Problem 48 - Calculus of a Single Variable

Question

Applying the First Derivative Test In Exercises \(41-48\) , 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, apply the First Derivative Test to identify all relative extrema, and (c) use a graphing utility to confirm your results. $$ f(x)=\frac{\sin x}{1+\cos ^{2} x} $$

Step-by-Step Solution

Answer

The function \(f(x)=\frac{\sin x}{1+\cos^{2} x}\) is increasing on the interval from 0 to \(\pi\), and decreasing on the interval from \(\pi\) to \(2\pi\). It has a relative maximum at \(x=\pi\). These results are confirmed by graphing the function.

Step 1: Finding the derivative

The first step is to find the derivative of the given function \(f(x)=\frac{\sin x}{1+\cos ^{2} x}\). Using the quotient rule, the derivative is given by:\[f'(x) = \frac{(1+\cos^{2}x)\cos x + \sin^2 x}{(1+\cos^{2}x)^2}\] Simplifying it further gives: \[f'(x) = \frac{\cos(x)+1}{(1+\cos^{2}x)^2}\]

Step 2: Applying the First Derivative Test

To find where the function is increasing or decreasing, we need to find where the function's derivative is positive or negative. For this, we need to find the critical points of the function, which are the points where the derivative is zero or undefined. Setting the derivative equal to zero we find: \[0=\cos(x)+1 \] And the solution is: \[x=\pi \]

Step 3: Interval Testing

To determine where the function is increasing or decreasing, plug in values from each interval defined by the critical points into the derivative. For \(x<\pi, 0<\cos(x)+1<2 \) so \(f'(x)>0\) and the function is increasing in this interval. For \(x>\pi, -2<\cos(x)+1<0 \) so \(f'(x)<0\) and the function is decreasing in this interval. Therefore, the function has a maximum at \(x=\pi\).

Step 4: Confirm with a Graphing Utility

Use a graphing utility to graph the function and verify the results obtained analytically. The function should be increasing on the interval from \(0 to \pi\) and decreasing on the interval from \(\pi to 2\pi\). The relative maximum should also be at \(x=\pi\).

Key Concepts

Relative Extrema

Relative extrema refer to the local minimums or maximums in a function within a specific interval.This concept is crucial as it helps identify key points on a graph where the function switches its behavior from increasing to decreasing or vice versa.

In the given problem, the relative extrema are found by applying the First Derivative Test.We first find where the derivative equals zero or is undefined. These points are called critical points.By evaluating the derivative before and after these critical points, we determine if the function has a relative maximum or minimum at these points.

In this exercise, it was found that at point \(x = \pi\), the function changes its direction from increasing to decreasing, identifying \(x = \pi\) as a relative maximum of the function.

Interval Analysis

Interval analysis involves examining different segments of a function to determine its behavior.By dividing the function into open intervals based on critical points, we can assess if the function is increasing or decreasing in these sections.

For this exercise, the critical point \(x = \pi\) creates two intervals:

• \((0, \pi)\)
• \((\pi, 2\pi)\)

These intervals help us apply the First Derivative Test more effectively, allowing us to confidently conclude that:- The function is increasing on \((0, \pi)\)- The function is decreasing on \((\pi, 2\pi)\).

This methodical breaking down of intervals simplifies the task of understanding complex functions and provides a structured approach to identifying relative extrema.

Critical Points

Critical points are vital in calculus as they represent potential locations for relative extrema.These are the points on the graph of a function where its derivative equals zero or is undefined.

For this exercise, we found the critical point by setting the derivative \( f'(x) \) to zero and solving for \( x \): \[0 = \cos(x) + 1 \]From this, \( x = \pi \) was obtained as a critical point. This single critical point is essential in identifying the intervals for analysis and determining the nature of relative extrema.

Identifying the critical points simplifies the mathematical analysis significantly and lays the groundwork for further interval testing and First Derivative Test applications.

Increasing and Decreasing Functions

Understanding when a function is increasing or decreasing is fundamental to analyzing its overall behavior.By examining the sign of the derivative, we can determine over which intervals a function is rising or falling.

The derivative \( f'(x) \) being positive indicates the function is increasing, while a negative derivative suggests it is decreasing.In this exercise,

• For \( x < \pi \): \( f'(x) > 0 \) means the function is increasing.
• For \( x > \pi \): \( f'(x) < 0 \) means the function is decreasing.

These changes in behavior perfectly align with the concept of relative extrema at \( x = \pi \), confirming it as a maximum point.

Analyzing increasing and decreasing intervals forms the cornerstone of understanding the graph’s shape and its extremities, providing core insights into its dynamics.