Problem 60 - Calculus

Question

Are the statements true or false? Give an explanation for your answer. $$\begin{aligned} &\text { The function } f(\theta)=\cos \theta-\sin \theta \text { is increasing on }\\\ &0 \leq 0 \leq \pi / 2 \end{aligned}$$

Step-by-Step Solution

Answer

False, the function is not increasing; it is decreasing or constant.

Step 1: Understand the Function and Interval

We are given the function \( f(\theta) = \cos \theta - \sin \theta \) and the interval \( 0 \leq \theta \leq \pi/2 \). The task is to determine whether this function is increasing on this interval.

Step 2: Find the Derivative

To check if the function is increasing, we first need to find the derivative \( f'(\theta) \). The derivative of \( \cos \theta \) is \( -\sin \theta \) and the derivative of \( -\sin \theta \) is \( -\cos \theta \). Therefore, \( f'(\theta) = -\sin \theta - \cos \theta \).

Step 3: Analyze the Derivative on the Interval

We need to analyze the sign of \( f'(\theta) = -\sin \theta - \cos \theta \) on the interval \( 0 \leq \theta \leq \pi/2 \). During this interval, both \( \sin \theta \) and \( \cos \theta \) are non-negative, thus \( -\sin \theta - \cos \theta \leq 0 \).

Step 4: Conclusion About the Function's Behavior

Since \( f'(\theta) \leq 0 \) for all \( \theta \) in the interval \( 0 \leq \theta \leq \pi/2 \), the function \( f(\theta) \) is not increasing on this interval. Instead, it is decreasing or constant, but not increasing.

Key Concepts

Derivative

In calculus, the derivative of a function provides critical information about the function's behavior. It represents the rate at which the function's value changes with respect to a change in its input variable. For instance, if we have a function \( f(x) \), the derivative, denoted as \( f'(x) \) or \( \frac{df}{dx} \), helps us understand whether the function is increasing, decreasing, or remaining constant at a point.

• If \( f'(x) > 0 \), the function is increasing at that point.
• If \( f'(x) < 0 \), the function is decreasing.
• If \( f'(x) = 0 \), the function could be at a local maximum, minimum, or plateau, indicating no instantaneous change.

For the exercise provided, we found the derivative\( f'(\theta) = -\sin \theta - \cos \theta \). Understanding this derivative was essential to evaluating if the function is increasing or decreasing on the given interval.

Function Behavior

Analyzing a function's behavior involves looking at how its values change as you move along its domain. This evaluation is mostly done using the derivative.

• An increasing function is one where as you increase your input, the output also increases.
• A decreasing function means that as the input increases, the output decreases.

In the context of the function \( f(\theta) = \cos \theta - \sin \theta \) on the interval \( 0 \leq \theta \leq \pi/2 \), we determined the function's derivative \( f'(\theta) = -\sin \theta - \cos \theta \). Since \( \sin \theta \) and \( \cos \theta \) are both non-negative within this interval, the derivative is negative throughout. Therefore, the function does not increase on this interval. Instead, its behavior is either decreasing or constant due to the non-positive derivative value.

Trigonometric Functions

Trigonometric functions like sine and cosine are fundamental in calculus and have specific properties and behaviors over different intervals. They are periodic and oscillate between -1 and 1.

• The sine function \( \sin(\theta) \) starts at 0, increases to 1 at \( \pi/2 \), then decreases back to 0 at \( \pi \).
• The cosine function \( \cos(\theta) \) starts at 1, decreases to 0 at \( \pi/2 \), then further reduces to -1 at \( \pi \).

In our problem, the function \( f(\theta) = \cos \theta - \sin \theta \) combines both sine and cosine functions. Throughout the interval \( 0 \leq \theta \leq \pi/2 \), \( \sin(\theta) \) is increasing, and \( \cos(\theta) \) is decreasing. By analyzing these behaviors together, particularly their derivatives, we can draw conclusions about the function formed from their combination. Understanding how each component of these trigonometric functions behaves individually allows us to deduce the behavior of the overall function on a given interval.