiation is the process of finding the derivative of a function. A derivative represents the rate at which a function is changing at any given point. Think of it as the slope of the curve of a function at a specific point. This concept is fundamental in calculus, reflecting how a change in one variable relates to the change in another.

To differentiate a function, students apply specific rules such as the power rule, product rule, and chain rule. In the context of trigonometric functions, certain derivatives become crucial, such as:

• The derivative of \( \sin x \) is \( \cos x \)
• The derivative of \( \cos x \) is \( -\sin x \)

By identifying these derivatives, you can determine if one function is an antiderivative of another by re-expressing the original function as a derivative. In our example, finding the derivatives of \( f(x) = -\sin x - \cos x \) and \( g(x) = \cos x - \sin x \) allows us to uncover their respective rates of change.

Sine function

The sine function, denoted as \( \sin x \), is a fundamental trigonometric function that appears frequently in calculus and various applications across mathematics. It represents the y-coordinate of a point on the unit circle as the angle x is swept from 0 to \( 2\pi \). This cyclical nature makes the sine function essential for modeling periodic phenomena such as sound waves.

• It is periodic with a period of \( 2\pi \)
• Its range is between -1 and 1
• Its maximum value is 1 and minimum value is -1

The sine function's derivative, as seen in calculus, is the cosine function. Understanding this derivative is key in identifying antiderivatives. For example, knowing the derivative of \( \sin x \) is \( \cos x \), allows us to understand part of the solution for whether \( f(x) = -\sin x - \cos x \) and \( g(x) = \cos x - \sin x \) are antiderivatives of one another.

Cosine function

The cosine function \( \cos x \) is another pivotal trigonometric function. It accompanies the sine function to form the basic building blocks of trigonometry. On the unit circle, \( \cos x \) gives the x-coordinate of the point as the angle x increases. Like sine, cosine is used extensively in mathematical applications to describe oscillatory behavior.

• It has a period of \( 2\pi \)
• The range is between -1 and 1
• The maximum value is 1 and the minimum value is -1

The derivative of the cosine function is \( -\sin x \). This negative sign is integral in solving exercises involving derivatives and antiderivatives, such as the example problem in hand. Recognizing the relationship between the derivatives of sine and cosine enables students to understand why the function \( g(x) = \cos x - \sin x \) is the antiderivative of \( f(x) = -\sin x - \cos x \). This concept highlights the interplay between these trigonometric functions in calculus.