Problem 53 - Calculus of a Single Variable

Question

Applying the First Derivative Test In Exercises \(41-48\) , consider the function on In Exercises \(49-54,\) (a) use a computer algebra system to differentiate the function, (b) sketch the graphs of \(f\) and \(f^{\prime}\) on the same set of coordinate axes over the given interval, (c) find the critical numbers of \(f\) in the open interval, and (d) find the interval(s) on which \(f^{\prime}\) is positive and the interval(s) on which it is negative. Compare the behavior of \(f\) and the sign of \(f^{\prime}\) $$ f(x)=-3 \sin \frac{x}{3}, \quad[0,6 \pi] $$

Step-by-Step Solution

Answer

The critical numbers of \(f(x) = -3 \sin \frac{x}{3}\) over the interval \([0,6 \pi]\) are \(\frac{3\pi}{2}, 3\frac{3\pi}{2}, 5\frac{3\pi}{2}\). The function's derivative, \(f'(x) = - \cos(\frac{x}{3})\), is positive over the intervals \([0, \frac{3\pi}{2}), [3\pi, 3\frac{3\pi}{2}), [6\pi, 6\frac{3\pi}{2})\] and negative over the intervals \([\frac{3\pi}{2}, 3\pi), [3\frac{3\pi}{2}, 6\pi), [6\frac{3\pi}{2}, 6 \pi]\]. The behavior of \(f\) and the sign of \(f'\) are consistent with each other: \(f\) increases where \(f'\) is positive and decreases where \(f'\) is negative.

Step 1: Differentiating the function

Differentiate \(f(x) = -3 \sin \frac{x}{3}\) by applying the chain rule. The derivative of \(\sin(x)\) is \(\cos(x)\) and the derivative of \(\frac{x}{3}\) is \(\frac{1}{3}\). Multiply these along with the original -3 to get the derivative, \(f'(x) = -3 \cdot \cos(\frac{x}{3}) \cdot \frac{1}{3} = -\cos(\frac{x}{3})\).

Step 2: Sketching the graph

To analyze the behavior of the function and its derivative, sketch the graphs of \(f\) and \(f'\) in the same graph over the interval \([0,6 \pi]\). Since we do not have a visualization tool here, this can be done on paper or through a graphing calculator.

Step 3: Finding critical numbers

A critical number of a function is a point in the domain of the function where its derivative is either equal to zero or undefined. Here, \(f'(x) = - \cos(\frac{x}{3})\). The function \(\cos(x)\) is equal to zero at \(x = (2n+1) \frac{\pi}{2}\), where \(n\) is an integer. Multiply this by 3 for the function \(\cos(\frac{x}{3}\) and we get that the derivative is zero for \(x = (2n+1)\frac{3\pi}{2}\). However, we are only considering the interval \([0,6 \pi]\), therefore, the critical numbers are \(x = \frac{3\pi}{2}, 3\frac{3\pi}{2}, 5\frac{3\pi}{2}\).

Step 4: Determining intervals

We can find the intervals on which \(f'(x)\) is positive or negative by testing the intervals between the critical numbers: if \(f'(x)\) is greater than 0 for a test point, the function is increasing, and if it’s less than 0, the function is decreasing. Take \(f'(x) = - \cos(\frac{x}{3})\), the intervals are given when \(2n\pi \leq x < (2n+1)\pi\), \(f'(x)\) is positive, and when \((2n+1)\pi \leq x < (2n+2)\pi\), \(f'(x)\) is negative, where \(n\) is an integer. Considering the interval \([0,6 \pi]\) we have multiple intervals: For \(x\) in \([0, \frac{3\pi}{2}), [3\pi, 3\frac{3\pi}{2}), [6\pi, 6\frac{3\pi}{2})\], \(f'(x)\) is positive. For \(x\) in \([\frac{3\pi}{2}, 3\pi), [3\frac{3\pi}{2}, 6\pi), [6\frac{3\pi}{2}, 6 \pi]\], \(f'(x)\) is negative.

Step 5: Comparing behaviors

Compare the behavior of \(f\) and the sign of \(f'\): When \(f'(x)\) is positive, \(f(x)\) is increasing; when \(f'(x)\) is negative, \(f(x)\) is decreasing. This matches up with the diagrams sketched in Step 2.

Key Concepts

Chain Rule

The Chain Rule is a fundamental concept in calculus, especially when dealing with composite functions. It helps us find the derivative of a function that is nested within another function. For example, consider the function \( f(x) = -3 \sin\left(\frac{x}{3}\right) \). It is composed of the sine function and a linear transformation \( \frac{x}{3} \).

To differentiate using the Chain Rule, follow these steps:

• Take the derivative of the outer function, which is \( \sin \), giving us \( \cos \).
• Then multiply it by the derivative of the inner function \( \frac{x}{3} \), which is \( \frac{1}{3} \).
• Don't forget to multiply everything by the constant factor originally outside, here \(-3\).

So, the derivative \( f'(x) = -3 \cdot \cos\left(\frac{x}{3}\right) \cdot \frac{1}{3} = -\cos\left(\frac{x}{3}\right) \).
Understanding and applying the Chain Rule is crucial for tackling more complex differentiation problems.

Critical Numbers

Critical numbers are vital points where a function's behavior changes. They occur at values of \( x \) where the derivative of the function \( f'(x) \) is zero or undefined.

For the function \( f(x) = -3 \sin\left(\frac{x}{3}\right) \), we found \( f'(x) = -\cos\left(\frac{x}{3}\right) \). To find critical numbers, set the derivative equal to zero: \( -\cos\left(\frac{x}{3}\right) = 0 \).

The cosine function equals zero at odd multiples of \( \frac{\pi}{2} \), i.e., \( \left(2n+1\right)\frac{\pi}{2} \), where \( n \) is an integer. For the transformed function, this becomes:

• \( x = (2n+1)\frac{3\pi}{2} \)

This gives us critical numbers within the interval \([0, 6\pi]\) of \( \frac{3\pi}{2}, 3\frac{3\pi}{2},\) and \(5\frac{3\pi}{2} \).
Finding critical numbers helps us understand where the function might switch from increasing to decreasing or vice versa.

Increasing and Decreasing Intervals

Knowing where a function increases or decreases reveals critical insights into its behavior. This is where the sign of the derivative, \( f'(x) \), plays a key role.

For the function \( f(x) = -3 \sin\left(\frac{x}{3}\right) \), we have the derivative \( f'(x) = -\cos\left(\frac{x}{3}\right) \). Understanding how cosine behaves helps us:

• \( f'(x) \) is positive on intervals where \( \cos\left(\frac{x}{3}\right) < 0 \).
• \( f'(x) \) is negative on intervals where \( \cos\left(\frac{x}{3}\right) > 0 \).

Considering our interval \([0, 6\pi]\):

• \( x \) in \([0, \frac{3\pi}{2}), [3\pi, 3\frac{3\pi}{2}), [6\pi, 6\frac{3\pi}{2})\) indicates \( f(x) \) is increasing because \( f'(x) > 0 \).
• \( x \) in \([\frac{3\pi}{2}, 3\pi), [3\frac{3\pi}{2}, 6\pi), [6\frac{3\pi}{2}, 6\pi]\) indicates \( f(x) \) is decreasing because \( f'(x) < 0 \).

These intervals help us understand how \( f(x) \) rises and falls graphically.

Differentiation

Differentiation is the process of finding the derivative of a function. The derivative tells us the rate at which a function changes with respect to one of its variables. In this exercise, differentiation was used to analyze the function \( f(x) = -3 \sin\left(\frac{x}{3}\right) \).

Key steps in differentiation include:

• Identifying the function type and applying the proper differentiation rules, such as the Chain Rule.
• Simplifying the result to understand the derivative's behavior across a given interval.

With \( f'(x) = -\cos\left(\frac{x}{3}\right) \), differentiation has provided critical insights into where our function increases or decreases. It helps us understand the slope at any point and guides us in graphing or analyzing the function.