Problem 49 - Essential Calculus

Question

Consider the function on the interval \((0,2 \pi)\) For each function, (a) find the open interval(s) on which the function is increasing or decreasing, (b) apply the First Derivative Test to identify all relative extrema, and (c) use a graphing utility to confirm your results. $$ f(x)=\sin x+\cos x $$

Step-by-Step Solution

Answer

The function increases on the intervals \(x<\pi/4\) and \(x > 5\pi/4\), and decreases on the interval between \(\pi/4\) and \(5\pi/4\). The function reaches a maximum at \(x=\pi/4\) and a minimum at \(x=5\pi/4\).

Step 1: Find the derivative

The derivative of \(f(x)=\sin x+\cos x\) is obtained using the chain rule. So, \(f'(x)=\cos x - \sin x\).

Step 2: Find the critical points

Set the derivative equal to zero and solve for \(x\). So, there are critical points are at \(x=\pi/4 + k\pi\), where \(k\) is any integer.

Step 3: Applying First Derivative Test

To find if there is a maximum or minimum at these points, apply the First Derivative Test. When \(x < \pi/4\), \(f'(x)>0\), implying that \(f(x)\) is increasing. When \(\pi/4 < x < 5\pi/4\), \(f'(x)<0\), implying that \(f(x)\) is decreasing and when \(x>5\pi/4\), \(f'(x)>0\), implying that \(f(x)\) is again increasing. Thus, at \(x=\pi/4\), \(f(x)\) is at a maximum and at \(x=5\pi/4\), \(f(x)\) is at a minimum.

Step 4: Graph the function

Use a graphing utility to graph the function \(f(x)=\sin x+\cos x\) to confirm the result of the critical points and extrema.

Key Concepts

Derivatives

In calculus, derivatives play a crucial role in understanding the behavior of functions. They measure the rate at which a function is changing at any given point. If you think of a curve on a graph, the derivative at a particular point helps us know the slope of the tangent to that curve at that point.
For functions like \( f(x) = \sin x + \cos x \), we use derivative rules to find their derivatives. The derivative of \( \sin x \) is \( \cos x \) and that of \( \cos x \) is \( -\sin x \). Therefore, the derivative of our function is \( f'(x) = \cos x - \sin x \).
Finding derivatives helps in solving problems related to motion, rates of change, and many more situations where variables are interdependent.

Critical Points

Critical points of a function are where the derivative is zero or undefined. These points are significant because they could indicate locations where the function might have a relative maximum, minimum, or a point of inflection.
For the function \( f(x) = \sin x + \cos x \), we find the critical points by solving \( f'(x) = \cos x - \sin x = 0 \). This occurs when \( \cos x = \sin x \). After solving this, we find that the critical points are given by \( x = \pi/4 + k\pi \), where \( k \) is any integer.
These points help in analyzing the areas on the graph where the function changes its increasing or decreasing behavior, and hence, they are extremely useful in optimization problems.

First Derivative Test

The First Derivative Test is a method used to classify critical points as relative maxima, minima, or neither. This test examines the sign of the derivative before and after each critical point.
For our function \( f(x) = \sin x + \cos x \), we apply the test at critical points. Around \( x = \pi/4 \), when \( x < \pi/4 \), \( f'(x) > 0 \), indicating the function is increasing. After this point, \( f(x) \) decreases between \( \pi/4 < x < 5\pi/4 \), as \( f'(x) < 0 \). After \( x = 5\pi/4 \), \( f(x) \) starts increasing again, as \( f'(x) > 0 \).
This confirms that at \( x = \pi/4 \), there is a relative maximum, and at \( x = 5\pi/4 \), there is a relative minimum. This test provides a systematic way to classify critical points without needing to plot them.

Trigonometric Functions

Trigonometric functions like \( \sin x \) and \( \cos x \) are fundamental in mathematics, especially in calculus, physics, and engineering. They describe the ratios of sides in right triangles and are periodic, meaning they repeat their values in regular intervals.
These functions are crucial in modeling wave-like phenomena, such as sound waves, light waves, and alternating current signals. In the problem \( f(x) = \sin x + \cos x \), both sine and cosine contribute to the behavior of the function across different intervals.
The periodic nature and derivatives of trigonometric functions make them an essential tool for analyzing periodic motion and cyclical phenomena. Understanding their properties and derivatives is key to many advanced applications in science and engineering.