Problem 54 - Thomas Calculus in SI Units

Question

a. Find the local extrema of each function on the given interval, and say where they occur. b. Graph the function and its derivative together. Comment on the behavior of \(f\) in relation to the signs and values of \(f^{\prime}\). $$f(x)=\sin x-\cos x, \quad 0 \leq x \leq 2 \pi$$

Step-by-Step Solution

Answer

Local maximum at \(x = \frac{3\pi}{4}\), local minima at \(x = 0, 2\pi\). \(f(x)\) is increasing when \(f'(x) > 0\) and decreasing when \(f'(x) < 0\).

Step 1: Find the Derivative

To find the local extrema, we need the derivative of the function.\[ f(x) = \sin x - \cos x \]The derivative, using basic differentiation rules, is:\[ f'(x) = \cos x + \sin x \].

Step 2: Set the Derivative to Zero

To find critical points, we set the derivative equal to zero:\[ \cos x + \sin x = 0 \].Rewrite this as \(\cos x = -\sin x\).This equation is true when \(\tan x = -1\).

Step 3: Solve for Critical Points

\(\tan x = -1\) occurs at angles where tangent has the values -1. These angles in the given interval \(0 \leq x \leq 2\pi\) are: \[ x = \frac{3\pi}{4}, \frac{7\pi}{4} \].

Step 4: Evaluate the Function at Critical Points and Endpoints

Find \(f(x)\) at these critical points and the endpoints \(x = 0\) and \(x = 2\pi\).\[ f(0) = \sin 0 - \cos 0 = -1 \] \[ f\left(\frac{3\pi}{4}\right) = \sin\left(\frac{3\pi}{4}\right) - \cos\left(\frac{3\pi}{4}\right) = \frac{\sqrt{2}}{2} + \frac{\sqrt{2}}{2} = \sqrt{2} \] \[ f\left(\frac{7\pi}{4}\right) = \sin\left(\frac{7\pi}{4}\right) - \cos\left(\frac{7\pi}{4}\right) = -\frac{\sqrt{2}}{2} + \frac{\sqrt{2}}{2} = 0 \] \[ f(2\pi) = \sin(2\pi) - \cos(2\pi) = -1 \].

Step 5: Determine Local Extrema

Compare the values of \(f(x)\) at the critical points and endpoints.\(f\left(\frac{3\pi}{4}\right) = \sqrt{2}\) is the maximum point, and \(f(x) = -1\) at both \(x=0\) and \(x=2\pi\) are lower values. At \(x=\frac{7\pi}{4}\), \(f(x)=0\), higher than endpoints but less than maximum.

Step 6: Graph and Analyze

Graph \(f(x) = \sin x - \cos x\) and its derivative \(f'(x) = \cos x + \sin x\). Notice where \(f'(x)\) crosses the x-axis (at the critical points), \(f(x)\) has horizontal tangents. When \(f'(x) > 0\), \(f(x)\) is increasing, and when \(f'(x) < 0\), \(f(x)\) is decreasing.

Key Concepts

Local Extrema

When dealing with functions, local extrema refer to points in an interval where the function reaches a local maximum or minimum. These are the 'peaks' and 'valleys' within a specified range.

To find these extrema, we use calculus, specifically derivative analysis. By taking the derivative of a function, we can determine points where the slope of the tangent line is zero. These are called critical points and can indicate potential local extrema.

For example, consider the function given, \( f(x) = \sin x - \cos x \). When we set its derivative \( f'(x) = \cos x + \sin x \) equal to zero, we find critical points at \( x = \frac{3\pi}{4} \) and \( x = \frac{7\pi}{4} \). Evaluating the function at these critical points helps to determine whether they are maxima or minima.

• Maxima occur where the function values transition from increasing to decreasing.
• Minima occur where the function transitions from decreasing to increasing.
• Endpoints can also be evaluated as potential extrema in a closed interval.

Derivative Analysis

The derivative function \( f'(x) \) plays a crucial role in understanding the behavior of any function. It measures the rate of change or the slope of the function at any point.

For the function \( f(x) = \sin x - \cos x \), the derivative \( f'(x) = \cos x + \sin x \) is derived through basic derivative rules for sine and cosine functions.

Analyzing this derivative tells us much about the original function:

• If \( f'(x) = 0 \), the function has a horizontal tangent (i.e., a critical point). This indicates a potential local extremum.
• If \( f'(x) > 0 \), the function is increasing. The slope is positive, showing a rise in the function's graph.
• If \( f'(x) < 0 \), the function is decreasing. The slope is negative, showing a fall in the function's graph.

By evaluating \( f'(x) \) across an interval, we can sketch the behavior of the original function, identifying where it rises and falls.

Trigonometric Functions

Trigonometric functions, like sine and cosine, are key players in many calculus problems. Understanding their properties and behaviors is crucial for dealing with problems involving periodic functions.

Sine and cosine functions have specific characteristics:

• They have a periodic nature, repeating every \( 2\pi \).
• Sine starts at 0 and reaches its maximum at \( \frac{\pi}{2} \), whereas cosine starts at 1 and decreases to 0 by \( \frac{\pi}{2} \).
• These functions have smooth curves, making them continually differentiable.

The exercise function, \( f(x) = \sin x - \cos x \), combines these two functions, creating a new curve. This combination modifies the amplitude and phase of the typical wave-like behavior.

Understanding the individual behaviors of \( \sin x \) and \( \cos x \), and how they interact when subtracted, allows us to predict critical points and extrema effectively. This understanding is vital in applying calculus techniques such as derivative analysis to these functions.