Problem 31 - Calculus Early Transcendentals: Pearson New International Edition

Question

Find a good bound for the maximum value of the given expression, given that \(c\) is in the stated interval. Answers may vary depending on the technique used. (See Example 5.) $$ \left|\frac{4 c}{\sin c}\right| ;\left[\frac{\pi}{4}, \frac{\pi}{2}\right] $$

Step-by-Step Solution

Answer

The maximum value of \( \left|\frac{4c}{\sin c}\right| \) on the interval is \( \frac{\pi}{\sqrt{2}} \).

Step 1: Understand the Expression

The expression we need to analyze is \( \left|\frac{4c}{\sin c}\right| \). We have to find its maximum value over the interval \( \left[\frac{\pi}{4}, \frac{\pi}{2}\right] \).

Step 2: Simplify Using Absolute Values

Since \( |x| \) represents the non-negative value of \( x \), and both the numerator and denominator in the given expression are positive over the interval, \( \left|\frac{4c}{\sin c}\right| \) simplifies to \( \frac{4c}{\sin c} \).

Step 3: Analyze the Expression

To find the maximum value, we consider the behavior of \( \frac{4c}{\sin c} \) as \( c \) changes from \( \frac{\pi}{4} \) to \( \frac{\pi}{2} \). \( \sin(c) \) increases and reaches its maximum value of 1 at \( \frac{\pi}{2} \), while \( 4c \) is linear.

Step 4: Find Critical Points

Find the derivative of \( \frac{4c}{\sin c} \), set it to zero, and solve for \( c \) to find any critical points in the interval. Using quotient rule, if necessary, analyze changes in the expression.

Step 5: Evaluate at the Endpoints

Evaluate \( \frac{4c}{\sin c} \) at the endpoints \( c = \frac{\pi}{4} \) and \( c = \frac{\pi}{2} \). At \( c = \frac{\pi}{4} \), the expression is \( \frac{\pi}{\sqrt{2}} \), and at \( c = \frac{\pi}{2} \), it simplifies to 2.

Step 6: Determine the Maximum

By comparing the endpoint values and any critical points, determine that the maximum value on the interval \( \left[\frac{\pi}{4}, \frac{\pi}{2}\right] \) is \( \frac{\pi}{\sqrt{2}} \).

Key Concepts

Derivative Analysis

When dealing with optimization problems in calculus, derivatives play an essential role. By differentiating a function, we can learn about its behavior, including where it might reach maximum or minimum values. In this exercise, we look at the expression \( \frac{4c}{\sin c} \) within the interval \( [\frac{\pi}{4}, \frac{\pi}{2}] \).To understand the function's behavior, we need its derivative. Differentiating can show us rates of change and help identify where the function is increasing or decreasing. This is crucial for

• Finding critical points, where the derivative is zero or undefined,
• Analyzing the intervals to find increasing or decreasing behavior,
• Determining where maxima and minima occur.

Derivative analysis gives us the mathematical tool to assess the peaks and valleys of a function, helping determine where its maximum value lies within an interval.

Quotient Rule

The Quotient Rule is a technique used to differentiate expressions that are fractions of two functions. For our function, \( \frac{4c}{\sin c} \), the Quotient Rule is applicable because it is the division of \(4c\) by \(\sin c\).The standard formula for the Quotient Rule is: if \( u \) and \( v \) are functions of \( x \), then the derivative of \( \frac{u}{v} \) is \( \frac{v(u') - u(v')}{v^2} \). Applying this to our function helps us find:

• \( u = 4c \) with \( u' = 4 \)
• \( v = \sin c \) with \( v' = \cos c \)
• The derivative \( \frac{\sin c(4) - 4c(\cos c)}{\sin^2 c} \)

Using the Quotient Rule is necessary to find the critical points and solve for potential maximum or minimum values.

Critical Points

Critical points are values of \( c \) where the derivative of our function is zero or undefined, indicating possible maxima or minima within the interval.To identify these points for \( \frac{4c}{\sin c} \), we set the derivative found using the Quotient Rule to zero. Solving \( \sin c(4) - 4c(\cos c) = 0 \) gives insights into where our function might reach critical values. Features to be aware of include:

• If the function is continuous over the interval, these points may mark a peak or trough.
• Critical points help us narrow down the range to test for maximum values.

Analyzing critical points, alongside evaluating the function at interval endpoints, is key to determining where the maximum value occurs within a given range.

Trigonometric Functions

Trigonometric functions, like \( \sin \) and \( \cos \), exhibit periodic behavior and are fundamental in calculus.For this problem, \( \sin c \) plays a crucial role since it is the denominator in our expression \( \frac{4c}{\sin c} \). Understanding its behavior over \( [\frac{\pi}{4}, \frac{\pi}{2}] \) is elemental. During this interval:

• \( \sin c \) increases from \( \sin(\frac{\pi}{4}) = \frac{\sqrt{2}}{2} \) to \( \sin(\frac{\pi}{2}) = 1 \).
• Trigonometric identities and values help simplify expressions and solve equations.
• As \( \sin c \) approaches its maximum at \( \frac{\pi}{2} \), the impact on \( \frac{4c}{\sin c} \) needs careful consideration.

Using trigonometric functions effectively allows us to dissect and manipulate the equation for maximum optimization. They inform us how the function behaves geometrically across the interval.